MEASUREMENTS OF THE TRANSVERSE SPIN-DEPENDENT TOTAL CROSS SECTION DIFFERENCE T FOR THE SCATTERING OF POLARIZED NEUTRONS FROM POLARIZED PROTONS by Wesley Scott Wilburn Department of Physics Duke University Date: Approved: N. Russell Roberson, Supervisor Lawrence E. Evans Daniel J. Gauthier Christopher R. Gould Werner Tornow Dissertation submitted in partial fulllment of the requirements for the degree of Doctor of Philosophy in the Department of Physics in the Graduate School of Duke University 1993 ABSTRACT (Physics-Nuclear) MEASUREMENTS OF THE TRANSVERSE SPIN-DEPENDENT TOTAL CROSS SECTION DIFFERENCE T FOR THE SCATTERING OF POLARIZED NEUTRONS FROM POLARIZED PROTONS by Wesley Scott Wilburn Department of Physics Duke University Date: Approved: N. Russell Roberson, Supervisor Lawrence E. Evans Daniel J. Gauthier Christopher R. Gould Werner Tornow An abstract of a dissertation submitted in partial fulllment of the requirements for the degree of Doctor of Philosophy in the Department of Physics in the Graduate School of Duke University 1993 MEASUREMENTS OF THE TRANSVERSE SPIN-DEPENDENT TOTAL CROSS SECTION DIFFERENCE T FOR THE SCATTERING OF POLARIZED NEUTRONS FROM POLARIZED PROTONS by Wesley Scott Wilburn Although the existence of a tensor component in the nucleon-nucleon interaction has been known since the discovery of the electric quadrupole moment of the deuteron, its strength has remained poorly determined, especially at low energies. The tensor force is also important as it is responsible for much of the binding energy of the triton, and it contributes signicantly to the binding of other few-nucleon systems. At low energies, the strength of the tensor interaction in neutron-proton scattering is characterized by the 1 phase-shift parameter which gives the mixing between the 3S1 and 3D1 states. Most observables which are sensitive to 1, such as the spin-correlation coecients Ayy () and Azz (), are also sensitive to other phase-shift parameters, particularly the phase shifts associated with the 1P1 and 3P1 states. The longitudinal and transverse spin-dependent dierences in total cross section, L and T , however, are very sensitive to 1 and insensitive to most other phase-shift parameters. In addition, L and T are predicted to cross through zero, allowing very accurate measurements to be made at these energies as they are unaected by most systematic errors. Measurements of T have been made at six energies in the region of the zerocrossing using a polarized proton target and a polarized neutron beam. The polarized proton target is of the brute-force type and consists of TiH2 cooled to below 17 mK by a 3He-4He dilution refrigerator in a 7 T magnetic eld. The polarized neutron beam is produced via the 3H(~p,~n)3He reaction using a polarized proton beam obtained from the Triangle Universities Nuclear Laboratory atomic beam polarized ion source and accelerated by a tandem Van de Graa accelerator. The beam polarization is monitored by a carbon-foil polarimeter. The temperature of the dilution refrigerator is measured by 3He melting curve thermometers. In addition, polarization transfer i coecients, Kyy , have been measured for the 3H(~p,~n)3He reaction at three energies using a 4He neutron polarimeter. A zero-crossing energy of Ezc = 5:08 0:10 MeV has been observed for T . A simple phase-shift analysis using this value results in 1 = 0:30 0.17 . This result is somewhat lower than predicted by the full Bonn potential, which has the smallest tensor force of all realistic nucleon-nucleon potential models. 0 ii ACKNOWLEDGMENTS The success of this project is due to the hard work of many people. I would like to rst thank my thesis advisor, Dr. Russell Roberson for teaching me the art of experiment and for his advice, encouragement, and patience. I am grateful to Dr. Chris Gould who has been a teacher and advisor, beginning when I was an undergraduate and continuing to this day. I have greatly enjoyed working with Dr. David Haase who has taught me cryogenic techniques and more. To Dr. Werner Tornow, who rst suggested this measurement, I am greatly indebted for his tireless eorts and considerable expertise. I would like to thank Chris Keith and Paul Human for helping to set up and run the experiments and for doing whatever was necessary to get the job done. Tim Murphy and Brian Raichle deserve thanks for the many midnight shifts they worked. Dr. Jim Koster's help in the early days of the experiment was invaluable. An experiment of this complexity would not be possible without a superb technical sta such as the one at TUNL. The machinists in the instrument shop, Al Lovette, Gene Harris, and Bob Hogan, could make anything I could draw, and almost always had a better way of doing it. Sidney Edwards and Pat Mulkey in the electronics shop deserve thanks for either building, xing, or modifying much of the electronics in this experiment, often on short notice. Paul Carter, John Dunham, and Ken Sweeton have my gratitude for xing what was broken and for keeping the lab running, despite vacuum failures, chiller woes, and water main breaks. I have beneted from many discussions on the theoretical aspects of this work with Dr. Alec Schramm. I appreciate the timely help of Dr. H. O. Klages at KFA in Karslruhe, Germany who provided the titanium hydride for the target. Without this material, the experiment would have been delayed for several months. Chris Westerfeldt has been a source of help and advice both before and during this project. Thanks are due also to Dr. Tom Clegg and Dr. Eric Crosson for their work with the TUNL Polarized Ion Source. I would like to thank my parents, Ed and Mary Wilburn, for encouraging my interest in science and for making my education possible. Finally, I would like to thank my son Grey for his smiles and laughs. I only hope he will be as happy in his pursuits as I have been in mine. This dissertation is dedicated to my wife Dianne whose constant love and encouragement has meant so much to me during this journey. iii Contents Abstract i Acknowledgments iii List of Figures vi List of Tables ix 1 Introduction 1 2 Theoretical Overview 3 2.1 2.2 2.3 2.4 Nuclear Tensor Interaction : : : : : : : : : : : : Phase-Shift Parameters : : : : : : : : : : : : : : Spin-Dependent Total Cross Section Dierences Potential Models : : : : : : : : : : : : : : : : : 3 Experimental Apparatus 3.1 Charged-Particle Beam : : : : : : : 3.1.1 Polarized Ion Source : : : : 3.1.2 Acceleration and Transport 3.1.3 Spin Transport : : : : : : : 3.2 Charged-Particle Polarimeter : : : 3.3 Neutron Beam : : : : : : : : : : : : 3.3.1 Neutron Beam Production : 3.3.2 Neutron Beam Collimation : 3.3.3 Neutron Detection : : : : : iv : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 : 5 : 9 : 10 : : : : : : : : : 32 32 32 34 38 41 46 46 54 57 3.4 Neutron Polarimeter : : : : : : : : : : : : : 3.5 Polarized Proton Target : : : : : : : : : : : 3.5.1 Dilution Refrigerator : : : : : : : : : 3.5.2 Titanium Hydride Target : : : : : : 3.5.3 Thermometry : : : : : : : : : : : : : 3.6 Data Acquisition Electronics : : : : : : : : : 3.6.1 T Measurements : : : : : : : : : : 3.6.2 Neutron Polarization Measurements : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 Experimental Procedure 4.1 T Measurements : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4.2 Kyy Measurements for the 3H(~p,~n)3He Reaction : : : : : : : : : : : : 0 5 Data Analysis 5.1 T Measurements : : : : : : : : : : : : : : : : : : : : : : : : 5.1.1 Calculation of the Neutron-Transmission Asymmetries 5.1.2 Calculation of the Average Neutron Beam Polarization 5.1.3 Calculation of T : : : : : : : : : : : : : : : : : : : : 5.1.4 Determination of the Zero-Crossing Energy of T : : 5.2 Kyy Measurements for the 3H(~p,~n)3He Reaction : : : : : : : : 5.2.1 Determination of Kyy : : : : : : : : : : : : : : : : : : : 5.2.2 Interpolation of Kyy Data : : : : : : : : : : : : : : : : 0 0 0 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 60 68 69 71 75 77 77 84 87 87 89 92 92 93 97 102 106 106 108 110 6 Comparison of Data with Potential Models 114 7 Conclusions and Summary 118 A Calculation of Neutron Depolarization Due to Magnetic Fields 120 B The Spin-Dependent Total Cross Section Dierence 151 tensor References 154 Biography 161 v List of Figures : : : : : : : : : : : : : : : : : : : : : 4 11 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 3.1 Energy Level Diagram for the Hydrogen Atom : : : : : : : : : : : : : 3.2 Low Energy Beam Transport : : : : : : : : : : : : : : : : : : : : : : : 3.3 High Energy Beam Transport : : : : : : : : : : : : : : : : : : : : : : 33 35 36 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 Feynman Diagram for One-Pion Exchange : : : : : : : : : : : : Potential-Model Predictions of the 1 Mixing Parameter : : : : : Triton Binding Energy versus Deuteron D -State Probability : : Potential-Model Predictions for L at Low Energies : : : : : : Potential-Model Predictions for T at Low Energies : : : : : : Sensitivity of L to a 1 Variation in 1S0 Using Arndt SP89 Sensitivity of T to a 1 Variation in 1S0 Using Arndt SP89 Sensitivity of L to a 1 Variation in 3P0 Using Arndt SP89 Sensitivity of T to a 1 Variation in 3P0 Using Arndt SP89 Sensitivity of L to a 1 Variation in 1P1 Using Arndt SP89 Sensitivity of T to a 1 Variation in 1P1 Using Arndt SP89 Sensitivity of L to a 1 Variation in 3S1 Using Arndt SP89 Sensitivity of T to a 1 Variation in 3S1 Using Arndt SP89 Sensitivity of L to a 1 Variation in 3P1 Using Arndt SP89 Sensitivity of T to a 1 Variation in 3P1 Using Arndt SP89 Sensitivity of L to a 1 Variation in 3D1 Using Arndt SP89 Sensitivity of T to a 1 Variation in 3D1 Using Arndt SP89 Sensitivity of L to a 1 Variation in 1 Using Arndt SP89 : Sensitivity of T to a 1 Variation in 1 Using Arndt SP89 : Zero-Crossing Predictions for L : : : : : : : : : : : : : : : : : Zero-Crossing Predictions for T : : : : : : : : : : : : : : : : : vi : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 Beam Current Integration Electronics : : : : : : : : : : : : : : : : : : Madison Convention Coordinate System : : : : : : : : : : : : : : : : Spin Precession for a Proton Beam, Transverse Horizontal on Target : Wien Filter Calibration : : : : : : : : : : : : : : : : : : : : : : : : : : Proton Beam Polarimeter : : : : : : : : : : : : : : : : : : : : : : : : Proton Beam Polarimeter Electronics : : : : : : : : : : : : : : : : : : Tritiated-Titanium Foil Holder : : : : : : : : : : : : : : : : : : : : : : ENDF/B-VI Total Neutron Cross Section of Carbon : : : : : : : : : ENDF/B-VI Total Neutron Cross Section of Oxygen : : : : : : : : : Measured Total Neutron Cross Section of Carbon : : : : : : : : : : : Measured Total Neutron Cross Section of Oxygen : : : : : : : : : : : Neutron Collimation and Shielding : : : : : : : : : : : : : : : : : : : Neutron Pre-Collimator : : : : : : : : : : : : : : : : : : : : : : : : : Neutron Detector Shield : : : : : : : : : : : : : : : : : : : : : : : : : Neutron Detector Assembly : : : : : : : : : : : : : : : : : : : : : : : Neutron Detector Electronics : : : : : : : : : : : : : : : : : : : : : : Neutron Polarimeter : : : : : : : : : : : : : : : : : : : : : : : : : : : High-Pressure Helium Gas Cell : : : : : : : : : : : : : : : : : : : : : Helium Cell Filling System : : : : : : : : : : : : : : : : : : : : : : : : Electronics for the Forward Neutron Detector Pair : : : : : : : : : : : Electronics for the Backward Neutron Detector Pair : : : : : : : : : : Coincidence Electronics for the Neutron Polarimeter : : : : : : : : : : Dilution Refrigerator : : : : : : : : : : : : : : : : : : : : : : : : : : : Dilution Refrigerator Cryostat : : : : : : : : : : : : : : : : : : : : : : Titanium Hydride Target : : : : : : : : : : : : : : : : : : : : : : : : : 3 He Melting Curve Thermometer : : : : : : : : : : : : : : : : : : : : Data Acquisition Electronics for T Measurements : : : : : : : : : : Data Acquisition Timing Diagram : : : : : : : : : : : : : : : : : : : : Beam Prole Monitor Feedback Steering Electronics : : : : : : : : : : Beam Prole Monitor Collector Spectrum : : : : : : : : : : : : : : : Beam Prole Monitor Fiducial Spectrum : : : : : : : : : : : : : : : : Neutron Polarimeter Data Acquisition Electronics (Part 1) : : : : : : Neutron Polarimeter Data Acquisition Electronics (Part 2) : : : : : : vii 38 38 40 42 44 45 47 49 50 51 52 55 55 57 59 60 61 62 63 64 65 67 70 72 74 76 78 80 81 82 83 85 86 5.1 Dead-Time Pulser Spectrum : : : : : : : : : : : : : : : : : : : : : : : 94 5.2 T Data with Fit : : : : : : : : : : : : : : : : : : : : : : : : : : : : 107 5.3 3H(~p,~n)3He Polarization-Transfer Coecient Data : : : : : : : : : : : 112 6.1 Comparison of T Measurements with Potential-Model Predictions : 115 6.2 Comparison of the Measured Ezc with Potential-Model Predictions : : 116 6.3 Comparison of 1 from T Data with Potential-Model Parameters : 117 viii List of Tables 2.1 Predicted Deuteron Properties for Various Potential Models : : : : : 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Wien Filter Settings : : : : : : : : : : : : : : : : : : : : : Analyzing Powers for 12C(p,p0)12C at lab = 40 : : : : : : Calculated Average Neutron Energies and Energy Widths : Measured Average Neutron Energies : : : : : : : : : : : : Asymmetry Dilution Factor : : : : : : : : : : : : : : : : : Calculated Neutron Depolarization : : : : : : : : : : : : : Eective Analyzing Powers for the Neutron Polarimeter : : : : : : : : : 43 46 48 53 53 54 66 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 Average Neutron-Transmission Asymmetries : : : : : : : : : : : : : : Analyzing Power Fits for 12C(p,p0)12C at lab = 40 : : : : : : : : : : Average Proton and Neutron Beam Polarizations : : : : : : : : : : : Asymmetries Normalized to the Proton Polarimeter Asymmetry : : : Asymmetries Normalized to the Proton Beam Polarization : : : : : : Asymmetries Normalized to Beam and Target Polarizations : : : : : : Calculated Values of T at a Neutron Energy of 1.94 MeV : : : : : Measured Values of T : : : : : : : : : : : : : : : : : : : : : : : : : Neutron Counts in the Neutron Polarimeter Detectors : : : : : : : : : Average Neutron Beam Polarizations : : : : : : : : : : : : : : : : : : Average Proton Beam Polarizations : : : : : : : : : : : : : : : : : : : Measured Values of the 3H(~p,~n)3He Polarization-Transfer Coecient : 3 H(~ p,~n)3He Polarization-Transfer Coecient Data : : : : : : : : : : : Interpolated Values of Kyy : : : : : : : : : : : : : : : : : : : : : : : : 98 99 100 103 104 105 105 105 108 109 109 110 111 113 0 ix : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 Chapter 1 Introduction Although the existence of a tensor component in the nucleon-nucleon interaction has been known since the discovery of the electric quadrupole moment of the deuteron, its strength is still poorly determined, especially at low energies. The tensor force is also important as it is responsible for much of the binding energy of the triton, and it contributes signicantly to the binding of other few-nucleon systems. At low energies, the strength of the tensor interaction in neutron-proton scattering is characterized by the 1 phase-shift parameter which gives the mixing between the 3S1 and 3D1 states. The notation used to describe the scattering states is 2S+1LJ , where S is the total spin, L refers to the orbital angular momentum, `, (S , ` = 0, P , ` = 1, D , ` = 2, etc.), and J = S + ` is the total angular momentum. Most observables which are sensitive to 1, such as the spin-correlation coecients Ayy () and Azz (), are also sensitive to other phase-shift parameters, particularly the phase shifts associated with the 1P1 and 3P1 states. The longitudinal and transverse spin-dependent dierences in total cross section, L and T , however, are very sensitive to 1 and insensitive to most other phase-shift parameters. In addition, L and T are predicted to cross through zero, allowing very accurate measurements to be made at energies where they are unaected by most systematic errors. Measurements of T have been made at six energies in the region of the zerocrossing using a polarized proton target and a polarized neutron beam. The polarized proton target is of the brute-force type and consists of TiH2 cooled to below 17 mK in a 7 T magnetic eld. The target is cooled by a 3He -4He dilution refrigerator and the magnetic eld is produced by a superconducting split-coil solenoid. The polar1 2 Chapter 1. Introduction ized neutron beam is produced via the 3H(~p,~n)3He reaction using a polarized proton beam. The proton beam polarization is monitored by a carbon-foil polarimeter. The temperature of the dilution refrigerator is measured by 3He melting curve thermometers. Transmitted neutrons are detected at 0 using two organic liquid scintillator cells mounted to photomultiplier tubes. Because of their importance in the analysis of the T data, polarization-transfer coecients, Kyy , have been measured for the 3H(~p,~n)3He reaction at three energies using a neutron polarimeter. A high-pressure 4He gas cell is used as an analyzer and the scattered neutrons are detected in four side detectors. The gas cell is an active target, scintillating from the recoiling alpha particles. Coincidences between the scattered neutrons and the recoiling alphas greatly reduce the background counts. A zero-crossing energy, Ezc , has been extracted from the T data and used in a simple phase-shift analysis to determine the value of 1 at that energy. The data and the resulting values of Ezc and 1 are compared to potential-model predictions. 0 Chapter 2 Theoretical Overview 2.1 Nuclear Tensor Interaction At present, the only satisfactory theories of the nuclear force are meson-exchange models. These models treat nucleons as fundamental particles which interact through the exchange of mesons with Yukawa-type potentials. Although nucleons are known to actually be composite, no realistic quark model of the nuclear interaction has as yet been constructed due to the computational diculties of such an approach. At the other extreme, phenomenological models exist, but they are unable to provide insight into the underlying mechanisms. Thus it is reasonable to look for the origins of a nuclear tensor interaction in meson exchange. The Yukawa potential arises from considering the wave equation for massive particles Yuk35]. ! 2c2 2 m 2 r ; h 2 = c12 @@t2 (2:1) Here, m is the mass of the exchanged particle. The solutions to this equation fall o exponentially with distance, with a decay constant proportional to the mass. The strength of the interaction is given by a coupling constant g. ;mcr=h = 4g e r (2:2) Since pions have the smallest mass of the mesons involved in the nuclear force, they will have the largest contribution at long range. Because pions have negative intrinsic 3 4 Chapter 2. Theoretical Overview E ,´ p´ u´ u´ E,´ –p´ m ps gps E ,p u gps u E,–p Figure 2.1: Feynman Diagram for One-Pion Exchange parity, a pseudoscalar coupling is chosen to conserve parity. The Lagrangian for onepion exchange (represented by the Feynman diagram in Figure 2.1) is then of the form Lps = gps i 5ps: (2:3) The matrix element for this interaction can be written 2 0 u02i 5u2) Mfi = gps((pu1;i 5pu0)12)( (2:4) + m2ps : p and p0 are the initial and nal momenta, and u and u0 are the initial and nal wave functions. In the non-relativistic limit, the wave functions have the form 1 0 s 1 (2:5) u1 = E2+MM B @ 1p CA 1 E +M where M is the nucleon mass, and p are the spin and momentum operators, and is the spin wavefunction. Evaluating the matrix element gives the result 2 Mfi = ;4Mgps2 (1q2q+)(m22 q) (2:6) ps 2.2. Phase-Shift Parameters 5 with q = p0 ; p. This interaction can be expressed as a momentum-space potential ; gps2 (1 k)(2 k) (2:7) Vps (k) = 4M 2 k2 + m2 : ps By performing a three-dimensional Fourier transform, the coordinate-space potential may be obtained Mac86]. ( " #) ;mpsr gps2 m2ps 3 3 Vps (x) = 4 12M 2 1 2 + S12(^x) 1 + m r + (m r)2 e r (2:8) ps ps S12 is the tensor operator S12(^x) = 3(1 x^ )(2 x^ ) ; 1 2: (2:9) The rst term of the potential is a spin-dependent central interaction while the second is a tensor interaction which mixes states with dierent orbital angular momentum. This type of interaction also arises from considering the tensor coupling of vector mesons such as the rho. (2:10) Lt = 2gMv @ ps This coupling results in a coordinate-space potential ( " #) ;mv r 2 2 g m 3 3 v v Vt(x) = 4 12M 2 21 2 ; S12(^x) 1 + m r + (m r)2 e r (2:11) v v containing a tensor term of the same form but opposite in sign compared to the pseudoscalar coupling. The tensor force is thus reduced at shorter distances where the heavier rho contributes to the nuclear interaction. 2.2 Phase-Shift Parameters The elastic scattering of one spinless particle from another can be described asymptotically as the superposition of an incident plane wave with a scattered spherical wave Sch68, DeB67]. If only central forces are involved, the orbital angular momentum, `, is conserved and the incident plane wave can be written as a partial-wave expansion. i = Aeikz = A 1 X `=0 i`(2` + 1)j` (kr)P` (cos ) (2.12) 6 Chapter 2. Theoretical Overview For large values of r, 1 sin( kr ; 2 ` ) j` kr ;i(kr;` ) ; e+i(kr;` ) : ie 2kr 2 2 (2.13) The incident plane wave can then be described (in the asymptotic limit) as the sum of incoming and outgoing spherical waves and is given by 1 i h X (2:14) i = 2Akr i`+1(2` + 1) e;i(kr;` 2 ) ; e+i(kr;` 2 ) P` (cos ): `=0 Since the scattered wave consists only of outgoing spherical waves, it can be included by modifying the second term of Equation 2.14 by a factor `. The total wave function is then 1 h i X = 2Akr i`+1(2` + 1) e;i(kr;` 2 ) ; ` e+i(kr;` 2 ) P` (cos ): (2:15) `=0 The scattered wave is then obtained by subtracting the incident wave from the total. s = ; i 1 X = 2Akr i`+1(2` + 1)(1 ; `)ei(kr;` 2 )P` (cos ) `=0 1 ikr X e i = A 2k r (2` + 1)(1 ; ei` )P` (cos ) `=0 (2.16) (2.17) Here ei` = ` and ` are the phase shifts. The total wave function can be written asymptotically in terms of the scattering amplitude, f (), as " ikz # e ikz = A e + f () (2:18) r : Comparing this result to Equation 2.15 gives 1 X i f () = 2k (2` + 1)(1 ; ei` )P` (cos ): (2:19) `=0 2.2. Phase-Shift Parameters 7 The dierential cross section is then () = jf ()j2 X 2 1 1 i (2.20) = k2 (2` + 1)e ` sin ` P` (cos ) : `=0 Integrating over gives the total cross section in terms of the phase shifts. 1 X 4 = k2 (2` + 1) sin2 ` (2:21) `=0 There exists one scattering eigenstate corresponding to each value of `. The contribution of each eigenstate is determined by its corresponding phase shift, `. The phase shifts are related to the scattering matrix, S, through the expression 8 > < e2i` `f = `i = ` Sfi = > (2:22) : 0 `f 6= `i when purely central forces act. The subscripts i and f refer to the initial and nal states. O-diagonal terms will occur in S if non-central interactions exist. In a phase-shift description of nucleon-nucleon scattering, the spin 1/2 nature of the nucleons must be considered. The principal quantum number is now the total angular momentum, J . Whereas before there was one state for each value of `, there are now four states for each value of the total angular momentum J , for J 1. There is a spin-singlet state with orbital angular momentum ` = J and three spin-triplet states with ` = J ; 1, ` = J , and ` = J + 1. Total spin S is conserved under the assumption of charge independence, while ` is conserved for pure central forces. Since the tensor force is manifestly non-central, mixing can occur for states with the same J , S , and parity, but with dierent ` the triplet states with ` = J ; 1 and ` = J + 1. For the simplest case in which mixing can occur, J = 1,1 the four scattering states are: 1P 1 3S 1 3 P1 3D 1 1 There is no mixing for J = 0 because there are no states with ` = J 1. 8 Chapter 2. Theoretical Overview and mixing of states can occur between 3S1 and 3D1. In the case of proton-proton and neutron-neutron scattering, only states with total isospin T = 1 exist, whereas in the case of neutron-proton scattering states with both T = 0 and T = 1 occur DeB67]. In addition, the Pauli principle requires (;1)S+T +` = ;1: (2:23) Thus, for identical particles, only singlet states with even ` and triplet states with odd ` are allowed. Mixing can then occur only for states with even J in the pp and nn cases. When a tensor interaction exists, the states with ` = J ; 1 and ` = J + 1 are no longer energy eigenfunctions. Instead, the new states are J and J which are a mixture of the two states of denite `. The part of the scattering matrix corresponding to these states has the form Bla52a, Bla52b] S = U;1e2i U where 0 U = [email protected] cos J ; sin J 0 = [email protected] J 0 0 J 1 sin J C A cos J 1 CA : (2:24) (2.25) (2.26) J and J are the phase shifts and J is the mixing parameter, in the notation of Blatt and Biedenharn. By convention, these states are referred to as 3S1 and 3D1, even though they are not states of pure `, except at E = 0. At low energies, 1 is the phase-shift parameter which most directly characterizes the strength of the tensor interaction. Because of the dierence in ` of the two states coupled by the tensor interaction, no mixing can occur at En = 0. At the lowest energies (below approximately 2 MeV), 1 is constrained by this kinematic consideration and by the properties of the deuteron. 2.3. Spin-Dependent Total Cross Section Dierences 9 2.3 Spin-Dependent Total Cross Section Dierences Phase-shift parameters are not directly measurable. Instead, experiments designed to measure the strength of the tensor force at low energies must measure observables sensitive to the 1 mixing parameter. Ideally, these observables will not be sensitive to any other phase-shift parameters. In order to be sensitive to a spin-spin interaction such as the tensor force, two polarizations must be measured by the observable. Such observables include the spin-correlation parameters Ayy () and Azz (). Unfortunately, these measurements are also very dependent upon the 1P1 phase shift except at = 90 which is experimentally very dicult. Another set of observables are the spin-dependent dierences in total cross section, , Bug80, Tor88] where = (0 I ;s) ; (0 I s): (2:27) I is the spin of the proton target, and s is the spin of the neutron projectile. In the simplest case, I and s are along the same axis and can either be longitudinal or transverse with respect to the direction of propagation. More general cases are discussed in Appendix B. The longitudinal and transverse cases are dened by L = (! (2.28) ) ; (! ! ) T = ("#) ; (""): (2.29) Where the top or rst arrow refers to the target spin and the bottom or second refers to the projectile spin. The total cross section can be decomposed into two parts = 0 + ss (2:30) where 0 is even under a reversal of projectile spin and ss is odd. The total cross section dierences depend only on the second term. = ( 0 + ss) ; ( 0 ; ss ) = 2 ss (2.31) The spin-spin cross section can then be divided into scalar and tensor parts (2:32) ss = s ( 1 2 ) + tS12 10 Chapter 2. Theoretical Overview where the tensor operator, S12, is dened in Equation 2.9. This gives L T = 2( s + 2 t) = 2( s ; t): (2.33) (2.34) L and T are then two dierent linear combinations of the scalar and tensor parts of the spin-spin cross section. These observables can be written in terms of the phase-shift parameters in the Stapp convention as Tor91] 2 3 ; 2 + cos 2 + cos 2 + 3 cos 2 1 S 0 3 P 0 1 P 1 66 77 66 ; cos 23S1 ; 3 cos 3P 1 + cos 23D1 77 6 7 p L = 2 666 ; 4 2 sin(3S1 + 3D1) sin 21 + 5 cos 21D2 777 (2.35) k 6 77 66 ; cos 23P 2 ; 5 cos 23D2 + cos 23F 2 75 4 p ; 4 6 sin(3P 2 + 3F 2) sin 22 + 2 3 cos 2 ; cos 2 + 3 cos 2 1 S 0 3 P 0 1 P 1 66 77 66 ; cos 23S1 ; 2 cos 23D1 77 6 77 p 77 : (2.36) T = k2 666 + 2 2 sin(3S1 + 3D1) sin 21 66 77 64 + 5 cos 21D2 ; 2 cos 23P 2 ; 3 cos 23F 2 75 p + 2 6 sin(3P 2 + 3F 2) sin 22 + In these equations, 1S0, for example, refers to the phase shift for the 1S0 state. 2.4 Potential Models Potential models of the nucleon-nucleon interactions vary considerably in the strength of the tensor force. In terms of the phase shifts, this variation can be seen in the 1 mixing parameter. Figure 2.2 shows 1 as a function of energy for several potential models as well as the phase-shift analysis, SM92, of Ardnt Mac87, Bra88, Mac89, Lac80, Nag78, Arn92]. This variation in tensor strength can also be observed in the predictions of the properties of the deuteron. Table 2.1 lists the deuteron D state probability, PD , magnetic dipole moment, d , and electric quadrupole moment, Qd, for various potential models Mac87, Bra88, Mac89, Nag78, Lac80, Rei68, Kel39, 2.4. Potential Models 11 1 (degrees) 5:0 Arndt SM92 Full Bonn Bonn A Bonn C Paris 4:0 3:0 2:0 1:0 0:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 Figure 2.2: Potential-Model Predictions of the 1 Mixing Parameter 30:0 12 Chapter 2. Theoretical Overview Potential Qd (fm2) d (N ) PD (%) Full Bonn 0.2807 0.8555 4.25 Bonn A 0.274 0.8548 4.4 Bonn B 0.278 0.8514 5.0 Bonn C 0.281 0.8478 5.6 Nijmegen 0.2715 5.4 Paris 0.279 0.853 5.8 Reid Soft Core 0.2796 6.5 Experiment 0.2859 0.857406 0.0003 0.000001 Table 2.1: Predicted Deuteron Properties for Various Potential Models Eri84]. The binding energy of the triton is also greatly aected by the choice of tensor strength. Figure 2.3 shows the strong correlation that exists between the deuteron D -state probability and the triton binding energy Bra88]. Models with a stronger tensor force, as indicated by a larger PD , underestimate the triton binding energy more than models with a weaker tensor force. Predictions of L and T can be calculated from the potential-model phase shifts. Figures 2.4 and 2.5 show these results for several models and for the Arndt phase-shift analysis in the energy range of interest. In addition, sensitivity calculations can be performed by varying the phase-shift parameters of one model. For this purpose, the Arndt phase-shift analysis, SP89, is chosen Arn89]. The phase shifts are varied by 1 and the resulting shifts in L and T are plotted in Figures 2.6{2.19 for J 1. Higher partial waves do not contribute signicantly to these observables. These gures show that L and T are more sensitive to 1 than to the other phase-shift parameters. L and T can be expressed to a good approximation in terms of only 1S0, 3S1, and 1. The sensitivity to 1 is a maximum at about 10 MeV: approximately 51 mb/degree for L and 26 mb/degree for T . Of particular interest from an experimental viewpoint are the zero-crossing energies of L and T , which also display large sensitivities to 1. Measurements made at the zero-crossing energies are insensitive to most systematic uncertainties. Predictions for the zero-crossings of L and T are shown in Figures 2.20 and 2.21. 2.4. Potential Models 13 EB (MeV) ;7:0 Experiment Argonne V14 Bonn A Bonn B Bonn C Paris Reid Soft-Core de Tourreil et al ;7:2 ;7:4 ? 4 ;7:6 b 3 3 + 2 4 ? b ;7:8 ;8:0 2 ;8:2 + ;8:4 ;8:6 ;8:8 ;9:0 7:0 6:5 6:0 5:5 PD (%) 5:0 4:5 4:0 Figure 2.3: Triton Binding Energy versus Deuteron D -State Probability for Various Potential Models 14 Chapter 2. Theoretical Overview L (mb) 1000:0 Arndt SM92 Full Bonn Bonn A/B/C Paris/Nijmegen 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 2:0 4 :0 6:0 8:0 En (MeV) 10:0 12:0 14:0 Figure 2.4: Potential-Model Predictions for L at Low Energies 2.4. Potential Models 15 T (mb) 1000:0 Arndt SM92 Full Bonn Bonn A/B/C Paris/Nijmegen 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 2 :0 4:0 6:0 8:0 En (MeV) 10:0 12:0 14:0 Figure 2.5: Potential-Model Predictions for T at Low Energies 16 Chapter 2. Theoretical Overview L (mb) 1000:0 S0 S ;1 S +1 1 1 0 1 0 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.6: Sensitivity of L to a 1 Variation in 1S0 Using Arndt SP89 2.4. Potential Models 17 T (mb) 1000:0 S0 S ;1 S +1 1 1 0 1 0 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.7: Sensitivity of T to a 1 Variation in 1S0 Using Arndt SP89 18 Chapter 2. Theoretical Overview L (mb) 1000:0 P0 P ;1 P +1 3 3 0 3 0 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.8: Sensitivity of L to a 1 Variation in 3P0 Using Arndt SP89 2.4. Potential Models 19 T (mb) 1000:0 P0 P ;1 P +1 3 3 0 3 0 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.9: Sensitivity of T to a 1 Variation in 3P0 Using Arndt SP89 20 Chapter 2. Theoretical Overview L (mb) 1000:0 P1 P ;1 P +1 1 1 1 1 1 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.10: Sensitivity of L to a 1 Variation in 1P1 Using Arndt SP89 2.4. Potential Models 21 T (mb) 1000:0 P1 P ;1 P +1 1 1 1 1 1 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.11: Sensitivity of T to a 1 Variation in 1P1 Using Arndt SP89 22 Chapter 2. Theoretical Overview L (mb) 1000:0 S1 S ;1 S +1 3 3 1 3 1 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.12: Sensitivity of L to a 1 Variation in 3S1 Using Arndt SP89 2.4. Potential Models 23 T (mb) 1000:0 S1 S ;1 S +1 3 3 1 3 1 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.13: Sensitivity of T to a 1 Variation in 3S1 Using Arndt SP89 24 Chapter 2. Theoretical Overview L (mb) 1000:0 P1 P ;1 P +1 3 3 1 3 1 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.14: Sensitivity of L to a 1 Variation in 3P1 Using Arndt SP89 2.4. Potential Models 25 T (mb) 1000:0 P1 P ;1 P +1 3 3 1 3 1 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.15: Sensitivity of T to a 1 Variation in 3P1 Using Arndt SP89 26 Chapter 2. Theoretical Overview L (mb) 1000:0 D1 D1 ; 1 D1 + 1 3 3 3 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.16: Sensitivity of L to a 1 Variation in 3D1 Using Arndt SP89 2.4. Potential Models 27 T (mb) 1000:0 D1 D1 ; 1 D1 + 1 3 3 3 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.17: Sensitivity of T to a 1 Variation in 3D1 Using Arndt SP89 28 Chapter 2. Theoretical Overview L (mb) 1000:0 1 1 ; 1 1 + 1 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.18: Sensitivity of L to a 1 Variation in 1 Using Arndt SP89 2.4. Potential Models 29 T (mb) 1000:0 1 1 ; 1 1 + 1 800:0 600:0 400:0 200:0 0:0 ;200:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 2.19: Sensitivity of T to a 1 Variation in 1 Using Arndt SP89 30 Chapter 2. Theoretical Overview L (mb) 100:0 Arndt SM92 Full Bonn Bonn A Bonn C Nijmegen Paris 50:0 0:0 ;50:0 ;100:0 5:6 5:8 6 :0 6 :2 6:4 6:6 En (MeV) 6:8 7:0 Figure 2.20: Zero-Crossing Predictions for 7:2 L 7:4 2.4. Potential Models 31 T (mb) 100:0 Arndt SM92 Full Bonn Bonn A Bonn C Paris/Nijmegen 50:0 0:0 ;50:0 ;100:0 4:0 4:2 4:4 4:6 4:8 5:0 5:2 En (MeV) 5:4 Figure 2.21: Zero-Crossing Predictions for 5:6 T 5:8 6:0 Chapter 3 Experimental Apparatus 3.1 Charged-Particle Beam Since it is not possible to accelerate neutral particles directly, polarized neutron beams must be produced as secondary beams from charged-particle reactions. In this experiment, the charged-particle beam consists of polarized protons. The beam is created by an ion source capable of producing nuclear spin-polarized beams of negative hydrogen ions. The beam must be transported to the accelerator where its energy is increased and then to the neutron production target where a polarized neutron beam is created via the 3H(~p,~n)3He reaction. In addition, the spin direction of the beam must be set to its proper orientation. 3.1.1 Polarized Ion Source The polarized proton beam used in this experiment is produced in the TUNL Atomic Beam Polarized Ion Source Cle89]. The ABPIS is capable of producing high-intensity beams of vector polarized protons, or vector or tensor polarized deuterons. It is an atomic beam source and it uses an electron cyclotron resonance (ECR) ionizer. Positive beams are extracted directly from the ionizer, while negative beams for acceleration by the tandem Van de Graa are produced by passing the positive beam through a cesium charge-exchange oven. The source can produce negative hydrogen beams with microampere intensities and polarizations of approximately 60%. The entire source rests on a high-voltage platform biased to ;72 kV to provide the beam with 32 3.1. Charged-Particle Beam 33 1 m I=+ 1/2 2 Energy –1/2 WF2 SF2 –1/2 3 + 1/2 4 Magnetic Field Figure 3.1: Energy Level Diagram for the Hydrogen Atom sucient energy to be transported to the accelerator. The polarized ion source produces nuclear polarization in a neutral atomic beam through a combination of two sextupole magnets and two adiabatic fast-passage transition units. The atomic beam is created by a dissociator which uses an RF discharge to break the molecular bonds of hydrogen. The atomic beam sprays out of a copper nozzle and passes through two sextupole magnets. The nozzle is cryogenically cooled and continuously coated with nitrogen to reduce molecular recombination. Atomic beam type polarized ion sources make use of the separation in energy of the magnetic substates of the hydrogen atom in a magnetic eld (Figure 3.1). The magnetic eld of a sextupole increases as a linear function of radial distance, thus focusing the hydrogen states labeled 1 and 2, and defocusing states 3 and 4. This eect is used to produce atomic polarization. Nuclear polarization is then obtained by using one 34 Chapter 3. Experimental Apparatus of the two transition units. The rst transition unit operates with a strong magnetic eld and is called SF2, while the second operates with a weak magnetic eld and is called WF2. SF2 consists of a DC magnetic eld of approximately 100 G and an oscillating eld with a frequency of 1450 MHz. This unit produces a transition from state 2 to state 4, giving a positive nuclear polarization along the z^-direction, Pz , dened by + ; N; (3:1) Pz = N N +N + ; where N+ and N; are the numbers of atoms with nuclear spin projections +1=2 and ;1=2. WF2 consists of a DC magnetic eld of approximately 10 G and an RF eld with a frequency of approximately 15 MHz. This unit causes a transition from state 1 to state 3, giving a negative Pz . The transition units can be rapidly toggled on and o by modulating the RF supplies, allowing the beam polarization to be quickly reversed. The spin state produced by SF2 will be referred to as spin up, while that produced by WF2 will be called spin down. After exiting from the sextupoles and transition units, the atomic beam is ionized by the ECR ionizer. This type of ionizer uses a plasma driven by microwave radiation to strip the electrons from the neutral hydrogen atoms. The plasma consists mostly of nitrogen introduced into the ionizer through a mass-"ow controller. Electrons from this buer gas are excited to high energy by the microwaves to perform the task of ionization. The plasma is conned in the radial direction by permanent magnets in a sextupole conguration and in the axial direction by two solenoids. Electrostatic lenses extract and focus the beam from the ionizer. Following the ionizer, a cesium oven converts the H+ beam to H; through double charge exchange with an eciency of approximately 10%. The beam is then accelerated to 26 keV after which it passes through the Wien lter where the spin direction is rotated (Section 3.1.3). Upon leaving the Wien lter, the beam receives an additional 46 keV acceleration. 3.1.2 Acceleration and Transport The beam produced by the polarized ion source, having an energy of approximately 72 keV,1 is analyzed by a magnet with a 30 de"ection. This angle aligns the direction Since charge exchange occurs in the cesium oven, the net acceleration increases the beam energy by 2eVCs , where VCs is the voltage applied to the Cs oven, typically a few hundred volts. 1 3.1. Charged-Particle Beam 35 Magnetic Quadrupole Polarized Ion Source Electrostatic Quadrupole Magnetic Steerer Wien Filter Einzel Inflection Magnet Tandem Accelerator Figure 3.2: Low Energy Beam Transport of the beam with the axis of the accelerator. Magnetic steerers are manually tuned to control the beam position, while a combination of three electrostatic quadrupole triplets, two einzels lenses, and one magnetic quadrupole doublet focuses the beam into the low energy end of the accelerator (Figure 3.2). Typically, beam currents of about 4 A are obtained on the low energy Faraday cup. The accelerator is a tandem Van de Graa, High Voltage Engineering Model FN. A tandem accelerator operates by accelerating a negative beam (in this case H;) toward a positively charged terminal, changing the charge of the beam, and then accelerating the positive beam (H+) away from the terminal VdG60], giving the beam an energy of 2eVTerminal . The charge state of the beam is changed by passing it through a thin carbon foil which strips away the electrons. In this experiment, a foil with a thickness of approximately 5 g/cm2 is used. The terminal of the accelerator is biased by two National Electrostatics Corporation (NEC) Pelletron charging chains. The chains consist of aluminum pellets joined by nylon links. The chains are mounted on pulleys driven by motors so that they run continuously between the terminal and each end of the accelerator. Charge is induced on the pellets by high-voltage DC supplies at the ground potential ends and is removed in the terminal. A corona current feedback system regulates the voltage of the terminal to keep the energy of the beam constant. The accelerator is contained in a pressure vessel lled with a mixture of carbon dioxide, nitrogen, and sulfur hexa"uoride. The beam is transmitted 36 Chapter 3. Experimental Apparatus Tandem Accelerator Steerer Feedback Analyzing Magnet Corona Feedback Proton Polarimeter Magnetic Quadrupole Magnetic Steerer Beam Current Slits Cryostat Leg Bending Magnet Beam Profile Monitor Neutron Production Target Figure 3.3: High Energy Beam Transport through a series of four evacuated acceleration tubes, consisting of alternating glass and stainless steel sections. The eciency for transporting the beam through the accelerator typically ranges from 55% at the lowest energies used in this experiment to 70% at the highest. Upon leaving the accelerator, the proton beam must be transported to the neutron production target (Figure 3.3). The beam is analyzed by passing through the switching magnet with a de"ection of 59 . The eld of the magnet is regulated by feedback from a nuclear magnetic resonance probe. Currents from horizontal slits placed at the exit of the magnet feedback to the corona circuit of the accelerator, keeping the 3.1. Charged-Particle Beam 37 energy of the beam constant. A nal bending magnet (cryostat leg bending magnet) directs the beam toward the neutron production target. The beam is focused by three magnetic quadrupole doublets. A carbon-foil polarimeter allows the measurement of the beam polarization (Section 3.2). Most of the beam steering between the exit of the accelerator and the cryostat leg bending magnet is accomplished by three sets of slit feedback loops. These feedback loops keep the beam centered on the slits in the horizontal plane by steering the beam such that the dierence in current from the left slit and the right slit is zero. Similarly, the beam is centered in the vertical plane using currents from the up and down slits Gou84]. The rst feedback loop keeps the beam centered at the entrance to the analyzing magnet. The second centers the beam at the entrance to the polarimeter, and the third centers the beam at the entrance to the cryostat leg bending magnet. The size of the slit openings can be adjusted to t the size of the beam. After the cryostat leg bending magnet, the beam is transported approximately 2 m to the neutron production target. A NEC beam prole monitor allows the shape of the beam to be viewed on an oscilloscope. In addition, signals from the beam prole monitor are read into the data acquisition system and used to steer the beam so that its centroid is centered on the beam pipe (Section 3.6). A nal slit feedback loop holds the beam centered on the neutron production target. This set of slits is a xed quad collimator and is described in Section 3.3.1. Since the proton beam must approach the cryostat quite closely, it must be shielded from the eld of the superconducting magnet. For this reason, the last 1.2 m of the beam pipe is made of soft iron with 5.6 mm walls. Inside of the pipe are two concentric cylinders of a high permeability alloy,2 1 mm thick. The beam current striking the neutron production target is integrated using a Brookhaven Nuclear Instruments 1000 whose output is a series of pulses with a frequency proportional to the current. These pulses are converted to a NIM level signal by a Ortec 416A gate and delay generator and then the width is set with a LeCroy 222 gate and delay before being sent to the data acquisition electronics (Figure 3.4). The analog output of the current integrator goes to a discriminator with upper and lower levels. The discriminator sends an inhibit signal to the computer to stop data acquisition when the beam current is outside of the window dened by the upper and lower levels. The beam current on target is 2 CO-NETIC AA, Magnetic Shield Corporation, Perfection Mica Company. 38 Beam Stop Chapter 3. Experimental Apparatus 1 µs Gate & Delay Current Integrator Gate & Delay 75 ns BCI Analog Discriminator INHIBIT Figure 3.4: Beam Current Integration Electronics y ˆ x ˆ ŝ ẑ k̂ Figure 3.5: Madison Convention Coordinate System typically 0.5{1.5 A. 3.1.3 Spin Transport Just as important as transporting the polarized proton beam to target is having it arrive with the proper spin orientation. The Madison convention Sat71] denes a coordinate system for describing the polarization of the beam (Figure 3.5). In order 3.1. Charged-Particle Beam 39 to produce the transversely polarized neutron beam required for the T measurements, the proton beam must arrive at the neutron production target with its spin oriented transverse to the beam direction (^z) and in the horizontal plane (^x direction). Although the proton beam emerges from the polarized ion source with a longitudinal spin orientation, its direction can be changed by the Wien lter spin precessor. Propagation of the beam through dipole bending magnets can aect the spin direction and must be taken in to account in setting the spin precessor. For beams polarized in the y^ direction (transverse vertical), spin precession does not occur in the bending magnets and the only consideration is to properly set the initial spin direction at the polarized ion source. For beams polarized in the x^ direction (transverse horizontal), however, spin precession does occur in the bending magnets and must be compensated for by setting the spin direction at the source. This is the case for the T measurements as the spin of the beam must be in the same direction as that of the polarized target, which is determined by the superconducting magnet. When a beam of spin 1=2 particles is bent by a magnet, the spin direction is not necessarily changed by the same angle as the propagation direction. A spin component normal to the magnetic eld will precess. If the angle of de"ection is , and the angle of spin precession is , it can be shown that the two angles are related by the g-factor of the particle Lew87]. = g (3:2) For the special case g = 1, the spin precession angle is the same as the de"ection angle, maintaining the relationship between the beam propagation direction and the spin direction. For g = 6 1, the angle between the propagation and spin directions, , changes. In the case of a proton beam with spin oriented in the x-z plane, increases according to + + final = initial + (gp ; 1) (3:3) where gp = 2:7927. For a negative hydrogen ion beam, however, the beam de"ection is in the opposite direction. ; ; final = initial ; (gp + 1) (3:4) Since the spin precession depends only on the angle of de"ection, the spin orientation at any point along the beam path can be calculated (Figure 3.6). Because the 40 Chapter 3. Experimental Apparatus ŝ IP IS ŝ 165° 30° 52° Accelerator 59° 54° 20° ŝ ŝ 90° Target Figure 3.6: Spin Precession for a Proton Beam, Transverse Horizontal on Target 3.2. Charged-Particle Polarimeter 41 cryostat leg bending magnet is located between the polarimeter and the target, there is a dierence in spin angle between the two. The de"ection of the magnet has been determined by surveying to be 19.82 0.04 . This gives a dierence in spin angle of 35.53 0.07 . All beam polarization measurements made with the carbon-foil polarimeter (Section 3.2) when the spin is transverse horizontal on target must then be corrected by cos(35.53 ). Since the beam produced by the polarized ion source is polarized along the beam direction, a spin precessor is needed to produce transverse polarization. The Wien lter3 performs this task with crossed electric and magnetic elds. The magnetic eld precesses the spin, while the electric eld compensates for the de"ection to keep the beam straight. Electrostatic quadrupole doublets reduce the size of the beam so that all particles traverse nearly the same magnetic eld. The Wien lter can be mechanically rotated about the beam direction, allowing any spin orientation to be achieved. The Wien lter has been calibrated by measuring the polarization in the x^ direction at the polarimeter as a function of magnetic eld (Figure 3.7). The non-linearity observed at the higher elds is due to a problem with the gaussmeter used to measure the magnetic eld.4 For this reason, only eld settings below 1200 G are used for spin precession. The portion of the curve from 0 to ;1200 G has been t to a sine function 2B max Px = Px sin B + 0 (3:5) 0 where B0 = 1653:12 G and 0 = 2:3224. From this data, Wien lter magnetic eld and mechanical angle settings, B and , can be determined for any spin orientation.5 Table 3.1 lists the settings which are important for this experiment. 3.2 Charged-Particle Polarimeter Since T is proportional to the neutron polarization, the beam polarization must be determined for each measurement. Although it is dicult to measure the neutron beam polarization directly, it is relatively easy to monitor the polarization of the ANAC model 2170 Spin Precessor. The gaussmeter has since been replaced. The new unit does not exhibit this behavior. It should be noted that a magnetic eld is in the y^-direction ( = 0 ) rotates the spin orientation in the x-z plane. 3 4 5 42 Px Chapter 3. Experimental Apparatus 1:0 0:8 0:6 b b b b b b b b 0:4 b 0:2 b b b b b b b 0:0 b b ;0:2 b b ;0:4 b b ;0:6 b b b ;0:8 ;1:0 0:0 ;500:0 ;1000:0 ;1500:0 ;2000:0 ;2500:0 B (G) Figure 3.7: Wien Filter Calibration for the 59 Beam Line as Measured by the Proton Polarimeter 3.2. Charged-Particle Polarimeter 43 B (G) Spin Orientation 0 ;861:17 ;x^ (Target) 0 ;1024:32 ;x^ (Polarimeter) 0 ;34:61 +^x (Target) 0 ;197:76 +^x (Polarimeter) +90 ;413:28 ;y^ +90 +413:28 +^y 0 +378:67 ;z^ (Target) 0 +215:52 ;z^ (Polarimeter) 0 ;447:89 +^z (Target) 0 ;611:04 +^z (Polarimeter) Table 3.1: Wien Filter Settings charged-particle beam and then relate this value to the neutron beam using known polarization-transfer coecients, Kyy . For this reason, a carbon-foil polarimeter is used to monitor the polarization of the proton beam during T measurements. This polarimeter has been calibrated by a neutron polarimeter in separate measurements (Section 4.2). The polarimeter consists of a carbon foil and two solid state charged-particle detectors contained within a small scattering chamber (Figure 3.8). The chamber is made of a cylindrical body with two arms as particle "ight paths in order to minimize the internal volume and thus reduce the need for vacuum pumping. The chamber is constructed of aluminum with the arms welded to the body. Dependex "anges are used at the entrance and exit of the chamber and at the ends of the arms. The detectors are mounted on blank dependex "anges at the ends of the arms with three sets of tantalum collimators dening the scattered beams with an angular acceptance of 3:46 . The carbon foil is 22 mm in diameter and 5 g/cm2 thick and is mounted on a plunger so that it can be removed from the beam when not in use. Slits in front of the polarimeter dene the beam to be 12.7 mm 12.7 mm. The slit currents are fed back to a steerer which keeps the beam centered on the entrance of the polarimeter. The detectors and all but the rst set of collimators are electrically isolated to reduce noise pickup. The detectors are silicon charged-particle detectors with 300 mm2 active 0 44 Chapter 3. Experimental Apparatus Detector 40° Beam Foil Tantalum Collimator Figure 3.8: Proton Beam Polarimeter areas and depletion depths of 1000 m. The angle of the arms is xed, for simplicity, at 40 . This angle is chosen to provide a relatively large analyzing power over a range of energies for protons elastically scattered from carbon. The signals from the detectors pass through Ortec 142 preampliers at the polarimeter and are then further amplied by Ortec 572 ampliers in the control room. The unipolar outputs of the ampliers are fed to Ortec 551 timing single-channel analyzers (TSCA). This setup is shown schematically in Figure 3.9. The TSCA windows are set to select one peak in the linear energy spectrum from the ampliers and the outputs are sent to the data acquisition electronics. The window is set by observing the delayed output of the amplier on an oscilloscope while triggering on the TSCA output. When a proton beam is used, only three peaks are observed. In order of increasing energy of the detected particle they are 1H(p,p)1H, 12C(p,p1)12C, and 12C(p,p0)12C. In the case of elastic scattering from hydrogen, it is the recoiling proton which is detected. Note that because of its excitation energy of 4.44 MeV, the rst excited state of carbon will not be observed at lower beam energies. The TSCA windows are 3.2. Charged-Particle Polarimeter Top Solid State Detector Bottom Solid State Detector Preamp 45 Amp Preamp TSCA Amp TSCA Gate & Delay Gate & Delay 75 ns TPOL 75 ns BPOL Figure 3.9: Proton Beam Polarimeter Electronics set around the 12C(p,p0)12C peak and the counts from both detectors are read by the data acquisition electronics into scalers. Asymmetries between the number of counts from each detector, "ppol, are then calculated for each spin state. L ; NR (3:6) "ppol = N NL + NR Here, NL and NR refer to the number of counts in the left and right detectors by convention, although the detectors are physically up and down for most measurements. The proton beam polarization, Pp, is given by (3:7) Pp = A"ppol y sin where Ay is the analyzing power at 40 for 12C(p,p0)12C (Table 3.2) and is the azimuthal angle of the spin axis in the polarimeter.6 The analyzing powers are obtained by tting published analyzing power data as a function of angle Mos65, Ter68] (Section 5.1.2). The neutron beam polarization is then obtained by knowing the polarization transfer coecient Kyy for the 3H(~p,~n)3He source reaction. 0 Pn = Kyy Pp 0 (3:8) Beam polarizations have been observed to remain very constant with time Kos90], allowing the polarization measurements to be made every 2{4 hours. Removing the Since the polarimeter is placed before the last bending magnet, the angle of the spin is not the same in the polarimeter and on target. For protons the dierence is 35.53 0.07 . 6 46 Chapter 3. Experimental Apparatus Ep 4.66 5.04 5.41 5.78 5.89 6.18 6.77 Ay ;0:496 ;0:724 ;0:535 ;0:775 ;0:812 ;0:851 ;0:520 Ay 0.022 0.009 0.024 0.015 0.022 0.009 0.009 Ep 7.21 7.55 7.99 8.66 8.90 9.15 9.60 Ay ;0:278 ;0:339 ;0:131 ;0:015 0.125 ;0:010 ;0:348 Ay 0.015 0.016 0.017 0.014 0.041 0.015 0.054 Table 3.2: Analyzing Powers for 12C(p,p0)12C at lab = 40 carbon foil between polarization measurements increases the beam current on target by approximately 40{100%, depending on the beam energy. 3.3 Neutron Beam The polarized neutron beam is produced as a secondary beam from the polarized proton beam. The beam is produced by an essentially unshielded source and is collimated after passing through the polarized proton target. The neutron detectors are placed at 0 inside a large shield. It is important to know the characteristics of the beam including its polarization, average energy, and energy spread. In addition, it is important to determine the amount of unpolarized background in the beam and the amount of depolarization caused by the superconducting magnet used in the polarized target. 3.3.1 Neutron Beam Production The polarized neutron beam is produced by the 3H(~p,~n)3He reaction Don71, Sim73]. This reaction is chosen because the threshold of 0.764 MeV makes the production of low energy neutron beams with an acceptable energy spread possible. By comparison, the 2H(~d,~n)3He reaction Lis75] has a positive Q value of 3.3 MeV making it dicult to produce neutron beams at 0 with energies less than 6 MeV. In addition, the 3H(~ p,~n)3He reaction has a large polarization transfer coecient at 0 (Section 5.2). The tritium is in the form of tritiated-titanium on a 0.51 mm thick 58Ni backing. A 3.3. Neutron Beam 47 Ceramic Insulator Suppressor Ring Proton Beam Tritiated Foil Havar Foil He Dependex Flange Neutron Beam He Fill Line Quad Collimator Figure 3.10: Tritiated-Titanium Foil Holder solid compound is used instead of gaseous tritium for safety considerations. The 58Ni backing stops the proton beam without producing neutrons due to the high threshold of 9.44 MeV for the 58Ni(p,n)58Cu reaction. To contain any tritium which might be released from the foil, as well as "akes of Ti 3H2, a 1 bar helium cell with a 2.5 m Havar entrance foil is placed in front of the tritium foil deR89]. A four-segment tantalum collimator denes the proton beam to be 4.76 mm 6.35 mm (vertical horizontal) and provides feedback to a magnetic beam steerer. The foil holder, shown in Figure 3.10, is electrically isolated to allow beam current integration and a suppressor voltage of ;300 V is applied to prevent leakage currents from secondary electrons. Since the number of neutrons produced is proportional to the number of protons incident on the tritiated-titanium foil, it is not necessary to measure the neutron "ux directly. Because tritiated-titanium foils cannot hold as much tritium as a gas cell, the neutron count rates are low, typically of order 100 s;1 after being attenuated by the target and with a detector eciency of approximately 25%. The proton beam loses energy in passing through the havar foil, the helium gas, and the tritiated-titanium, making the energy of the neutron beam dicult to determine exactly. In addition, these losses determine the energy spread of the neutron beam. For these reasons, it is important to both estimate the energy losses for the proton beam and to directly measure the energy of the neutron beam. The energy losses are calculated using the FORTRAN code BABEL Bow82]. The average energy of the beam is taken to be that of the neutrons produced half way through the tritiated- 48 Chapter 3. Experimental Apparatus Ep (MeV) En (MeV) En (keV) 3.02 2.00 114.5 4.66 3.71 80.6 5.41 4.48 71.7 5.89 4.97 67.1 6.18 5.27 64.6 6.77 5.87 60.2 7.21 6.31 57.3 Table 3.3: Calculated Average Neutron Energies and Energy Widths titanium. The energy width is taken to be the dierence in energy between neutrons produced at the start and the end of the tritiated-titanium added in quadrature to the straggling of the proton beam as it passes through the havar and the helium. The straggling is estimated to be one tenth the total energy loss in these materials. The results are shown for the energies of interest in Table 3.3. To determine the neutron energy more accurately, direct measurements are needed. For this purpose, yield curves have been measured for carbon and oxygen targets for neutron energies in the vicinity of known resonances using an unpolarized beam Nat91, Nat90]. Figure 3.11 shows the total cross section for carbon in this region, while Figure 3.12 shows the total cross section for oxygen. The carbon target is a block of graphite approximately 130 mm thick and the oxygen target is a plastic cylinder of water approximately 127 mm thick. The targets are placed between the cryostat and the collimator and the number of neutrons at 0 counted for a xed amount of proton beam charge. The measurement is repeated with the target removed in the case of carbon, and with the target replaced by an empty plastic cylinder in the case of oxygen. The total cross section is then given by Nin (3:9) Total = ;x ln Nout where Nin and Nout are the number of neutrons counted with the target in and out and x is the target thickness. The results of this measurement are shown in Figure 3.13 for carbon and in Figure 3.14 for oxygen. Since only the positions of the resonances are important, it is not necessary to know x accurately. Comparing the positions of the resonances with the known values gives a calibration for the relationship between 3.3. Neutron Beam 49 (mb) 10:0 8:0 6:0 4:0 2:0 0:0 0:0 2:0 4:0 6:0 8:0 10:0 En (MeV) Figure 3.11: ENDF/B-VI Evaluation for the Total Neutron Cross Section of Carbon 50 Chapter 3. Experimental Apparatus (mb) 10:0 8:0 6:0 4:0 2:0 0:0 0:0 2:0 4:0 6:0 8:0 10:0 En (MeV) Figure 3.12: ENDF/B-VI Evaluation for the Total Neutron Cross Section of Oxygen 3.3. Neutron Beam 51 ; ln(Nin =Nout ) 2:5 b b b 2:0 b b b b b b b b b b b b b b 1:5 b b b b b b b b b b b b b b b b b b b b 1:0 0:5 0:0 2:0 3:0 4:0 5:0 Ep (MeV) 6:0 7:0 Figure 3.13: Measured Total Neutron Cross Section of Carbon 8:0 52 Chapter 3. Experimental Apparatus ; ln(Nin =Nout ) 3:0 2:5 b b b b b b 2:0 b b b b b b b b b b b b b b b 1:5 1:0 0:5 0:0 5:0 5:2 5:4 5:6 5:8 6:0 6:2 Ep (MeV) 6:4 6:6 6:8 Figure 3.14: Measured Total Neutron Cross Section of Oxygen 7:0 3.3. Neutron Beam 53 Ep (MeV) En (MeV) 3.02 1.94 4.66 3.65 5.41 4.42 5.89 4.91 6.18 5.21 6.77 5.81 7.21 6.25 Table 3.4: Measured Average Neutron Energies En (MeV) 1.94 1.000 3.65 1.077 4.42 1.450 4.91 1.500 5.21 1.530 5.81 1.330 6.25 1.430 Table 3.5: Asymmetry Dilution Factor Due to the Unpolarized Neutron Background the incoming proton energy and the outgoing neutron energy (Table 3.4). Time-of-"ight studies using an unpolarized beam have shown that although no neutrons are produced by the 58Ni beam stop, background neutrons are produced by the slits, the havar foil, and titanium in the Ti 3H2, as well as by 13C deposited on the havar. No polarization has been observed of these neutrons in measurements using the polarized beam and the neutron polarimeter. The net eect of this background is to dilute the measured asymmetries by a factor . Values of as determined from the time-of-"ight spectra are listed in Table 3.5. The values of do not increase monotonically with energy as they depend upon the detector thresholds. Since the neutron beam must pass through the eld of the superconducting magnet, there is the possibility of spin precession. Fortunately, the neutron spins are 54 Chapter 3. Experimental Apparatus En (MeV) P=P0 1.94 0.978 3.65 0.984 4.42 0.986 4.91 0.987 5.21 0.988 5.81 0.989 6.25 0.989 Table 3.6: Calculated Neutron Depolarization Due to the Superconducting Magnet parallel to the direction of the magnetic eld and ideally are not eected. There are, however, regions where neutrons not exactly on the beam axis, due to the nite extent of the beam, experience small transverse eld components. This eect is greatest as the beam passes between the two coils of the split-coil magnet. The precession caused by these elds acts in opposite directions for neutrons on either side of the beam axis so that the average spin direction is preserved and the net eect is to depolarize the beam by some amount. In order to calculate this eect the FORTRAN code NEUTRONS is used (Appendix A). This program performs a Monte Carlo simulation of the beam passing through the magnetic eld. The trajectories chosen are a random sampling of those allowed by the collimator. The spin precession of each neutron is tracked and averages calculated over all trajectories. The results for 10,000 neutrons, tabulated in Table 3.6, are used in determining the neutron polarization during T measurements. 3.3.2 Neutron Beam Collimation In order to obtain a large solid angle, the neutron source must be placed as close as possible to the polarized target as its size limits that of the beam. For this reason, the collimator is placed after the target instead of in front of it as in a shielded source arrangement. Having a completely unshielded neutron source would, however, illuminate the superconducting magnet and other parts of the cryostat, greatly increasing the background. To alleviate this problem, a small pre-collimator is used. In addition, the neutron detectors are placed within a large shield (Figure 3.15) to further 3.3. Neutron Beam 0 m 55 1 m 2 m 3 m 4 m Cryostat Detectors Neutron Beam Copper Polyethylene Pre-Collimator Collimator Polyethylene Detector Shield Figure 3.15: Neutron Collimation and Shielding Figure 3.16: Neutron Pre-Collimator suppress the background. The pre-collimator, shown in Figure 3.16, is constructed of a copper cone with a rectangular double-truncated bore. The entrance opens at a total included angle of 30 in each plane, while the exit opens at an angle of 1.0 in the horizontal plane and 3.0 in the vertical plane. The size of the opening at the narrowest point is 5.33 mm 8.76 mm. The total included angle of the copper cone is 70 and is 197 mm in diameter at its base. The thickness of the cone in the beam direction is 133 mm, giving a neutron attenuation factor of 44 at 10 MeV Che87]. The precollimator is mounted on three stainless-steel rods 25.4 mm in diameter. Each rod 56 Chapter 3. Experimental Apparatus has a turnbuckle in the center and a dial indicator, allowing vertical adjustment to 25 m. The rods are connected to a rotating stage which sits on an x-y translation table. This arrangement allows precise control of the axial, horizontal, and vertical position of the pre-collimator as well as the horizontal and vertical angles. The positioning mechanism is constructed such that the pre-collimator tilts and rotates about its narrowest opening. This denes a beam spot at the center of the polarized target having dimensions 9.35 mm 25.65 mm with a solid angle of 0.53 msr.7 The collimator is a rectangular block of polyethylene with a tapered rectangular bore. It is 500 mm high, 400 mm wide, and 660 mm long. As in the case of the pre-collimator, the bore opens with a total included angle of 1.0 in the horizontal plane and 3.0 in the vertical plane. The opening at the entrance is 12.70 mm 38.10 mm and at the exit is 24.23 mm 72.69 mm. The collimator is mounted on precision linear bearings which ride on cylindrical rails. This allows the collimator to be rolled away from the cryostat, allowing access for maintenance. The frame supporting this mechanism is made of stainless-steel angle stock welded together. Non-magnetic materials are used whenever possible in order to minimize the force on the superconducting magnet. The collimator is designed to provide an attenuation factor of at least 400 for neutron energies up to 14 MeV. The entrance and exit of the collimator are used as reference points in aligning all other devices along the neutron beam path. The alignment of the collimator with the target has been veried with the target at 4 K using X-ray lm. The lm is exposed by bombarding a deuterium gas cell with 5 MeV deuterons. Gammas produced in the gas and in the tantalum beam stop are believed to be responsible for exposing the lm. Two lms are used, one between the cryostat and the collimator and one after the collimator with an exposure time of 30 minutes. Rare-earth radiators are placed in front of the lm to improve the sensitivity. The detector shield (Figure 3.17) is a rectangular block of polyethylene measuring 806 mm 806 mm 1016 mm. It has a straight rectangular bore, 57 mm 190 mm. The two neutron detectors are inserted from the top and bottom into a cylindrical hole. They are attached to aluminum plates on the top and bottom of the shield and contained within an iron pipe to provide magnetic shielding. This arrangement allows Since the pre-collimator is not very thick at the narrowest point, the actual solid angle is closer to the value of 0.9 msr dened by the collimator. 7 3.3. Neutron Beam 57 Magnetic Shield Beam Detectors Figure 3.17: Neutron Detector Shield the detectors to be easily removed for sighting through the collimation assembly. The shield is constructed of stacked 25.4 mm thick sheets of polyethylene with aluminum plates on the top and bottom and with vertical rods compressing the entire stack. The bore of the shield extends all the way through, allowing the neutron beam to exit without scattering back into the detectors. The iron pipe also has windows to reduce the attenuation of the beam. Additional shielding is placed around the beam entrance and around the detector access holes on the top and bottom. 3.3.3 Neutron Detection In order to measure T , the transmitted neutron "ux at 0 must be detected. Unlike charged particles, neutrons are not detected directly, but through secondary reactions. Recoil protons from a hydrogen-rich target provide one such method, and in the case of organic liquid scintillators, the scintillation "uid is also the hydrogen target. The neutron detectors in this experiment consist of such scintillators attached to photomultiplier tubes which detect the "ashes of light from the neutron events. These detectors are also sensitive to gammas, and a means of discriminating the two particles is necessary. Pulse-shape discrimination provides this information and is implemented using commercially available electronic modules. Two detectors are used to cover the entire solid angle of the neutron beam, using readily available equipment. 58 Chapter 3. Experimental Apparatus Organic liquid scintillators operate by detecting the recoil protons from 1H(n,n)1H reactions. The protons then interact with the scintillator, either by excitation or ionization, which emits light during de-excitation or recombination Swa60]. This light strikes the photocathode of the photomultiplier tube, producing electrons. The number of electrons is increased through secondary emission by the dynodes of the electron multiplier. The nal electron current is collected by the anode. As the light has dierent decay times for neutrons and gammas, the current appears as a pulse with dierent rise and fall times. This property is exploited by the pulse-shape discrimination electronics. The scintillator cells are aluminum cylinders 127 mm in diameter and 127 mm in length with one face made of glass to allow the light to exit. The insides of the cells are coated with a white re"ector paint to reduce light loss. The scintillator "uid is Bicron BC-501 and an expansion tube is provided to accommodate thermal expansion of the liquid without bubbles. The photomultiplier tubes are Hamamatsu type R1250, which have 127 mm diameter photocathodes. As photomultiplier tubes are greatly aected by magnetic elds, shields made of a high permeability material are placed around them. The scintillator-photomultiplier assembly (Figure 3.18) is mounted to a "ange which bolts to the detector shield. The electronics for the neutron detectors, shown in Figure 3.19, take the anode signals and produce a NIM level output pulse for each neutron detected. Signals used for determining dead-time corrections are also produced. The Link 5020 modules perform the functions of amplication, pulse-shape discrimination, and single-channel energy analysis Ada78, Lin]. This module distinguishes between neutrons and gammas by measuring the signal decay-times. The neutron output of the Link 5020 is converted to a fast NIM signal using a LeCroy 222 gate and delay generator before being sent to the data acquisition electronics. A second LeCroy 222 is congured as a 100 kHz pulser. Its output is ANDed with the live-time signal from the Link 5020. This signal and the ungated pulser signal are sent to the data acquisition system for calculating a dead-time correction, . = NNpulser (3:10) gated Although this calculation only takes into account the 400 ns per event dead-time of the Link 5020, this is a valid assumption since the dead-time of the detector is approximately 25 ns per event. 3.3. Neutron Beam 59 Mounting Flange Aluminum Plate Stainless Steel Support Rod Polyethylene Detector Shield Phototube Clamp Scintillator Support Ring Beam Figure 3.18: Neutron Detector Assembly 60 Chapter 3. Experimental Apparatus Top Anode Neutron Detector PSD n out Gate & Delay Live Time 75 ns TOP 75 ns AND TGAT 100 kHz Pulser 75 ns AND BGAT Live Time Bottom Anode Neutron Detector PSD n out Gate & Delay 75 ns BOT Figure 3.19: Neutron Detector Electronics 3.4 Neutron Polarimeter Although the polarization of the neutron beam is usually determined by measuring the polarization of the charged-particle beam, it is important to be able to measure the neutron polarization directly. A neutron polarimeter provides a means of calibrating such secondary polarimeters and measuring polarization transfer coecients. The polarimeter consists of a 4He target and neutron detector pairs placed symmetrically about the beam direction (Figure 3.20). Since the analyzing power of neutrons scattering elastically from helium can be calculated with an accuracy of approximately 0:01, the neutron beam polarization can be extracted from the measured left-right asymmetry. The polarimeter is placed between the neutron collimator and the detector shield when in use and is oriented to measure polarization in the x^ direction (horizontal plane). The distance from the neutron source to the center of the polarimeter target is 1.79 m. The target consists of helium gas at a pressure of 100 bar in a steel cylinder with 1 mm walls Tor74]. The active volume of the cell has a diameter of 44.6 mm and a height of 158.2 mm. The gas has approximately 5% xenon added in order 3.4. Neutron Polarimeter 61 Neutron Detectors Collimator Beam Target Cell Figure 3.20: Neutron Polarimeter to make it scintillate with the recoiling alpha particles. This arrangement makes it an active target and allows the measurement of timing coincidences and alpha energy, reducing the background from the neutron detectors. The gas cell, shown in Figure 3.21, has windows at both ends to allow the light from scintillations to be detected in photomultiplier tubes. The inside of the gas cell is coated with 0.8 mm of magnesium oxide as a re"ector. Evaporated on top of the re"ector is 120 g/cm2 of p -quaterphenyl which acts as a wavelength shifter, improving the response of the photomultiplier tubes. The photomultiplier tubes are Hamamatsu type H1161 having a diameter of 51 mm. The cell is lled using the system shown in Figure 3.22. The xenon is loaded rst by liquefying the gas in the condensation chamber and then allowing it to evaporate and expand into the target cell until a pressure of approximately 5 bar is attained. This method is necessary as the xenon is stored at too low of a pressure to ll the cell directly. Then, 99.999% helium is bled into the cell in stages until the nal pressure of 100 bar is reached. The gas lling system is constructed completely of stainless-steel with viton o-rings used as valve stem seals and is designed to withstand pressures of 330 bar. A sintered stainless-steel lter 62 Chapter 3. Experimental Apparatus Photomultiplier Tube Glass Window He – Xe Gas Fill Line Figure 3.21: High-Pressure Helium Gas Cell 3.4. Neutron Polarimeter 63 Pressure Gauge Target Cell Rupture Disc Xe Turbo-Molecular Pump He Backing Pump Trap Condensation Chamber Figure 3.22: Helium Cell Filling System cooled to liquid nitrogen temperatures is used to trap contaminants during the helium ll. The lter must be warmed above the melting point of xenon when lling with this gas. As small amounts of contaminants, such as nitrogen and oxygen, greatly reduce the performance of the cell by absorbing the light of scintillation, the cell and gas lling system are pumped by a turbo-molecular pump before use. The neutron detectors are placed symmetrically about 0 at angles where the product of the square of the analyzing power and the cross section is a maximum for 4He(n,n)4He. There are two such angle pairs, one forward and one backward, and two detector pairs are used. At the lowest energy measured, however, one maximum is too low in magnitude to be useful and only one detector pair is used. The detectors are organic liquid scintillators coupled to photomultiplier tubes through light guides. The light guides match the geometry of the 45 mm 158 mm 76 mm thick rectangular scintillator cells to the 51 mm diameter phototubes. The photomultiplier tubes are Hamamatsu type H1161. The neutron detectors are mounted to a detector ring having 1 graduations. The entire apparatus has been aligned optically with the neutron beam collimator. The forward detector pair uses an electronic setup (Figure 3.23) similar to that used for the main neutron detectors (Section 3.3.3). Since there are 64 Chapter 3. Experimental Apparatus 40 ns Timing L1 Anode Neutron Detector PSD n out Gate & Delay Live Time L1TIM 75 ns L1PSD 75 ns AND L1GAT 100 kHz Pulser 75 ns AND R1GAT Live Time R1 Anode Neutron Detector PSD n out Gate & Delay 75 ns R1PSD 40 ns Timing R1TIM Figure 3.23: Electronics for the Forward Neutron Detector Pair not enough Link 5020 PSD modules for all four detectors, a dierent setup must be used for the backward angle pair (Figure 3.24). In this setup, a Canberra 2160A PSD module is used to determine the zero-crossing of the anode signal after the peak of the pulse Can83]. By measuring the time between the start of the pulse, determined by a constant-fraction discriminator (CFD), and this zero-crossing, neutrons can be distinguished from gammas. Monte Carlo techniques have been used to calculate the eective analyzing power of the neutron polarimeter Tor75]. These calculations make corrections for the nite geometry of the target cell and neutron detectors, and for double-scattering events. Double scattering which includes neutrons scattering from materials in the target cell other than helium must be considered. The double-scattering events which are considered are He-He, He-Xe, Xe-He, He-Fe, and Fe-He. Since only events which 3.4. Neutron Polarimeter L2 Anode Neutron Detector 65 Stop PSD Strobe Start TAC 250 ns CFD 75 ns AND L2 20 ns 2 ns L2TIM Amp & TSCA L2TAC R2 Anode Neutron Detector Stop PSD Strobe Start TAC 250 ns CFD 75 ns R2 AND 20 ns 2 ns Amp & TSCA R2TIM R2TAC Figure 3.24: Electronics for the Backward Neutron Detector Pair 66 Chapter 3. Experimental Apparatus En (MeV) d (mm) Ay 1.94 107 408 0:764 0:010 5.21 50 408 ;0:629 0:010 121 338 0:919 0:015 5.81 51 408 ;0:632 0:010 121 308 0:916 0:015 Table 3.7: Eective Analyzing Powers for the Neutron Polarimeter include the detection of an alpha are recorded, single scattering events from elements other than helium need not be considered. Cross sections and analyzing powers are obtained from phase-shift data sets. The phase shifts for helium come from experimental data Sta72] as do the phase shifts for xenon and iron at the higher energies. At the lowest energy measured, 1.94 MeV, experimental cross section data for xenon are used with analyzing powers assumed to be zero. The calculated eective analyzing powers, Ay , are shown in Table 3.7 for the neutron energies at which measurements have been made. The length of the "ight paths, d, from the center of the target cell to the center of the detector is also given. For an event to be considered valid, three criteria must be satised. First, there must be a coincidence between the top and bottom photomultiplier tubes of the target cell, reducing the noise from the tubes. Second, the signal from the neutron detector must meet pulse-shape discrimination requirements, eliminating signals from gammas. Third, there must be a coincidence between the target cell signal and one of the neutron detector signals, in order to reject neutron background events. Figure 3.25 shows the electronic setup which performs these functions. The anode signals from the target cell photomultiplier tubes are used to form the coincidences, while the dynode signals provide the alpha recoil energy information. Summing the dynode signals improves the energy resolution of the detector as only part of the light of scintillation is deposited in each photomultiplier tube. The gains of the two tubes are matched using a 137Cs source. Since the alpha recoil energy is quite dierent for the forward and backward angles, the summed dynode signal is split into two signals which are amplied and delayed by dierent amounts before being recombined. This process creates two pulses for each alpha, one large in amplitude and one small. The large 3.4. Neutron Polarimeter 67 High Gain Linear Fan In/Out 250 ns Linear Fan In/Out 270 ns Linear Gate Amp E Low Gain Top Photo Tube Bottom Photo Tube Dynode Anode Anode CFD 200 ns AND CFD Dynode 20 ns 100 ns Start AND L1TIM 20 ns Stop L1TOF TAC 20 ns L1 Start AND R1TIM 20 ns Stop R1TOF TAC 20 ns R1 Start AND L2TIM Stop TAC L2TOF SCA L2TAC 20 ns R2TIM Start AND Stop TAC R2TOF SCA R2TAC OR OR 300 ns Figure 3.25: Coincidence Electronics for the Neutron Polarimeter 68 Chapter 3. Experimental Apparatus amplitude pulse is gated through to the data acquisition system only when an event occurs in one of the backward angle detectors. Conversely, the small amplitude pulse is gated through only when an event occurs in one of the forward angle detectors. This arrangement allows a single ADC to digitize the alpha recoil energy for both detector pairs. Neutron time-of-"ight from the target cell to the neutron detectors is determined using Ortec 467 time-to-amplitude converters (TAC). The start signal for each TAC comes from the timing signal of the corresponding neutron detector after the coincidence with the target cell anode coincidence signal. The coincidences are formed such that the timing signal from the neutron detector determines the timing of the TAC start signal. The stop signal comes from a delayed version of the target cell anode signal. This delay is adjusted to give a suitable range for the neutron timeof-"ight signals (typically 100{200 ns). Because of the coincidence requirements, the count rates are low, typically a few per second. 3.5 Polarized Proton Target The polarized proton target uses the brute-force technique SP86] of cooling the target material to low temperatures in a large magnetic eld. This method has the advantage of producing very thick targets with fewer polarized impurities than other methods such as laser pumping. The brute-force method makes use of the energy dierence between the dierent nuclear magnetic substates. The population pm of a substate with spin projection m measured in the direction of the magnetic eld B is given by emgB=kT pm = mX (3:11) =+I mgB=kT e m=;I where g is the nuclear g-factor and k is the Boltzmann constant. For protons, I = 1=2 and the target polarization PT is given by p;1=2 : (3:12) PT = pp+1=2 ; +1=2 + p;1=2 For a given temperature T and magnetic eld B , the polarization is then gB PT = tanh kT : (3:13) 3.5. Polarized Proton Target 69 The magnetic eld is produced using a 7 T superconducting magnet.8 The magnet is a split-coil solenoid with Nb-Ti windings. The magnetic eld in the sample region is uniform, having a homogeneity of 0.1% over 1 cm. The magnet is charged to its rated current of 86.2 A and placed in persistent mode, causing the eld to be essentially constant over the course of the measurement. The inductance of the magnet is approximately 30 H, storing over 100 kJ of energy at full eld. For protons in a 7 T eld, the polarization is given by PT = tanh(T0=T ) (3:14) with T0 =7.15 mK. The sample must then be cooled to a temperature of order T0 to produce signicant polarization. A 3He -4He dilution refrigerator is used to achieve these temperatures. 3.5.1 Dilution Refrigerator Since measurements of T require that the proton target remain polarized for several days at a time, a means of continuous cooling at temperatures of 10{20 mK is required. The only continuous method for cooling below 300 mK is the 3He -4He dilution refrigerator Lou74]. The principle of operation of the dilution refrigerator Lon51, Lon62] relies on a phase separation which occurs in 3He -4He mixtures at temperatures below several hundred millikelvin. One phase (dilute phase) is a mixture of approximately 6% 3He and 94% 4He, while the other (concentrated phase) is essentially pure 3He. Below 500 mK, the 4He, obeying Bose-Einstein quantum statistics because of its 0 spin, is super"uid in its ground state and has no entropy. The 3He, however, is spin 1/2 and obeys Fermi-Dirac statistics. Thus it has a non-zero heat capacity and entropy even as the temperature approaches absolute zero. A diusion of 3He from the concentrated phase to the dilute phase is analogous to evaporation and produces cooling. A dilution refrigerator (Figure 3.26) maintains an osmotic pressure gradient across the phase boundary by continuously pumping 3He from the dilute phase and restoring it to the concentrated phase. The phase separation occurs in the mixing chamber. 3He which is to be returned to the mixing chamber is rst condensed by the cold plate and is then cooled as much as possible by heat exchangers 8 American Magnetics, Incorporated, Oak Ridge, Tennessee. 70 Chapter 3. Experimental Apparatus He Out He In Cold Plate He Out Condenser Still Heat Exchanger Still Continuous Heat Exchanger Discrete Heat Exchangers Concentrated Phase Mixing Chamber Dilute Phase Figure 3.26: Dilution Refrigerator 3.5. Polarized Proton Target 71 in thermal contact with the dilute phase. 3He is removed from the dilute phase in the still. A heater in the still increases the recirculation rate by raising the vapor pressure of the 3He. The 4He, having a much lower vapor pressure remains behind. The cold plate is cooled to approximately 1 K by pumping on liquid 4He obtained from the surrounding liquid helium bath. In principle, the ultimate temperature of a dilution refrigerator is limited only by the eciency of the heat exchangers and temperatures of a few millikelvin have been reached by this method. The dilution refrigerator used in this experiment is an S. H. E. Model 430. The refrigerator has an ultimate temperature of 4.5 mK and a cooling power of 1.5 W at 10 mK. The 3He is recirculated at a rate of 500 mol/s by an Edwards 9B3 diffusion pump backed by a hermetically sealed mechanical pump. A copper cold nger is connected to the mixing chamber for attaching the target. The dilution refrigerator and the superconducting magnet are contained within a liquid helium cryostat (Figure 3.27). The cryostat holds the liquid helium necessary for operating the refrigerator and cooling the magnet, and provides thermal insulation. The cryostat requires 100 liters of liquid helium for the initial cooldown, and approximately 30 liters each day thereafter. Hollow brass cylinders are inserted in the magnet bore as spacers to remove most of the liquid helium from the neutron beam, reducing the attenuation of the beam. 3.5.2 Titanium Hydride Target The criteria for choosing a suitable material for use as a polarized proton target are very stringent. The hydrogen in the material must be polarizable by a magnetic eld and it must have a large heat conductivity at millikelvin temperatures. All other nuclei contained in the material must not be polarized by the magnetic eld. Molecular hydrogen, H2, is not suitable since the two nuclei in the molecule pair o with opposite spins in the para-hydrogen conguration. The heat capacity associated with converting the para-hydrogen to ortho-hydrogen is too large to be overcome in practice. Most hydrocarbons can be ruled out because of their low heat conductivities and because of the presence of 13C, which becomes polarized. Many other hydrogen compounds do not have a suciently high hydrogen content or have unwanted contaminants. One material which is suitable for use as a polarized proton target is 72 Chapter 3. Experimental Apparatus 0 m Liquid Nitrogen Shield Liquid Helium Vacuum Can Still 1 m 0.5 K Shield Mixing Chamber Superconducting Magnet Beam Spacer Target 2 m Figure 3.27: Dilution Refrigerator Cryostat 3.5. Polarized Proton Target 73 titanium hydride, TiH2, as demonstrated by Aures et al. Aur84]. Titanium hydride is made by heating very pure titanium in a hydrogen atmosphere. The TiH2 is then in the form of a powder which must be compressed to form a solid sample. The proton density of such a target is 9:0 1022 cm;3 as compared to 5:3 1022 cm;1 for solid H2. It has been shown that heat conductivities of approximately 5 10;5 W/cmK at temperatures of 10{20 mK can be attained with this type of sample Hee85]. This value is an order of magnitude below that of pure titanium Chi76]. Recent work indicates that the conductivity depends greatly on the purity of the material Li92]. The rst attempts to produce a target for this experiment used commercially available TiH2 with a purity of 99%. The powder was pressed into a silver box under a pressure of 0.4 GPa. Silver is chosen because of its excellent heat conductivity and because it does not become polarized. The polarization was determined to be only 15% by measuring the asymmetry in the neutron transmission (Section 4.1). This value corresponds to a temperature of approximately 50 mK. Since polarizing a sample and measuring its polarization is an expensive and time consuming process, a simpler method of determining the quality of the TiH2 target is needed. The Wiedemann-Franz law = LT (3:15) states that below 1 K the thermal conductivity, , of a metal is proportional to its electrical conductivity, , where L = 25 nW#/K2 is the Lorentz constant and T is the temperature. Because of this relation, a measurement of the electrical conductivity as a function of temperature gives an indication of the thermal properties of the material. Electrical resistance measurements of the rst TUNL TiH2 target at 300 K and at 80 K indicate that the resistance decreases as a function of increasing temperature. This behavior indicates the material is behaving as an electrical insulator and suggests that its thermal conductivity will be low as well. Similar measurements for the target used by Aures et al at KFA Karlsruhe, Germany Hee85] indicate that their sample behaves electrically as a metal, implying a relatively high heat conductivity. The dierence is thought to be due to impurities in the commercial TiH2. The nal TiH2 target is made from the same material as the Karlsruhe target.9 The powder is pressed into a rectangular copper box with the front and back faces We gratefully acknowledge the assistance of H. O. Klages at KFA Karlsruhe, Germany in providing the material for this target. 9 74 Chapter 3. Experimental Apparatus Silver Alloy Solder Copper Beam Titanium Hydride Figure 3.28: Titanium Hydride Target open, since the copper becomes polarized. Copper is chosen to more closely approximate the Karlsruhe design. The box, made from 2 mm copper plate, is placed within a hardened steel die. The powder is pressed into the box under a pressure of 1.5 GPa. A 0.5 mm silver foil was placed at the bottom of the die to allow the connection of thermometers to the TiH2, but good thermal contact was not established. As a result, the target polarization must be determined by nuclear measurements (Section 4.1). The rear face, which suered from mechanical defects, has been machined to give a smooth surface. A bored copper rod is soldered to the top of the box for attachment to the dilution refrigerator cold nger. The TiH2 target (Figure 3.28) has a density of 3.75 g/cm3 and a thickness of 22.4 mm giving a proton thickness of 0.203 b;1. The area of the TiH2 face is 14.0 mm 34.1 mm. The total amount of hydrogen contained in the target is 1.6 mol. Electrical resistivity measurements indicate a metallic behavior similar to the Karlsruhe target. Proton polarizations obtained from this target are in the range 40{50%, corresponding to temperatures of 13{17 mK. 3.5. Polarized Proton Target 75 3.5.3 Thermometry Although attempts to accurately measure the temperature of the TiH2 directly have been unsuccessful, it is still important to monitor various temperatures inside the cryostat as an indication of how well the dilution refrigerator is working. The simplest thermometers are resistors whose resistance changes with temperature. Approximately twelve carbon or thick lm resistors are attached to various parts of the refrigerator, including the cold nger, mixing chamber, still, and the discrete heat exchangers. In addition, two calibrated germanium resistors are used, one attached to the cold nger, and one to the mixing chamber. The resistance measurements are made using an AC four-lead resistance bridge. Although the resistance thermometers are the easiest to use, they are not useful below approximately 50 mK as the resistance saturates. In the temperature range 5{40 mK, nuclear orientation thermometry may be used. 54 Mn and 60Co are the most common isotopes used for this purpose Mar83]. A cobalt nuclear orientation thermometer consists of a small amount of 60Co embedded in a single crystal of natural cobalt. Below about 100 mK, the cobalt nuclei become aligned along the crystal direction, due to the large eective magnetic eld (several tesla) of the ferromagnetic material. The 60Co decays with a half-life of 5.3 years, emitting gammas with energies of 1333 keV and 1173 keV. For an unpolarized crystal, the radiation is emitted isotropically. As the crystal becomes more polarized, the radiation becomes anisotropic with a minimum "ux along the crystal axis. Such a crystal is mounted inside the cryostat in thermal contact with the cold nger. The cobalt is located outside of the bore of the superconducting magnet to minimize the eect of the magnetic eld on its polarization. An intrinsic germanium gamma detector is placed outside of the cryostat along the crystal axis. The gamma count rate is measured and compared to the rate at temperatures above 40 mK to determine the temperature. Measurements to an accuracy of 1 mK can be obtained by counting for approximately 300 s. Since the presence of a neutron beam makes accurate measurements impossible, this method is not always practical. In addition, since warm counts are required, the germanium detector cannot be moved once the refrigerator is cold. A third type of thermometer which is used to measure the temperature of the dilution refrigerator is the 3He melting curve thermometer (MCT) Kei90, Kei92]. 76 Chapter 3. Experimental Apparatus Capacitor Plates Epoxy He Chamber Fill Line BeCu Mounting Screw OFHC Copper Ag Powder 11.4 mm Figure 3.29: 3He Melting Curve Thermometer The MCT relies on the precise knowledge of the pressure versus temperature of 3He along the melting curve at low temperatures Gre86]. The MCT contains a mixture of solid and liquid 3He in what is essentially a constant volume (Figure 3.29). The cell is lled at 1 K through a capillary tube leading to a room temperature gas handling system. As the temperature decreases, a plug of solid 3He forms in the tube, sealing the cell at a constant average density. The top of the 3He chamber is a 250 m thick diaphragm to which one plate of a capacitor is attached. Changes in pressure are observed as changes in the capacitance, which can be measured to an accuracy of 1 10;5 pF out of a total capacitance on the order of 10 pF. This corresponds to a temperature sensitivity of approximately 1.5 K at 10 mK. By calibrating the capacitance versus pressure characteristics of the cell at 1 K with a precision Bourdon tube gauge, the absolute accuracy of the temperature measurement is estimated to be 0.36 mK. Two such devices are installed in the cryostat, one attached to a silver foil which touches the TiH2 target and one attached to the cold nger. 3.6. Data Acquisition Electronics 77 3.6 Data Acquisition Electronics Data from the measurements are collected by CAMAC modules controlled by a MBD11 Multiple Branch Driver Rob81]. The MBD is connected to a Digital Electronics Corporation VAXStation 3200 which supervises the data acquisition, stores the data, and performs online analysis. In addition, this computer performs part of the feedback steering of the proton beam. The data acquisition software runs under the TUNL XSYS data acquisition and analysis system Gou81]. 3.6.1 T Measurements For the T measurements it is necessary to measure count-rate asymmetries to an accuracy of order 1 10;4 . Instrumental asymmetries due to the data acquisition system must therefore be of order 1 10;5 or less. To achieve this level of stability, several methods are employed. First, the spin of the neutron beam is rapidly reversed at a rate of 10 Hz to reduce the eects of detector gain drifts and other instrumental shifts. Precision timing is used to insure that the same amount of time is spent counting in each spin state. Finally, the data are collected and stored in 800 ms time slices, allowing asymmetries to be calculated for very short periods of time before being averaged together. All the data are collected by counting pulses in scalers since all particle discrimination takes place in hardware (Section 3.3.3). In addition to the neutron detector and polarimeter counts, pulses from the beam current integration and dead-time measurements are recorded. The fast spin-reversal is controlled by a NIM Spin State Control (SSC) module constructed in the TUNL electronics shop. The module produces an eight-step spin sequence: + ; ; + ; + +; which cancels the eects of detector drifts to second order in time Kos90]. All data fed to the data acquisition system pass through a Phillips 706 sixteen-channel discriminator. Gate and delay generators are used to create a veto signal which blocks the data during a spin reversal (Figure 3.30). One pair of gate and delay generators determine the exact timing, vetoing the data starting 2 ms before a spin reversal until 5 ms past. This amount of time is needed to insure that the polarization of the source has stabilized Kos90]. A second pair of gate and delay generators blank the output of the rst when an inhibit signal is received by the SSC. This arrangement allows the timing circuits of the rst set to 78 Chapter 3. Experimental Apparatus 3 µs Gate & Delay Gate Linear Gate Stretcher ADC Gate CAMAC Preset Scaler 250 ns 250 ns Gate & Delay ADC Interface SCAN ROUT INHIBIT SF + Start CAMAC Spin State Output – Reset Controller Register WF IPIS Spin Control Gated Clock Del Gate & Delay 10 Hz Clock Del Gate & Delay Blank Gate & Delay Del Gate & Delay 7 ms 100 ms Veto 75 ns 125 ns TOP BOT TGAT BGAT PULS CAMAC Scalers BCI TPOL BPOL Figure 3.30: Data Acquisition Electronics for T Measurements 3.6. Data Acquisition Electronics 79 run continuously, avoiding drifts which occur when they are turned on and o. Data collection is inhibited when the beam current drops below a preset level and while writing the data to disk at the end of a run. The SSC determines the spin state of the ion source by sending TTL level signals to the ABPIS through ber optic cables. The signals modulate the RF supplies of the two transition units. The SSC provides duplicates of these signals, labeled \+" and \;", for routing information. The \+" signal is read into an analog-to-digital converter (ADC) to signal the computer that a spin reversal has occurred. The computer reads all the CAMAC scalers and stores them according to spin state, a high value in the ADC indicating spin up , and a low spin down . At the end of an eight-step sequence, the CAMAC preset is decremented. A timing diagram is shown in Figure 3.31. The data are stored in the computer as spectra of counts versus time. Each time channel corresponds to one 800 ms eight-step spin sequence. There are two spectra for each scaler, corresponding to the two spin states. A run is comprised of 1023 such spin sequences. At the end of a run, the data are written to disk and asymmetries are calculated for display on the control-room monitor. The end of a run is signaled by the CAMAC preset scaler reaching zero. The beam prole monitor (BPM) operates by rotating a helical wire through the proton beam. The geometry is such that the wire travels across the beam rst horizontally and then vertically Nat84]. The current collected by the wire is preamplied before being sent to the control room. In addition, a magnetic pickup on the rotating shaft picks up signals from three magnets mounted on the chamber. The rst magnet produces the strongest signal and is used as a trigger to indicate the beginning of a scan. The other magnets generate ducial marks which indicate the centers of the x and y passes, respectively. The signals from the collector and the ducials pass through isolation ampliers since they are referenced to dirty ground. In addition, the ducial signal is split with one branch going through a broad-band isolation transformer to preserve timing information. The trigger signal (the largest of the ducial marks) res an EG&G T140/N zero-crossing detector. The output of the zero-crossing detector starts a linear ramp. The collector signal, the ducial signal, and the ramp are sampled by linear gate stretchers before being sent to ADC's. The gate signals are generated by a variable frequency clock which runs during the ramp. This setup is shown in Figure 3.32. The beam prole information (Figures 3.33 and 3.34) is stored in two sets of 80 Chapter 3. Experimental Apparatus 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 s Clock + – Veto Blank Clock 5 ms 2 ms Veto 3 ms Blank 0.0 0.1 s Figure 3.31: Data Acquisition Timing Diagram 3.6. Data Acquisition Electronics 81 Beam Collector Iso Profile Amp Monitor Linear Gate Stretcher Fiducials ADC Gate ADC Interface Isolation Amplifier Linear Gate Stretcher Iso Amp Zero Crossing Detector ADC Gate Gate Or ADC Interface 10 ns 75 ns Gate & Delay Gate Linear Gate Stretcher Ramp Generator Period ADC Gate Gate Or Gate Gated Clock ADC Interface 1 µs Bit 14 2.5µs SCAN ROUT Del Gate & Delay +x CAMAC Output Register –x +y –y 1 µs Gate & Delay Beam Steerer Control Figure 3.32: Beam Prole Monitor Feedback Steering Electronics 82 Chapter 3. Experimental Apparatus Counts 1000000 800000 600000 400000 200000 0 0 100 200 300 400 500 Channel Figure 3.33: Beam Prole Monitor Collector Spectrum spectra. The rst set has the ramp as the x-axis and the collector as the y-axis. The second again has the ramp as the x-axis, and the ducials as the y-axis. Each set has one spectrum for spin up , one for spin down , and one which is the sum of the two. A FORTRAN code running as a subprocess calculates the centroids of the two collector peaks and the centers of the x and y ducial marks. Bits are sent at 1 s intervals to an automatic steerer according to the relative position of the centroids and ducial marks. The output signals are capacitively coupled to preserve the ground isolation. 3.6. Data Acquisition Electronics 83 Counts 10000000 8000000 6000000 4000000 2000000 0 0 100 200 300 400 Channel Figure 3.34: Beam Prole Monitor Fiducial Spectrum 500 84 Chapter 3. Experimental Apparatus 3.6.2 Neutron Polarization Measurements For the neutron polarization measurement, it is necessary to measure asymmetries to an accuracy of order 5 10;3 . Although this requirement is somewhat relaxed compared to the T measurements, the fast spin reversal techniques described above are used. There are three major dierences, however. First, the signals obtained from the polarimeter are in the form of energy and time-of-"ight information and must be read by ADC's. In contrast, the detector signals for the T measurements are simply read by scalers. Second, the routing of the scaler signals is performed by fanning out the signals to two sets of scalers and selectively enabling just one set according to the spin state (Figures 3.35 and 3.36) instead of using software to perform the routing. Third, the ADC and scaler data is accumulated for an entire run instead of being stored in 800 ms time slices. Two ADC's are used for the neutron polarimeter. The rst records the alpha recoil energy signal from the center detector. Gates for this ADC can come from coincidences with any of the four neutron detectors (Section 3.4). The gates pass through the Phillips 706 discriminator module so that they are vetoed during spin reversal as are the scaler signals. Next, coincidences are formed between gates from the forward detectors and their corresponding PSD signals. The PSD coincidences for the backward detectors are generated at an earlier stage. Routing information is created by the ADC interface according to the neutron detector and the spin state. Delays are applied to the linear signal and the gates to allow a proper time relation to be established. The second ADC records the neutron time-of-"ight signals from the four neutron detectors. The linear signals are are fanned together using sum and invert ampliers. The gate for this ADC comes from the interface for the rst ADC so that the ADC's are triggered together and both alpha energy and neutron time-of"ight are recorded for each event. The events are stored in two-dimensional spectra with neutron time-of-"ight as the x-axis and alpha recoil energy as the y-axis. There are two spectra for each of the four neutron detectors, corresponding to the two spin states. 3.6. Data Acquisition Electronics 85 3 µs Gate & Delay Gate Linear Gate Stretcher ADC Gate CAMAC Preset Scaler 250 ns 250 ns Gate & Delay ADC Interface ROUT + INHIBIT ROUT – SCAN ROUT + Start CAMAC Spin State Output – Reset Controller Register SF WF IPIS Spin Control Gated Clock Del Gate & Delay 10 Hz Clock Del Gate & Delay Blank Gate & Delay Del Gate & Delay 7 ms 100 ms Veto 75 ns 125 ns TGAT SCAL1 BGAT SCAL2 PULS SCAL3 BCI SCAL4 TPOL SCAL5 BPOL SCAL6 L1PSD GATE1 R1PSD GATE2 L2TIM GATE3 R2TIM GATE4 Figure 3.35: Neutron Polarimeter Data Acquisition Electronics (Part 1) 86 Chapter 3. Experimental Apparatus L1TOF R1TOF Del Amp L2TOF ADC Gate R2TOF ADC Interface Del Amp E 125 ns Gate & Delay GATE4 GATE3 GATE2 250 ns Gate & Delay R1TIM 1 µs AND Gate & Delay AND Gate & Delay 125 ns 250 ns GATE1 Gate & Delay Gate & Delay L1TIM ADC Gate ADC Interface Or Bit 15 125 ns ROUT + ROUT – 125 ns Enable SCAL1 SCAL2 SCAL3 CAMAC Scalers SCAL4 SCAL5 SCAL6 Enable CAMAC Scalers Figure 3.36: Neutron Polarimeter Data Acquisition Electronics (Part 2) Chapter 4 Experimental Procedure 4.1 T Measurements As the total cross section is related to the forward scattering amplitude by the optical theorem, T can be measured by observing the change in transmitted neutron "ux through the polarized target upon reversing the beam polarization. The transmitted "ux, N , is given by N = N0e;x (4:1) where N0 is the incident "ux, x is the target thickness, and is the cross section. For the case of a polarized beam and a polarized target, the cross section can be expressed as 1 (4:2) = 0 ; Pn PT 2 where 0 is the unpolarized cross section, PT is the proton target polarization, and Pn are the neutron beam polarizations for the two spin-states. The beam polarizations can be expressed in a more convenient form as Pn = Pn Pn where + ; Pn = Pn +2 Pn + ; Pn = Pn ;2 Pn : 87 (4:3) (4.4) 88 Chapter 4. Experimental Procedure The neutron transmissions for the transverse geometry are then N = N0e;x(0 ; 2 Pn PT T )e 12 Pn xPT T : 1 An asymmetry can be formed using the two beam transmissions N+ ; N; " = N + + N; (4:5) (4.6) + 12 Pn PT xT ; e; 12 Pn PT xT e (4.7) = + 1 Pn PT xT ; 1 PnPT xT +e 2 e 2 (4.8) = tanh 21 Pn PT x T which is independent of the incident "ux and the unpolarized cross section. Since the argument of the hyperbolic tangent is small ( 0:025), the approximation tanh(x) x can be made " = 21 Pn PT x T : (4:9) Then in order to determine T , the neutron transmission asymmetry, ", must be measured as well as the two polarizations, Pn and PT , and the target thickness, x. The neutron beam polarization, Pn, can be inferred from the proton beam polarization measured by the carbon-foil polarimeter. This measurement is relatively fast, with a sucient statistical accuracy being reached in a few minutes, allowing it to be repeated every few hours. The product of the proton target polarization and target thickness, PT x, is determined by measuring the neutron transmission asymmetry at an energy below 2 MeV where the uncertainty in T is small. This measurement, however, is slow, taking on the order of 12 hours to reach a sucient statistical accuracy. Because of the length of time required, this measurement is usually only performed once, at the beginning of an experiment. In addition, the instrumental asymmetry of the system due to errors such as beam misalignment must be measured. This requires taking neutron transmission asymmetry data with the target warmed to a high enough temperature (> 800 mK) to depolarize the protons. The true asymmetry is then taken to be the dierence between the cold and warm values. Because the cold and warm measurements contribute equally to the nal uncertainty, the most ecient use of time is to measure them to equal statistical precision. 4.2. Kyy Measurements for the 3H(~p,~n)3He Reaction 0 89 The cooling of the target begins 3{5 days before the start of measurements, which typically last for 7 days. This allows the dilution refrigerator to run for at least 48 hours before the target is needed, giving it time to reach thermal equilibrium. Because of the long time required to polarize the target, the cold measurements are made rst. The beam polarization is measured at an energy where the 12C(p,p0)12C analyzing power is known, and the energy is then lowered to En = 1:94 MeV to determine PT x. TSCA windows are set for the carbon-foil polarimeter and PSD parameters are adjusted for the neutron detectors. The gas cell in front of the tritiated-titanium foil is evacuated and then lled with helium. Helium is added to the cell approximately once per day to replace the gas which diuses through the beam entrance foil. Neutron transmission asymmetries are measured for approximately 12 hours at this energy. Beam polarization measurements are made every few hours to monitor the state of the polarized ion source. Once the target polarization has been determined, measurements are made at higher neutron energies where T is not known. With each energy change, the polarimeter TSCA windows and neutron detector PSD settings must be reset. Data is collected for 18{48 hours at each energy, with beam polarization measurements made every few hours. After making measurements at all energies of interest, the target is warmed to approximately 1 K. The asymmetry measurements are then repeated at each energy, including 1.94 MeV. 4.2 Measurements for the 3H(~p,~n)3He Reaction Kyy 0 The polarization-transfer coecient, Kyy , can be written as the ratio of outgoing neutron polarization to incoming proton polarization 0 Kyy = PPn : p 0 (4:10) The measurement of the proton beam polarization is discussed in Section 3.2 (see Equation 3.7). The neutron polarization is measured by scattering the neutrons from helium. The scattered neutron "uxes observed by a pair of detectors placed 90 Chapter 4. Experimental Procedure symmetrically about the beam direction can be written as Hai72] NL = N0(1 + PnAy )L NR = N0(1 ; PnAy )R (4.11) where refers to the spin state, N0 are the neutron "uxes into the detector for an unpolarized beam, Ay is the analyzing power, and contains the detector eciency and solid angle. Since the "ux from the polarized ion source is dierent for the two spin states, N0+ and N0; must be specied separately. N0 and can be eliminated by forming the following ratio of count rates. NL+ NR; = N0+L(1 + Pn+ Ay )N0;R(1 ; Pn; Ay ) (4.12) NL; NR+ N0;L(1 + Pn; Ay )N0+R(1 ; Pn+ Ay ) + Pn+ Ay )(1 ; Pn; Ay ) (4.13) = (1 (1 + Pn; Ay )(1 ; Pn+ Ay ) Substituting for Pn using Equation 4.3 gives (1 + Pn Ay )(1 ; Pn Ay ) = 1 + (Pn Pn )Ay ]1 ; (Pn Pn )Ay ] (4.14) = (1 Pn Ay )2 ; Pn2A2y (4.15) (1 Pn Ay )2 (4.16) where the approximation Pn2A2y 1 has been made. Since Pn+ and Pn; are opposite in sign and equal in magnitude to within 20%, this is a valid approximation. The neutron polarimeter asymmetry, "npol, is then dened as r NL+ NR ; 1 N N (4.17) "npol = r NL NR+ L+ R + 1 NL NR+ ; ; ; ; = Pn Ay : (4.18) The neutron detectors are placed at an an angle corresponding to a maximum in the product of the square of the 4He analyzing power and the cross section, A2y . 4.2. Kyy Measurements for the 3H(~p,~n)3He Reaction 0 91 At most energies, two detector pairs are used to take advantage of both the forward and backward maxima. The ght path from the helium cell to the neutron detectors is set as a compromise between time resolution and counting rate. The coincidence timing between the center and side detectors is set using a 22Na source. The neutron polarization is measured continuously for a period of approximately 24 hours. Such a long time is necessary due to the low counting rate. The proton polarization is measured approximately every 2 hours. With each change in energy, the detector angles are changed and the amplier gains and coincidence timing is checked. In addition, PSD must be set for each detector at each energy. Chapter 5 Data Analysis 5.1 T Measurements Following Equation 4.8, as T can be written in terms of the experimental parameters (5:1) T = xP2"P T n where " is the observed asymmetry in neutron transmission, Pn is the neutron beam polarization, PT is the proton target polarization, and x is the target thickness. As discussed in Section 2.2, T is constrained at low energies by kinematic considerations and by the properties of the deuteron. At energies below 2 MeV, the values of T obtained from potential models, phase-shift analysis, and eective-range parameters are in good agreement. Thus it is possible to write " " (E ) # T (E ) = "Pn(E ) T (E0) (5:2) Pn 0 where E0 2 MeV is the calibration energy at which T can be calculated, and E is a higher energy at which T is to be determined. The asymmetry at E0 eectively measures the product of target thickness and polarization, xPT . Normalizing to the asymmetry at E0 eliminates all energy-independent quantities from the analysis. Additionally, measurements made with the target unpolarized allow the subtraction of instrumental asymmetries due to eects such as target misalignment. The data has been taken in four sets, labelled A, B, C, and D, each having a dierent target polarization. In two cases, a measurement has been repeated within the same set. 92 5.1. T Measurements 93 These measurements are distinguished by the subscripts 1 and 2. In the analysis of the data, statistical and systematic errors will be treated separately. 5.1.1 Calculation of the Neutron-Transmission Asymmetries The data is collected in time-ordered spectra with each channel giving the number of counts collected during one eight-step spin sequence lasting 800 ms. Since the data is sorted by spin state, there are two spectra for each scaler value corresponding to data taken with spin up and with spin down . Because the amount of time spent collecting data in each spin state is determined by hardware, interrupting the data collection will make these times unequal during the current spin sequence. This problem occurs when the beam current falls outside of the discriminator window and an inhibit signal is sent to the data acquisition system. These events must be detected by the o-line analysis and the data from these spin sequences rejected. This task is accomplished by analyzing the dead-time pulser spectra for irregularities. The dead-time pulser produces regular pulses with a frequency of approximately 100 kHz, equivalent to 40,000 counts per channel in each spin state. The pulser is stable to approximately 5 10;4 over the time spanned by an entire spectrum and any deviations larger than this value indicate that the time spent collecting data was incorrect. When data collection is interrupted, one or both of the pulser spectra will show a spike or dip in an otherwise "at spectrum (Figure 5.1). These irregularities are used to identify the channels to be rejected by the FORTRAN program CHAY. This program makes three passes through each set of spectra in evaluating the data. The rst pass generates a histogram of the number of channels versus counts per channel with a bin width of 250 counts. The rst-pass average counts per channel is then taken to be that of the bin with the most number of channels. This method of estimating the average removes most of the spikes and dips which would otherwise bias the result, as they are not statistically distributed. The second pass calculates the average pulser counts per channel, excluding all channels which are more than 200 counts away from the rst-pass average. This value is essentially the average which would be measured if no interruptions in the data collection occur. In addition, a standard deviation is calculated. The third and nal pass extracts the data, excluding 94 Chapter 5. Data Analysis Counts 50000 45000 40000 35000 30000 25000 20000 15000 10000 5000 0 0 200 400 600 Channel Figure 5.1: Dead-Time Pulser Spectrum 800 1000 5.1. T Measurements 95 all channels where the pulser counts in either spectrum are more than four standard deviations away from the average. Since the dead-time pulser is independent of the detectors, this process in no way aects the statistical distribution of the neutron data. Typically, less than 1% of the data must be rejected. The data which is not rejected is used to calculate asymmetries in the transmitted neutron "ux on a channel-by-channel basis. First, neutron yields, N~ , are calculated by normalizing the detector counts, N , to the integrated beam current, I , and multiplying by the dead-time correction, . (5:3) N~ = NI The asymmetry for either of the two neutron detectors is then ~ ~ " = N~+ ; N~; (5.4) N+ + N; (5.5) = ; + where the subscripts \+" and \;" refer to the spin state and + + (5.6) = N N;; = II+ : (5.7) ; Since the detector counts obey a Poisson distribution, the standard uncertainty is simply given by the square-root of the number of counts. p N = N (5:8) The counts from the beam current integration, however, are not normally distributed. Instead, the uncertainty arises from the nite resolution due to digitization. An uncertainty of 1/2 count occurs when changing spin state. Because of the order of the eight-step sequence (+ ; ; + ; + +;), three such changes occur for each spin state.1 Adding the uncertainties in quadrature gives the uncertainty in the integrated Since the time spent vetoing data during spin reversal is short compared to the time between BCI pulses, ;; and ++ are treated as if they are one continuous measurement in this analysis. 1 96 current for one spin state. Chapter 5. Data Analysis p I = 23 From these results and can be found. q = 1=N+ + 1=N; (5:9) (5.10) p q 3 = 2 1=I+2 + 1=I;2 The uncertainty in the asymmetry for one detector is then s 2 " = ( + )2 1=N+ + 1=N; + 34 (1=I+2 + 1=I;2 ): (5.11) (5:12) The asymmetries for the two detectors can be combined in two ways. Averaging the two asymmetries gives the same result as a single detector located at 0 . It is this value which will be used to obtain T . Taking the dierence of the two asymmetries gives essentially a measure of the analyzing power for the 3H(~p,~n)3He source reaction and is proportional to the neutron beam polarization. The sum and dierence asymmetries are dened as (5.13) "S = "T +2 "B "D = "T ;2 "B : (5.14) The uncertainties for these combined asymmetries, however, cannot be obtained simply by adding the uncertainties for the individual asymmetries in quadrature. Since "T and "B are calculated using the same beam current values, I+ and I;, a correlation exists. The uncertainties in "S and "D are given properly as s (5.15) "S = 14 ("T 2 + "B 2) + ( + 2)2T(B + )2 2 T B s "D = 14 ("T 2 + "B 2) ; ( + 2)2T(B + )2 2: (5.16) T B The correlation term increases the uncertainty in "S and decreases it in "D. 5.1. T Measurements 97 Once the asymmetries have been calculated for each 800 ms channel and for all runs, they are averaged and standard deviations are calculated. n " ! X i "2 (5.17) " = 2 " i i=1 v u 1 u " = u (5.18) n u X t 1="2i i=1 v u n u X u ("i ; ")2="2 u u i=1 (5.19) " = t n(n ; 1) Here, n is the total number of 800 ms measurements and " is the reduced standard deviation. The results for the neutron detector asymmetries for all measurements are tabulated in Table 5.1. In all subsequent calculations, the standard deviations will be used as the uncertainties in these asymmetries. 5.1.2 Calculation of the Average Neutron Beam Polarization The neutron polarization is determined by measuring the polarization of the proton beam with a carbon analyzer as described in Section 3.2. These measurements are typically made every 2{4 hours, with the exception of data at the calibration energy E0 = 1:94 MeV. At this energy the analyzing power is small (approximately ;0:15), making beam polarization measurements susceptible to systematic eects. For this reason, polarization measurements are made at a higher energy either before or after the data is taken. In some cases, the polarization is measured both before and after and the results averaged. Polarimeter asymmetries are measured for spin up and spin down and the dierence taken to cancel systematic eects. (5:20) "ppol = "ppol+ ;2 "ppol; Runs taken between two polarization measurements are assigned a polarimeter asymmetry equal to the average of the two. Runs at the beginning or end of a T measurement are assigned an asymmetry by a linear extrapolation from the two closest 98 Chapter 5. Data Analysis En (MeV) Set n 1.94 A 47,174 B 48,651 C 49,428 D1 39,448 D2 7,127 3.65 C 57,865 D 20,263 4.42 A 63,316 4.91 A 66,302 5.21 D1 44,931 D2 185,087 5.81 A 47,236 6.25 B 42,061 C 142,101 En (MeV) Set n 1.94 A 37,624 C 54,789 D 47,699 3.65 C 67,179 D 23,126 4.42 A 60,624 4.91 A 64,152 5.21 D 221,298 5.81 A 68,713 6.25 B 173,181 C 128,196 "S 179.60 ;115:00 ;177:10 ;107:20 ;114:40 ;33:81 ;25:99 7.45 1.54 0.72 ;0:06 ;4:83 6.24 5.06 Target Polarized "S ("S ) 4.36 (4:52) 5.06 (5:27) 6.26 (6:89) 4.18 (4:42) 9.97 (11:03) 3.22 (3:31) 6.29 (6:46) 2.74 (2:75) 2.83 (2:85) 5.08 (5:14) 1.39 (1:41) 3.00 (3:03) 5.54 (5:59) 2.33 (2:36) "D ;70:44 60.72 68.83 56.15 52.53 ;26:09 ;23:64 ;44:47 ;26:10 17.75 20.96 ;10:59 15.19 16.24 "D 4.08 4.74 6.00 3.52 8.44 3.03 5.38 2.52 2.64 4.37 1.31 2.84 5.16 2.24 ("D ) (4.21) (4:90) (6:73) (3:62) (8:50) (3:13) (5:39) (2:52) (2:64) (4:40) (1:31) (2:85) (5:25) (2:26) Unit 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 "S ;7:21 20.21 4.54 4.76 3.60 2.93 ;3:03 ;3:32 ;1:08 ;3:11 ;4:88 Target Unpolarized "S ("S ) 4.94 (5:12) 5.92 (6:70) 4.06 (4:80) 2.92 (2:98) 4.51 (4:71) 2.51 (2:50) 2.53 (2:56) 1.38 (1:39) 2.44 (2:47) 2.57 (2:59) 2.35 (2:38) "D ;68:81 60.86 60.62 ;16:61 ;29:15 ;51:89 ;24:47 23:87 ;4:51 23.55 22.04 "D 4.60 5.65 3.50 2.76 4.07 2.37 2.39 1.30 2.32 2.40 2.27 ("D ) (4:77) (6:40) (3:55) (2:85) (4:04) (2:37) (2:39) (1:29) (2:31) (2:44) (2:30) Unit 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 Table 5.1: Average Neutron-Transmission Asymmetries 5.1. T Measurements Ep (MeV) 4.66 5.04 5.41 5.78 5.89 6.18 6.77 7.21 7.55 7.99 8.66 8.90 9.15 9.66 n `max 2 13 7 1.09 13 10 1.56 9 8 2.34 9 5 0.99 13 8 1.12 11 7 9.53 13 10 3.34 13 9 3.28 13 9 1.16 13 9 0.72 13 6 1.48 16 8 1.53 20 6 1.29 12 8 0.77 99 d d (mb) 158.0 162.0 132.8 182.5 178.6 195.0 218.0 177.0 125.0 148.0 133.0 102.3 129.9 85.0 Aexp y ;0:50 ;0:72 ;0:521 ;0:779 ;0:849 ;0:82 ;0:53 ;0:30 ;0:36 ;0:14 +0:00 +0:124 ;0:011 ;0:342 Aexp y 0.03 0.01 0.026 0.041 0.034 0.01 0.01 0.02 0.02 0.02 0.02 0.041 0.044 0.058 Afit y ;0:496 ;0:724 ;0:535 ;0:775 ;0:812 ;0:851 ;0:520 ;0:278 ;0:339 ;0:131 ;0:015 +0:125 ;0:010 ;0:348 Afit y 0.022 0.009 0.024 0.015 0.022 0.009 0.009 0.015 0.016 0.017 0.014 0.041 0.015 0.054 Table 5.2: Analyzing Power Fits for 12C(p,p0)12C at lab = 40 polarimeter measurements. The average polarimeter asymmetry for an entire T measurement is then calculated, weighting each asymmetry by the number of runs which it spans. Analyzing powers for the polarimeter are determined from published data Mos65, Ter68] for the elastic scattering of protons from carbon. The products of dierential cross section, dd , and analyzing power, Ay , are t as a function of center-of-mass angle using rst-order associated Legendre polynomials. The number of terms used in the t, `max, is chosen at each energy to minimize the reduced chi-square, 2 . The exp results of the t, Afit y , as well as the experimental value, Ay , at the angle of interest (lab = 40 ) are shown in Table 5.2, where n is the number of experimental data used in each t. Proton beam polarizations are calculated using Equation 3.7 and neutron beam polarizations using Equation 3.8 and the depolarization factor listed in Table 3.6. Kyy Pp Pn = P=P (5:21) 0 0 Table 5.3 lists the average beam polarizations, Pp and Pn, for each measurement. The uncertainties, Pp and Pn , come from uncertainties in the values of the carbon analyzing power and in the polarization-transfer coecient and will be treated as 100 Chapter 5. Data Analysis En (MeV) Ep (MeV) 1.94 5.89 7.21 4.66 4.66 6.18 3.65 4.66 4.42 4.91 5.21 5.41 5.89 6.18 5.81 6.25 6.77 7.21 Set A B C D1 D2 C D A A D1 D2 A B C Target Polarized "ppol Pp ;0:499 0.755 0.151 ;0:667 0.279 ;0:691 0.245 ;0:607 0.314 ;0:453 0.270 ;0:669 0.213 ;0:528 ;0:276 0.634 ;0:496 0.751 0.290 ;0:419 0.412 ;0:595 ;0:285 0.673 0.153 ;0:676 0.162 ;0:716 Target Unpolarized Pp En (MeV) Ep (MeV) Set "ppol 1.94 5.41 A ;0:291 0.668 7.21 C 0.150 ;0:663 4.66 D 0.273 ;0:675 3.65 4.66 C 0.268 ;0:664 D 0.273 ;0:676 4.42 5.41 A ;0:294 0.675 4.91 5.89 A ;0:440 0.666 5.21 6.18 D 0.430 ;0:621 5.81 6.77 A ;0:256 0.605 6.25 7.21 B 0.149 ;0:659 C 0.154 ;0:681 Pp 0.020 0.036 0.031 0.027 0.005 0.030 0.023 0.028 0.020 0.004 0.006 0.012 0.036 0.039 Pn 0.496 ;0:438 ;0:454 ;0:399 ;0:298 ;0:471 ;0:372 0.439 0.532 ;0:310 ;0:440 0.547 ;0:559 ;0:592 Pn 0.030 0.034 0.032 0.028 0.017 0.033 0.025 0.028 0.023 0.011 0.015 0.020 0.036 0.038 Pp 0.030 0.036 0.030 0.029 0.030 0.030 0.018 0.007 0.010 0.036 0.037 Pn 0.439 ;0:436 ;0:444 ;0:468 ;0:476 0.468 0.472 ;0:460 0.491 ;0:545 ;0:563 Pn 0.031 0.034 0.031 0.032 0.033 0.030 0.021 0.016 0.017 0.035 0.036 Table 5.3: Average Proton and Neutron Beam Polarizations 5.1. T Measurements 101 systematic uncertainties. The analysis of the polarization-transfer coecient, Kyy , is discussed in Section 5.2. Knowledge of the beam polarizations allows neutron asymmetry measurements made at the same energy and target polarization to be combined. In order to simplify the calculation of uncertainties, whenever possible constants containing systematic uncertainties are not included in the normalization. Instead, they are factored in at a later stage. Thus, the neutron asymmetries are normalized to the proton polarimeter asymmetries whenever the analyzing power is the same for all measurements at that energy. This is possible at all energies except E0 where the polarization has been determined at a variety of beam energies. In this case the neutron asymmetries are normalized to the proton beam polarization. Since the polarization-transfer coefcients are the same for all measurements at a given energy, Kyy is not included until later in the analysis. Once normalized, the values are weighted by their statistical variance and averaged. "X # n y i 2 y = ( (5.22) stat y) 2 i=1 (stat yi ) v u 1 u (5.23) stat y = u n u t X1=(stat yi)2 0 0 i=1 # sys yi sys y = (stat y)2 (5.24) 2 ( y ) i=1 stat i In these expressions yi refers to either "="ppol or "=Pp . Note that unpolarized target data taken in dierent sets can be combined as the target polarization is the same in all cases, namely zero. This is not true, however, for the polarized target data, and asymmetries from dierent data sets cannot be combined at this point. Finally, instrumental asymmetries are removed by subtracting the warm data (target unpolarized) from the cold data (target polarized). "X n "="ppol = ("="ppol)c ; ("="ppol)w (5.25) "=Pp = ("=Pp)c ; ("=Pp )w (5.26) The normalized asymmetries for the cases cold, warm, and cold minus warm as well 102 Chapter 5. Data Analysis as the appropriate averages are shown in Tables 5.4, and 5.5. Here, stat and sys refer to the statistical and systematic uncertainties, respectively. 5.1.3 Calculation of T The next step in the analysis of the data is to combine measurements made at the same energy, but in dierent data sets which have dierent target polarizations. Since the target polarization information is contained in the calibration data, the other energies are normalized to these values, forming the ratio . " "ppol (E ) (5:27) = " (E ) Pp 0 The averaging is performed as before and the results shown in Table 5.6.2 T values can now be determined by including the constants from Equation 5.2 and multiplying by the calculated value of T at E0. 8 9 3 " #" y #2 < = Ky (E0) 4 PP0 (E0) 5 ( E ) T = :Ay (E ) (E ) (5:28) ( E ) T 0 Kyy (E ) PP0 (E ) 0 0 0 The 12C(p,p0)12C analyzing power for the calibration measurement does not appear in this expression as it was included at an earlier stage of the analysis. The corrections for unpolarized neutron background, , are obtained from Table 3.5 and values for the polarization-transfer coecient, Kyy from Table 5.14. T (E0) is obtained by taking the average of predictions from potential models, phase-shift analysis, and eective-range parameter calculations (Table 5.7).3 The full Bonn is a relativistic meson exchange model. Bonn A, B, and C are one-boson exchange approximations to the full Bonn. The three dier only in the strength of the tensor interaction. A systematic uncertainty is assigned equal to the standard deviation of the predicted values. Table 5.8 lists the values of T obtained from the analysis of the data. The total uncertainty, , is given by q (5:29) = (stat)2 + (sys )2: 0 There exists a correlation in the values being averaged due to using the same warm asymmetries in both cases. The eect, however, is small in practice and will be ignored in this analysis. 3Predictions from the Paris and Nijmegen Potentials are not included as they use 1S phase shifts 0 obtained from p-p scattering. 1 S0 for n-p scattering is known to be dierent by several degrees. 2 5.1. T Measurements 103 Target Polarized En (MeV) Set ("="ppol )c stat 3.65 C ;125:22 12.26 D ;122:02 30.33 4.42 A ;26:99 9.96 4.91 A ;3:11 5.75 5.21 D1 2.48 17.72 D2 ;0:15 3.42 Average ;0:06 3.36 5.81 A 16.95 10.63 6.25 B 40.78 36.54 C 31.24 0.00 sys 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Unit 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 Target Unpolarized En (MeV) Set ("="ppol )w stat 3.65 C 17.76 11.11 D 13.19 17.25 Average 16.42 9.34 4.42 A ;9:97 8.50 4.91 A 6.89 5.82 5.21 D ;7:72 3.23 5.81 A 4.22 9.65 6.25 B ;20:87 17.38 C ;31:69 15.46 Average ;26:91 11.55 sys 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Unit 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 Target Polarized Minus Target Unpolarized En (MeV) Set "="ppol stat sys 3.65 C ;141:64 16.55 0.00 D ;138:44 31.74 0.00 4.42 A ;17:02 13.09 0.00 4.91 A ;10:00 8.18 0.00 5.21 D 7.66 4.66 0.00 5.81 A 12.73 14.36 0.00 6.25 B 67.69 38.32 0.00 C 58.15 18.59 0.00 Unit 10;4 10;4 10;4 10;4 10;4 10;4 10;4 10;4 Table 5.4: Neutron-Transmission Asymmetries Normalized to the Proton Polarimeter Asymmetry 104 Chapter 5. Data Analysis Target Polarized En (MeV) Set ("=Pp )c stat 1.94 A 237.88 5.99 1.94 B 172.41 7.90 1.94 C 256.30 9.97 1.94 D1 176.61 7.28 D2 252.54 24.35 Average 182.84 6.98 sys 6.30 9.31 11.50 7.86 2.79 7.44 Unit 10;4 10;4 10;4 10;4 10;4 10;4 Target Unpolarized En (MeV) Set ("=Pp )w stat 1.94 A ;10:79 7.67 C ;30:48 10.11 D ;6:72 7.11 Average ;13:20 4.63 sys 0.47 1.66 0.30 0.65 Unit 10;4 10;4 10;4 10;4 Target Polarized Minus Target Unpolarized En (MeV) Set "=Pp stat sys 1.94 A 251.08 9.73 6.33 1.94 B 185.61 9.16 9.33 1.94 C 269.50 10.99 11.52 1.94 D 196.04 8.38 7.47 Unit 10;4 10;4 10;4 10;4 Table 5.5: Neutron-Transmission Asymmetries Normalized to the Proton Beam Polarization 5.1. T Measurements 105 En (MeV) Set stat sys Unit 3.65 C ;52:56 6.50 2.25 10;2 D ;70:62 16.48 2.69 10;2 Average ;54:99 6.05 2.31 10;2 4.42 A ;6:78 5.22 0.17 10;2 4.91 A ;3:98 3.26 0.10 10;2 5.21 D 3.91 2.39 0.15 10;2 5.81 A 5.07 5.72 0.13 10;2 6.25 B 36.47 20.73 1.83 10;2 C 21.58 6.96 0.92 10;2 Average 23.09 6.60 1.01 10;2 Table 5.6: Neutron-Transmission Asymmetries Normalized to Beam and Target Polarizations T (E0) (mb) Full Bonn 899.8 Bonn A 958.5 Bonn B 959.2 Bonn C 964.0 Arndt SM92 Analysis 879.2 Eective Range Parameters 972.7 Average 938.9 39.1 Table 5.7: Calculated Values of T at a Neutron Energy of 1.94 MeV Model En (MeV) T (mb) stat sys 3.65 260.2 28.6 29.1 4.42 45.9 35.3 4.8 4.91 42.1 34.5 3.9 5.21 ;39:3 24.1 3.9 5.81 ;26:5 29.9 3.0 6.25 ;67:8 19.4 7.6 Table 5.8: Measured Values of T 40.8 35.6 34.7 24.4 30.1 20.8 106 Chapter 5. Data Analysis 5.1.4 Determination of the Zero-Crossing Energy of T In order to determine the zero-crossing energy, a continuous function must be t to the T data. For this purpose, the prediction of the Bonn potential is used with the transformation E ! a + bE: (5:30) The coecients a and b are allowed to vary to nd the smallest total chi-square. " i Bonn(a + bE ) #2 X ( E ) ; T T 2 = (5:31) i )(E ) ( T i The uncertainties used are the total uncertainties of Table 5.8. The best t is obtained with a = ;0:800 MeV, b = 1:127, resulting in a total chi-square of 2 = 3:98 and a reduced chi-square of 2 = 0:995. The data and t are shown in Figure 5.2. The zero-crossing energy obtained from the t is Ezc = 5:08 0:10 MeV: The errors on the zero-crossing energy are obtained by shifting a by an amount sucient to increase 2 by 1.0. The zero-crossing energy determined by the shifted t gives the one-sigma bound on Ezc . A simple phase-shift analysis has been performed, allowing only the 1 mixing parameter to vary4 in order to reproduce Ezc . The result obtained from the analysis is 1 = 0:30 0.17 : These results will be discussed further in Chapter 6. 5.2 Measurements for the 3H(~p,~n)3He Reaction Kyy 0 Since the determination of T depends upon the polarization-transfer coecients, Kyy , for the 3H(~p,~n)3He neutron source reaction, it is important to know these parameters well. In particular, it is essential to know the polarization transfer at the calibration energy, E0. Existing data only extends down to a neutron energy of 0 4 All phase-shift parameters except 1 are obtained from the full Bonn potential. 5.2. Kyy Measurements for the 3H(~p,~n)3He Reaction 107 0 T (mb) 600:0 500:0 400:0 300:0 200:0 100:0 0:0 ;100:0 ;200:0 3:0 3:5 4:0 4:5 5:0 En (MeV) 5:5 Figure 5.2: T Data with Fit 6:0 6:5 7:0 108 Chapter 5. Data Analysis Forward Angle Backward Angle En (MeV) NL+ NL; NR+ NR; NL+ NL; NR+ NR; 1.94 | | | | 1316 2192 1757 1117 5.21 1133 676 632 1006 204 422 353 197 5.81 198 361 303 157 81 29 41 107 Table 5.9: Neutron Counts in the Neutron Polarimeter Detectors 2.12 MeV, requiring an extrapolation to E0 = 1:94 MeV. In addition, it is useful to conrm the earlier data in the region of the T zero-crossing. For these reasons, Kyy has been measured at neutron energies of 1.94, 5.21, and 5.81 MeV. The new data is combined with the existing data to predict the polarization transfer at all energies at which T measurements are made. 0 5.2.1 Determination of Kyy 0 The neutron polarimeter asymmetry is calculated for each pair of detectors by forming the asymmetry for the two detectors as dened in Section 4.2. rN N L+ R NL NR+ ; 1 "npol = r N N (5.32) L+ R + 1 NL NR+ q NL+ NR;(NL; + NR+) + NL; NR+(NL+ + NR;) "npol = (5.33) p 2 p NL+NR; + NL; NR+ ; ; ; ; NL and NR refer to the number of counts in the left and right detectors respectively. The subscripts \+" and \;" refer to the spin-state of the beam. This method of averaging reduces the sensitivity of the asymmetry to beam misalignments and detector eciency dierences. In addition, since two detectors are used, it is not necessary to know the incident neutron "ux. Table 5.9 lists the total number of counts in each detector and spin-state combination summed over all runs. The average neutron polarizations can be calculated from the average asymmetries using (5:34) Pn = A "(npol y 4He) 5.2. Kyy Measurements for the 3H(~p,~n)3He Reaction 0 En (MeV) "npol "npol Pn stat Pn 1.94 107 ;0:2362 0.0122 ;0:309 0.016 5.21 50 0.2405 0.0166 ;0:382 0.026 121 ;0:3163 0.0277 ;0:344 0.030 Average ;0:366 0.020 5.81 51 ;0:3045 0.0300 0.482 0.048 121 0.4594 0.0560 0.502 0.061 Average 0.490 0.038 Table 5.10: Average Neutron Beam Polarizations 109 sys Pn 0.004 0.006 0.006 0.006 0.008 0.008 0.008 En (MeV) Ep (MeV) "ppol Pp Pp 1.94 6.18 0.401 ;0:471 0.005 5.21 6.18 0.395 ;0:464 0.005 5.81 6.77 ;0:316 0.608 0.011 Table 5.11: Average Proton Beam Polarizations where the 4He(n,n)4He eective analyzing powers, Ay (4He), are obtained from Table 3.7. The uncertainties in the analyzing powers are treated as systematic uncertainties. The results for each angle pair at each energy as well as the weighted averages over both angles are listed in Table 5.10. The average proton polarimeter asymmetries are obtained by taking the dierence between the spin up and spin down asymmetries as in Equation 5.20 and averaging over all runs. The average proton polarization is given by (5:35) Pp = A "(pol y 12C) where the 12C(p,p0)12C analyzing powers, Ay (12C), are obtained from Table 5.2. Again, the uncertainties in the analyzing powers are treated as systematic uncertainties. As in the T measurements, the proton beam polarization for the En = 1:94 MeV measurement is obtained at a higher energy. Table 5.11 lists the average proton polarizations at each energy. Once the neutron and proton beam polarizations are known, the polarization- 110 Chapter 5. Data Analysis En (MeV) 1.94 5.21 5.81 Kyy 0.656 0.789 0.806 stat Kyy 0.034 0.043 0.063 0 sys Kyy 0.011 0.015 0.020 0 0 Kyy 0.036 0.046 0.066 0 Table 5.12: Measured Values of the 3H(~p,~n)3He Polarization-Transfer Coecient transfer coecient is simply the ratio of the two. Kyy = PPn p (5:36) 0 Table 5.12 lists the values of Kyy obtained, as well as statistical and systematic uncertainties. The total uncertainty, Kyy , is given by q (5:37) Kyy = (stat Kyy )2 + (sys Kyy )2: 0 0 0 0 0 5.2.2 Interpolation of Kyy Data 0 The present Kyy measurements are combined with data from an earlier experiment at Los Alamos National Laboratory (LANL) Don71, Hai72] in order estimate the values at the energies of interest. These data are listed in Table 5.13 and plotted in Figure 5.3. The value of Kyy is estimated at other energies by linear interpolation between the two nearest measurements. The uncertainty is estimated by a linear interpolation of the variances. Estimates of Kyy are made by this method at the neutron energies used in the T measurements. These values are listed in Table 5.14. 0 0 0 5.2. Kyy Measurements for the 3H(~p,~n)3He Reaction 111 0 En (MeV) 1.94 2.12 3.12 4.15 4.66 5.16 5.21 5.67 5.81 6.17 6.67 7.17 7.68 8.18 9.18 10.19 11.18 12.18 13.18 14.19 15.19 Kyy 0.656 0.632 0.661 0.719 0.681 0.724 0.789 0.788 0.806 0.828 0.808 0.789 0.769 0.785 0.670 0.681 0.567 0.551 0.444 0.509 0.393 0 Kyy 0.036 0.048 0.033 0.052 0.031 0.036 0.046 0.037 0.066 0.045 0.040 0.047 0.048 0.043 0.050 0.048 0.043 0.056 0.046 0.039 0.036 0 Source TUNL LANL LANL LANL LANL LANL TUNL LANL TUNL LANL LANL LANL LANL LANL LANL LANL LANL LANL LANL LANL LANL Table 5.13: 3H(~p,~n)3He Polarization-Transfer Coecient Data Measured at LANL and TUNL 112 Kyy Chapter 5. Data Analysis 0 1:0 TUNL LANL 2 0:9 0:8 0:7 0:6 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0:5 2 0:4 2 2 0:3 0:2 0:1 0:0 0:0 2:0 4 :0 6:0 8:0 10:0 En (MeV) 12:0 14:0 16:0 Figure 5.3: 3H(~p,~n)3He Polarization-Transfer Coecient Data Measured at LANL and TUNL 5.2. Kyy Measurements for the 3H(~p,~n)3He Reaction 113 0 En (MeV) 1.94 3.65 4.42 4.91 5.21 5.81 6.25 Kyy 0.656 0.691 0.699 0.703 0.789 0.806 0.825 0 Kyy 0.036 0.044 0.042 0.034 0.046 0.066 0.044 0 Table 5.14: Interpolated Values of the 3H(~p,~n)3He Polarization-Transfer Coecient Chapter 6 Comparison of Data with Potential Models The measured values of T in Table 5.8 can now be compared with potential-model predictions. Figure 6.1 shows the measured values of T along with the t obtained in Section 5.1.4, the phase-shift analysis SM92 of Arndt, and the predictions of several potential models. The best agreement is obtained with the Bonn potentials, which have a relatively weak tensor force. The region of the zero-crossing is expanded in Figure 6.2 to compare the t to the potential models. In this plot, the error bars indicate the uncertainty in the zero-crossing energy. The value of the 1 phaseshift parameter obtained by performing a simple phase-shift analysis at the zerocrossing energy (Section 5.1.4) is plotted in Figure 6.3. Also included in this plot are the values from potential models and the Arndt SM92 phase-shift analysis. Results obtained from recent experiments sensitive to 1 are also shown Ock91b, Ock91a, Sch88]. The Bonn results are from measurements of the neutron-proton polarization transfer coecient, Kyy .1 The Erlangen/T)ubingen result is from a measurement of the neutron-proton spin-correlation parameter, Ayy (), at cm = 90 0 A value of 1 was not reported for the Bonn measurement at 17.4 MeV. Instead, a phase-shift analysis similar to the one for the TUNL data was performed by W. Tornow to obtain the plotted value Tor92]. 1 114 115 T (mb) 600:0 TUNL (tpn) TUNL (ddn) ? Fit Arndt SM92 Full Bonn Bonn A/B/C Paris/Nijmegen 500:0 400:0 300:0 200:0 100:0 0:0 ;100:0 ;200:0 3:0 3:5 4:0 4:5 5:0 En (MeV) 5:5 6:0 6:5 7:0 Figure 6.1: Comparison of T Measurements with Potential-Model Predictions 116 Chapter 6. Comparison of Data with Potential Models T (mb) 100:0 Ezc + Fit Arndt SM92 Full Bonn Bonn A Bonn C Paris/Nijmegen 50:0 + + + 0:0 ;50:0 ;100:0 4:0 4:2 4:4 4:6 4:8 5:0 5:2 En (MeV) 5:4 5:6 5:8 6:0 Figure 6.2: Comparison of the Measured Zero-Crossing of T with Potential-Model Predictions 117 1 (degrees) 5:0 TUNL Erlangen/Tubingen 2 Bonn 4 Arndt SM92 4:0 Full Bonn Bonn A Bonn C Paris 3:0 4 2:0 1:0 0:0 2 4 ;1:0 ;2:0 0:0 5:0 10:0 15:0 En (MeV) 20:0 25:0 30:0 Figure 6.3: Comparison of 1 Obtained from T Measurements with PotentialModel Parameters Chapter 7 Conclusions and Summary Measurements of the spin-dependent dierence in total cross section, T for the scattering of polarized neutrons from polarized protons have been made at six energies from 3.65 to 6.25 MeV. T crosses through zero in this energy range, and a zerocrossing energy has been extracted from the data: Ezc = 5:08 0:10 MeV. Since a zero-crossing measurement is unaected by most systematic errors, the uncertainty is due almost entirely to counting statistics. A phase-shift analysis has been performed at the zero-crossing energy in which the mixing parameter 1 is allowed to vary, while the other parameters are xed to the full Bonn potential values. The result of this analysis is 1 = 0:30 0.17 : A complete phase-shift analysis allowing all phase-shift parameters to vary and including all relevant data will be performed at a later date in collaboration with R. A. Arndt at Virginia Polytechnic Institute and State University. However, no signicant changes are expected from such an analysis. The present result suggests that the tensor force is relatively weak in this energy range. Comparison of the data with potential models shows agreement with potentials such as Bonn A which has a weak tensor force and comes the closest to predicting the correct triton binding energy. The result is also consistent with recent theoretical work suggesting that three-body forces and relativistic eects are not signicant in the binding of the triton Pic92, Sam92], 118 119 leaving a weak tensor force as the most likely solution of the triton binding energy problem. It is important to extend the present measurements to higher energies in order to verify these conclusions. Because of their importance in the analysis of the T data, polarization-transfer coecients, Kyy , have been measured for the 3H(~p,~n)3He reaction at three energies using a neutron polarimeter. The results are in agreement with earlier measurements made at Los Alamos National Laboratory. 0 Appendix A Calculation of Neutron Depolarization Due to Magnetic Fields A beam of neutrons passing through a magnetic eld has the spins of the neutrons precessed about the eld component transverse to the spin direction. Since the beam has a non-zero spatial extent, the trajectories of the particles are not identical. Thus, particles with dierent trajectories traverse dierent magnetic elds and experience diering spin precessions. This process results in an eective depolarization of the beam which must be calculated by Monte Carlo techniques. The FORTRAN code NEUTRONS has been developed to calculate the spin precessions of a beam of neutrons passing through an arbitrary magnetic eld. The program is based on the computer code SPINX Bow91]. The program represents the wavefunctions of the neutrons as two-component Dirac spinors (the lower two components are neglected). 0 1 =B (A:1) @ 1 CA 2 The precession is then represented by a 2 2 rotation matrix, T, which gives the relationship between the initial and nal wavefunctions. f = Ti 120 (A:2) 121 The precession is calculated as a product of small precessions which can be treated as innitesimal. T = TN TN ;1!TN ;2 : : : T3!T2T1 Tn = I cos 2 + i sin 2 B^ (A.3) (A.4) where I is the 2 2 identity matrix, is the angle of precession, B^ is a unit vector in the direction of the magnetic eld, and is the Pauli spin vector = xx^ + y y^ + z z^ : The Pauli spin matrices are dened as x y z 0 0 = B @ 1 0 0 = B @ i 0 1 = B @ 0 1 1C A 0 1 ;i C A 0 1 0C A: ;1 (A:5) (A.6) (A.7) (A.8) The spin precession angle for an innitesimal rotation, , is given by = #pdB (A:9) E where d is the distance travelled, B is the magnitude of the magnetic eld, E is the energy of the neutron and # = ;1:325 10;5 (MeV) 21 /(Gcm) is a constant. The step size is made small enough such that the dierence between calculating the rotation matrix in one step, Tn , and in two half-steps, TLn and TUn , is less than a specied amount, , for each component. 2 Tij ; X TikU TkjL (i = 1 2 j = 1 2) (A:10) k=1 122 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields The initial wavefunction is calculated from the angles i and i from the input le which give the initial spin direction according to the Madison convention (Section 3.1.3). ! ! ! ! (A.11) 1i = cos 2 cos 2 ; i cos 2 sin 2 ! ! ! ! 2i = sin 2 cos 2 + i sin 2 sin 2 (A.12) Trajectories are chosen by picking a point at random within the initial rectangular beam spot, and picking a direction from a cosine distribution, with the constraint that the trajectory ends inside the nal beam spot. The dimensions of the rectangular initial and nal beam spots are specied in the input le. Spin projections are calculated for the nal wavefunction along each cartesian axis, and along an axis specied in the input le. The spin projection along an axis specied by a unit vector, n^ , is given by hP (^n)i = hf j P (^n) j f i (A.13) ^ I + ( n ) P (^n) = 2 : (A.14) The spin projections are averaged over all trajectories and standard deviations are calculated. The angles f and f which give the nal spin direction are also calculated. The magnetic eld used by NEUTRONS is calculated in a separate program on a three-dimensional grid. The components Bx, By , and Bz are calculated at each point. For this experiment, the magnetic eld of the superconducting solenoid has been calculated by summing over all windings. The eld of one winding can be calculated from the vector potential of a current loop in cylindrical coordinates Jac75] " # 2 (2 ; k ) K ( k ) ; 2 E ( k ) 4 Ia (A.15) A = q k2 c (a + )2 + z2 k2 = (a + 4a )2 + z2 where a is the radius of the winding, I is the current, c is the speed of light in vacuum, and K (k) and E (k) are elliptic integrals. The solution is given as a series and is in 123 cgs units. " #2 1 X 4 Iaz n (2 n + 1) (2 n ; 1)!! 2n B = c(a + )2 + z2]3=2 k (A.16) n 2 n! n=0 n + 1 " (2n ; 1)!! #2 1 2 X n 2 Ia 2n k (A.17) Bz = c(a + )2 + z2]3=2 (2n + 1) 1 ; n + 1 a n 2 n! n=0 124 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields PROGRAM NEUTRONS C C Written 1/8/91 by W. Scott Wilburn, TUNL C Based on the FORTRAN program SPINX by J. D. Bowman, LANL C C *********************************************************** C This program uses three files which must be assigned to C FORTRAN units before execution. C to FOR001. The format of this file can be obtained from C reading the comments about the input parameters given C below. C created by this program. C assigned to FOR003. C declared as BFIELD below must match that of the array C stored in this file. C *********************************************************** The input file is assigned The output file is assigned to FOR002 and is The B field array file is Note that the size of the array C C This program performs a Monte Carlo simulation of the C precession of the spins of a neutron beam as it passes C through a magnetic field. C SPINPROP.FOR in that it obtains the B field at each step C from an array previously generated by BFIELD.FOR instead of C calculating the values as they are needed. C trajectories are chosen to begin within a specified C rectangular area in the x-y plane and an initial z position C and to end within another rectangular area in the x-y plane C at a final z position. C equal probability within the specified plane, and the C direction of travel is chosen with a cosine probability C distribution such that a trajectory parallel to the z axis C is most probable. C function is calculated from the initial values of the C angles theta and phi given. It differs from the program Random The initial position is chosen with The initial-two component spinor wave Each neutron is transported 125 C along its chosen trajectory in small enough steps that the C spin precession can be treated as an infinitesimal C rotation. C the product of all the rotation C is computed along the way. C rotation matrix representing the entire precession has been C generated. C criteria which compares the result of taking a whole step C with that of taking two half steps. C the rotation matrix from the two step sizes differ by more C than a specified amount, the step size is cut in half and C we try again. C spinor wavefunction is calculated by operating on the C initial wavefunction with the rotation operator. C projection expectation values are calculated along the x, C y, and z axes and along the target spin direction and C written to output. C squares of sums for all neutrons are kept up with so that C the statistical mean and standard deviation can be C calculated. C calculate the mean theta and phi spin angles with standard C deviations. A rotation matrix is generated for each step and matrices (for one neutron) At the end of the trajectory, a The step size is determined with a convergence If the components of At the end of each trajectory, the final Spin Also, the sums of spin projections and Finally, the mean spin projections are used to C C Input Variables (read from FOR001): C C E Neutron Energy (MeV) (common block DATA) C ETA Magnet Angle (deg converted to rad) (common C block DATA) C N Number of neutrons C PHII Initial azimuthal spin angle (deg conv. to rad) C THETAI Initial polar spin angle (deg conv. to rad) C XMAXF Maximum final x position (cm) C XMAXI Maximum initial x position (cm) 126 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields C YMAXF Maximum final y position (cm) C YMAXI Maximum initial y position (cm) C ZF Final z position (cm) C ZI Initial z position (cm) C C Input Variables (read from FOR003): C C BFIELD Array of B field vectors (G) (common block C FIELD) C NBMAX Dimensions of BFIELD C RMAX Maximum x, y, and z covered by BFIELD (cm) C RMIN Minimum x, y, and z covered by BFIELD (cm) C (common block FIELD) C C Output Variables (written to FOR002): C C PAVE Mean values of spin projections C PDEV Standard deviations of spin projections C PHIF Mean azimuthal angle of final spin (deg) C PHIFDEV Standard deviation of final phi (deg) C RF Final position (cm) C RI Initial position (cm) C THETAF Mean polar angle of final spin (deg) C THETAFDEV Standard deviation of final theta (deg) C P Spin projections on cartesian axes and C target spin direction C C Internal Variables: C C CVG C C C Indicates whether convergence has occurred in calculating rotation matrix D Step size in fraction of total distance to be travelled 127 C DELTA_T C Difference in component of rotation matrices calculated using step D and D/2 C DR Magnitude of step (cm) C DX Interval between points in BFIELD (cm) C (common block Field) C F Fraction of total distance travelled C GAMMA Relativistic gamma (common block DATA) C J Counting variable C K Counting variable C L Counting variable C M Counting variable C PSIF Final spinor wavefunction C PSII Initial spinor wavefunction C PSQSUM Sum of squares of spin projections C PSUM Sum of spin projections C R Position vector (cm) C R_B Position vector of B field for full step (cm) C RD Position difference vector, RF-RI (cm) C RL_B Position vector of B field for lower half C C step (cm) RU_B C Position vector of B field for upper half step (cm) C S Spin direction vector C SIN_ALPHA Sine of horizontal trajectory angle C SIN_AMAX Sine of maximum alpha C SIN_AMIN Sine of minimum alpha C SIN_BETA Sine of vertical trajectory angle C SIN_BMAX Sine of maximum beta C SIN_BMIN Sine of minimum beta C T Rotation matrix for a full step C TL Rotation matrix for lower half step C TT Matrix product TU*TL C TU Rotation matrix for upper half step 128 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields C U Rotation matrix for entire trajectory C W Unit vector along target spin C C Constants: C C D0 Initial step-size in fraction of total distance C to be moved C E0 Neutron rest energy (common block CONSTANTS) C EPS Convergence criteria on rotation matrices for C determining step-size C I Imaginary unit (common block CONSTANTS) C PI Pi (common block CONSTANTS) C SEED Seed for random number generation for choosing C trajectories C SI 2x2 identity matrix (common block CONSTANTS) C SIGMA 2x2x3 Pauli spin vector (common block CONSTANTS) C SX \ C SY } Pauli spin matrices (common block CONSTANTS) C SZ / C X Unit vector along x (common block CONSTANTS) C Y Unit vector along y (common block CONSTANTS) C Z Unit vector along z (common block CONSTANTS) C C ***************************** C *** Variable Declarations *** C ***************************** C IMPLICIT NONE C LOGICAL*1 CVG INTEGER*2 J,K,L,M,N C C 129 INTEGER*4 NBMAX(3),SEED REAL*4 D,DX(3),D0,DELTA_T,DR,E,E0,EPS,ETA,F,GAMMA C > ,PAVE(4),PDEV(4),PHIF,PHIFDEV,R(3),R_B(3),RD(3) > ,RF(3),RI(3),RL_B(3),RMIN(3),RU_B(3),RMAX(3) > ,SIN_ALPHA,SIN_AMAX,SIN_AMIN,SIN_BETA,SIN_BMAX > ,SIN_BMIN,THETAF,THETAFDEV,W(3),X(3),XMAXF > ,XMAXI,Y(3),YMAXF,YMAXI,Z(3),ZF,ZI C REAL*8 > BFIELD(11,27,407,3),P(4),PHII,PI,PSQSUM(4) ,PSUM(4),S(3),THETAI C COMPLEX*8 I,SI(2,2),SIGMA(2,2,3),SX(2,2),SY(2,2),SZ(2,2) COMPLEX*16 PSIF(2),PSII(2),T(2,2),TL(2,2),TU(2,2),TT(2,2) C > ,U(2,2) C EQUIVALENCE (SIGMA(1,1,1),SX),(SIGMA(1,1,2),SY) > ,(SIGMA(1,1,3),SZ) C COMMON /CONSTANTS/ E0,I,PI,SI,SIGMA,X,Y,Z COMMON /DATA/ E,ETA,GAMMA COMMON /FIELD/ BFIELD,DX,RMIN C C C DATA D0 /1.0/ DATA E0 /939.573/ DATA EPS /1.0E-5/ DATA I /(0.0,1.0)/ DATA SEED /67459873/ DATA SI /(1.0,0.0),(0.0,0.0),(0.0,0.0),(1.0,0.0)/ DATA SX /(0.0,0.0),(1.0,0.0),(1.0,0.0),(0.0,0.0)/ 130 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields DATA SY /(0.0,0.0),(0.0,-1.0),(0.0,1.0),(0.0,0.0)/ DATA SZ /(1.0,0.0),(0.0,0.0),(0.0,0.0),(-1.0,0.0)/ DATA X /1.0,0.0,0.0/ DATA Y /0.0,1.0,0.0/ DATA Z /0.0,0.0,1.0/ C C ******************************** C *** Preliminary Calculations *** C ******************************** C C Calculate value of pi C PI=DACOS(-1.0D0) C C Calculate relativistic gamma C GAMMA=1+E/E0 C C ****************** C *** Read Input *** C ****************** C C Get input parameters: C C ZI Initial z position of neutrons (cm) (R*4) C ZF Final z position of neutrons (cm) (R*4) C XMAXI Maximum initial x position (cm) (R*4) C YMAXI Maximum initial y position (cm) (R*4) C XMAXF Maximum final x position (cm) (R*4) C YMAXF Maximum final y position (cm) (R*4) C N Number of neutrons to be tracked (I*2) C E Energy of neutrons (MeV) (R*4) C THETAI Initial spin angle theta of neutron (deg) (R*4) 131 C PHII Initial spin angle phi of neutron (deg) (R*4) C ETA Angle of magnet (deg) (0 is longitudinal) (R*4) C C Open input file C OPEN (UNIT=1,STATUS='OLD',ACCESS='SEQUENTIAL',READONLY) C C Read data C READ (1,*) ZI,ZF READ (1,*) XMAXI,YMAXI READ (1,*) XMAXF,YMAXF READ (1,*) N,E READ (1,*) THETAI,PHII READ (1,*) ETA C C Close input file C CLOSE (UNIT=1) C C Open output file C OPEN (UNIT=2,STATUS='NEW',ACCESS='SEQUENTIAL') C C Write input data and header for trajectory information to C output file C WRITE (2,100) ZI WRITE (2,110) ZF WRITE (2,120) XMAXI WRITE (2,130) YMAXI WRITE (2,140) XMAXF WRITE (2,150) YMAXF 132 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields WRITE (2,160) N WRITE (2,170) E WRITE (2,180) THETAI WRITE (2,190) PHII WRITE (2,200) ETA WRITE (2,210) C 100 FORMAT (' Initial z position (cm) ',F7.2) 110 FORMAT (' Final z position (cm) ',F7.2) 120 FORMAT (' Maximum initial x position (cm) ',F7.2) 130 FORMAT (' Maximum initial y position (cm) ',F7.2) 140 FORMAT (' Maximum final x position (cm) ',F7.2) 150 FORMAT (' Maximum final y position (cm) ',F7.2) 160 FORMAT (' Number of neutrons ',I7) 170 FORMAT (' Neutron energy (MeV) 180 FORMAT (' Initial spin angle theta (degrees) 190 FORMAT (' Initial spin angle phi (degrees) ',F7.2) 200 FORMAT (' Magnet angle eta (degrees) ',F7.2) 210 FORMAT (/,' NUMBER > ,'Px Xi Py Yi Pz ',F5.2) Xf ',F6.2) Yf Pt') C C Open file containing B field array C OPEN (UNIT=3,FORM='UNFORMATTED',ACCESS='SEQUENTIAL' > ,STATUS='OLD') C C Read header information C READ (3) NBMAX READ (3) RMIN READ (3) RMAX READ (3) C ' 133 C Read B field array C READ (3) BFIELD C C Close file C CLOSE (UNIT=3) C C ******************************** C *** Preliminary Calculations *** C ******************************** C C Convert angles from degrees to radians C THETAI=THETAI*PI/180 PHII=PHII*PI/180 ETA=ETA*PI/180 C C Calculate unit vector along target spin C W(1)=SIN(ETA) W(2)=0.0 W(3)=COS(ETA) C C Calculate interval between points in BFIELD C DO K=1,3 DX(K)=(RMAX(K)-RMIN(K))/(NBMAX(K)-1) END DO C C Calculate initial spin wavefunction, PSII, from initial C theta and phi C 134 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields PSII(1)=DCOS(THETAI/2)*DCOS(PHII/2)-I*DCOS(THETAI/2) > *DSIN(PHII/2) PSII(2)=DSIN(THETAI/2)*DCOS(PHII/2)+I*DSIN(THETAI/2) > *DSIN(PHII/2) C C C C Initialize variables. PSUM are the spin projection C expectation values for each axis summed over all C trajectories calculated. PSQSUM are the sums of the C squares of these values. They are used at the end to C calculate averages and standard deviations. C DO K=1,3 PSUM(K)=0.0 PSQSUM(K)=0.0 END DO C C ************************** C *** Propagate Neutrons *** C ************************** C C Calculate N random trajectories, propagate a neutron spin C wavefunction along each trajectory, and calculate C expectation values of the spin projections on the x, y, and C z axes. C DO J=1,N C C ************************* C *** Choose Trajectory *** C ************************* C 135 C Choose a random trajectory for a neutron starting at C z=ZI and ending at z=ZF. C randomly chosen in the range -XMAXI<=x<=XMAXI and C -YMAXI<=y<=YMAXI respectively. C chosen by picking random trajectories with a cosine C probability distribution such that the pass through the C region -XMAXF<=x<=XMAXF, and -YMAXF<=y<=YMAXF. C RF are the initial and final vectors so chosen and RD is C RF-RI. C in the x and y planes respectively. C SIN_AMIN are the sines of maximum and minimum values of C alpha which allow the trajectory to pass through the C specified region. C for beta. C randomly chosen values of alpha and beta for this C trajectory. Starting x and y values are Final x and y values are RI and Alpha and beta are the angles of the trajectory SIN_AMAX and SIN_BMAX and SIN_BMIN work the same SIN_ALPHA and SIN_BETA are the sines of the C RI(1)=XMAXI*(2*RAN(SEED)-1) RI(2)=YMAXI*(2*RAN(SEED)-1) RI(3)=ZI C SIN_AMAX=(XMAXF-RI(1))/(ZF-ZI) SIN_AMIN=(-XMAXF-RI(1))/(ZF-ZI) SIN_BMAX=(YMAXF-RI(2))/(ZF-ZI) SIN_BMIN=(-YMAXF-RI(2))/(ZF-ZI) C SIN_ALPHA=(SIN_AMAX-SIN_AMIN)*RAN(SEED)+SIN_AMIN SIN_BETA=(SIN_BMAX-SIN_BMIN)*RAN(SEED)+SIN_BMIN C RF(1)=(ZF-ZI)*SIN_ALPHA+RI(1) RF(2)=(ZF-ZI)*SIN_BETA+RI(2) RF(3)=ZF C 136 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields DO K=1,3 RD(K)=RF(K)-RI(K) END DO C C ************************* C *** Transport Neutron *** C ************************* C C Transport a neutron spin along the trajectory from RI to C RF starting with initial spin wave function PSII. PSIF C is the calculated final spin wave function. C C Set initial values of position (R), fraction of C trajectory travelled (F), and incremental distance(D). C DO K=1,3 R(K)=RI(K) END DO C D=D0 F=0.0 C C Initialize U, a 2x2 matrix corresponding to the C cumulative rotation of the neutron spin C DO K=1,2 DO L=1,2 U(K,L)=SI(K,L) END DO END DO C DO WHILE (F.LT.1.0) C 137 C Check to see if we are at the last step and it will C take us past the end of the trajectory. C adjust the last step to end at the right point. If so, C IF ((F+D).GT.1.0) THEN D=1-F END IF C C Calculate the rotation matrix for a whole step (T) C and for the interval taken as two half steps (TL and C TU). C more than EPS, it is not a good approximation to C approximate the step with an infinitesimal rotation, C and we cut the step size in half and try again. If T differs from the product of TL and TU by C DO K=1,3 R_B(K)=R(K)+(D/2)*RD(K) RL_B(K)=R(K)+(D/4)*RD(K) RU_B(K)=R(K)+(3*D/4)*RD(K) END DO C DR=D*SQRT(RD(1)**2+RD(2)**2+RD(3)**2) C CALL ROT(R_B,DR,T) CALL ROT(RL_B,DR/2,TL) CALL ROT(RU_B,DR/2,TU) C C Calculate TT and the difference between it and T, C DELTA_T C CALL MULT(TL,TU,TT) C CVG=.TRUE. 138 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields C DO K=1,2 DO L=1,2 DELTA_T=ABS(T(K,L)-TT(K,L)) IF (DELTA_T.GT.EPS) THEN CVG=.FALSE. END IF END DO END DO C C If the convergence criteria is not met, cut the step C size in half. C of the distance travelled, F, by the step taken, D, C multiply the cumulative rotation matrix, U, by the C rotation performed in this step, T, and set the step C size back to its original value. If it is met, increment the fraction C IF (CVG.EQ..FALSE.) THEN D=D/2 ELSE F=F+D DO K=1,3 R(K)=R(K)+D*RD(K) END DO CALL MULT(U,T,U) D=D0 END IF C END DO C C ******************************************* C *** Calculate Final Spinor Wavefunction *** C ******************************************* 139 C DO K=1,2 PSIF(K)=(0.0,0.0) END DO C DO K=1,2 DO L=1,2 PSIF(K)=PSIF(K)+U(K,L)*PSII(L) END DO END DO C C ***************************** C *** Calculate Projections *** C ***************************** C C Calculate spin projection expectation values for at the C end of the trajectory and write to output file C CALL PROJ(PSIF,X,P(1)) CALL PROJ(PSIF,Y,P(2)) CALL PROJ(PSIF,Z,P(3)) CALL PROJ(PSIF,W,P(4)) C WRITE (2,300) J,RI(1),RI(2),RF(1),RF(2),P C 300 FORMAT (2x,I5,4(3x,F5.2),4(2x,F6.4)) C C ***************************** C *** Accumulate Statistics *** C ***************************** C C C Add spin projections to statistical accumulations 140 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields DO K=1,4 PSUM(K)=PSUM(K)+P(K) PSQSUM(K)=PSQSUM(K)+P(K)**2 END DO C END DO C C ************************************* C *** Calculate Statistical Results *** C ************************************* C C Calculate averages and and standard deviations for the spin C projections. C DO K=1,4 PAVE(K)=PSUM(K)/FLOAT(N) IF (N.NE.1) THEN PDEV(K)=SQRT(ABS((PSQSUM(K)-FLOAT(N)*PAVE(K)**2) > /FLOAT(N-1))) ELSE PDEV(K)=0.0 END IF END DO C C Calculate the average theta and phi for final spin C wavefunction, with standard deviations C C Get unit vector for spin direction. C DO K=1,3 S(K)=2*PAVE(K)-1 END DO C 141 THETAF=ACOS(S(3)) IF (ABS(S(3)).EQ.1.0) THEN THETAFDEV=0.0 ELSE THETAFDEV=(2/SQRT(1-S(3)**2))*PDEV(3) END IF C IF ((S(1).EQ.0.0).AND.(S(2).EQ.0.0)) THEN PHIF=0.0 PHIFDEV=0.0 ELSE IF (S(2).GE.0.0) THEN PHIF=ACOS(S(1)/SQRT(S(1)**2+S(2)**2)) ELSE PHIF=2*PI-ACOS(S(1)/SQRT(S(1)**2+S(2)**2)) END IF PHIFDEV=(2/(S(1)**2+S(2)**2))*SQRT(S(2)**2*PDEV(1)**2 > +S(1)**2*PDEV(2)**2) END IF C C Convert angles from radians to degrees C THETAF=THETAF*180/PI THETAFDEV=THETAFDEV*180/PI PHIF=PHIF*180/PI PHIFDEV=PHIFDEV*180/PI C C ******************** C *** Write Output *** C ******************** C WRITE (2,400) PAVE(1),PDEV(1) WRITE (2,410) PAVE(2),PDEV(2) 142 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields WRITE (2,420) PAVE(3),PDEV(3) WRITE (2,430) PAVE(4),PDEV(4) WRITE (2,440) THETAF,THETAFDEV WRITE (2,450) PHIF,PHIFDEV C 400 FORMAT (//,' Averages',/,' PX = ',F6.4,' +/- ',F6.4) 410 FORMAT (' PY = ',F6.4,' +/- ',F6.4) 420 FORMAT (' PZ = ',F6.4,' +/- ',F6.4) 430 FORMAT (' PT = ',F6.4,' +/- ',F6.4,/) 440 FORMAT (' THETA = ',F6.2,' +/- ',F6.2) 450 FORMAT (' PHI = ',F6.2,' +/- ',F6.2,/) C END C C C SUBROUTINE MULT(U1,U2,U3) C C Multiplies two spin rotation operators to obtain a third, C U3=U2*U1 C C Input Variables: C C U1 Rotation operator acting first C U2 Rotation operator acting second C C Output Variable: C C U3 Matrix product U2*U1 C C Internal Variables: C C K Counting variable 143 C L Counting variable C M Counting variable C UT Matrix product U2*U1 (temporary) C IMPLICIT NONE C INTEGER*2 K,L,M COMPLEX*16 U1(2,2),U2(2,2),U3(2,2),UT(2,2) C C C Initialize UT, which will temporarily be the product. C allows U3 to be the same variable as U1 or U2, if desired. C DO K=1,2 DO L=1,2 UT(K,L)=(0.0,0.0) END DO END DO C C Calculate UT=U2*U1 C DO K=1,2 DO L=1,2 DO M=1,2 UT(K,L)=UT(K,L)+U2(K,M)*U1(M,L) END DO END DO END DO C C Set U3=UT C DO K=1,2 DO L=1,2 This 144 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields U3(K,L)=UT(K,L) END DO END DO C RETURN END C C C SUBROUTINE ROT(R1,DR1,T1) C C Constructs the approximate rotation operator for an C incremental step by assuming it is infinitesimal C C Input Variables: C C DR1 Length of step (cm) C R1 Position at which field is to be calculated (cm) C (center of step) C C Output Variable: C C T1 Rotation matrix for step C C Internal Variables: C C B Magnetic field vector at R1 (G) C BMAG Magnitude of B (G) C BUN Unit normal vector along B C FB Fraction R is away from nearest grid point in C each dimension C K Counting variable C L Counting variable 145 C M Counting variable C P Counting variable C THETA Angle of rotation (rad) C C Constants: C C BFIELD Array of B field vectors (G) (common block C C FIELD) DX Interval between points in BFIELD (cm) (common C block FIELD) C E Energy of neutron (MeV) (common block DATA) C E0 Neutron rest energy (MeV) (common block C CONSTANTS) C ETA Angle of magnet (rad) (common block DATA) C GAMMA Relativistic gamma (common block DATA) C I Imaginary unit (common block CONSTANTS) C NB Array coordinates of R1 in BFIELD C OMEGA Precession of 1 MeV neutron after travelling 1 C cm in 1 G field C PI Pi (common block CONSTANTS) C RMIN Minimum x, y, and z covered by BFIELD (cm) C (common block FIELD) C SIGMA Pauli spin vector (common block CONSTANTS) C SI 2x2 identity matrix (common block CONSTANTS) C X Unit vector along x (common block CONSTANTS) C Y Unit vector along y (common block CONSTANTS) C Z Unit vector along z (common block CONSTANTS) C IMPLICIT NONE C INTEGER*2 K,L,M,NB(3),P REAL*4 DR1,DX(3),E,E0,ETA,FB(3),GAMMA,OMEGA,R1(3) C 146 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields > ,RMIN(3),X(3),Y(3),Z(3) C REAL*8 B(3),BFIELD(11,27,407,3),BMAG,BUN(3),PI,THETA COMPLEX*8 I,SI(2,2),SIGMA(2,2,3) COMPLEX*16 T1(2,2) C C C COMMON /CONSTANTS/ E0,I,PI,SI,SIGMA,X,Y,Z COMMON /DATA/ E,ETA,GAMMA COMMON /FIELD/ BFIELD,DX,RMIN C DATA OMEGA /-1.325E-5/ C C Calculate magnetic field vector by interpolating between C points in BFIELD array C C Calculate coordinates of R in BFIELD array. NB(K) are the C integer parts while FB(K) is are the fractional parts minus C 1/2. C C Initialize B. C DO K=1,3 NB(K)=IINT((R1(K)-RMIN(K))/DX(K))+1 FB(K)=(R1(K)-RMIN(K))/DX(K)+1.0/2-NB(K) B(K)=0.0 END DO C C Linearly interpolate between nearest 8 points in BFIELD. C DO K=0,1 DO L=0,1 147 DO M=0,1 DO P=1,3 B(P)=B(P)+(1.0/2-(-1)**K*FB(1))*(1.0/2-(-1)**L > *FB(2))*(1.0/2-(-1)**M*FB(3)) > *BFIELD(NB(1)+K,NB(2)+L,NB(3)+M,P) END DO END DO END DO END DO C C Calculate magnitude of B C BMAG=DSQRT(B(1)**2+B(2)**2+B(3)**2) C C Calculate unit vector in direction of B C DO K=1,3 IF (BMAG.EQ.0.0) THEN BUN(K)=0.0 ELSE BUN(K)=B(K)/BMAG END IF END DO C C Calculate precession of spin during step C THETA=(OMEGA/(GAMMA*SQRT(E)))*DR1*BMAG C C Initialize T1, rotation matrix C DO K=1,2 DO L=1,2 T1(K,L)=(0.0,0.0) 148 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields END DO END DO C C Calculate T1=Icos(THETA/2)+isin(THETA/2)(SIGMA*BUN) C DO K=1,2 DO L=1,2 T1(K,L)=T1(K,L)+DCOS(THETA/2)*SI(K,L) DO M=1,3 T1(K,L)=T1(K,L)+I*DSIN(THETA/2)*BUN(M) > *SIGMA(K,L,M) END DO END DO END DO C RETURN END C C C SUBROUTINE PROJ (PSI1,N1,P1) C C Calculates the projection of the spin wavefunction PSI1 on C the normal vector N1. P1 is the result C C Input Variables: C C PSI1 Spinor wavefunction C N1 Unit vector C C Output Variable: C C P1 Spin projection of PSI1 on N1 149 C C Internal Variables: C C J Counting variable C K Counting Variable C L Counting variable C PHI Result of operating on PSI1 with projection C operator C C Constants: C C E0 Neutron rest energy (MeV) (common block C CONSTANTS) C I Imaginary unit (common block CONSTANTS) C PI Pi (common block CONSTANTS) C SIGMA Pauli spin vector (common block CONSTANTS) C SI 2x2 identity matrix (common block CONSTANTS) C X Unit vector along x (common block CONSTANTS) C Y Unit vector along y (common block CONSTANTS) C Z Unit vector along z (common block CONSTANTS) C IMPLICIT NONE C INTEGER*2 J,K,L REAL*4 E0,N1(3),X(3),Y(3),Z(3) REAL*8 P1,PI COMPLEX*8 I,SI(2,2),SIGMA(2,2,3) COMPLEX*16 PHI(2),PSI1(2) C C C C C 150 Appendix A. Calculation of Neutron Depolarization Due to Magnetic Fields COMMON /CONSTANTS/ E0,I,PI,SI,SIGMA,X,Y,Z C C Initialize PHI, result of operating on PSI1 with projection C operator C PHI(1)=(0.0,0.0) PHI(2)=(0.0,0.0) C C Calculate PHI=1/2I+(SIGMA*N1)]PSI1 C DO J=1,2 DO K=1,2 PHI(J)=PHI(J)+SI(J,K)*PSI1(K)/2 DO L=1,3 PHI(J)=PHI(J)+SIGMA(J,K,L)*N1(L)*PSI1(K)/2 END DO END DO END DO C C Initialize P1, the expectation value of the projection of C PSI1 along N1 C P1=0.0 C C Calculate P1=<PSI1|PHI> C DO J=1,2 P1=P1+REAL(CONJG(PSI1(J))*PHI(J)) END DO C RETURN END Appendix B The Spin-Dependent Total Cross Section Di erence tensor Following Equation 2.31, a general expression for the spin-dependent total cross section dierence can be written (1 2) = 2(1 2) s + S12 t] (B:1) where 1 and 2 are the angles of the proton target spin and neutron beam spin with respect to the beam direction, x^ . As before, s and t are the scalar and tensor parts of the spin-spin cross section, ss. The equation can be expressed in terms of 1 and 2 to give1 = 2(sin 1 sin 2 + cos 1 cos 2) s ; (sin 1 sin 2 ; 2 cos 1 cos 2) t]: (B:2) Thus, it is possible to measure dierent linear combinations of s and t by changing the spin directions of the target and beam. The measurement of two dierent linear combinations is sucient to completely determine the two spin-spin cross sections. Although the measurement of both L and T fullls this requirement, they are not the most desirable for two reasons. First, changing between the longitudinal and transverse geometries requires that the magnet be rotated. This cannot be accomplished while the target is polarized and precludes measuring both T and L in the same experiment. Second, since the 1 An identical result is obtained using the statistical polarization tensor formalism Kei93]. 151 152 Appendix B. The Spin-Dependent Total Cross Section Dierence tensor goal of these measurements is to determine the strength of the tensor interaction, the most useful measurement is one in which the linear combinations which are diagonal in s and t. Instead, L and T are mixtures of these cross sections. There is, however, a pair of geometries which avoids both of these problems. These special cases of will be referred to as scalar and tensor . tensor is the case in which 1 2 = 0. The result is then given by tensor = 3 sin 21 t (B:3) where the upper sign applies in the case 1 ; 2 = + 2 , and the lower sign in the case 1 ; 2 = ; 2 . A maximum is obtained for 1 = 4 , 34 , giving max = 3 : tensor (B:4) t scalar is the case in which S12 = 0. As a result, scalar = p 3 sin 221 s (B:5) 1 + 3 cos 1 with the constraint tan 1 tan 2 = 2. The upper sign refers to the case ; 2 2 + 2 . Fixing the target spin angle at 45 (1 = + 4 ) then gives the result tensor = 3 t 2 = ;45 (B.6) = p6 s 2 63 : (B.7) 10 Thus, the experiment can be switched between tensor and scalar by simply resetting the spin direction of the neutron beam. The target is left unchanged. It is often useful to have expressions for these cases in terms of the simpler longitudinal and transverse cross section dierences. 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Bulletin of the American Physical Society, 33(1988) 1582. Tor91] W. Tornow. Recent Developments in Low-Energy Nucleon-Nucleon Interaction Studies. In S. W. Wissink, C. D. Goodman, and G. E. Walker, editors, Spin and Isospin in Nuclear Interactions, pages 461{487, New York, 1991. Plenum Press. Proceedings of an international conference held March 11{ 15, 1991 in Telluride, Colorado. Tor92] W. Tornow, 1992. Private Communication. VdG60] R. J. Van de Graa. Tandem Electrostatic Accelerators. Nuclear Instruments and Methods, 8(1960) 195{202. Yuk35] H. Yukawa. Interaction of Elementary Particles. Proceedings of the Physico-Mathematical Society of Japan, 17(1935) 48{57. Biography Wesley Scott Wilburn Personal Born in Pensacola, Florida, March 9, 1964 Married Dianne Williams, August 11, 1989 Education B.S. Physics, North Carolina State University, Raleigh, North Carolina, 1987 B.S. Electrical Engineering, North Carolina State University, Raleigh, North Carolina, 1987 A.M. Physics, Duke University, Durham, North Carolina, 1990 Academic Positions President, NCSU Society of Physics Students, 1986{1987 Townes-Perkins Fellow, Duke University, 1987{1990 Teaching Assistant, Duke University, 1987{1989 Research Assistant, Duke University, 1989{1993 Memberships American Physical Society Sigma Pi Sigma 161 162 Biography Publications W. Tornow, O. K. Baker, C. R. Gould, D. G. Haase, N. R. Roberson, and W. S. Wilburn. Experimental Determination of the NN Tensor Force. In J. Sowinski and S. E. Vigdor, editors, Physics with Polarized Beams on Polarized Targets. World Scientic, Singapore, 1990. J. E. Koster, E. D. Davis, C. R. Gould, D. G. Haase, N. R. Roberson, L. W. Seagondollar, S. Wilburn, and X. Zhu. Direct Reaction Test of T Violation in 2 MeV Neutron Scattering from Aligned 165 Ho. Physics Letters B, 267(1991) 23{26. C. D. Keith, C. R. Gould, D. G. Haase, N. R. Roberson, W. Tornow, and W. S. Wilburn. 3He Melting Curve Thermometry in a Nuclear Polarization Experiment. Hyperne Interactions, 75(1992) 525{532. Contributed Abstracts W. S. Wilburn, N. R. Roberson, K. A. Sweeton, E. W. Hoen, and T. B. Clegg. Computer Control System for a Polarized Ion Source. Bulletin of the American Physical Society, 33(1988) 2201. W. S. Wilburn, N. R. Roberson, W. Tornow, C. R. Gould, and D. G. Haase. Experimental Determination of the Zero-Crossing of the Spin-Dependent NeutronProton Total Cross-Section Dierences T and L and its Bearing on the NN Tensor Force. In Proceedings of the 7th International Conference on Polarization Phenomena in Nuclear Physics, Paris, 1990, page 27A. W. S. Wilburn, P. R. Human, J. E. Koster, N. R. Roberson, W. Tornow, C. R. Gould, D. G. Haase, and C. D. Keith. Measurement of T in Polarized Neutron/Polarized Proton Scattering. Bulletin of the American Physical Society, 37(1992) 902. Biography 163 W. S. Wilburn, P. R. Human, J. E. Koster, N. R. Roberson, W. Tornow, C. R. Gould, D. G. Haase, and C. D. Keith. The NN Tensor Force from ~n-~p Scattering. In I. R. Afnan and R. T. Cahill, editors, Proceedings of the XIII International Conference on Few-Body Problems in Physics, Adelaide, 1992, page 124. Co-Authored Abstracts T. Clegg, K. Felder, W. Hooke, J. Hunn, H. Lewis, A. Lovette, H. Middleton, H. Pf)utzner, R. Roberson, H. Robinson, K. Sweeton, and S. Wilburn. Design of an Atomic Beam Polarized Ion Source with an ECR Ionizer. Bulletin of the American Physical Society, 33(1988) 905. K. Felder, H. G. Robinson, H. W. Lewis, W. S. Wilburn, and T. B. Clegg. Development of New RF Cavities for Use in a Polarized Ion Source. Bulletin of the American Physical Society, 33(1990) 2200. J. E. Koster, C. R. Gould, D. G. Haase, W. Seagondollar, N. R. Roberson, W. S. Wilburn, and X. Zhu. Holmium Deformation Eect Measured with a Rotating Aligned Target. Bulletin of the American Physical Society, 35(1990) 926. C. D. Keith, D. G. Haase, C. R. Gould, W. S. Wilburn, N. R. Roberson, and W. Tornow. Application of 3 He Melting Curve Thermometry in a Nuclear Orientation Cryostat. Bulletin of the American Physical Society, 35(1990) 2361. T. B. Clegg, D. J. Abbott, T. C. Black, E. R. Crosson, R. K. Das, K. A. Fletcher, C. R. Howell, H. J. Karwowski, J. E. Koster, S. Lemieux, E. J. Ludwig, M. AlOhali, N. R. Roberson, K. A. Sweeton, W. Tornow, W. S. Wilburn, and J. Z. Williams. Report on Operation of a New Intense Polarized Ion Source and ECR Ionizer. In Proceedings of the 7th International Conference on Polarization Phenomena in Nuclear Physics, Paris, 1990, page 3E. 164 Biography W. Tornow, N. R. Roberson, W. S. Wilburn, C. R. Gould, and D. G. Haase. Experimental Determination of the Nucleon-Nucleon Tensor Force via T and L Measurements. In Proceedings of the 7th International Conference on Polarization Phenomena in Nuclear Physics, Paris, 1990, page 29A. C. D. Keith, C. R. Gould, D. G. Haase, P. R. Human, N. R. Roberson, W. Tornow, and W. S. Wilburn. Level Structure of 4 He and the Spin Dependence of the n3 He Total Cross Section. Bulletin of the American Physical Society, 37(1992) 1257.

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