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Detailed balance study of time reversal invariance with interfering resonances*
NIOMI B
Nuclear Instruments and Methods in Physics Research B79 (1993) 290-292 North-Holland
Beam Interactions with Materials&Atoms
Detailed balance study of time reversal invariance with interfering resonances * G.E. Mitchell a, E.G. Bilpuch b, C.R. Bybee a, J.M. Drake a and J.F. Shriner Jr. ’ a North Carolina State Universify, Raleigh, North Carolina 27695-8202, USA, and Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708-0308, b Duke University, Durham, North Carolina 27708-0305, USA, and Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708-0308, ’ Tennessee Technological University, Cookeville, Tennessee 38505, USA
USA USA
Bunakov and Weidenmiiller suggested that large enhancement of time reversal invariance violation may be observed near two interfering resonances via a test of detailed balance. In our (p, a) resonance data on =Na 27Al, 31P, 35Cl, and 39K, there are 33 pairs of adjacent resonances which have the same spin and parity. The difference in the differential cross sections for the (p, (Y,) and (a, p,,) reactions was calculated for these resonance pairs using experimental values for the partial widths. The collision matrix elements were obtained for a Hamiltonian H = H,, + iH ‘, following the approach of Moldauer. T’he differences show striking dependence on energy and angle and on the particular pair of resonances, with the relative sensitivity of the detailed balance test varying by many orders of magnitude. These preliminary results indicate that this class of experiments may be more sensitive than previous detailed balance tests.
1. Introduction
Recently there has been increased interest in tests of symmetry breaking in the compound nucleus [l]. The measurement of large parity (P) violation in polarized neutron transmission measurements stimulated this revival of interest. Following the observation of parity violation at the several percent level at JINR, Dubna [2], an extensive study of parity violation was initiated by the TRIPLE collaboration [3,4]. A variety of tests of time reversal (T) invariance in the compound nucleus are under active consideration for both P-even T-odd and P-odd T-odd symmetry violations. The large enhancements observed experimentally for parity violation (due to very close-lying levels and to the relatively long lifetime of the compound nuclear resonances) are also expected to be present in tests of T-violation. This general approach to symmetry violation in the compound nucleus treats the system as chaotic and assumes that the fluctuation properties are described by random matrix ensembles. There is also revived interest in the so-called traditional detailed balance test: the comparison of the reaction rates for the two-body reactions (a, b) and * Work supported in part by the U.S. Department of Energy, Office of High Energy, Office of High Energy and Nuclear Physics under grants No. DE-FG05-88ER40441, DE-FGOS91ER40619, and DE-FGO5-87ER40353. 0168-583X/93/$06.00
(b, a), where the entrance particles are unpolarized and the exit particles are in their ground states. Following the early detailed balance tests [5] for a single compound nuclear level, emphasis shifted to the regime of strongly overlapping resonances. (See Hamey et al. 161for a summary and analysis of detailed balance data in the fluctuation regime.) Recently there has been renewed interest in detailed balance tests in regions of weakly overlapping resonances [7,8]. Bunakov and Weidenmiiller [7] argue that the enhancement factors can be several orders of magnitude for two close-lying compound nuclear resonances in a region where the average resonance spacing is much larger than the average resonance width. Traditional detailed balance measurements have the advantage of not requiring polarized targets or beams. (Many of the practical problems in the performance of T-violation tests arise from effects due to target polarization.) In their general evaluation of the detailed balance test for a pair of close-lying resonances, Bunakov and Weidenmiiller made simplifying approximations for the values of the partial widths and also considered only total reaction cross sections. To apply to a specific experimental case, one needs explicit expressions for the two cross sections (and therefore the difference), as well as experimental values for the resonance parameters. We derived expressions for the differential cross sections for a variety of target and resonance angular momenta. Calculations were performed for differential
0 1993 - Elsevier Science Publishers B.V. All rights reserved
291
G.E. Mitchell et al. / Time reversal invariance
rather than total cross sections, since the differential cross sections seemed more directly related to experiment. We followed the approach of Moldauer [9] to calculate the collision matrix elements for a Hamiltonian H = H,, + i H’, where H,, and H’ are real. The resonance parameters were taken from earlier measurements by our group on (p, p) and (p, a) reactions in the mass region A = 24-40. In section 2 the results are discussed. In our data there are 33 pairs of adjacent resonances which have the same spin and parity and have widths in both the proton and alpha channels. The difference in cross sections for the (p, a) and (a, p) reactions shows a striking dependence on energy and angle and on the particular pair of resonances. A brief summary is given in the final section.
p + “P
-
OL +
'*Si
E,
= 3.1490 E, = 3.1570
MeV MeV
Fig. 1. Values of A / W as a function of center-of-mass energy and angle for a pair of compound resonances in 32S.The two resonances are located at E, = 3.1490 and 3.1570 MeV. See text for discussion.
2. Results The traditional detailed balance test involves unpolarized projectiles and observation of the exit particles in their ground state. We specialize to A(p, a& and C(a, p&4 reactions, and consider differential cross sections. We define a quantity ACE, 0): A(E, 0) k;
= 2
da
-(E, g(p, a) dQ,p,,, k;
da
-(E,
g(p, a) d&w,
k* dcr O)- a -(E, da, PI df&,, k* O)+ DL
-
0)
da
da, PI dn,,,p,
(E, 0) ’
(1) where k = 2a/A and g is the-statistical weight factor. Calculations were performed for target angular momenta A = l/2, 3/2, and 572 and C= 0, since we have experimental data for these cases. Natural parity resonances J” = l-, 2+, 3-, and 4+ were considered. A Hamiltonian of the form Ho + i H’ was assumed. We followed the approach of Moldauer [9], simplified to two interfering states, and assumed only internal mixing. To first order A is proportional to the matrix element W of H’. Experimental values of the resonance parameters were then used to calculate the cross sections. Our group has measured and analyzed high resolution (p, p) and (p, a> data on five odd-Z even-N targets in the nuclear 2s-ld shell: 23Na [lo], 27Al [ll], 3lP [12], 35C1 [13], and 39K [14]. Evaluation of these results yielded 33 pairs of adjacent resonances which are thought to have the same spin and parity and have measured widths in both the pa and a,, channels. As an example consider a specific pair of 2+ resonances in 32S. The resonances have the following (laboratory) parameters: E, = 3.2515 MeV, Es = 3.2597 MeV, F,, = 1.65 keV; F,, = 0.05 keV, F, = 6.8 keV, r,,, = 0.02 keV, I’,, = 0.08 keV, r,, = 2.5 keV. The
labels a and b refer to the two resonances, while 1, 2, 3 refer to the channels: 1 denotes the proton channel with channel spin s = 0 and orbital angular momentum I= 2, 2 denotes the proton channel with s = 1 and I= 2, and 3 denotes the alpha channel with s = 0 and I= 2. The ratio A/W is plotted in fig. 1 for this pair of resonances. For this resonance pair, A(E, 0) is a maximum near one of the two resonances and is very strongly peaked at 90” and 180”. The plots of A versus E and 0 display a variety of patterns for different pairs of resonances: the maximum value of A may be near one of the resonances or anywhere in between. The angular dependence shows a broad range of shapes, ranging from isolated peaks to wide (or narrow) ridges to very complicated patterns. The shapes (the energy and angular dependence) and the absolute value of A vary enormously. Therefore, in order to evaluate the merits of a detailed balance test of time reversal invariance using this approach, one should consider a specific pair of resonances. For our examples the maximum value of A/W varies from 2 X 10v4 to 2 X lo-‘, a range of 3 orders of magnitude. This provide the first major result: the relative enhancements vary strongly with energy, angle, and resonance pair. One should choose very carefully the particular pair of resonances to be used in a detailed balance test. (Naturally, the size of the term A/W is not the only factor to be considered in evaluating the merits of a particular experiment. In practice counting rates, target quality, beam stability, etc. are extremely important. In any case, such spikelike behavior as shown in fig. 1 is probably not as suitable for an actual experiment as a case where A changes more slowly with energy and angle. The key point is that large enhancements are present, but only at certain energies and angles.) IV. NUCLEAR PHYSICS
292
G.E. Mitchell et al. / Time reversal invariance
Since one expects to observe only a null value, the theoretical question becomes this: from a given null measurement (or set of null measurements), what is the value of the bound which will be set on the Tviolating matrix element? The matrix element is assumed to be drawn from a random distribution. Setting a bound and a confidence limit is an involved procedure, as discussed at length in [7,8]. However, an order of magnitude estimate can be obtained by assuming that the matrix element W is equal to its root-meansquare value W,,. In our calculations a typical value of A/W is of order 10e3. Assuming a null value A_ measured at the lo-* level seems reasonable, since that is the level of sensitivity attained in ref. [5]. With A/W= 10m3 and A, < lo-‘, this places an upper limit on W,, of order 10 eV. Since the strength of the strong interaction matrix element is of order MeV, a very rough estimate of 10e5 is obtained for an upper limit on the ratio (or (the ratio of the T-violating to the T-conserving interaction). This very small value should not be taken literally. However, this estimate does suggest the possible advantages of such experiments. This approximate limit on CQ is much better than the limits obtained in previous detailed balance measurements. These calculations suggest that detailed balance tests of time reversal invariance with two interfering levels should be seriously considered.
3. Summary Bunakov and Weidenmiiller suggested that large enhancement of T-violation in detailed balance experiments should occur near two interfering resonances. We have obtained explicit expressions for differential cross sections for the A(p, a,)C and C(a, p&l reactions for the target angular momenta C = 0, A = l/2, 3/2, and 5/2, and for the natural parity resonances l-, 2+, 3- and 4+. The approach of Moldauer (utilizing the R-matrix formalism) was followed to calculate the collision matrix elements for a Hamiltonian H = H,, + iH’; the matrix element of H’ is denoted by W. The resonance parameters were taken from earlier
measurements by our group on (p, p) and (p, a) reactions; there are 33 possible pairs of adjacent resonances with the same angular momentum and parity. The quantity A (the difference between the cross sections for the inverse reactions, divided by their average) was calculated for each resonance pair. A shows striking dependence on angle and energy, with maximum values of A/W ranging over several orders of magnitude. These preliminary results suggest that a detailed balance test involving two interfering resonances may be much more sensitive than previous detailed balance tests.
Acknowledgements The authors would like to thank V.E. Bunakov, C.R. Gould, H.L. Harney, and H.A. Weidenmiiller for valuable discussions. One of us (J.M.D.) acknowledges the US Department of Education for a Patricia Roberts Harris Fellowship.
References [l] N.R. Roberson, C.R. Gould and J.D. Bowman (eds.),
Tests of Time Reversal Invariance in Neutron Physics (World Scientific, Singapore, 1987). [2] V.P. Alfimenkov et al., Nucl. Phys. A398 (1983) 93. [3] J.D. Bowman et al., Phys. Rev. L&t. 65 (1990) 1192. [4] C.M. Frankle et al., Phys. Rev. Lett. 67 (1991) 564. [5] H. Driller et al., Nucl. Phys. A317 (1979) 300. [6] H.L. Hamey, A. Hiipper and A. Richter, Nucl. Phys. A518 (1990) 35. [7] V.E. Bunakov and H.A. Weidenmiiller, Phys. Rev. C39 (1989) 70. [8] E.D. Davis and U. Hartmann. Ann. Phys. (New York) 211 (1991) 334. [9] P.A. Moldauer, Phys. Rev. 165’0968) 1136. [lo] J.R. Vanhoy et al., Phys. Rev. C36 (1987) 920. [ll] R.O. Nelson et al., Phys. Rev. C29 (1984) 1656; Phys. Rev. C30 (1984) 755. [12] D.F. Fang et al., Phys. Rev. C37 (1988) 28. [13] WK. Brooks, Ph. D. dissertation, Duke University (1988). [14] B.J. Warthen, Ph. D. dissertation, Duke University (1987).
Nuclear Instruments and Methods in Physics Research B79 (1993) 290-292 North-Holland
Beam Interactions with Materials&Atoms
Detailed balance study of time reversal invariance with interfering resonances * G.E. Mitchell a, E.G. Bilpuch b, C.R. Bybee a, J.M. Drake a and J.F. Shriner Jr. ’ a North Carolina State Universify, Raleigh, North Carolina 27695-8202, USA, and Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708-0308, b Duke University, Durham, North Carolina 27708-0305, USA, and Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708-0308, ’ Tennessee Technological University, Cookeville, Tennessee 38505, USA
USA USA
Bunakov and Weidenmiiller suggested that large enhancement of time reversal invariance violation may be observed near two interfering resonances via a test of detailed balance. In our (p, a) resonance data on =Na 27Al, 31P, 35Cl, and 39K, there are 33 pairs of adjacent resonances which have the same spin and parity. The difference in the differential cross sections for the (p, (Y,) and (a, p,,) reactions was calculated for these resonance pairs using experimental values for the partial widths. The collision matrix elements were obtained for a Hamiltonian H = H,, + iH ‘, following the approach of Moldauer. T’he differences show striking dependence on energy and angle and on the particular pair of resonances, with the relative sensitivity of the detailed balance test varying by many orders of magnitude. These preliminary results indicate that this class of experiments may be more sensitive than previous detailed balance tests.
1. Introduction
Recently there has been increased interest in tests of symmetry breaking in the compound nucleus [l]. The measurement of large parity (P) violation in polarized neutron transmission measurements stimulated this revival of interest. Following the observation of parity violation at the several percent level at JINR, Dubna [2], an extensive study of parity violation was initiated by the TRIPLE collaboration [3,4]. A variety of tests of time reversal (T) invariance in the compound nucleus are under active consideration for both P-even T-odd and P-odd T-odd symmetry violations. The large enhancements observed experimentally for parity violation (due to very close-lying levels and to the relatively long lifetime of the compound nuclear resonances) are also expected to be present in tests of T-violation. This general approach to symmetry violation in the compound nucleus treats the system as chaotic and assumes that the fluctuation properties are described by random matrix ensembles. There is also revived interest in the so-called traditional detailed balance test: the comparison of the reaction rates for the two-body reactions (a, b) and * Work supported in part by the U.S. Department of Energy, Office of High Energy, Office of High Energy and Nuclear Physics under grants No. DE-FG05-88ER40441, DE-FGOS91ER40619, and DE-FGO5-87ER40353. 0168-583X/93/$06.00
(b, a), where the entrance particles are unpolarized and the exit particles are in their ground states. Following the early detailed balance tests [5] for a single compound nuclear level, emphasis shifted to the regime of strongly overlapping resonances. (See Hamey et al. 161for a summary and analysis of detailed balance data in the fluctuation regime.) Recently there has been renewed interest in detailed balance tests in regions of weakly overlapping resonances [7,8]. Bunakov and Weidenmiiller [7] argue that the enhancement factors can be several orders of magnitude for two close-lying compound nuclear resonances in a region where the average resonance spacing is much larger than the average resonance width. Traditional detailed balance measurements have the advantage of not requiring polarized targets or beams. (Many of the practical problems in the performance of T-violation tests arise from effects due to target polarization.) In their general evaluation of the detailed balance test for a pair of close-lying resonances, Bunakov and Weidenmiiller made simplifying approximations for the values of the partial widths and also considered only total reaction cross sections. To apply to a specific experimental case, one needs explicit expressions for the two cross sections (and therefore the difference), as well as experimental values for the resonance parameters. We derived expressions for the differential cross sections for a variety of target and resonance angular momenta. Calculations were performed for differential
0 1993 - Elsevier Science Publishers B.V. All rights reserved
291
G.E. Mitchell et al. / Time reversal invariance
rather than total cross sections, since the differential cross sections seemed more directly related to experiment. We followed the approach of Moldauer [9] to calculate the collision matrix elements for a Hamiltonian H = H,, + i H’, where H,, and H’ are real. The resonance parameters were taken from earlier measurements by our group on (p, p) and (p, a) reactions in the mass region A = 24-40. In section 2 the results are discussed. In our data there are 33 pairs of adjacent resonances which have the same spin and parity and have widths in both the proton and alpha channels. The difference in cross sections for the (p, a) and (a, p) reactions shows a striking dependence on energy and angle and on the particular pair of resonances. A brief summary is given in the final section.
p + “P
-
OL +
'*Si
E,
= 3.1490 E, = 3.1570
MeV MeV
Fig. 1. Values of A / W as a function of center-of-mass energy and angle for a pair of compound resonances in 32S.The two resonances are located at E, = 3.1490 and 3.1570 MeV. See text for discussion.
2. Results The traditional detailed balance test involves unpolarized projectiles and observation of the exit particles in their ground state. We specialize to A(p, a& and C(a, p&4 reactions, and consider differential cross sections. We define a quantity ACE, 0): A(E, 0) k;
= 2
da
-(E, g(p, a) dQ,p,,, k;
da
-(E,
g(p, a) d&w,
k* dcr O)- a -(E, da, PI df&,, k* O)+ DL
-
0)
da
da, PI dn,,,p,
(E, 0) ’
(1) where k = 2a/A and g is the-statistical weight factor. Calculations were performed for target angular momenta A = l/2, 3/2, and 572 and C= 0, since we have experimental data for these cases. Natural parity resonances J” = l-, 2+, 3-, and 4+ were considered. A Hamiltonian of the form Ho + i H’ was assumed. We followed the approach of Moldauer [9], simplified to two interfering states, and assumed only internal mixing. To first order A is proportional to the matrix element W of H’. Experimental values of the resonance parameters were then used to calculate the cross sections. Our group has measured and analyzed high resolution (p, p) and (p, a> data on five odd-Z even-N targets in the nuclear 2s-ld shell: 23Na [lo], 27Al [ll], 3lP [12], 35C1 [13], and 39K [14]. Evaluation of these results yielded 33 pairs of adjacent resonances which are thought to have the same spin and parity and have measured widths in both the pa and a,, channels. As an example consider a specific pair of 2+ resonances in 32S. The resonances have the following (laboratory) parameters: E, = 3.2515 MeV, Es = 3.2597 MeV, F,, = 1.65 keV; F,, = 0.05 keV, F, = 6.8 keV, r,,, = 0.02 keV, I’,, = 0.08 keV, r,, = 2.5 keV. The
labels a and b refer to the two resonances, while 1, 2, 3 refer to the channels: 1 denotes the proton channel with channel spin s = 0 and orbital angular momentum I= 2, 2 denotes the proton channel with s = 1 and I= 2, and 3 denotes the alpha channel with s = 0 and I= 2. The ratio A/W is plotted in fig. 1 for this pair of resonances. For this resonance pair, A(E, 0) is a maximum near one of the two resonances and is very strongly peaked at 90” and 180”. The plots of A versus E and 0 display a variety of patterns for different pairs of resonances: the maximum value of A may be near one of the resonances or anywhere in between. The angular dependence shows a broad range of shapes, ranging from isolated peaks to wide (or narrow) ridges to very complicated patterns. The shapes (the energy and angular dependence) and the absolute value of A vary enormously. Therefore, in order to evaluate the merits of a detailed balance test of time reversal invariance using this approach, one should consider a specific pair of resonances. For our examples the maximum value of A/W varies from 2 X 10v4 to 2 X lo-‘, a range of 3 orders of magnitude. This provide the first major result: the relative enhancements vary strongly with energy, angle, and resonance pair. One should choose very carefully the particular pair of resonances to be used in a detailed balance test. (Naturally, the size of the term A/W is not the only factor to be considered in evaluating the merits of a particular experiment. In practice counting rates, target quality, beam stability, etc. are extremely important. In any case, such spikelike behavior as shown in fig. 1 is probably not as suitable for an actual experiment as a case where A changes more slowly with energy and angle. The key point is that large enhancements are present, but only at certain energies and angles.) IV. NUCLEAR PHYSICS
292
G.E. Mitchell et al. / Time reversal invariance
Since one expects to observe only a null value, the theoretical question becomes this: from a given null measurement (or set of null measurements), what is the value of the bound which will be set on the Tviolating matrix element? The matrix element is assumed to be drawn from a random distribution. Setting a bound and a confidence limit is an involved procedure, as discussed at length in [7,8]. However, an order of magnitude estimate can be obtained by assuming that the matrix element W is equal to its root-meansquare value W,,. In our calculations a typical value of A/W is of order 10e3. Assuming a null value A_ measured at the lo-* level seems reasonable, since that is the level of sensitivity attained in ref. [5]. With A/W= 10m3 and A, < lo-‘, this places an upper limit on W,, of order 10 eV. Since the strength of the strong interaction matrix element is of order MeV, a very rough estimate of 10e5 is obtained for an upper limit on the ratio (or (the ratio of the T-violating to the T-conserving interaction). This very small value should not be taken literally. However, this estimate does suggest the possible advantages of such experiments. This approximate limit on CQ is much better than the limits obtained in previous detailed balance measurements. These calculations suggest that detailed balance tests of time reversal invariance with two interfering levels should be seriously considered.
3. Summary Bunakov and Weidenmiiller suggested that large enhancement of T-violation in detailed balance experiments should occur near two interfering resonances. We have obtained explicit expressions for differential cross sections for the A(p, a,)C and C(a, p&l reactions for the target angular momenta C = 0, A = l/2, 3/2, and 5/2, and for the natural parity resonances l-, 2+, 3- and 4+. The approach of Moldauer (utilizing the R-matrix formalism) was followed to calculate the collision matrix elements for a Hamiltonian H = H,, + iH’; the matrix element of H’ is denoted by W. The resonance parameters were taken from earlier
measurements by our group on (p, p) and (p, a) reactions; there are 33 possible pairs of adjacent resonances with the same angular momentum and parity. The quantity A (the difference between the cross sections for the inverse reactions, divided by their average) was calculated for each resonance pair. A shows striking dependence on angle and energy, with maximum values of A/W ranging over several orders of magnitude. These preliminary results suggest that a detailed balance test involving two interfering resonances may be much more sensitive than previous detailed balance tests.
Acknowledgements The authors would like to thank V.E. Bunakov, C.R. Gould, H.L. Harney, and H.A. Weidenmiiller for valuable discussions. One of us (J.M.D.) acknowledges the US Department of Education for a Patricia Roberts Harris Fellowship.
References [l] N.R. Roberson, C.R. Gould and J.D. Bowman (eds.),
Tests of Time Reversal Invariance in Neutron Physics (World Scientific, Singapore, 1987). [2] V.P. Alfimenkov et al., Nucl. Phys. A398 (1983) 93. [3] J.D. Bowman et al., Phys. Rev. L&t. 65 (1990) 1192. [4] C.M. Frankle et al., Phys. Rev. Lett. 67 (1991) 564. [5] H. Driller et al., Nucl. Phys. A317 (1979) 300. [6] H.L. Hamey, A. Hiipper and A. Richter, Nucl. Phys. A518 (1990) 35. [7] V.E. Bunakov and H.A. Weidenmiiller, Phys. Rev. C39 (1989) 70. [8] E.D. Davis and U. Hartmann. Ann. Phys. (New York) 211 (1991) 334. [9] P.A. Moldauer, Phys. Rev. 165’0968) 1136. [lo] J.R. Vanhoy et al., Phys. Rev. C36 (1987) 920. [ll] R.O. Nelson et al., Phys. Rev. C29 (1984) 1656; Phys. Rev. C30 (1984) 755. [12] D.F. Fang et al., Phys. Rev. C37 (1988) 28. [13] WK. Brooks, Ph. D. dissertation, Duke University (1988). [14] B.J. Warthen, Ph. D. dissertation, Duke University (1987).