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The two-body photodisintegration of the 3He nucleus
I 1.B:2.I i i
Nuclear Physics A211 (1973) 533--540; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
THE TWO-BODY PHOTODISINTEGRATION OF THE 3He NUCLEUS J. A. H E N D R Y
Department of Computational and Statistical Science, University of Liverpool
and A. C. PHILLIPS Department of Physics, University of Manchester Received 26 February 1973 (Revised 8 June 1973) Abstract: The two-body photodisintegration of 3He is calculated using SHe and proton-deuteron wave functions obtained by assuming a separable interaction for the two-nucleon t-matrix. We show that the isotropic component of the cross section is unlikely to yield useful information on the D-state probability densities of the deuterort and 3He. A detailed comparison is made with the data on the cross section at 90 °. The separable approximation can account for some of the experimental results, but other experiments suggest that a more sophisticated treatment of the nuclear interactions and wave functions is necessary.
1. Introduction The photodisintegration of 3He can provide useful information on the structure of the 3He nucleus provided that the effects of the scattering in the final state are fully taken into account. This paper describes calculations of the cross sections for twobody photodisintegration of 3He and for the inverse reaction, radiative protondeuteron capture. We concentrate on the low energy region where theory is most reliable, and where the reactions are mainly sensitive to the low momentum components of the 3He wave function. For photon energies below 30 MeV the differential cross section for 3He photodisintegration can be parameterized by dadiJdg2 = a + b sin 2 0(1 + 8 cos
0+? cos 2 0),
(1)
where 0 is the angle between the outgoing proton and deuteron in the c.m. The corresponding cross section for radiative proton-deuteron capture can be obtained using detailed balance. We have
d~oa~ _ do'~i~ dr2
dO
E~
(2)
3pc2E '
where/~ is the nucleon-deuteron reduced mass, E~ the c.m. ?-ray energy and E the c.m. energy of the proton-deuteron system. The calculations reported here are based on the use of Faddeev equations and the 533
534
J.A. HENDRY AND A. C. PHILLIPS
separable approximation for the two-nucleon t-matrix. We adopt a procedure which is similar to that used in earlier work 1- 3). In particular the calculations of ref. 3) are extended so as to include the effects of tensor forces in the nucleon-deuteron state and to deal with energies above the three-nucleon break-up threshold. In brief, Yamaguchi interactions are used to generate numerical 3He wave functions, an analytic fit is made and the parameters governing the asymptotic behaviour are modified so as to correspond to the actual value of the 3He binding energy. We use 3He wave functions which correspond to deuteron D-state probability densities of 4 and 7 %; a detailed description of these can be found in appendix A of ref. 3). The nucleon-deuteron scattering corrections are evaluated by the numerical solution of an integral equation. The numerical methods used are described in the appendix of ref. 2). Finally, using the prescription described in ref. 3) a correction is made to account for the Coulomb repulsion in the proton-deuteron state. In comparing with experiment, it is useful to consider separately the isotropic component of the cross section, 4zra, and the magnitude of the cross section at a proton-deuteron angle of 90 °. There is also experimental information on the complete angular distribution 4). This is of limited theoretical interest since it is sensitive to electric and magnetic multipoles of high order and is not particularly sensitive to the structure of the 3He bound state 4, 5). The main interest in the isotropic cross section stems from the speculation that this component could provide useful information on the D-state probability density of 3He [refs. 6, 7)]. However to explore this possibility one has to estimate the magnetic dipole (M1) contribution to the cross section. This cannot be evaluated without reference to models for meson exchange or interaction currents. However, the magnitude of the M1 amplitude is known at low energy and, for energies below the threenucleon break-up threshold, the energy dependence is insensitive to the particular form of the interaction currents 3). These calculations also indicate that the M1 contribution to the isotropic cross section becomes increasingly insignificant as the energy increases and that the major contribution comes from electric dipole (El) transitions involving the D-state components of the 3He and 2H nuclei. In sect. 2 the calculations of ref. 3) are extended by including the effects of tensor forces in the proton-deuteron state and by considering energies above and below the three-nucleon break-up threshold. It is shown that accurate measurements of the isotropic cross section are unlikely to yield useful information on the D-state components of 3He and 2H. The photodisintegration cross section at 90 ° is given by a+b. Except for energies near to the proton-deuteron threshold, the 90 ° cross section is insensitive to the uncertainties in the value of a. The dominant term, b, is due to electric dipole transitions and depends mainly on the S-state components of 3He. As a consequence the theoretical analysis of the 90 ° cross section is relatively straightforward. For low energies there is a clear-cut separation of the known electromagnetic interaction and the uncertain nuclear interaction, since interaction currents do not contribute to
3He PHOTODISINTEGRATION
535
electric multipole transitions in the long wavelength approximation. Furthermore a detailed knowledge of the short range behaviour of the wave functions is not necessary because the form of the E1 operator, (E" r), emphasises the long-range properties of the system. The E1 contribution to the photodisintegration at energies above the three-nucleon break-up threshold has been calculated by including the effects of tensor forces in the 3He and proton-deuteron states, and the effects of the Coulomb repulsion between proton and deuteron. The results are described in sect. 3.
2. The isotropic component of the photodisintegration and capture cross sections The theoretical and experimental values for the isotropic cross section for radiative proton-deuteron capture are illustrated in fig. I. The corresponding photodisintegrai
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Fig. 1. The isotropic cross section for radiative proton deuteron capture as a function of the c.m. energy of the p-d state. The E1 cross sections, obtained using Yamaguchi interaction which give 4 and 7% for the deuteron D-state, and an estimate for the M1 cross section are compared with the data o f refs. 4. 9. to).
tion cross sections can be obtained using detailed balance. The figure shows the E1 cross section obtained when Yamaguchi interactions with different tensor components are used to calculate the 3He and proton-deuteron wave functions; two cases were considered, corresponding to 4 % and 7 9/o D-state probability densities for the deuteron. The estimate obtained in ref. 3) for the M1 cross section at low energies is also included. It is apparent that the larger D-state probability gives a smaller cross section for c.m. energies less than 12 MeV, but a higher cross section at energies above 12 MeV. We can interpret this result in the following way: The interactions used have been adjusted to fit the quadrupole moment of the deuteron. This is mainly sensitive to the long-range part of the D-state wave function.
536
J.A. HENDRY AND A. C. PHILLIPS
If the D-state probability is reduced, the fit to the quadrupole moment requires an increase in the range of the D-state wave function. [In realistic models of the deuteron the minimum value of the D-state probability is set by requiring the range of the interaction to be less than that of one pion exchange s). ] Because of the inflexibility of the Yamaguchi model, the smaller D-state corresponds to a smaller D-state wave function at medium distances but a larger wave function at large distances. When the corresponding interactions are used to generate 3He wave functions a similar behaviour in the D-state components is produced. Thus i f a reaction is sensitive to the asymptotic part of the D-state wave function, the model with the smaller D-state probability will give the larger cross section. We conclude from fig. 1, that the isotropic cross section depends mainly on the asymptotic D-state wave functions at low energies but begins to be sensitive to the medium-range behaviour of the D-states, for energies higher than 12 MeV. In realistic models the uncertainties in the D-state probabilities of the deuteron and 3He are a reflection of the unknown behaviour of the short- and medium-range part of the wave functions. Thus in principle the photodisintegration reaction at energies above 12 MeV could give useful information on the D-states. In practice this is not possible because of the uncertainties in the theory at these energies; higher-order multipole transitions, retardation, recoil and, possibly, interaction current corrections would have to be evaluated. For completeness the experimental data 4, 9,1 o) on the isotropic cross section are also included in fig. 1. The large error bars are an indication of the difficulty in measuring this part of the cross section. It is possible to estimate the M1 contribution at low energies 3), which when combined with the E1 contribution gives a cross section in agreement with the low energy data of Griffiths et al. 9) t. It is not possible to obtain a reliable estimate for the M1 cross section at higher energies. However even in the absence of such an estimate, it is clear that there is a discrepancy between theory and experiment in the region of 10 MeV, since the E1 cross section alone is considerably larger than the experimental value. 3. The photodisintegration and capture cross sections at 90 °
In this section we compare experimental and theoretical results on the photodisintegration reaction in a region where the E1 transition amplitude is dominant. There are several measurements of the photodisintegration cross section 11-14) and of the directly related capture cross section 4,.1o). In addition the electrodisintegration of 3He provides useful information on the photodisintegration reaction 15). The photodisintegration diffential cross section has a broad peak at a photon energy E~ ~ 12 MeV, which at 90 ° is dominated by E1 transitions. In this region the data of refs. lO-13) are in good agreement with each other and give a 90 ° cross section 1. Near the three-rmcleon break-up energy the cross section obtained is 4-0~ bigger than that reported in ref. a). This is because of the inclusion of tensor forces in the p-d state.
3He PHOTOD1SINTEGRATION
537
which is approximately 86#b/sr at Er ~ 12 MeV. However the experiments of refs. 4,1~, ~5) are also mutually consistent but imply that the cross section at 12 MeV is 15 to 20 % bigger; i.e. approximately 106 #b/sr. In fig. 2 we compare the theoretical calculations with the experimental data on the i
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E IS (MeV} Fig. 2. The two-body photodisintegration at 90 ° as a function of the photon energy. The E1 cross sections, obtained using ¥amaguchi interactions which give 0, 4 and 7 ~ for the deuteron D-state, are compared with the data of refs. 4. ,,, lz).
90 ° cross section; the data of Berman et al. 11) and Stewart et al. 13) are included to represent the experiments which imply a smaller cross section at the peak, and the experiments which give the larger cross section are represented by the data of Belt et al. 4). The curves labelled 0, 4 and 7 % were obtained by using Yamaguchi interactions which yield 0, 4 and 7 % for the deuteron D-state; these interactions were used to evaluate the 3He and proton-deuteron wave functions. In addition these curves also include a proton-deuteron Coulomb correction which decreases the cross section at the peak by 7 %. We note the tendency for the cross section to decrease when the D-state probability density increases, but even for the largest D-state, 7 ~ , the theoretical cross section is greater than the experimental values obtained by Berman and Stewart. This is significant since, within the context of simple separable approximations, the theoretical results in fig. 2 are most likely to be too small. This is because the interactions used neglect the short-range repulsion between the nucleons. As a result the 3He wave functions used give a charge form factor which falls off too slowly with increasing q 2 A similar situation occurred in the calculations described in ref. 2). Wave function I of ref. 2) corresponds to the 0 ~ D-state wave function used here to evaluate the cross section in fig. 2, This wave function can be modified to give a better charge form factor. First, the overall range parameters of the wave function can be increased; in ref. 2)
538
J.A. HENDRY AND A. C. PHILLIPS
this was done to obtain wave function II from I. Second, short-rangerepulsion between the nucleon pairs may be introduced to obtain the wave function I I I of ref. 2). We have followed an identical procedure for the aHe wave function given by the 4 ~o deuteron D-state. A change in the overall range parameters gives a photodisintegra-
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Fig. 3. The two-body photodisintegration at 90°. The effects of increasing the radius of the wave function derived from the 4~ deuteron D-state are illustrated by curves A and B. The curve A corresponds to changing the overall size parameter of the wave function and the curve B to including short-range repulsion between the nucleons.
tion cross section illustrated by curve A in fig. 3. The introduction o f short-range repulsion gives curve B. The latter calculation is not completely consistent since shortrange repulsion is not included in the proton-deuteron state. I f this were done the overlap of the deuteron and SHe wave functions would increase and as a result the cross section would be larger than that given by curve B. We conclude that the photodisintegration cross section at the peak is unlikely to be much smaller than that given by curve B if a simple separable interaction is assumed t However we note that it is conceivable that a separable interaction, which is merely constructed to give a unitary two-nucleon t-matrix with the reasonable behaviour near the deuteron and singlet-deuteron poles, could give a nucleon-deuteron component in the three-nucleon bound state which is too big. Indeed dispersion relation calculations on neutron-deuteron scattering 16,17) provide evidence to support this conjecture. Neutron-deuteron scattering can provide information on the overlap of the triton and deuteron wave functions; this overlap is roughly related to the t This conclusion is unlikely to be changed if higher-order multipole transitions are included. The E3 transition lowers the 90° cross section by about IX, a decrease which is cancelled by the M1 contribution.
3He PHOTODISINTEGRATION
539
magnitude of the two-body photodisintegration of 3H. We define the function
F(q2) = (B+ 3hZq2~4M / (aHlndq)"
(2)
Here B is the energy needed to remove a neutron from all, M the nucleon mass, the vector I3H) represents the triton state, and Indq) a plane wave state of a neutron and deuteron with momentum q and angular momentum ½. This function is related to the residue of the triton pole in neutron-deuteron scattering, by the equation exp
(i6)sin 6 _ R/( q2+ 4MB~
R=
(3)
3h 2 ] '
q
(4M 22= J r ( - 4MB I
(4)
Clearly R is a measure of the normalization of the asymptotic neutron-deuteron component of the triton. All wave functions derived from the separable Yamaguchi interactions give R = - 4.6 f m - ~,
(5)
if the asymptotic behaviour is adjusted so as to correspond to the actual value of the 3H binding energy. The residue R may also be extracted by applying dispersion relations to neutron-deuteron scattering 16, t T). All these calculations give a smaller magnitude for R. The most reliable calculation is due to Locher which uses forward dispersion relations and gives R = -3.2__.0.4 fm -1,
(6)
The error of +_0.4 is probably underestimated in view of the uncertainties in the phase shift analysis used in Locher's calculation and also in view of the small contribution of the triton pole in the physical region. However this result does suggest that the separable interaction wave functions tend to overemphasise the asymptotic neutron-deuteron component of the three-nucleon bound state. If this is the case, the two-body photodisintegration cross sections predicted by these wave functions are likely to be too large. We conclude that within the context of the separable approximation the photodisintegration cross section at the peak is unlikely to be smaller than that given by curve B of fig. 3. The data of Belt, Kundu, and Fetisov suggests that the separable approximation is sufficient to account for the photodisintegration reaction. But the data of Berman, Stewart, Finckh and W61fli indicate that the reaction requires a more sophisticated treatment of the nuclear interactions and wave functions. However, we note that dispersion relation calculations on neutron-deuteron scattering suggest that the separable interaction wave functions may overestimate the photodisintegration
540
.I.A. HENDRY AND A. C. PHILLIPS
cross section an d that m o r e realistic wave functions co u l d give results closer to the d a t a o f B e r m a n et al. We are i n d e b t e d to the D a r e s b u r y N u c l e a r Physics L a b o r a t o r y f o r c o m p u t i n g facilities.
References
1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)
A. C. Phillips, Phys. Rev. 170 (1968) 952 I. M. Barbour and A. C. Phillips, Phys. Rev. C1 (1970) 165 A. C. Phillips, Nucl. Phys. A184 (1972) 337 B.D. Belt, C. R. Bingham, M. L. Halbert and A. van der Woude, Phys. Rev. Lett. 24B (1970) 1120 I. M. Barbour and J. A. Hendry, Phys. Lett. 38B (1972) 151 G. M. Bailey, G. M. Grifliths and T. W. Dolmelly, Phys. Lett. 24B (1967) 222 S. K. Kundu, Y. M. Shin and G. D. Wait, Nucl. Phys. A171 (1971) 384 N. K. Glendenning and G. Kramer, Phys. Rev. 126 (1962) 2159 G. M. Grifl~ths, M. Lal and C. D. Scarfe, Can. J. Phys. 41 (1963) 724 W. W01fli, R. Bosch, J. Lang, R. Muller and P. Marmier, Phys. Lett. 22 (1966) 75; Helv. Phys. Acta 40 (1967) 946 B. Berman, L. Koester, Jr., and J. Smith, Phys. Rev. 133 (1963) BlI7 J. Stewart, R. Morrison and J. O'Connel, Phys. Rev. 138 (1964) B372 E. Finckh, R. Kosiek, K. H. Lindenberger, U. Meyer-Berkhout, N. Nucker and K. Schlupmann, Phys. Lett. 7 (1963) 271 V. N. Fetisov, A. N. Gorbunov and A. T. Varfolomeev, Nucl. Phys. 71 (1965) 305 S. K. Kundu, Y. M. Shin and G. D. Wait, Nucl. Phys. AI71 (1971) 384 M. P. Locher, Nucl. Phys. B23 (1970) 116 R. Bower, Ann. of Phys. 73 (1972) 372; L. P: Kok and A. S. Rinat, Nucl. Phys. A156 (1970) 593
Nuclear Physics A211 (1973) 533--540; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
THE TWO-BODY PHOTODISINTEGRATION OF THE 3He NUCLEUS J. A. H E N D R Y
Department of Computational and Statistical Science, University of Liverpool
and A. C. PHILLIPS Department of Physics, University of Manchester Received 26 February 1973 (Revised 8 June 1973) Abstract: The two-body photodisintegration of 3He is calculated using SHe and proton-deuteron wave functions obtained by assuming a separable interaction for the two-nucleon t-matrix. We show that the isotropic component of the cross section is unlikely to yield useful information on the D-state probability densities of the deuterort and 3He. A detailed comparison is made with the data on the cross section at 90 °. The separable approximation can account for some of the experimental results, but other experiments suggest that a more sophisticated treatment of the nuclear interactions and wave functions is necessary.
1. Introduction The photodisintegration of 3He can provide useful information on the structure of the 3He nucleus provided that the effects of the scattering in the final state are fully taken into account. This paper describes calculations of the cross sections for twobody photodisintegration of 3He and for the inverse reaction, radiative protondeuteron capture. We concentrate on the low energy region where theory is most reliable, and where the reactions are mainly sensitive to the low momentum components of the 3He wave function. For photon energies below 30 MeV the differential cross section for 3He photodisintegration can be parameterized by dadiJdg2 = a + b sin 2 0(1 + 8 cos
0+? cos 2 0),
(1)
where 0 is the angle between the outgoing proton and deuteron in the c.m. The corresponding cross section for radiative proton-deuteron capture can be obtained using detailed balance. We have
d~oa~ _ do'~i~ dr2
dO
E~
(2)
3pc2E '
where/~ is the nucleon-deuteron reduced mass, E~ the c.m. ?-ray energy and E the c.m. energy of the proton-deuteron system. The calculations reported here are based on the use of Faddeev equations and the 533
534
J.A. HENDRY AND A. C. PHILLIPS
separable approximation for the two-nucleon t-matrix. We adopt a procedure which is similar to that used in earlier work 1- 3). In particular the calculations of ref. 3) are extended so as to include the effects of tensor forces in the nucleon-deuteron state and to deal with energies above the three-nucleon break-up threshold. In brief, Yamaguchi interactions are used to generate numerical 3He wave functions, an analytic fit is made and the parameters governing the asymptotic behaviour are modified so as to correspond to the actual value of the 3He binding energy. We use 3He wave functions which correspond to deuteron D-state probability densities of 4 and 7 %; a detailed description of these can be found in appendix A of ref. 3). The nucleon-deuteron scattering corrections are evaluated by the numerical solution of an integral equation. The numerical methods used are described in the appendix of ref. 2). Finally, using the prescription described in ref. 3) a correction is made to account for the Coulomb repulsion in the proton-deuteron state. In comparing with experiment, it is useful to consider separately the isotropic component of the cross section, 4zra, and the magnitude of the cross section at a proton-deuteron angle of 90 °. There is also experimental information on the complete angular distribution 4). This is of limited theoretical interest since it is sensitive to electric and magnetic multipoles of high order and is not particularly sensitive to the structure of the 3He bound state 4, 5). The main interest in the isotropic cross section stems from the speculation that this component could provide useful information on the D-state probability density of 3He [refs. 6, 7)]. However to explore this possibility one has to estimate the magnetic dipole (M1) contribution to the cross section. This cannot be evaluated without reference to models for meson exchange or interaction currents. However, the magnitude of the M1 amplitude is known at low energy and, for energies below the threenucleon break-up threshold, the energy dependence is insensitive to the particular form of the interaction currents 3). These calculations also indicate that the M1 contribution to the isotropic cross section becomes increasingly insignificant as the energy increases and that the major contribution comes from electric dipole (El) transitions involving the D-state components of the 3He and 2H nuclei. In sect. 2 the calculations of ref. 3) are extended by including the effects of tensor forces in the proton-deuteron state and by considering energies above and below the three-nucleon break-up threshold. It is shown that accurate measurements of the isotropic cross section are unlikely to yield useful information on the D-state components of 3He and 2H. The photodisintegration cross section at 90 ° is given by a+b. Except for energies near to the proton-deuteron threshold, the 90 ° cross section is insensitive to the uncertainties in the value of a. The dominant term, b, is due to electric dipole transitions and depends mainly on the S-state components of 3He. As a consequence the theoretical analysis of the 90 ° cross section is relatively straightforward. For low energies there is a clear-cut separation of the known electromagnetic interaction and the uncertain nuclear interaction, since interaction currents do not contribute to
3He PHOTODISINTEGRATION
535
electric multipole transitions in the long wavelength approximation. Furthermore a detailed knowledge of the short range behaviour of the wave functions is not necessary because the form of the E1 operator, (E" r), emphasises the long-range properties of the system. The E1 contribution to the photodisintegration at energies above the three-nucleon break-up threshold has been calculated by including the effects of tensor forces in the 3He and proton-deuteron states, and the effects of the Coulomb repulsion between proton and deuteron. The results are described in sect. 3.
2. The isotropic component of the photodisintegration and capture cross sections The theoretical and experimental values for the isotropic cross section for radiative proton-deuteron capture are illustrated in fig. I. The corresponding photodisintegrai
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Fig. 1. The isotropic cross section for radiative proton deuteron capture as a function of the c.m. energy of the p-d state. The E1 cross sections, obtained using Yamaguchi interaction which give 4 and 7% for the deuteron D-state, and an estimate for the M1 cross section are compared with the data o f refs. 4. 9. to).
tion cross sections can be obtained using detailed balance. The figure shows the E1 cross section obtained when Yamaguchi interactions with different tensor components are used to calculate the 3He and proton-deuteron wave functions; two cases were considered, corresponding to 4 % and 7 9/o D-state probability densities for the deuteron. The estimate obtained in ref. 3) for the M1 cross section at low energies is also included. It is apparent that the larger D-state probability gives a smaller cross section for c.m. energies less than 12 MeV, but a higher cross section at energies above 12 MeV. We can interpret this result in the following way: The interactions used have been adjusted to fit the quadrupole moment of the deuteron. This is mainly sensitive to the long-range part of the D-state wave function.
536
J.A. HENDRY AND A. C. PHILLIPS
If the D-state probability is reduced, the fit to the quadrupole moment requires an increase in the range of the D-state wave function. [In realistic models of the deuteron the minimum value of the D-state probability is set by requiring the range of the interaction to be less than that of one pion exchange s). ] Because of the inflexibility of the Yamaguchi model, the smaller D-state corresponds to a smaller D-state wave function at medium distances but a larger wave function at large distances. When the corresponding interactions are used to generate 3He wave functions a similar behaviour in the D-state components is produced. Thus i f a reaction is sensitive to the asymptotic part of the D-state wave function, the model with the smaller D-state probability will give the larger cross section. We conclude from fig. 1, that the isotropic cross section depends mainly on the asymptotic D-state wave functions at low energies but begins to be sensitive to the medium-range behaviour of the D-states, for energies higher than 12 MeV. In realistic models the uncertainties in the D-state probabilities of the deuteron and 3He are a reflection of the unknown behaviour of the short- and medium-range part of the wave functions. Thus in principle the photodisintegration reaction at energies above 12 MeV could give useful information on the D-states. In practice this is not possible because of the uncertainties in the theory at these energies; higher-order multipole transitions, retardation, recoil and, possibly, interaction current corrections would have to be evaluated. For completeness the experimental data 4, 9,1 o) on the isotropic cross section are also included in fig. 1. The large error bars are an indication of the difficulty in measuring this part of the cross section. It is possible to estimate the M1 contribution at low energies 3), which when combined with the E1 contribution gives a cross section in agreement with the low energy data of Griffiths et al. 9) t. It is not possible to obtain a reliable estimate for the M1 cross section at higher energies. However even in the absence of such an estimate, it is clear that there is a discrepancy between theory and experiment in the region of 10 MeV, since the E1 cross section alone is considerably larger than the experimental value. 3. The photodisintegration and capture cross sections at 90 °
In this section we compare experimental and theoretical results on the photodisintegration reaction in a region where the E1 transition amplitude is dominant. There are several measurements of the photodisintegration cross section 11-14) and of the directly related capture cross section 4,.1o). In addition the electrodisintegration of 3He provides useful information on the photodisintegration reaction 15). The photodisintegration diffential cross section has a broad peak at a photon energy E~ ~ 12 MeV, which at 90 ° is dominated by E1 transitions. In this region the data of refs. lO-13) are in good agreement with each other and give a 90 ° cross section 1. Near the three-rmcleon break-up energy the cross section obtained is 4-0~ bigger than that reported in ref. a). This is because of the inclusion of tensor forces in the p-d state.
3He PHOTOD1SINTEGRATION
537
which is approximately 86#b/sr at Er ~ 12 MeV. However the experiments of refs. 4,1~, ~5) are also mutually consistent but imply that the cross section at 12 MeV is 15 to 20 % bigger; i.e. approximately 106 #b/sr. In fig. 2 we compare the theoretical calculations with the experimental data on the i
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E IS (MeV} Fig. 2. The two-body photodisintegration at 90 ° as a function of the photon energy. The E1 cross sections, obtained using ¥amaguchi interactions which give 0, 4 and 7 ~ for the deuteron D-state, are compared with the data of refs. 4. ,,, lz).
90 ° cross section; the data of Berman et al. 11) and Stewart et al. 13) are included to represent the experiments which imply a smaller cross section at the peak, and the experiments which give the larger cross section are represented by the data of Belt et al. 4). The curves labelled 0, 4 and 7 % were obtained by using Yamaguchi interactions which yield 0, 4 and 7 % for the deuteron D-state; these interactions were used to evaluate the 3He and proton-deuteron wave functions. In addition these curves also include a proton-deuteron Coulomb correction which decreases the cross section at the peak by 7 %. We note the tendency for the cross section to decrease when the D-state probability density increases, but even for the largest D-state, 7 ~ , the theoretical cross section is greater than the experimental values obtained by Berman and Stewart. This is significant since, within the context of simple separable approximations, the theoretical results in fig. 2 are most likely to be too small. This is because the interactions used neglect the short-range repulsion between the nucleons. As a result the 3He wave functions used give a charge form factor which falls off too slowly with increasing q 2 A similar situation occurred in the calculations described in ref. 2). Wave function I of ref. 2) corresponds to the 0 ~ D-state wave function used here to evaluate the cross section in fig. 2, This wave function can be modified to give a better charge form factor. First, the overall range parameters of the wave function can be increased; in ref. 2)
538
J.A. HENDRY AND A. C. PHILLIPS
this was done to obtain wave function II from I. Second, short-rangerepulsion between the nucleon pairs may be introduced to obtain the wave function I I I of ref. 2). We have followed an identical procedure for the aHe wave function given by the 4 ~o deuteron D-state. A change in the overall range parameters gives a photodisintegra-
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£ (MeV)
Fig. 3. The two-body photodisintegration at 90°. The effects of increasing the radius of the wave function derived from the 4~ deuteron D-state are illustrated by curves A and B. The curve A corresponds to changing the overall size parameter of the wave function and the curve B to including short-range repulsion between the nucleons.
tion cross section illustrated by curve A in fig. 3. The introduction o f short-range repulsion gives curve B. The latter calculation is not completely consistent since shortrange repulsion is not included in the proton-deuteron state. I f this were done the overlap of the deuteron and SHe wave functions would increase and as a result the cross section would be larger than that given by curve B. We conclude that the photodisintegration cross section at the peak is unlikely to be much smaller than that given by curve B if a simple separable interaction is assumed t However we note that it is conceivable that a separable interaction, which is merely constructed to give a unitary two-nucleon t-matrix with the reasonable behaviour near the deuteron and singlet-deuteron poles, could give a nucleon-deuteron component in the three-nucleon bound state which is too big. Indeed dispersion relation calculations on neutron-deuteron scattering 16,17) provide evidence to support this conjecture. Neutron-deuteron scattering can provide information on the overlap of the triton and deuteron wave functions; this overlap is roughly related to the t This conclusion is unlikely to be changed if higher-order multipole transitions are included. The E3 transition lowers the 90° cross section by about IX, a decrease which is cancelled by the M1 contribution.
3He PHOTODISINTEGRATION
539
magnitude of the two-body photodisintegration of 3H. We define the function
F(q2) = (B+ 3hZq2~4M / (aHlndq)"
(2)
Here B is the energy needed to remove a neutron from all, M the nucleon mass, the vector I3H) represents the triton state, and Indq) a plane wave state of a neutron and deuteron with momentum q and angular momentum ½. This function is related to the residue of the triton pole in neutron-deuteron scattering, by the equation exp
(i6)sin 6 _ R/( q2+ 4MB~
R=
(3)
3h 2 ] '
q
(4M 22= J r ( - 4MB I
(4)
Clearly R is a measure of the normalization of the asymptotic neutron-deuteron component of the triton. All wave functions derived from the separable Yamaguchi interactions give R = - 4.6 f m - ~,
(5)
if the asymptotic behaviour is adjusted so as to correspond to the actual value of the 3H binding energy. The residue R may also be extracted by applying dispersion relations to neutron-deuteron scattering 16, t T). All these calculations give a smaller magnitude for R. The most reliable calculation is due to Locher which uses forward dispersion relations and gives R = -3.2__.0.4 fm -1,
(6)
The error of +_0.4 is probably underestimated in view of the uncertainties in the phase shift analysis used in Locher's calculation and also in view of the small contribution of the triton pole in the physical region. However this result does suggest that the separable interaction wave functions tend to overemphasise the asymptotic neutron-deuteron component of the three-nucleon bound state. If this is the case, the two-body photodisintegration cross sections predicted by these wave functions are likely to be too large. We conclude that within the context of the separable approximation the photodisintegration cross section at the peak is unlikely to be smaller than that given by curve B of fig. 3. The data of Belt, Kundu, and Fetisov suggests that the separable approximation is sufficient to account for the photodisintegration reaction. But the data of Berman, Stewart, Finckh and W61fli indicate that the reaction requires a more sophisticated treatment of the nuclear interactions and wave functions. However, we note that dispersion relation calculations on neutron-deuteron scattering suggest that the separable interaction wave functions may overestimate the photodisintegration
540
.I.A. HENDRY AND A. C. PHILLIPS
cross section an d that m o r e realistic wave functions co u l d give results closer to the d a t a o f B e r m a n et al. We are i n d e b t e d to the D a r e s b u r y N u c l e a r Physics L a b o r a t o r y f o r c o m p u t i n g facilities.
References
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A. C. Phillips, Phys. Rev. 170 (1968) 952 I. M. Barbour and A. C. Phillips, Phys. Rev. C1 (1970) 165 A. C. Phillips, Nucl. Phys. A184 (1972) 337 B.D. Belt, C. R. Bingham, M. L. Halbert and A. van der Woude, Phys. Rev. Lett. 24B (1970) 1120 I. M. Barbour and J. A. Hendry, Phys. Lett. 38B (1972) 151 G. M. Bailey, G. M. Grifliths and T. W. Dolmelly, Phys. Lett. 24B (1967) 222 S. K. Kundu, Y. M. Shin and G. D. Wait, Nucl. Phys. A171 (1971) 384 N. K. Glendenning and G. Kramer, Phys. Rev. 126 (1962) 2159 G. M. Grifl~ths, M. Lal and C. D. Scarfe, Can. J. Phys. 41 (1963) 724 W. W01fli, R. Bosch, J. Lang, R. Muller and P. Marmier, Phys. Lett. 22 (1966) 75; Helv. Phys. Acta 40 (1967) 946 B. Berman, L. Koester, Jr., and J. Smith, Phys. Rev. 133 (1963) BlI7 J. Stewart, R. Morrison and J. O'Connel, Phys. Rev. 138 (1964) B372 E. Finckh, R. Kosiek, K. H. Lindenberger, U. Meyer-Berkhout, N. Nucker and K. Schlupmann, Phys. Lett. 7 (1963) 271 V. N. Fetisov, A. N. Gorbunov and A. T. Varfolomeev, Nucl. Phys. 71 (1965) 305 S. K. Kundu, Y. M. Shin and G. D. Wait, Nucl. Phys. AI71 (1971) 384 M. P. Locher, Nucl. Phys. B23 (1970) 116 R. Bower, Ann. of Phys. 73 (1972) 372; L. P: Kok and A. S. Rinat, Nucl. Phys. A156 (1970) 593