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Polarization in elastic nucleon-deuteron scattering
Volume
31B, number 5
POLARIZATION
PHYSICS
IN
ELASTIC
of Physics,
Case
Western
2 March 1970
NUCLEON-DEUTERON
J. KRAUSS Department
LETTERS
SCATTERING
t
and K. L. KOWALSKI Reserve
Received
University,
Cleveland,
Ohio 44106,
USA
10 January 1970
Two-nucleon tensor forces have been included in an approximate calculation of elastic nucleon-deuteron scattering at 14.4 and 22.7 MeV. Polarizations in reasonable agreement with experiment are obtained.
All calculations of low energy nucleon-deuteron (N-d) scattering to date * have been carried out under the assumption of central two-nucleon interactions. As a consequence all of these computations have been intrinsically incapable of determining the polarization, P(e), of the scattered nucleon in the elastic N-d reaction among other effects depending upon doublet-quartet transitions. The primary inhibiting feature here has been the prohibitive number of integral equations which necessarily appear with the introduction of non-central forces. However, there are several indications from the observed behavior of P(Q) [5] which suggest that the study of the spin-correlation properties of the three-nucleon system may be a more sensitive probe of the underlying dynamics than, for example, the cross section [6]. First, we note the rather rapid change in the characteristic shape of P(Q) between 14 to 23 MeV [5]. Secondly, we recall the gross failure of the simple impulse approximation in predicting P(Q) at an energy (40 MeV) where the cross section is not fitted too badly, at least in the forward scattering hemisphere [7]. In this notewe report the results of a calculation of P(B) at 14.4 and 22.7 MeV using an approximation procedure suggested by Sloan [8]** $ in order to reduce the mathematical complexity inherent in the ‘exact’ approaches referred to pret This work was supported in part by the US Atomic Energy Commission. * Hefs.? and 2 contain rather complete bibliographies of the calculations done through 1968. In this connection we refer to those works which employ exact three-particle dynamics. The semi-phenomenological analysis of the 9.0 MeV data by Purrington [3] includes tensor forces in Born approximation for high partial waves and as a guide for the phenomenological fit of the lowest partial waves. See ref. 4 for a recent approximate approach to N-d scattering.
viously * 4. Our results for P(Q) are displayed in fig. 1 and the corresponding cross sections are given in fig. 2. Before commenting on these results we will outline the particular features of our application of Sloan’s method tt. Although the essential practicability of this technique in no way depends upon the structure of the two-particle interaction, we assumed as a matter of mathematical convenience the simplest, separable, Yamaguchi [ 111 forms for the S-wave singlet and the S- and D-wave spin triplet N-N partial-wave states $1. The lack of any higher two-particle partial waves was almost certainly responsible for part of the suppression of the forward diffraction peak in the differential cross section (cf. fig. 2) #.
** For a discussion of the connection between the formalisms of refs. 8 and 9 and an interpretation of the specific approximation due to Sloan see Kowalski [lo]. z In Sloan’s approach [B-lo] the three-particle scattering integral equations are placed into the form of a Heitler equation. This last was solved trivially via partial-wave decomposition. The approximation involved consists of representing the source term, which also appears in the kernel, of this Heitler equation by a sum of the impulse graph and the nucleon-exchange terms. 4 The tectmique employed in ref. 4 also shows considerable promise for carrying out practical calculations with tensor forces. II The details of this calculation will be published elsewhere. $$ For the S-wave singlet the parameters of Phillips [12] corresponding to an effective range of 2.50 fm were employed. In the case of the S- and D-wave triplet the parameters of Brady [13] corresponding to a deuteron D-state probability of 7% were used. # This effect is characteristically present in earlier calculations at about these energies [l-2].
263
Volume 31B, number 5
P HY SI C S L E T T E R S
Using this input for the N-N interaction the partial-wave projections of the impulse graph terms as well as the nucleon-exchange term were computed numerically without further approximation and employed as source functions in the three-particle equations $. The N-d amplitude was approximated by the first four total J states; the contribution of the highest two, however, was undoubtedly relatively small. We see from fig. 1 that the predicted polarizations are in fair quantitative agreement with the experimental results at these energies [5]. All of the qualitative features of the data are reproduced including the presence of a negative dip in the 22.7 MeV case and the absence of one at 14.4 MeV. This last quality is probably the most significant aspect of the polarization data over this relatively small energy band [5]. In view of the approximations to the three-particle dynamics built into our approach [8, I0] :~ and the rather crude representation of the N-N interaction, the agreement is perhaps remarkable. The predicted cross sections are in reasonable agreement with the data [14, 15] at the intermediate angles. The discrepancies in the forSee foregoingpage.
2 M a r c h 1970
E= 14.4 MEV
0.2
~} {~}
0.1
0.0
Lo
0.6
o.2
-0.2
-o.6
E= 22.7 MEV
0.2
-Lo
$0
0.1 Q_
0.0 ,o
-o
j
cos eCMO ~, -0.1
_
-,o
,~0
~+ %0
Fig. 1. N-d p o l a r i z a t i o n at 14.4 and 22.7 MeV. The solid line indicates the r e s u l t s of the p r e s e n t c a l c u l a lation. The data a r e taken f r o m F a i v r e et al. [5].
120
E = 22.7 M~=V
0 0
E = 14.4 MEV
0
I00
rl ALLRED ET AL. o KIKUCHI ET AL.
o BUNKER
I001
0
O, oi
~;~80-60. ~ °
4O
4O
20t
20
O
1.0
~
0.6
0.2 -0.2 col eCM
-0.6
-I.0
0
I
1.0
I
; I
0.6
I
I
I
I
0.2 -0.2 co= eCM
I
I
-0.6
I
-hO
Fig. 2. N-d c r o s s s e c t i o n s at 14.4 and 22.7 MeV. The solid line indicates the r e s u l t s of the p r e s e n t calculation. The data a r e taken f r o m r e f s . 14 and 15.
264
Volume 31B, n u m b e r 5
PHYSICS
ward direction were commented upon previously in connection with the number of two-particle angular momentum states included in the calculation. The disagreement in the backward d i r e c tion cannot be explained in such a manner and most probably r e f l e c t s the lack of exact unitarity in Sloan's method; of course, part of the trouble at cos 8 ~ +i certainly a r i s e s from this fact as well. This effect, for cos e ~ - I , is r a t h e r unusual in that this technique s e e m s to over-damp the (dominant) nucleon-exchange, or pickup, diagram ##. ## The calculations of Sloan [8] also show the s a m e b e h a v i o r at cos ~ - 1 .
References 1. I. Duck, in: Advances in N u c l e a r P h y s i c s , eds. M. B a r a n g e r and E. Vogt (Plenum P r e s s , New York, 1968) Vol. I, p. 341. 2. A. Mitra, in: Advances in Nuclear P h y s i c s , eds. M. B a r a n g e r and E. Vogt (Plenum P r e s s , New York, 1969), Vol. III, p. 1. 3. R. D. P u r r i n g t o n and J. L. G a m m e l , Phys. Rev. 168 (1968) 1174.
LETTERS
2 M a r c h 1970
4. W. Ebenh~h, A. S. R i n a t - R e i n e r and Y. Avishai, Phys. L e t t e r s 29B (1969) 638; Ann. Phys. (N.Y.) 55 (1969) 341. 5. J.C. Faivre et al., Nucl. Phys. A127 (1969) 169; H. E. Conzett, G. Igo and W. J. Knox, Phys. Rev. Letters 12 (1964) 222; J. L. Malanify et al., Phys. Rev. 146 (1966) 632; R. L. Walter and C. A. Kelsey, Nuel. Phys. 46 (1963) 66. 6. For further commentary on this point see H. P. Noyes, in: Polarized targets and ion sources (Direction de la Physique Centre d'Etudes Nucl~aires de Saelay, 1966) p. 309. 7. H. Kottler and K. L. Kwalski, Phys. Rev. 138 (1965) B619. 8. I.H. Sloan, Phys. Rev. 165 (1968) 1587; Phys. Letters 25B (1967) 84. 9. R.W. Finkel and L. Rosenberg, Phys. Rev. 168 (1968) 1841. 10. K. L. Kowalski, Quasi-unitary three-particle approximations, Phys. Rev., to be published. 11. Y. Yamaguchi, Phys. Rev. 95 (1954) 1628, 1635. 12. A.C. Phillips, Nucl. Phys. A107 (1968) 209. 13. T. Brady et al., to be published. 14. S.Kikuehi et al., J. Phys. Soc. Japan 15 (1960) 9; J. C. Allred, A. H. Armstrong and L. Rosen, Phys. Rev. 91 (1953) 90. 15. S.N. Bunker et al., Nucl. Phys. Al13 (1968) 461.
265
31B, number 5
POLARIZATION
PHYSICS
IN
ELASTIC
of Physics,
Case
Western
2 March 1970
NUCLEON-DEUTERON
J. KRAUSS Department
LETTERS
SCATTERING
t
and K. L. KOWALSKI Reserve
Received
University,
Cleveland,
Ohio 44106,
USA
10 January 1970
Two-nucleon tensor forces have been included in an approximate calculation of elastic nucleon-deuteron scattering at 14.4 and 22.7 MeV. Polarizations in reasonable agreement with experiment are obtained.
All calculations of low energy nucleon-deuteron (N-d) scattering to date * have been carried out under the assumption of central two-nucleon interactions. As a consequence all of these computations have been intrinsically incapable of determining the polarization, P(e), of the scattered nucleon in the elastic N-d reaction among other effects depending upon doublet-quartet transitions. The primary inhibiting feature here has been the prohibitive number of integral equations which necessarily appear with the introduction of non-central forces. However, there are several indications from the observed behavior of P(Q) [5] which suggest that the study of the spin-correlation properties of the three-nucleon system may be a more sensitive probe of the underlying dynamics than, for example, the cross section [6]. First, we note the rather rapid change in the characteristic shape of P(Q) between 14 to 23 MeV [5]. Secondly, we recall the gross failure of the simple impulse approximation in predicting P(Q) at an energy (40 MeV) where the cross section is not fitted too badly, at least in the forward scattering hemisphere [7]. In this notewe report the results of a calculation of P(B) at 14.4 and 22.7 MeV using an approximation procedure suggested by Sloan [8]** $ in order to reduce the mathematical complexity inherent in the ‘exact’ approaches referred to pret This work was supported in part by the US Atomic Energy Commission. * Hefs.? and 2 contain rather complete bibliographies of the calculations done through 1968. In this connection we refer to those works which employ exact three-particle dynamics. The semi-phenomenological analysis of the 9.0 MeV data by Purrington [3] includes tensor forces in Born approximation for high partial waves and as a guide for the phenomenological fit of the lowest partial waves. See ref. 4 for a recent approximate approach to N-d scattering.
viously * 4. Our results for P(Q) are displayed in fig. 1 and the corresponding cross sections are given in fig. 2. Before commenting on these results we will outline the particular features of our application of Sloan’s method tt. Although the essential practicability of this technique in no way depends upon the structure of the two-particle interaction, we assumed as a matter of mathematical convenience the simplest, separable, Yamaguchi [ 111 forms for the S-wave singlet and the S- and D-wave spin triplet N-N partial-wave states $1. The lack of any higher two-particle partial waves was almost certainly responsible for part of the suppression of the forward diffraction peak in the differential cross section (cf. fig. 2) #.
** For a discussion of the connection between the formalisms of refs. 8 and 9 and an interpretation of the specific approximation due to Sloan see Kowalski [lo]. z In Sloan’s approach [B-lo] the three-particle scattering integral equations are placed into the form of a Heitler equation. This last was solved trivially via partial-wave decomposition. The approximation involved consists of representing the source term, which also appears in the kernel, of this Heitler equation by a sum of the impulse graph and the nucleon-exchange terms. 4 The tectmique employed in ref. 4 also shows considerable promise for carrying out practical calculations with tensor forces. II The details of this calculation will be published elsewhere. $$ For the S-wave singlet the parameters of Phillips [12] corresponding to an effective range of 2.50 fm were employed. In the case of the S- and D-wave triplet the parameters of Brady [13] corresponding to a deuteron D-state probability of 7% were used. # This effect is characteristically present in earlier calculations at about these energies [l-2].
263
Volume 31B, number 5
P HY SI C S L E T T E R S
Using this input for the N-N interaction the partial-wave projections of the impulse graph terms as well as the nucleon-exchange term were computed numerically without further approximation and employed as source functions in the three-particle equations $. The N-d amplitude was approximated by the first four total J states; the contribution of the highest two, however, was undoubtedly relatively small. We see from fig. 1 that the predicted polarizations are in fair quantitative agreement with the experimental results at these energies [5]. All of the qualitative features of the data are reproduced including the presence of a negative dip in the 22.7 MeV case and the absence of one at 14.4 MeV. This last quality is probably the most significant aspect of the polarization data over this relatively small energy band [5]. In view of the approximations to the three-particle dynamics built into our approach [8, I0] :~ and the rather crude representation of the N-N interaction, the agreement is perhaps remarkable. The predicted cross sections are in reasonable agreement with the data [14, 15] at the intermediate angles. The discrepancies in the forSee foregoingpage.
2 M a r c h 1970
E= 14.4 MEV
0.2
~} {~}
0.1
0.0
Lo
0.6
o.2
-0.2
-o.6
E= 22.7 MEV
0.2
-Lo
$0
0.1 Q_
0.0 ,o
-o
j
cos eCMO ~, -0.1
_
-,o
,~0
~+ %0
Fig. 1. N-d p o l a r i z a t i o n at 14.4 and 22.7 MeV. The solid line indicates the r e s u l t s of the p r e s e n t c a l c u l a lation. The data a r e taken f r o m F a i v r e et al. [5].
120
E = 22.7 M~=V
0 0
E = 14.4 MEV
0
I00
rl ALLRED ET AL. o KIKUCHI ET AL.
o BUNKER
I001
0
O, oi
~;~80-60. ~ °
4O
4O
20t
20
O
1.0
~
0.6
0.2 -0.2 col eCM
-0.6
-I.0
0
I
1.0
I
; I
0.6
I
I
I
I
0.2 -0.2 co= eCM
I
I
-0.6
I
-hO
Fig. 2. N-d c r o s s s e c t i o n s at 14.4 and 22.7 MeV. The solid line indicates the r e s u l t s of the p r e s e n t calculation. The data a r e taken f r o m r e f s . 14 and 15.
264
Volume 31B, n u m b e r 5
PHYSICS
ward direction were commented upon previously in connection with the number of two-particle angular momentum states included in the calculation. The disagreement in the backward d i r e c tion cannot be explained in such a manner and most probably r e f l e c t s the lack of exact unitarity in Sloan's method; of course, part of the trouble at cos 8 ~ +i certainly a r i s e s from this fact as well. This effect, for cos e ~ - I , is r a t h e r unusual in that this technique s e e m s to over-damp the (dominant) nucleon-exchange, or pickup, diagram ##. ## The calculations of Sloan [8] also show the s a m e b e h a v i o r at cos ~ - 1 .
References 1. I. Duck, in: Advances in N u c l e a r P h y s i c s , eds. M. B a r a n g e r and E. Vogt (Plenum P r e s s , New York, 1968) Vol. I, p. 341. 2. A. Mitra, in: Advances in Nuclear P h y s i c s , eds. M. B a r a n g e r and E. Vogt (Plenum P r e s s , New York, 1969), Vol. III, p. 1. 3. R. D. P u r r i n g t o n and J. L. G a m m e l , Phys. Rev. 168 (1968) 1174.
LETTERS
2 M a r c h 1970
4. W. Ebenh~h, A. S. R i n a t - R e i n e r and Y. Avishai, Phys. L e t t e r s 29B (1969) 638; Ann. Phys. (N.Y.) 55 (1969) 341. 5. J.C. Faivre et al., Nucl. Phys. A127 (1969) 169; H. E. Conzett, G. Igo and W. J. Knox, Phys. Rev. Letters 12 (1964) 222; J. L. Malanify et al., Phys. Rev. 146 (1966) 632; R. L. Walter and C. A. Kelsey, Nuel. Phys. 46 (1963) 66. 6. For further commentary on this point see H. P. Noyes, in: Polarized targets and ion sources (Direction de la Physique Centre d'Etudes Nucl~aires de Saelay, 1966) p. 309. 7. H. Kottler and K. L. Kwalski, Phys. Rev. 138 (1965) B619. 8. I.H. Sloan, Phys. Rev. 165 (1968) 1587; Phys. Letters 25B (1967) 84. 9. R.W. Finkel and L. Rosenberg, Phys. Rev. 168 (1968) 1841. 10. K. L. Kowalski, Quasi-unitary three-particle approximations, Phys. Rev., to be published. 11. Y. Yamaguchi, Phys. Rev. 95 (1954) 1628, 1635. 12. A.C. Phillips, Nucl. Phys. A107 (1968) 209. 13. T. Brady et al., to be published. 14. S.Kikuehi et al., J. Phys. Soc. Japan 15 (1960) 9; J. C. Allred, A. H. Armstrong and L. Rosen, Phys. Rev. 91 (1953) 90. 15. S.N. Bunker et al., Nucl. Phys. Al13 (1968) 461.
265