Is the relativistic distorted-wave impulse approximation useful to nuclear reaction?

Transcription

1 Is the relativistic distorted-wave impulse approximation useful to nuclear reaction? K. Miyazaki Abstract e have reconsidered the nonrelativistic distorted-wave t-matrix approximation (NR-DTA) for proton knockout (p; p) reaction using modern high-quality phenomenological optical potentials and NN t-matrix. e have calculated 4 Ca(p; p) reactions at T LAB = MeV and compared the results with the relativistic distorted-wave impulse approximation (RDIA) calculations. It is found that the NR-DTA is superior to the RDIA in consistent description of the cross section and the analyzing power. An immediate relativistic extension of the DIA to the nuclear reaction has a problem. 1 Introduction It seems that the success of the relativistic impulse model for proton-nucleus (p-a) elastic scattering [1,] inspires us to apply it to other p-a scatterings and reactions. Recent highquality phenomenological relativistic optical model potentials [3,4] and nucleon-nucleon (NN) t-matrix [5] have made such attempts possible. One of the most thoroughgoing investigations among them is the RDIA description of the (p; p) reaction by Cooper and Maxwell [6]. The result of their enormous numerical calculations is however somewhat discouraging. It is not obviously superior to the old calculation [7] based on the NR- DTA, which used the nonrelativistic nonglobal optical potentials by interpolating them and the nonrelativistic Love-Franey NN t-matrix. It is therefore worthwhile to perform the NR-DTA calculation using new relativistic inputs of the optical potential and the NN t-matrix. Such a calculation becomes an extensive test of them. Furthermore the comparison of it with the result of Ref. [6] provides us an insight on the validity of the relativistic approach to nuclear reactions. In the next section we review the NR-DTA formalism for (p; p) reaction. The relevant factorized NN t-matrix is evaluated explicitly in section 3. The numerical results are shown and discussed in comparison with RDIA results in section 4. Finally, our conclusion is drawn in section 5. This paper is the revised version of CDS ext-3-7. Some texts and mistypes have been corrected. 1

2 The formalism of NR-DTA Although there are several review articles [8-1] and textbooks [13,14] for (p; p) reaction, we here review the NR-DTA brie y for the discussion in 4. The t-matrix for (p; p) reaction in the nonrelativistic distorted-wave impulse approximation (NR-DIA) is T (p;p) = D 1 E (+) t pp t ; (1) where (+) and 1() are the distorted waves of the incoming () and two outgoing (1 and ) protons, t is a single-particle bound-state wave function of the knocked-out proton (t) in the target nucleus (or more precisely speaking, an overlap function between the wave functions of target and residual nucleus) and t pp is the t-matrix operator for proton-proton (p-p) scattering in free space. For simplicity, the spin-orbit distortions are neglected. Equation (1) is represented in momentum space as T (p;p) = D 1 Introducing the reduced p-p t-matrix we then factorize it as E D E D p 1 p h p 1 p j t pp j p p t i p (+) E h p t j t i : () h p 1 p j t pp j p p t i = 3 (p 1 + p p p t ) h p 1 p k t pp k p p t i ; (3) h p 1 p k t pp k p p t i! h k 1 k k t pp k k k t i ; (4) where k, k 1 and k are the asymptotic momentums of the incoming and two outgoing protons in the p-a c.m. frame and k t is restricted by the momentum conservation: k t = k 1 + k k : (5) As a result, T (p;p) = h k 1 k k t pp k k k t i hd 1 E D E D E (+) p 1 p p h p 1 + p i p j t i : (6) This is an essential expression of the NR-DTA for (p; p) reaction. The actual calculation is performed in coordinate space: hd 1 T (p;p) = h k 1 k k t pp k k k t i D E (+) h p j r i r E D E r 1 h r 1 j p 1 i r h r j p i i h p 1 + p p j r t i h r t j t i : (7) Using h p 1 + p p j r t i = h p 1 j r t i h p j r t i h r t j p i ; (8)

3 e nally obtain T (p;p) = h k 1 k k t pp k k k t i hd E 1 r 1 h r 1 jr t i Z = h k 1 k k t pp k k k t i D E D r h r jr t i h r t jr i d 3 r 1 (r) r (+) E i h r t j t i ; (9) (r) (+) (r) t (r) : (1) The integral of Eq. (1) is called as the distorted momentum distribution. Usually, the argument r in the initial distorted-wave function is multiplied by a factor 1 1=A [14], where A is the mass number of the target nucleus. In this work, such a factor is ignored because other 1=A e ects, the coupling term between the coordinates of two outgoing protons [14] and the recoil term [9], have already been neglected from the rst. The DTA is often referred as a zero-range approximation. This term is however misleading. In the model of NN t-matrix using one-boson exchange potentials, the zerorange means in nite masses of the mesons. Such a situation is unphysical while the DTA is a reasonable physical prescription as will be discussed in section 4. 3 The factorized NN t-matrix Here we show explicit expression of the factorized NN t-matrix in Eq. (4). The relativistic model of Ref. [5] enables us to calculate it directly: h k 1A k A k t pp k k A k ta i = u 1 (k 1A ) u (k A ) ^t as pp u 1 (k A ) u (k ta ) ; (11) where the Dirac spinor u 1() (k ia ) (i =, 1, and t) is and 1 u 1() (k ia ) = N ia ; (1) 1() k ia = (E ia + M) 1= EiA + M p N ia = : (13) E ia M p is the mass of a proton, E ia (i =, 1 and ) are the total energies of the incoming and two outgoing protons and E ta = E 1A + E A E A. In this section, the momentums in the p-a c.m. and laboratory frame are attached by subscripts A and L respectively. It is well known that the target proton (t) in nucleus is o the mass shell, E ta k ta 6= M p. However, we use free proton mass because the phenomenological NN t-matrix is valid only for it. Anti-symmetric p-p t-matrix operator ^t as pp in Eq. (11) is given by ^t as pp = i=s;v;t;a;p f as i ^ 1i ^i ; (14) 3

4 f as i = f i q j=s;v;t;a;p ji f j Q ; (15) where ^ i are the complete sets of 4 4 matrix, ji is the Fierz matrix, f i is the invariant Fermi amplitude, and q = (E 1A E A ; k 1A k A ) and Q = (E A E A ; k A k A ) are the direct and exchange four-momentum transfers. Equation (11) is expanded in Pauli spinor space as u 1 (k 1A ) u (k A ) ^t as pp u 1 (k A ) u (k ta ) = M + im y 1y + im y y + M xx 1x x + M yy 1y y + M zz 1z z + M xz 1x z + M zx 1z x : (16) In this work we suppose coplanar experiment. The incident direction is taken as z-axis and y-axis is perpendicular to the scattering plane. It is noted that the factorized NN t-matrix is evaluated in the p-a c.m. frame not in the p-p c.m. one. Thus, there are 8 olfenstein amplitudes M in Eq. (16). They are given by M = fs as S + f V as ( V V 3 ) + f T as ( T + T 3 ) f A as A3 ; (17) for () = (), (y), (y) and (yy), while for () = (xx), (zz), (xz) and (zx), M = fv as V 3 + f T as ( T + T 3 ) + f as A ( A A3 ) + f P as P : (18) The coe cients S for the scalar amplitudes in Eq. (17) are S = N t 1 + t 1 ; (19) y S = N t 1 + t 1 ; () y S = N t t ; (1) yy S = N 1 1 t t ; () with N = N A N 1A N A N ta : (3) The quantities j i are given by 1 j k L j = c A ; (4) E A + M p 1 1 = c 1A j k Lj + cos 1L j k 1L j E 1A + M p ; (5) 4

5 1 = c A j k Lj + cos L j k L j E A + M p ; (6) 1 t = sin 1L j k 1L j sin L j k L j ; (7) E ta + M p 1 = sin 1L j k 1L j E 1A + M p ; (8) = sin L j k L j E A + M p ; (9) t = c ta j k Lj + cos 1L j k 1L j + cos L j k L j E ta + M p ; (3) where k il are the momentums of the incoming and two outgoing protons in the laboratory frame, the two scattering angles 1L (> ) and L (> ) are on opposite sides of the incident direction. The quantities c ia (i =, 1 and ) are de ned in the Lorenz transformation of the momentums from the laboratory to p-a c.m. frame: k A = c A k L ; (31) c A = pam A E L + M A ; (3) c 1A = k 1A = k 1L + c 1A k L ; (33)! 1 pa 1 k E L + M A vpa 1L v pa pa E 1L ; (34) and c A = k A = k L + c A k L ; (35)! 1 pa 1 k E L + M A vpa L v pa pa E L ; (36) c ta = c 1A + c A c A ; (37) where M A is the mass of the target nucleus, v pa = k L = (E L + M A ) is the velocity of the 1=. p-a c.m. in the laboratory frame and pa = 1 vpa The detailed expressions of the other coe cients in Eqs. (17) and (18) are presented in Appendix. 5

6 e have derived the factorized NN t-matrix relevant to the NR-DTA of (p; p) reaction by calculating it directly in terms of the Dirac spinor and the invariant Fermi amplitudes. This is an essential di erence of the present calculation from old ones [7,1,16]. If we start at olfenstein NN amplitude in Pauli spin space as Eq. (16), we have to rotate the quantization axis in p-p c.m. frame into that in p-a c.m. frame and perform igner rotation further. The old NR-DTA calculations did not take into account igner rotation and so are not accurate. Our method in this section can avoid such complicated procedure. 4 Numerical analyses Although the spin-orbit distortions have been ignored for simplicity in section, they are necessary to calculate polarization observables. The detailed expression of the NR-DTA t-matrix for (p; p) reaction with spin-orbit distortions can be found in Ref. [15]. The method of 3-dimensional integration is also according to Ref. [15], in which the distortedwave functions of two out-going protons are rotated to avoid enormous summations over the z-components of angular momentums in their partial wave expansions. This clever method saves CPU time considerably so that we are able to perform the calculations by personal computer in terms of C language. Figures 1-4 display the energy sharing distributions of cross sections and analyzing powers for proton knockout from 1d 3= and 1 d 5= state of 4 Ca at T LAB = MeV for outgoing proton angles ( 1L ; L ) = (9 ; 47 ) and (54 ; 54 ). Experimental data are from Ref. [16]. The present calculations are limited to a 4 Ca target because the several 1=A e ects are ignored as mentioned in section. The distorted waves are calculated using the optical potential GOP1 of Ref. [3]. The bound-state wave functions are calculated by the same way as Ref. [7]. The spectroscopic factor is a simple shell-model value j +1 except for normalization to the data by.77 for 1d 3= at (54 ; 54 ). Our results are compared with Figs. 1 and in Ref. [6]. It is noted that our cross section is half of that in Ref. [6], d 5 d 1L d L d (E 1L E L ) = 1 d 5 d 1L d L de 1L : (38) Typical features of our NR-DTA results are apparent in Fig. 1. The experimental data of cross section and analyzing power are well reproduced below E 1L E L = 5 MeV while above 5 MeV the calculated cross section is smaller and the analyzing power is much larger than the data. The RDIA results are quite di erent. Its cross section shows much narrower distribution than the experimental data and our result. Its analyzing power shows a dip around E 1L E L = 6 MeV that is absent both in the data and ours. However the data of analyzing powers above 5 MeV are well reproduced. In Fig. the NR-DTA calculation reproduces the cross section relatively well, but there still 6

7 remain small discrepancies with the data over the range of E 1L E L. The discrepancies however become much larger in the analyzing power. To the contrary, the calculated cross section by the RDIA has narrow distribution again, while the data of analyzing power in E 1L E L > are fairly well reproduced. The same features as Figs. 1 and are also seen in (54 ; 54 ) scattering. In Fig. 3, both the cross section and analyzing power are well reproduced in all the range of E 1L E L except for overall normalization mentioned above. However the RDIA calculation reproduces neither the cross section nor the analyzing power. In Fig. 4 the small di erences between our calculation and the data in the cross section at larger values of je 1L E L j are revealed as the large di erences in the analyzing power as in Fig.. The RDIA cannot reproduce the cross section at all. It is also noted that our NR-DTA calculations in Figs. 1-4 improve old NR-DTA results [7,16] considerably. In the NR-DTA calculations, if the cross section is well reproduced, the analyzing power is also reproduced. Inversely, even small discrepancies with the data of cross section lead to large discrepancies in the analyzing power. This consistency between the calculated cross section and analyzing power is not found in the RDIA calculations. e consider that absence of the consistency is an essential defect of the RDIA. Reference [6] also calculates the analyzing powers at T LAB = 3 MeV for outgoing proton angle (3 ; 55 ). However there are no data of the cross sections, which are necessary for systematic analyses as shown above. Although we do not show gures, our calculations cannot reproduce the data. This is not discouraging because the NR-DTA does not work well at the asymmetric kinematics of 1L 6= L and E 1L E L > as already seen in Figs. 1 and. To the contrary the behavior of RDIA is inconsistent. It reproduces the analyzing powers for such kinematics at T LAB = MeV, but cannot so at T LAB = 3 MeV. As a result of the factorization prescription of Eq. (4), the energy sharing distributions of the cross sections in the NR-DTA calculation are mainly determined by the distorted momentum distribution of Eq. (1). The success of the NR-DTA but the failure of the RDIA to reproduce the cross sections suggests that the factorization is reasonable. In the relativistic approach, the intermediate state jpi hpj in Eq. () is replaced by jp+i hp+j + jp i hp j, where + and indicate the positive and negative energy states. The contribution of jp i hp j is just the virtual NN pair e ect. As also pointed out in Ref. [6], it cannot be determined without ambiguity. In this respect we have to note Refs. [17] and [18]. The modi ed relativistic mean eld (RMF) models, in which the e ect of negative-energy propagation or the Z-graph contribution is reduced, could reproduce the quasielastic electron scattering [17] and p+a optical potential [18] much better than the original RMF models. Moreover, the ambiguity is not intrinsic to the negative energy state, but intrinsic to the intermediate state itself regardless of positive or negative energy. This is because that the phenomenological NN t-matrix has a physical meaning only for asymptotic positive energy state. Now, the traditional view, which 7

8 regards the factorization as a crude approximation [19-1] of the full nite-range DIA, has to be revised. e want to advocate that the factorization is not an approximation but a reasonable prescription to extract physically meaningful NN t-matrix element relevant to the (p; p) reaction from all the matrix elements in the intermediate states. 5 Conclusion e have reconsidered the NR-DTA description of the (p; p) reaction using modern high-quality phenomenological optical potential and NN t-matrix. The factorized NN t-matrix is directly evaluated in terms of the Dirac spinor and the invariant Fermi amplitude unlike old NR-DTA calculations. e have calculated 4 Ca(p; p) reactions at T LAB = MeV and compared the results with the RDIA calculations. It is found that the NR-DTA results are better than the RDIA ones especially for the energy sharing distributions of cross sections. More importantly, the NR-DTA has the consistency between the calculated cross sections and analyzing powers in reproducing experimental data. However it is absent in the RDIA calculations. Here we present a question. Is the relativistic model really useful to nuclear reactions? Our answer is two-hold. The relativistic description of the phenomenological optical potentials and the NN t-matrix are really useful. These energy-dependent global ttings became possible only after the development of the Dirac approach. To the contrary, the immediate relativistic extension of the NR-DIA cannot be justi ed. This is because that the RDIA assumes unphysical extrapolation of the NN t-matrix to the intermediate state given by elastic distortions. However, for a de nitive conclusion, further improvements of both the relativistic and nonrelativistic models are necessary. Appendix: expressions of Here we present explicit representations of s in Eqs. (17) and (18). First, V in Eq. (17) is given by changing the signs of 1 i and j in S of Eqs. (19)-(), while V 3 are V = S 1 i! 1 i ; j! j ; (A1) V 3 = N 1 1 t t ; (A) y V 3 = N t t ; (A3) y V 3 = N 1 1 t t 1 + ; (A4) yy V 3 = N t + 1 t 1 + ; (A5) 8

9 xx V 3 = N t ; (A6) zz V 3 = N 1 t 1 + ; (A7) xz V 3 = N t 1 + ; (A8) zx V 3 = N 1 1 t : (A9) T is given by changing the signs of 1 i and j in V 3 while T 3 are of Eqs. (A)-(A9), T = V 3 1 i! 1 i ; j! j ; (A1) T 3 = N t 1 + t ; (A11) y T 3 = N t 1 + t ; (A1) y T 3 = N t t 1 ; (A13) yy T 3 = N t t 1 ; (A14) xx T 3 = N t + t t 1 t ; (A15) zz T 3 = N t 1 t t t ; (A16) T xz 3 = N t 1 1 t t 1 t ; (A17) T zx 3 = N t 1 1 t t + 1 t : (A18) P in Eq. (18) are xx P = N 1 t 1 + ; (A19) zz P = N t 1 ; (A) xz P = N 1 1 t ; (A1) zx P = N t 1 + : (A) A is given by changing the signs of 1 i and j in P of Eqs. (A19)-(A), A = P 1 i! 1 i ; j! j ; (A3) while A3 is given by changing the signs of 1 i and j in T 3 of Eqs. (A11)-(A18), A3 = T 3 1 i! 1 i ; j! j : (A4) 9

10 References [1] J.A. McNeil, J.R. Shepard and S.J. allace, Phys. Rev. Lett. 5, 1439 (1983); J.R. Shepard, J.A. McNeil and S.J. allace, Phys. Rev. Lett. 5, 1443 (1983). [] B.C. Clark, S. Hama, R.L. Mercer, L. Ray and B.D. Serot, Phys. Rev. Lett. 5, 1644 (1983). [3] S. Hama, B.C. Clark, E.D. Cooper, H.S. Sherif, R.L. Mercer, Phys. Rev. C41, 737 (199). [4] E.D. Cooper, S. Hama, B.C. Clark and R.L. Mercer, Phys. Rev. C47, 97 (1993). [5] O.V. Maxwell, Nucl. Phys. A6, 59 (1996); A638, 747 (1998). [6] O.V. Maxwell and E.D. Cooper, Nucl. Phys. A63, 441 (1996). [7] Y. Kudo and K. Miyazaki, Phys. Rev. C34, 119 (1986). [8] G. Jacob and Th.A.J. Maris, Rev. Mod. Phys. 38, 11 (1966); 45, 6 (1973). [9] T. Berggren and H. Tyrén, Annu. Rev. Nucl. Sci. 16, 153 (1966). [1] D.F. Jackson, in Advances in Nuclear Physics, edited by M. Baranger and E. Vogt (Plenum, New York, 1971), Vol.4, p.1. [11] R. Bengtsson, T. Berggren, Ch. Gustafsson, Phys. Rep. 41, 191 (1978). [1] P. Kitching,.J. McDonald, Th.A.J. Maris and C.A.Z. Vasconcellos, in Advances in Nuclear Physics, edited by J.. Negele and E. Vogt (Plenum, New York, 1985), Vol.15, p.43. [13] I.E. McCarthy, Introduction to Nuclear Theory, Ch.14 (John iley, New York, 1968). [14] D.F. Jackson, Nuclear Reactions (Methuen and Co LTD, London, 197). [15] N.S. Chant and P.G. Roos, Phys. Rev. C7, 16 (1983). [16] L. Antonuk, et al. Nucl. Phys. A37, 389 (198). [17] K. Miyazaki, CERN Document Server (CDS) ext--49 revised by Mathematical Physics Preprint Archive (mp_arc) [18] K. Miyazaki, CERN Document Server (CDS) ext--68 revised by Mathematical Physics Preprint Archive (mp_arc) [19] N. Austern, Phys. Rev. Lett. 41, 1696 (1978). 1

11 [] D.F. Jackson, Physica Scripta, 5, 514 (198). [1] C. Samanta, N.S. Chant, P.G. Roos, A. Nadasen, J. esick and A.A. Cowley, Phys. Rev. C34, 161 (1986). 11

12 d 5 σ /d Ω 1L d Ω L d(e 1L E L ) ( µ b/sr MeV) (a) E 1L E L (MeV).4 (b). A y E 1L E L (MeV) Figure 1: Cross section (a) and analyzing power (b) for proton knockout from the 1d 3= state of 4 Ca at MeV for outgoing proton angles (9 ; 47 ). The solid curves are the results of NR-DTA calculation. Experimental data are from Ref. [16]. 1

13 d 5 σ /d Ω 1L d Ω L d(e 1L E L ) ( µ b/sr MeV) (a) E 1L E L (MeV).4 (b). A y E 1L E L (MeV) Figure : Same as Fig. 1 for proton knockout from the 1d 5= state. 13

14 d 5 σ /d Ω 1L d Ω L d(e 1L E L ) ( µ b/sr MeV) 3 1 (a) E 1L E L (MeV).4 (b). A y E 1L E L (MeV) Figure 3: Same as Fig. 1 for outgoing proton angles (54 ; 54 ). 14

15 d 5 σ /d Ω 1L d Ω L d(e 1L E L ) ( µ b/sr MeV) (a) E 1L E L (MeV).3. (b).1 A y E 1L E L (MeV) Figure 4: Same as Fig. 3 for proton knockout from the 1d 5= state. 15