Study of the approximations used in calculations on the (p, 2p) reaction

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Nuclear Physics A129 (1969) 145--155; (~) North-HollandPublishing Co., Amsterdam

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Not to be reproduced by pbotoprint or microfilm without written permission from the publisher

STUDY OF T H E APPROXIMATIONS USED IN CALCULATIONS ON T H E (p, 2 p ) R E A C T I O N B. K. JA1N *

Department of Physics, University of Surrey, Guildford, Surrey, England Received 4 November 1968 Abstract: The predictions of the di-proton model and the kinetic energy approximation for the (p, 2p) reaction are compared and then, using the di-proton model, uncertainties introduced in these calculations due to various approximations and ambiguities are estimated. The localization of the (p, 2p) reaction in configuration space is also studied.

I. Introduction

In recent years there has been a growing interest in the study of quasi-elastic knockout reactions such as the (p, 2p) reaction to investigate the individual proton states in nuclei. The spectroscopic analyses of this reaction on the I p shell nuclei and some 2s-ld shell nuclei have shown ~,2) that the (p, 2p) reaction can be used to study precisely the single-particle aspect of the atomic nuclei. In order to determine the reliability of these calculations, in this paper we consider the uncertainties due to various approximations and ambiguities involved in these analyses. The differential cross section for a symmetric co-planar (p, 2p) reaction in distortedwave impulse approximation can be written as 2.3) daa

- = V(O) Y, B,,i(O).

d(2o dQ 1dE

.~j

(l)

The first factor, F(O), contains the kinematic factors and the free proton-proton cross section, and is defined as

F(O) = 4n~ q,qo po+Q i da_(Eo,90o), h2 Po qo-ql d~C2pp

(2)

for non-relativistic kinematics, and

F(O) = 4q2 mZc2+h2qZsinO da h2po (,n-Zc~'-+h-Zc2Q2)~ d ~ (E'o, 90°),

(3)

for relativistic kinematics. The second factor, B, tj(0 ), contains the nuclear structure information and can be reduced essentially to the spectroscopic factor and the t Present Address: N.P. Division, Bhabha Atomic Research Ccnter, Trombay, India. 145

146

B.

distorted have the into four involved

K.

JAIN

momentum distribution which we denote by gnu. All the symbols used here same meaning as in the previous papers I* “). We have divided this paper sections. In sect. 2 we discuss the calculation of g,,j and the uncertainties in it while in sect. 3 we discuss the treatment of F(0).

2. Calculation of gR1, 2.1. COMPARISON OF THE Dl-PROTON PROXIMATION MODEL

MODEL

AND THE

KINETIC

ENERGY

AP-

The final system in the (p, 2p) reaction is a three-body system; hence, it is quite complicated to solve the Schrodinger equation for the final state of the reaction exactly. Therefore, in order to get some physically useful results in a reasonable

400

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Fig. I. Comparison of the angular distribution for Ip knock-out from 12C using the di-proton model approximation (full line) and the kinetic energy approximation (dashed line).

amount of computational time the final system has been reduced to a two-body system either by approximating the potential energy of the system (the di-proton model “)) or by approximating the kinetic energy of the system 2*4). The di-proton model is exact in the plane-wave limit of the scattering states while the kinetic energy approximation model is exact in the limit of an infinitely heavy nucleus. Also from the formalisms of these two model we observe that the computational time involved in the di-proton model is much smaller than that in the other approximation. In order to compare the predictions of these two models, we have plotted in fig. I the differential cross section for the “C(p, 2p) reaction at an incident energy of 460

(p, 2p) RRACTION

147

MeV. For the sake of convenience in computation we have used the trapezoidal shape for the optical potential for the outgoing waves, and have estimated the parameters for this potential from those of available Saxon-Woods potentials. As we see from the figure the difference between the two curves is small. This resemblance of the two predictions assures us that the computational advantage in the di-proton model is achieved while still maintaining the accuracy of the results at least to the extent of that in the kinetic energy approximation. For this reason we have used the di-proton model for the remainder of the work of this paper.

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Fig. 2. Localization o f F ( r t c ) (full line) a n d ( - - I ) G ( r t c ) 2.2. L O C A L I Z A T I O N IN C O N F I G U R A T I O N

(dashed line) in configuration space.

SPACE

We know that the (p, 2p) reaction, in principle, contains the infocmation about the momentum distribution of the single particle in the atomic nucleus 5). The ac-

148

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JAIN

curacy of this information extracted from the study of the angular correlation distribution of the outgoing protons or the recoil m o m e n t u m distribution of the residual nucleus, very much depends upon the localization of the reaction in configuration space. To discuss this spatial distribution we have chosen the (p, 2p) reaction on a representative nucleus, 12C. In the representation of the di-proton model the m o m e n t u m distribution gn~i is given by 3)

m =

.

r,c)lgp,,i(r,c)7~o

(koA, a r l c ) ) ,

(4)

where 7~B and ;(p denote the distorted waves for the di-proton and the incoming proton respectively, On~i is the bound state wave function of the knocked out proton, a = (1 - l/A), rtc refers to the position vector of the knocked out proton with respect to the centre of mass of the residual nucleus, and koA and kBc are the channel momenta of the incoming proton and the outgoing " d i - p r o t o n " channels respectively. In fig. 2 we have plotted the real and imaginary part of the integral

fx~-)*(kBC, rlc)Xp~+~(koA, arlc)Pl(cos

O) sin 0d0d~b = F ( r l c ) + iG(rlc ),

for two values of the recoil m o m e n t u m Q at an incident energy of 460 MeV. In this integral (0, tk) are the polar angles of rlc. We have also plotted the bound state wave function R~tj for a lp~ state. As we see from the figure, the curves corresponding to IQJ --- 100 MeV/c peak around 6 fm, and considering their overlap with the bound state wave function it is clear that the shape and the magnitude of the cross section at this value of Q are mainly sensitive to the behaviour of the wave function at large radii. At the higher recoil momentum, i.e. 229 MeV/c, in addition to a peak around 6 fro, there is another peak around 2 fm. Hence, as expected from the classical considerations, the cross section at this value of Q depends upon the details of the wave function inside the mean radius (rms radius) of the nucleus. But, as we can see from fig. 2 and also from the plane-wave limit of distorted waves 5.6), the product (Tta-~*;(~+)) oscillates with Irlcl and the frequency of oscillations increases with IOl; therefore, the cross section at the large values of IQ[ is very small and is not very sensitive to the behaviour cf the wave function at small radii. Extrapolating from the conclusions at these two values of Q in fig. 2 we can say that the cross section for small and moderate values of Q depends sensitively upon the tail of the overlap integral while that for the large values of Q depends upon the overlap integral inside the nuclear mean radius but not very sensitively. Thus, in order to extract any detailed information on the behaviour of the overlap integral inside the nucleus one has to perform a very careful analysis of very accurate experiments. As we have seen in the previous analyses 1) the peaks in the angular distribution for the protons knocked out with angular momentum 1 = 1 from the nuclei in lp shell occur around IQI = 100 MeV/c. Therefore, from the observations of the preceding paragraph, in order to extract the reliable spectroscopic information from the experimental data in lp shell nuclei it is very important to describe the wave

(p, ')p) REACTION

200

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dnodn,dE (u b.l,,leV-t sr-,~) IOO

~xXX

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43

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eLABlde) l

Fig. 3. Comparison of the angular distribution for Ip knock-out from 12C using Saxon-Woods (full line) and harmonic oscillator (dashed line) potentials for the bound state wave function.

400

300

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)

.-

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~)LAE(dcg ) Fig. 4. Comparison o f the angular distribution for Ip knock-out from 6Li using Saxon-Woods (full line) and harmonic oscillator (dashed line) potentials for the bound state wave function.

150

B.K.

JAIN

function accurately in the surface region. Due to the considerable success of the oscillator potential and the convenience in handling it in nuclear structure calculations it is sometimes tempting to use it in direct reaction calculations. Recalling that the oscillator wave functions at large radii go as r t e x p ( - r 2 / 2 b 2) (where b is the oscillator parameter), whereas the required behaviour is e-~'/r, (where c~ is determined by the separation energy 7)), we may expect that for weakly bound protons the cross section is considerably underestimated, although the effect may be less serious for protons with large separation energies. To demonstrate this, in fig. 3, we have plotted the differential cross section for the t2C(p, 2p) reaction at an incident energy of 160 MeV using both the Saxon-Woods and oscillator potentials to generate the bound state wave function. In fig. 4 we have plotted the corresponding curves for the 6Li(p, 2p) reaction for an incident energy c f 185 MeV. The separation energies in the two cases are 15.958 MeV and 4.665 MeV respectively. The parameters for these potentials are taken from the analysis of elastic electron scattering. The oscillator parameter b for 12C is taken as 1.65 fm and for 6Li as 1.70 fm, while the SaxonWoods potential parameters are taken from ref. to). in 12C, as we see from fig. 3, the difference between two curves is not alarming while in 6Li, besides the change in the shape of angular distribution, the magnitude of the cross section at the lefthand peak obtained using the oscillator wave function is about half that obtained using the Saxon-Woods wave function. 2.3. FINITE RANGE EFFECTS In the di-proton model the transition matrix element Tfi is factorized into the function #~ti and the matrix element of the two-nucleon interaction by neglecting the dependence of the distorted waves on r o t, the relative position vector of the two outgoing protons 3). This factorization is exact in the plane-wave limit. In the distorted-wave approximation this factorization is achieved by assuming the interaction Vol between the two protons to be short range. In fact, the distortion introduces local momenta which are different from the asymptotic momenta and hence affects the two nucleon interaction; this intimate connection between the distortion and the two-nucleon interaction can only be taken into account accurately in a finite range calculation. Therefore, in order to estimate the uncertainties introduced by the zero range approximation in computing the matrix element Trl, in this section, following Jackson 3), we apply the local-energy approximation (LEA) of Buttle and Goldfarb 1t) to the di-proton model of the (p, 2p) reaction. Eventually we can write, in DWBA

Tfi

=

c)xp" f V(kop)Z(B-)*(kBcr, c)¢.,j(r, ,

~+' (koA, at, ¢)d~, c.

For a two nucleon potential of Yukawa or Gaussian shape

(5)

(1)) 2p) REACTION

| 51

where ro represents the range of the potential and D is a constant involving the characteristic constants of the potential V(ro~). The expectation value of kop may be written as

-= ~

2 2 = k~, +½koA-¼kec m

+ h-~ [Voc(koA , a r l c ) + V~c(r~c) - Ua(kBc , r l c ) - E s ] ,

(6)

where V,c is the (real) potential which gives the single-particle wave function for the bound proton with a separation energy of E~, Voc and Ua are the optical potentials for the incoming proton and the outgoing "di proton", and ko~ is the momentum conjugate to ro,. In the usual zero range DWIA formalism we have Tf,

=f

, (-)* jto,(K)XB (kBc, rtc)Zp(+ '(koA,

m ar,c)flP.,j(r,c)dric,

(7)

where r'=

koA--½kac--kol.

(8)

In the LEA the expression analogous to (7) is r,c)Z(p+)(koA, arlc)C~,,~j(r,c)drlc.

=

(9)

The difference between the expressions (7) and (9) for Tfl lies in the definitions (6) and (8) of the momenta r and r', and it is twofold. Firstly, r depends upon r,c whereas r ' does not, secondly, the magnitude and variation of K with the scattering angle are different from those of r'. To investigate this precisely, in fig. 5 we have plotted m

~2 Voc(koA' a r ' c ) + V ' c ( r ' c ) -

Ua(kec' r'c)l

-= V(r ' c ) +

iW(r'c)

against r~c for ~2C(p, 2p) reaction at an incident energy of 160 MeV and at a scattering angle of 28 °. For the same reaction, in fig. 6, we have plotted the function k21 + ½koA--¼ksc2 2 ~m Es - K2

against the scattering angle. In subsection 2.2 we have noticed that the major contributions to Tf~ comes from the region of r,c around 6 fm. Now from figs. 5 and 6 it can be seen that in this region U(r~c)+iW(r,c ) is much smaller than K 2. Hence we can neglect the dependence of I'll on r~c. Therefore, in LEA the matrix element Tpv(K') =

i'ei""o't(ro,)dro,

is replaced by Tpp(K) = f e '~" "°'t(ro,)dro,.

B. K, JAIN

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s

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Fig. 5. Radial behaviour of the function

U(rlc)+iW(rlc ) for

arC(p, 2p) reaction.

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(rob/st) 6

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Fig. 6. Comparison of the angular variation of ,¢2 and 21k0~]= and the free p - p cross section c o r responding to K 2 (full line) and 21k0t[ z (dashed line).

(p, ;2p) REACTION

153

For scattering on the energy shell, the energy of the two protons in their centre-of-mass system before and after the collision is the same, i.e. Ikoll = IkoA-½kncl; therefore from eq. (8) I~'l = 2"~lkot[ = 2~lkoA--½kBc[ •

(9)

In fig. 6 we have also plotted 2[kol I. As we note from this figure the difference between the two curves decreases with the scattering angle. To observe its effect on the free p-p cross section which appears in the expression for the cross section (eq. (1)), in fig. 6 we have also plotted da/df2po (Eo, 90 °) corresponding to K 2 and Ir'l 2 defined by eq. (9) at the incident energy of 160 MeV. From this we note that the correction due to the finite-range effect decreases with the scattering angle. Consequently the ratio of the left-hand maximum to the right-hand one in the angular distribution for ~2C(p, 2p) reaction changes by about 15 %. At a higher incident energy, e.g. 460 MeV, although K z and IK'I 2 differ, they do not produce any uncertainty in the cross section because da/df2pp remains nearly constant at these energies. 3. Calculation of F(O) 3.1. TREATMENT OF THE KINEMATICS

In order to get an idea about the uncertainty introduced in the cross section due to the treatment of the kinematics, in fig. 7 we have plotted F(O) for 12C(p, 2p) reaction at an incident energy of 160 MeV for the relativistic kinematics (Fa(0)) and the non-relativistic kinematics (F,,~a(0)). As we observe from this figure, FR(0) in contrast to FNR(0) remains constant over the angular range of interest and also its value is slightly larger than that of FNa(0 ). Consequently, in the computation of cross section with the hen-relativistic kinematics, the ratio of the right-hand maximum and the left-hand maximum in the angular distribution is slightly underestimated. In the particular case of t2C(p, 2p) reaction at the incident energy of 160 MeV this underestimation is about 24 %. This uncertainty mainly arises from the fact that do'/dQpp (Eo, 90 °) has a large gradient at lower energies (Eo). For the same reason at higher incident energies the difference between Fa(0 ) and FNa(0) is negligible over the whole angular range of interest. 3.2. COMPUTATION OF dn/d-Qpp(E'o, 90 :)

In the transition matrix Tfl for the (p, 2p) reaction the two nucleon matrix element Tpp is an off-energy-shell matrix element, therefore the choice of the relative energy Eo at which do'/df2pp would be calculated is quite ambiguous. Non-relativistically the energy E~ before the scattering of two protons is , E°

h2 2m [P°+ Q[2'

B.K.

154

JAIN

and after scattering is ,

/l 2

Eo =

~m

]q°-ql]2"

In order to compare the difference between the cross sections predicted by these two choices for E~, in fig. 8 we have plotted d~/dQpp (E~, 90 °) for these two energies 0.9

F(! }

-

~

........................

Relot ivistl¢

i - b M¢V-i sr -2 fro-3}

Non-~¢lativistic

0.3

I 23

I 33

i 43

l 53

OL,,.et'~,~) Fig. 7. C o m p a r i s o n o f

F(O) for

relativistic and non-relativistic kinematics.

I0

da i o ~(Eo,90} d PP (mb/lrl S ................. ....

30

;o

s'o

r

_~ 460McV ..... 160MeV

6'o

O Ag[ de! )

Fig. 8. C o m p a r i s o n o f the free p r o t o n - p r o t o n cross section at 90 ° for relative energies E'0 before (dashed line) and after (full line) the scattering.

(!~, '2p) REACTION

155

at the incident energies of 160 MeV and 460 MeV for 12C(p, 2p) reaction. As we see from the figure the difference between two curves at 160 MeV is considerable while that at 460 MeV is negligible. 4. Conclusions

In conclusion we can make the following points: (i) All the uncertainties involved in the calculation of the cross section for the (p, 2p) reaction are negligible at higher energies ( > 400 MeV). At lower energies, however, these uncertainties are appreciable; for example, the uncertainties introduced in the cross section, at the incident proton energy of 160 MeV, due to the zero range approximation and the non-relativistic treatment of kinematics are 15 ~ and 24 respectively. Also, at lower energies due to the large gradient of da/dt2pr at the corresponding E o at these energ_ies, the position of tbe left-hand peak in the angular distribution for / ¢: 0 knock-out proton is influenced by the choice of E 0 considerably. And since the relative energy Eo at which da/dQpp should be calculated is quite ambiguous, the procedure of extracting the spectroscopic factor from the normalization of the computed left-hand peak to the corresponding experimental peak is liable to uncertainties. (ii) The localization of the (p, 2p) reaction in contiguration space is such that it can be used as a sensitive tool to investigate the low momentum components of the single-particle states. For the investigation of the higher momentum components, however, one has to do a very careful analysis of the accurate high-energy ( > 400 MeV) experiments. (iii) The computational advantage in the di-proton model representation is achieved while still maintaining the accuracy of the results at least to the extent of that in the kinetic energy approximation. The author wishes to express his thanks to Dr. D. F. Jackson for her kind supervision and the careful reading of the manuscript. The facilities provided by the University of Surrey Computer Unit and the S.R.C. Atlas Computer Laboratory, and the grant of the University of Surrey Research Studentship are gratefully acknowledged. References 1) B. K. Jain and D. F. Jackson, Nucl. Phys. A99 (1967) ll3; B. K. Jain, Nucl. Phys., to be submitted 2) H. Tyr6n, S. Kullander, O. Sundberg, R. Ramachandran, P. Isacsson and T. Berggren, Nucl. Phys. 79 (1966) 321 3) D. F. Jackson, Nucl. Phys. A90 (1967) 209 4) K. L. Lim and I. E. McCarthy, Phys. Rev. 133 (1964) 1006 5) T. Berggren and H. Tyr6n, Ann. Rev. Nucl. Sci. 16 (1966) 153 6) D. F. Jackson and B. K. Jain, Phys. Lett. 27B (1-968) 147 7) T. Berggren, Nucl. Phys. 72 (1965) 337 8) U. Meyer-Berkhout et al., Ann. of Phys. 8 (1959) 119 9) D.L. Lin and S. S.M. Wong, The University of Rochester, Department of Physics and Astronomy, Report No. UR-875-176 10) L. R. B. Elton and A. Swift, Nucl. Phys. A94 (1967) 52 11) P. Buttle and L. Goldfarb, Proc. Phys. Soc. 83 (1964) 701