Calculation of the nucleon-nucleus optical potential

I

2.E

I

Nuclear

Physics

Al69

Not to be reproduced

CALCULATION

(1971) 417-425;

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OF THE NUCLEON-NUCLEUS

OPTICAL POTENTIAL

H. R. KIDWAI t and J. R. ROOK Nuclear Physics Laboratory,

Oxford

Received 25 September 1970 (Revised 22 February 1971) We use the Greenlees formula to calculate the value and energy variation of the real part of the nucleon-nucleus optical potential. We find reasonable agreement with potentials obtained from analysis of experimental data. We discuss the validity of the Greenlees formula and show that it is expected to be satisfactory for bombarding energies less than about 50 MeV.

Abstract:

1. Introduction Greenlees ‘) has obtained good agreement with experimental data by using a very simple formula for the real part of the nucleon-nucleus optical potential V(Y). His formula is obtained by summing all the individual internucleon potentials and is

V(r) =

/v(r- v’)p(;)dr’,

where P(Y) is the density of nucleons in the nucleus and v(r-r’) is the internucleon potential suitably averaged over the spin and isospin states. Eq. (1) is inadequate phenomenologically because it gives an energy-independent potential and theoretically because the presence of a hard core in v would cause a divergence of the integral. Both of the difficulties are avoided if we follow ref. ‘) and use the t-matrix in place of v giving V(r) = s t(r -r’)p(r’)dr’.

(2)

Although Greenlees writes down eq. (1) he uses an empirical form for v and this is equivalent to using eq. (2). Hence we shall call eq. (2) the Greenlees formula. Eq. (2) has been used at high energies in ref. “) but it has previously been considered inadequate at the energies considered by Greenlees, i.e. bombarding energies less than 50 MeV, because multiple scattering effects are apparently neglected. On the other hand eq. (2) is fairly satisfactory for the bound-state problem except for nonlocal terms arising from the anti-symmetry of the wave function. These terms are neglected in eq. (2) but it has been shown “) that this is a valid approximation for positive energies. The reason why multiple scattering effects are not large in the bound + Now at the Indian Institute of Technology,

Kanpur, India. 417

418

H. R. KIDWAI

AND

J. R. ROOK

state case is that the true wave function heals to the model wave function within a distance less than the internucleon separation and so we have a series of separate two-body collisions. It will be our argument that eq. (2) is valid for positive energies for the same reason. We have investigated the validity of eq. (2) in two different ways. Firstly we have calculated f from the Hamada-Johnston ‘) internucleon potential. From this we have used eq. (2) to calculate Vat the centre of “‘Pb and compared our calculated values with those obtained by fitting scattering data on medium and heavy nuclei. We obtain good agreement for the value of I’ and its variation with bombarding energy and we think this is strong evidence for the validity of eq. (2). Secondly we have investigated the importance of the three-body terms and shown that they are small. In sect. 2 we present the details of our calculation of V while in sect. 3 we discuss the information which our calculation gives concerning the validity of eq. (2). We discuss our calculation of the contribution arising from three-body terms in an appendix.

2. Calculation of the optical potential We consider an incident equation for t is “)

nucleon

colliding

with a nucleon

in the nucleus.

t = v-vGt,

The (3)

where v is the internucleon potential and G the Green function for the nucleon pair in the presence of the remaining nucleons. We let 1c/denote the wave function of the pair while C$denotes the wave function of the pair in the absence of v, it is essentially a plane wave characterized by the momentum of the pair. We have “) tf#l = II*

(4)

J/ = 4-G@.

(5)

and eqs. (3) and (4) give

We have already stated that we shall calculate the potential at the centre of ‘08Pb and we shall approximate this by considering an infinite system with proton and are neutron densities appropriate to the centre of “‘Pb. The steps in the calculation firstly to calculate G and then to approximate it by a simple analytical form, then to use this simple form to solve eq. (5), giving t from eq. (4) and finally V from eq. (2). 2.1. CALCULATION

OF G

Since we are using the approximation of an infinite system it is convenient plane waves as our basic states. Then following Schiff “) we obtain

s

to use

eiQ'. x

G(x) = -%-

___

(27C)3h2 Qf2 - Q2

dQ',

(6)

NUCLEON-NUCLEUS

419

OPTICAL POTENTIAL

where nr is the nucleon mass, x is the internucleon separation, Q is the initial relative momentum of the nucleon pair and the Q’ integration extends only over those values allowed by the Pauli principle. In the absence Green function for a free nucleon pair G(x) = Gr,,,(x)

of the Pauli principle

= ;zz

eq. (6) gives the

e;.

(7)

In order to obtain the constraints imposed by the Pauli principle we let k, and k, be the momenta of the incident and struck nucleons at the centre of the nucleus. Then Q = + (k, - k2) and the total momentum is K = k, + k,. We similarly define K’, Q’, k; and ki but we note that the total momentum is conserved so that K’ = K. We let k, be the Fermi momentum so that the Pauli principle gives k; > k,; If 9 is the angle between

(8)

k; > kF.

K’ and Q’ we obtain

from (8) the restrictions

k;-+K’2-Q’2

< 0,

Pa)

(cos 01 < a = (&K’” +Qf2 - k;)/(K’Q’).

Pb)

Eq. (6) can now be evaluated but one troublesome point is that G(x) is not spherically symmetric. We have investigated this point and found that for incident energies less than 50 MeV the non-spherical terms are not large and so we shall ignore them. Then we obtain sin (Q’x) I(a)Q”dQ’ + fi sin Qx~(~ )Q2

Q’x where a, is obtained

by inserting

Qt2_Q2

2Q

QX

’

1’ (10)

Q’ = Q in eq. (9b) and

Z(a) = 0

a<0

= ha

O~a~l

= 4n

a > 1.

(11)

G(x) = ‘2

Gr,,,(x).

(12)

For large x eq. (10) gives

2.2. APPROXIMATE

FORM

FOR G

In principle eq. (10) can be inserted into eq. (3) and t obtained but this is lengthy and we look for an approximate scheme of calculation. Our first approximation is to take average values of K and Q rather than consider each nucleon in the nucleus separately. We put for these average values (K) = 2 (Q) = k, and we calculate k, from the formula k; = ‘$(E+V),

(13)

420

H. R. KIDWAI

where E is the incident

AND

J. R. ROOK

energy and Y the real part of the optical potential.

In fig. 1 we show the dependence of Z(a,)/4n of eq. (12) on the total energy E+ V of eq. (13). We see that Z (a,)/4rr is small for total energies less than about 100 MeV so our next approximation is to put Z (uO) = 0. We should note that (E+ V) < 100

Fig. 1. Dependence

of I &)/(472)

on the total energy.

I.5 -

T e * 0.5 -

0

I

I

I

I

50

100

150

200

ENERGY(Wl)

Fig. 2. Dependence

of y on the total

energy.

MeV corresponds to E 5 50 MeV and covers the region considered by Greenlees ‘). With this approximation eq. (12) tells us that as x --f co we have G(x) --, 0. This suggests we approximate G(x) by the form (14)

NUCLEON-NUCLEUS

when m* and y are parameters

OPTICAL

which we determined

421

POTENTIAL

by comparing

eqs. (14) and (12)

numerically. In particular we determine y from the ratio G(0.5)/G(O.l) and m* from the values of y and G(O.l). We find that IN* = 1 to within 2 o/owhile the dependence of y on the total energy is shown in fig. 2. 2.3.

CALCULATION

OF

t AND

V

If we insert the Green function

of eq. (14) into eq. (5) and operate

v2x- y2x+

by Vz we obtain

q$#+x)=0,

(15)

where $ = 4 - x. This is the equation of the reference spectrum method “). It has been solved ‘) using the Hamada-Johnston potential “) for u to give the appropriate spin-isospin mixture of the t-matrix elements.

I

I loo

I

I

300

E dco) Fig. 3. Energy

variation

At the centre of a nucleus, by writing

of the real part of the proton

optical

the only region we consider,

l’ = p

t(x)dx. s

potential

for 208Pb.

eq. (2) is well approximated

(16)

Since V appears in the definition of 4 in eq. (15) it follows that t is a function of V and y while, from fig. 2, y itself is a function of V. This interdependence gives us rather an awkward self-consistency problem since it implies that eq. (15) needs to be solved many times. To avoid this difficulty we take y = 0.7 fm-’ a value suggested as an average by fig. 2. The remaining self-consistency involving just u is easily handled.

422

H. R. KIDWAI

AND J. R. ROOK

We show the dependence of the calculated potential on the incident energy in fig. 3 compared with some points obtained from analyses of experimental data “). We do not obtain a good fit at the higher energies nor do we expect a fit above about 50 MeV in view of our neglect of I(a,). For energies less than 50 MeV, V is approximated by T/ = (63-0.28E)

MeV.

(17)

The energy variation in eq. (17) is a little less than required to fit experiment ‘) but it should be remembered that the non-local term considered in ref. “) also produces a small energy variation so we should not expect complete agreement. 802;;Pb

60 -

F 8405

ZO-

0 Fig. 4.

I 05

I 1.0 Y (fm-ll

I I.5

The real part of the proton optical potential at the centre of 208Pb with a bombarding energy of 30 MeV.

We should finally mention the y dependence of V. Our value of y was originally chosen as an average over all bombarding energies, but for bombarding energies less than 50 MeV a more reasonable value might be y = 0.8 fm- ‘. The detailed y dependence is shown in fig. 4 at a bombarding energy of 30 MeV and we see that our calculated V is reduced by about 10 % if we choose y = 0.8 fm-l rather than y = 0.7 fm-‘. 3. Discussion The main purpose

of our work was to investigate

the validity

of eq. (2) by checking

whether it gives the correct depth and energy variation for V. The reasonable agreement we have obtained suggests that eq. (2) is indeed a good first approximation. Our calculations have given some information about why this is the case. We discuss first of all the contribution to the potential arising from the coupling between the elastic and inelastic channels. In general this is a complex term, the imaginary part is the imaginary part of the optical potential while the real part

NUCLEON-NUCLEUS

OPTICAL

423

POTENTIAL

contributes to the real part of the potential but is neglected in eq. (2). We might expect a priori that the real and imaginary parts of this term are comparable to each other in which case the empirical results lo) suggest a 20 o/, contribution at 50 MeV, falling at lower energies. In terms of our calculation the presence of elastic scattering in the two-nucleon system implies inelastic processes in the nucleon-nucleus system. Thus the imaginary part of the potential and the contribution to the real part under consideration are connected with the pole in eq. (10) and hence to the non-vanishing of Z(a,). However our calculations have indicated that Z(a,) is small below bombarding energies of 50 MeV. The whole discussion of this term has been qualitative but no quantitative calculations appear to have been made. At present it seems that this term is not very small and represents a major limitation upon the usefulness of eq. (2). It has previously been assumed that eq. (2) would not be valid because multiple scattering is neglected. It is our contention that after a collision the true wave function heals to the optical model wave function before the next nucleon is reached giving a series of two-body collisions. We see in eq. (15) that outside the potential u we have x Me -y*/r and hence $ heals to 4 in a distance of the order of y-i = 1.4 fm compared with an internucleon separation of roughly 1.7 fm. Although y-l is less than this separation distance the difference is not marked and so we have estimated the importance of the three-body terms using a simple model. We give the details in an appendix but the eventual conclusion is that the three-body terms make only a 3 “/L contribution to the real part of the potential. It thus appears that it is a good approximation to neglect multiple scattering.

Appendix CALCULATION

OF THE THEEE-BODY

TERMS

We follow Bethe I’) and Day I’) when the potential V = pSf(x)dx+p’S~;,(u,-r,)t(r,

is

-v,)Z’3’(v,,

t-2, rd.

(A.1)

A detailed discussion of the functions y13 and ZC3) is given in refs. I’, 12), here we quote the forms we have used. We have sin k,, r (-4.2) 1];3cr) = v23cr)-? k,, r where r23Cr) =

=e

1, -I(r-C)

c 0_

(A.3)

r
c,

(A-4)

r

and k23 is a value for the relative momentum of nucleons 2 and 3. We thus see that rl’ represents a correlation function between nucleons 2 and 3 inside the nucleus. The function ZC3’ is obtained by solving the Faddeev equations inside the nucleus.

424

H. R. KIDWAI

An approximate z’3’ =

solution

AND

J. R. ROOK

is r2)

r;l(l-532+3532512)+r;2(1-531+3531512)+~;2(531(32-531-532)

t325121. (A.5)

+@[531532+&2(531+t32)-$531

Here 5 is a two-body correlation function as in 5 we replace the parameter 1 of eq. (A.4) by indicate which nucleon pair is involved, qjl for and the corresponding value of k in eq. (A.5) is

eqs. (A.3) and (A.4). For the case of y. The suffixes attached to r,~’and 5 example is a function of r3 - rl = r3 1 k31. For @ we take the form

G = sin (kr, rr2) sin (kr3 r13) sin (k23 r23) kkkrrr 12 13 23 12 13 23

(A-6)

The above approximations are forced upon us if we are to obtain simple answers but we could still take the t calculated in sect. 2. We are here concerned with obtaining TABLE

1

Values of R for the indicated parameters

A (fmz-3l) 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.9 0.7

a rough estimate for t

1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22 1.22

k,z = k,, (fm-‘) 1.1 1.5 1.9 2.3 2.7 1.1 1.1 1.1 1.1 1.1

of the three-body

(fn? ‘)

(fit?)

0.71 0.77 0.77 0.71 0.77 0.77 0.77 0.77 0.77 0.77

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.2

0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7

terms and in consequence

t(x) = VoevpX/x, = 0

1.0 1.0 1.0 1.0 1.0 1.0 1.5 0.5 1.0 1.0

3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3

(f& 0.29 0.27 0.26 0.26 0.25 0.27 0.27 0.30 0.28 0.54

we take a simple form

X>S

(A-7)

X
(A-8)

where s is the separation distance 13). We calculate the value of R defined as the ratio of the second integral in eq. (A.l) to the first and present results in table 1 for a range of values of the parameters defined above. The essential points made by this table are that R is not very sensitive to any of the parameters and the value of R is about 0.3 fm3. The value of the second term in eq. (A.l) compared with the first is obtained by multiplying by p which is about 0.2 fmd3 giving an overall ratio of 0.06. This is not quite the complete answer since we have not performed the spin-isospin sums. A

NUCLEON-NUCLEUS

OPTICAL

POTENTIAL

425

detailed analyses depends on the precise form of t and since we have established that the second term in eq. (A.l) is comparatively small it does not seem worthwhile to present it here. The general effect is to reduce the value of R roughly by a factor of two. We thus conclude that the three-body terms make only a 3 % contribution to the optical potential as stated in sect. 3. References 1) 2) 3) 4) 5) 6) 7) 8)

9) 10) 11) 12) 13)

G. W. Greenlees, G. J. Pyle and Y. C. Tang, Phys. Rev. 171 (1968) 1115 A. K. Kerman, H. McManus and R. M. Thaler, Ann. of Phys. 8 (1959) 551 D. Slanina and H. McManus, Nucl. Phys. All6 (1968) 271 T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962) 382 L. I. Schiff, Quantum mechanics (McGraw Hill, New York, 19S5) H. A. Bethe, B. H. Brandow and A. G. Petschek, Phys. Rev. 129 (1963) 225 H. R. Kidwai, to be published A. E. Glassgold and P. J. Kellog, Phys. Rev. 109 (1958) 1291; P. G. Roos and N. S. Wall, Phys. Rev. 140 (1965) B1237; P. Jastrow and I. Harris, Nucl. Phys. 9 (1958) 437; F. Bjorklund, I. Blandford and S. Fernbach, Phys. Rev. 108 (1957) 795 F. G. J. Perey, Phys. Rev. 131 (1963) 745 P. E. Hodgson, The optical model of elastic scattering (Oxford University Press, 1963) H. H. Bethe, Phys. Rev. 138 (1965) B804 B. D. Day, Phys. Rev. 151 (1966) 826 S. A. Moszkowski and B. L. Scott, Ann. of Phys. 11 (1960) 65