Optical model with polynomial potential

[

2.E

I

Nuclear IPhysics 35 (1962) 71--84; @) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

OPTICAL MODEL WITH POLYNOMIAL

POTENTIAL

A. V. LUKYANOV, Yu. V. ORLOV and V. V. TUROVTSEV

Nuclear Research Institute, Moscow State University, Moscow, USSR Received 13 June 1961 The form of the potential of the optical model with surface absorption is investigated. The parameters making it possible to describe well the experimental data on the scattering of 14 MeV neutrons are selected for the potential in the form of polynomials. Different potentials are compared.

Abstract:

1. I n t r o d u c t i o n

The optical model with the potential, the real part of which is taken in the form proposed b y Woods and Saxon 1), has gained wide currency at present. It was shown, however, in ref. 2) in which the polynomial potential with volume absorption and without taking into account spin-orbital interaction was investigated that the polynomial potential leads to similar results, other things being equal. The volume absorption optical model without taking into account spin-orbit interaction has a number of drawbacks, though it gives the correct dependence of the integral cross sections (the total at, elastic scattering crs and absorption ~r) on the mass number A for medium and heavy nuclei, and describes the qualitative peculiarities of the elastic scattering differential cross sections at 0 < 90 °. For light nuclei the cross section ratio as/~r, according to this model, proves to be overestimated as compared with the experimental data, the calculated differential cross sections have deep minima which are not observed in experiment, and the backward scattering turns out to be too large. From ref. 4) it follows that these defects can be eliminated if the potential used takes account of the spin-orbit interaction and has the maximum imaginary part on the surface of the nucleus (surface absorption 3)). In contrast to ref. 4) in which the real part is the Woods-Saxon potential and the imaginary part is a Gaussian curve, the optical model with surface absorption and spinorbital interaction is investigated in the present paper, the potential being described by polynomials. 2. P o l y n o m i a l P o t e n t i a l

The parameters obtained in the present work lead to agreement with the experimental data on the scattering of 14 MeV neutrons for a nuclear potential 71

A . V . LUKYANOV et al.

72

of the following form t v

=

Vc

1 dp(r) - r dr

o(r)+iVc, q ( r ) - V s R

-

(o.z),

O)

where

p(r)=}~_gg[r--R~Vl[r--R~"

2[ r - R \ 2

1-'1

~0 i q(r)=

1-- (r--R~72 \ dl ]-1

for

0~_ r ~ R--dr,

for

R--dj<~r<_R+d t,

for

r ~ R + dt .

for

O<_r<_ R--dr,

for

R--dr<--r<~R+dr,

for

r>=R+d r

Here R is given b y R = (roAi+8) fm (1 fm = 1 femtometer = 10-is cm) and /, is the n-meson mass at rest. The phase shifts and cross sections were determined b y the well-known formulae through the numerical solution of the Schroedinger equation with potential (1) in the region R--d<_ r <_ R+d, where d = max {dr, dl}. The method does not differ in principle from that described in ref. ~). The pair of differential equations of second order corresponding to the two directions of the neutron spin was reduced to a set of differential equations of first order which was solved b y the R u n g e - K u t t a method. For the initial data use was made of the function and its first derivative at r = R--d where their analytical expressions are known. The simultaneous variation of all potential parameters (VcR, Vci, VSR, dr, dr, r 0 and 8) is quite a problem. The solution is, however, facilitated b y the fact that the region of the values of the parameters and several dependencies in the behaviour of cross sections as functions of some of them have been studied fairly well. In particular, this enables the investigator to make a plausible preliminary choice of the parameters VCR and VSR. The fixing of the parameter VCR under the condition that the parameters r 0 and 8 determining the nuclear radius remain free is quite permissible owing to the well-known ambiguity of the type VcR--R. The value VCR = 44"MeV is therefore quite acceptable since it certainly lies in the necessary region (this value coincides with that used in ref. 4)). The value VSR = 7.7 MeV was taken from ref. 6). The change of this parameter within sensible limits does not change appreciably the cross sections (especially integral cross sections). This parameter can be * W h e n t h e p r e s e n t i n v e s t i g a t i o n h a d b e e n c o m p l e t e d a n article b y G r e e n et al. s) a p p e a r e d . A non-local optical m o d e l in w h i c h t h e real p a r t of t h e p o t e n t i a l is also described b y a p o l y n o m i a l of t h e fifth o r d e r is u s e d in t h i s i n v e s t i g a t i o n .

OPTICAL MODEL

7~

m a d e more accurate b y the d a t a on the polarization of s c a t t e r e d neutrons, b u t u n f o r t u n a t e l y t h e r e are y e t no d a t a for En ---- 14 MeV. I n the choice of the other p a r a m e t e r s use was m a d e of the calculated d a t a describing the b e h a v i o u r of the cross sections with the change of each of the p a r a m e t e r s dr, d t and Vc~ separately. T h e dependence of the cross sections at and ar on the parameters d r a n d d t is close to linear in a b r o a d region of values, which essentially facilitates the choice of these parameters. W i t h good a c c u r a c y the cross sections at and ar as functions of dr, with the other p a r a m e t e r s fixed, can be r e p r e s e n t e d in the form a = a~°~+~dr.

(2)

/.0

,~ 05

~-~ Fig. 1. Real parts of the potentials. The solid line shows the potential (1) with the parameters (3). The dotted line designates the potential (4) and the dot-and-dash line the potential (1) approximating the potential (4) by the least squares method.

T h e q u a n t i t y a ~°~ does not d e p e n d on dr; the coefficient ~ is almost c o n s t a n t for m e d i u m and h e a v y nuclei (~ ~ 33 fm for a r and x ~ 40 fm for at) a n d s o m e w h a t less for light nuclei (x ~ 15 fm for ar and x ~ 30 fm for at). T h e

74

A . V . LUKYANOV et al.

comparative p r o x i m i t y of the values of a for ar and at in the first case points to a weaker dependence of the elastic scattering integral cross section on the parameter dr. Yet the form of angular distribution of elastically scattered neutrons proves to be quite sensitive to the change of the p a r a m e t e r dr. As dr increases, the oscillations in the diffractive p a t t e r n become less sharp a n d the m i n i m a J

6_

I

J

3_

• t

~

/

/

/ •

//

r#I#~. . . . -

o

1

3

I

~

]

5

I

A~

6

]Fig. 9. Comparison of theoretical and e x p e r i m e n t a l d e p e n d e n c e of t h e cross sections at (upper curves) and ar (lower curves) on the mass n u m b e r . A The solid lines show the theoretical cross sections calculated for t h e p o t e n t i a l (1) with the p a r a m e t e r (3). The d o t t e d lines show t h e theoretical cross sections of ref. s). The e x p e r i m e n t a l d a t a designated b y vertical dashes are t a k e n from refs. 12, 12).

in the cross sections shift somewhat at v~ > 90 ° towards smaller angles (this shift is different for different nuclei). The choice of the parameters dr, di, Vci, r 0 and 6 can be made b y the m e t h o d described in ref. 2) b y considering several

OPTICAL MODEL

75

fixed values of a parameter, Vc~ for example. It has proved, however, that the value ~TCI = 11 MeV accepted in ref. 4) makes it possible to find the region of values of the parameters dr, dr, r0 and ~ for which the cross sections at and ar are in good agreement with experiment. We have accepted the following values of the parameters: VCR = 44 MeV, d r = 3.36 fm,

VcI = 11 MeV, d1 :

1.62 fro,

VSR = 7.7 MeV, r o = 1.25 fro,

(5 = 0.

(3)

In fig. 1 the potential (1) is shown, making use of the above parameters. The cross sections ar and at calculated ~dth these parameters are represented, with experimental data, in fig. 2. Similar curves from ref. 2) are also given for comparison. The results demonstrate considerably better agreement with experimental data, especially for the light nuclei. Though the parameters were chosen b y the integral cross sections, the neutron elastic scattering differential cross sections agree with experiment well on the whole (fig. 3). This is additional proof of the correctness of the choice of parameters made. There being no deep minima and large backward scattering in the calculated angular distributions indicates that the defects which were mentioned in the introduction are actually eliminated if the potential (1) is used, the decrease at 0 ~ n of the elastic scattering cross section being attributable to taking into account the spin orbital interaction (fig. 4) *. The introduction of surface absorption weakens considerably the oscillations in the diffractive picture and makes it possible to describe well the experimental data on the integral cross sections for the nuclei of nearly all periodic system, with parameters independent of the mass number A. There is no precision agreement between calculated and experimental angular distributions. Yet evidently the optical model cannot at present claim to precision agreement since it describes only general averaged properties of nuclei. It is a known fact 8), for example, that the differential cross section m a y change appreciably from one isotope to another, i.e., when A changes b y unity. The model under consideration, however, with the only dependence of the potential on A determined b y the dependence of the nuclear radius (R : 1.25 A~ fm), can predict only a smooth dependence on A and is unable to account for such changes. The increase of VSR from Levintov's value of 7.7 MeV up to 10.35 MeV does not practically change the agreement of the theory with experimental data (the value of VSR = 8.3 MeV is accepted in ref. 4)). Only the elastic scattering in the area nearest to the angle 0 ----~ decreases essentially thereby; unfortunately, no experimental data for the scattering are available. This dependence of the angular distribution on the spin-orbit coupling is demonstrated in fig. 5 which represents the curves of differential cross sections for different values of VSR. Thus, our preliminary choice of the parameter VSR is t This effect is explained qualitatively b y I. S. Shapiro ~).

'16

A . V . LUKYANOV st at.

10

r~ F

,

lO

Bi

{'~

%

,.

$

1.0

0.1

0.01

0,0Ol

- -

0

I

60

I

I

120

18() e (degrees)

F i g . 3a. C o m p a r i s o n of t h e t h e o r e t i c a l a n d e x p e r i m e n t a l a n g u l a r d i s t r i b u t i o n of e l a s t i c a l l y s c a t t e r e d n e u t r o n s . T h e s o l i d l i n e s : t h e d i f f e r e n t i a l c r o s s s e c t i o n s c a l c u l a t e d w i t h t h e p o t e n t i a l (1), for t h e v a l u e s of t h e p a r a m e t e r s (3) for a l l n u c l e i , e x c e p t for M g for w h i c h t h e c a l c u l a t i o n is p e r f o r m e d w i t h t h e p o t e n t i a l (4) a n d t h e p a r a m e t e r a = 0.74 f m i n s t e a d of a = 0.65 fm. (The p a r a m e t e r a = 0.74 fin g i v e s a g r e e m e n t of t h e i n t e g r a l c r o s s s e c t i o n s w i t h t h e e x p e r i m e n t . ) T h e d o t t e d l i n e d e s i g n a t e s t h e r e l e v a n t d a t a f r o m refs. 4, la). T h e e x p e r i m e n t a l d a t a for t h e n u c l e i Mg, Ca a n d Cs (En = 14.6 M e V ) a r e t a k e n f r o m ref. la), for t h e n u c l e i S m , S b a n d t3i ( E n = 14 NIeV) f r o m ref. 14), a n d for t h e n u c l e u s Cu (En = 14 MeV) f r o m ref. 4). T h e c a l c u l a t i o n d a t a c o r r e s p o n d t o t h e n e u t r o n e n e r g y E n = 14 M e V for a l l n u c l e i e x c e p t for C a a n d Cd for w h i c h E n = 14.6 M e V (the d i f f e r e n c e of r e s u l t s for E . = 14.6 M e V a n d E n = 14 M e V is of n o p r a c t i c a l i m p o r t a n c e ) .

OPTICAL

77

MODEL

quite plausible and can be made more accurate only if there are data on polarization. It is noteworthy that the calculated cross sections, including the differential ones, practically do not change when the sign of this parameter is altered. The lack of sensitivity to the sign of the spin-orbital interaction constant can easily be understood. Indeed, the spin-orbit terms entering into the equations corresponding to the two values of the total neutron moment (1" = lq-½) differ only in factors which are equal to l and - - ( l + 1) respectively.

v b

0.01

\ 0.001

0

I 20

I

I

I

I

I

40

60

80

100

120

Fig.

3b. S e e c a p t i o n

I

I

140 160 180 e (degrees)

fig . 3 a.

At large l the one can be ignored compared with l. As a result the amplitudes of scattered waves calculated from the solution of the equations with/" = l + ½ will be close to the amplitudes corresponding to the equations with ~"= l~:½ in which the sign of VSR is altered. In the case of large l the cross sections prove

78

A.V.

LUKYANOV et arl.

1

"R ~X

t.0

al

x\

/I

~x\ \

\ \

/ \

/

D.Ol

0ool 0

30

60

9O

I70

150

t80

Fig. 4. T h e e f f e c t of t a k i n g i n t o a c c o u n t t h e s p i n - o r b i t i n t e r a c t i o n o n t h e e l a s t i c s c a t t e r i n g differe n t i a l cross s e c t i o n . T h e solid line is for VsR = 8.28 M e V a n d t h e d o t t e d line for VsR = 0.

t0

0t

0DJ

°°°'t D

3Z)

GO

.90

~20

150

1

180

Fig, 5. Dependence of angular distribution on spin-orbit coupling. The solid line is for VsR ~ 10.35 MeV, t h e d o t - a n d - d a s h line for VsR = 8.3 3 f o r , a n d t h e d o t t e d line for VsR = 7.7 MeV.

OPTICAL

79

MODEL

symmetrical with respect to the permutation of the amplitudes obtained from the solution of the equations corresponding to the two different values of the total neutron moment. Therefore the corresponding partial cross sections are not sensitive to the change of the sign of Vs~. At small l the change of this sign is inessential because of the smallness of the spin-orbit potential itself as compared with the other terms in the equation. Thus the lack of sensitivity of the cross section to the sign of Vs~ is conditioned on the one hand by the smallness of the spin-orbital coupling constant, which makes it possible to regard the spin-orbital term as perturbation in specific cases, and on the other hand, by the smallness of the neutron spin as compared with the orbital m o m e n t u m when l is sufficiently large. For the same reasons the change of the polarization of elastically scattered neutrons practically reduces to the change of the sign. The picture is different if we consider the bound states by the shell model in which the sign of the spin-orbit interaction constant is highly essential since it determines the sequence of levels with different j. Experiments show that the levels with y"---- l + ½ always lie below those with ~"---- l--½, i.e., in the first case the potential is bound to be deeper. This is what determines the sign of the spin-orbit interaction constant in the shell model. The calculations in the present work were performed with the sign which gives the correct level sequence.

3. Potential of W o o d s - S a x o n Type and its C o m p a r i s o n w i t h P o l y n o m i a l Potential With the exception of the parameters dr and VSR all parameters of the potential (1) with which it is possible to describe well the experimental cross sections with the data of ref. 4). As was shown above the difference in the values of the parameter VsR (7.7 MeV and 8.3 MeV) cannot be essential. At the same time the value dr = 3.36 fm leads to the surface thickness A ~ 3.44 fm, differing appreciably from the respective value A m 2.84 fm following from ref. 4) (A is the size of the region in which the real part of the potential decreases in its absolute value from 0.9 to 0.1 of its maximum). To track down this discrepancy calculations were performed for the potential and parameters coinciding with those accepted in ref. 4): r--R

-i

p(r, -----[ l + e x p ( - - ~ ) 1 a --- 0.65 fro, VcR : 44 MeV,

,

q(r, : exp [ - - (~b-~)z] ,

b ---- 0.98 fm, Vcl :

11 MeV,

R ---- 1.25A½ fm, VSR = 8.3 MeV.

The potential defined by these formulae is shown also in fig. 1.

(4)

80

A.V. LUKYANOV et al.

The calculations with the potential (1) were performed for the nuclei A1 and Cu. In both cases the calculated intergral cross sections prove to be lower than the experimental ones and consequently the theoretical cross sections from ref. 4). The difference in the total cross sections, for example, comes up to 6 to 8 %. The results obtained are represented together with experimental data in table 1. The same table lists the results of calculation for the nucleus TABLE 1 Experimental and theoretical cross sections for A1 and Cu at A1 Experiment of refs. n.l*)

1.73±0.03

ar Cu 2.964-0.06

A1 0.97=k0.02

Cu 1.49=k0.02

Theory with potential (1) and parameters (3)

1.77

2.96

1.01

1.49

Theory with potential (4)

1.63

2.73

0.97

1.40

Theory with potential (1) approximating the potential (4) Bjorklund ref. 1~)

and

Fernbach

theory

2.69

1.33

of 1.02

1.50

Cu with the potential (1) approximating the potential (4) b y the least squares method (the surface thicknesses are very close when this approximation is used). The cross sections obtained in the present investigation for these two potentials differ comparatively little. In particular the total cross sections differ b y less than 2 %. A somewhat larger difference between the respective cross sections ar is evidently attributable to the insufficiently adequate approximation of the Gaussian curve b y a polynomial of the fourth order. The comparison of the relevant differential cross sections is represented in fig. 6. The angular distribution curve for the potential in the form of polynomials is shifted to the left in the region of large scattering angles as compared with the similar curve for the potential (4). The comparison of the results for the potentials (1) and (4) approximating each other b y the least squares method shows that the potentials with the same zl lead to nearly equivalent results regardless of the analytical form of the functions describing the potential. The Runge-Kutta method was used for solving the Schroedinger equations in the calculations with the potential (4). Yet the solution at the initial point is unknown analytically in this case. Therefore we obtained it b y two methods: either representing the solution as a power series and determining the ex-

OPTICAL MODEL

81

pansion coefficients from the condition of the finiteness of the solution at zero, or b y choosing the initial point at such a distance from the centre of the nucleus for which the change of the potential can still be neglected and where the solution can therefore be o b t a i n e d in the analytical form.

1.0

\

aoo~ o

jo

so

so

ao

cso

~o

Fig. 6. Comparison of the differential cross sections for the nucleus Cu calculated with the potential (4) (solid line) and with the potential (1) approximating the potential (4) by the least squares method (dotted line). T h e solution of the equation

__d2u

{ dx 2 + s + p ( x ) 4

5 dp(x)l(/+l) x dx x2

}

+iKq(x)

=

O,

u

with

x = kor, K = VcdVcR,

k0 :

~¢/2MVcR/~ 2,

~ ---- E/VcR ,

VsR(hko] 2 { l VCR\7/ × --(l+l)

d--

for for

(5)

i = l + 1, i = 1--½,

as a power series can be written as u= Re

~t =

Reu+ilmu,

r n, y,~x

x/+1 n=0

Imu=

xl+l

~ y ~mx . n=0

n

(6)

8 9.

A.V. ¢

LUKYANOV

et

al.

tt

The coefficients ~ ,~ a n d )J. were calculated b y the following recurrence formulae: '. =

-

( 2 / + 1)-" ~

( d . . , ~., .._ , , ., - a ,., , 7 .. _ , , , ) ,

ra=l gg

~

tt

t

v

t!

y . = - - ( 2 / + 1 ) - " ~ (d,~, . _ m + d ..~._,~), t¢

~'o = 1/(2/+1)!!,

ro = 0

(7)

(n > 0).

In eqs. (7) use was made of the notation d', ---- e+%2+2Oxz,

d'z = ~ 1 ,

d'. = ~ . _ ~ + ~ n ~ .

(n > 2),

d" = ~/3._~

(n > 1),

(8)

where ~¢. and/3, are the respective coefficients of the expansion of the functions p(x) and q(x) in the power series of the form p(x) =

anx",

(9)

q(x) = e-(R/b) 2 / 3 , , X "~.

"a~O

"t'l~O

The coefficients ~. and /3. were also obtained with the recurrence relations 1

n--1

°on - - n k oa {m=o ~ am0%--m-Z--an-1}

for

n_>--1,

% =

/3~ = 2{koR/3n_l--/3~_z}/n(koh) ~

for

n >_ 1,

/3-1 =

[ 1 + exp 0,

(-- R/a) 1-1,

(10)

/30 ----- 1.

In finding the initial d a t a b y the second m e t h o d we used the tables of ref. 9) for the Bessel spherical functions ?'z(x), since the application of the recurrence relation ~'~(x) ---- { ( 2 l - - 1 ) / x } j l _ l ( X ) - - ] z _ 2 ( x ) , when x ~< 1, leads to a larger loss of accuracy. The choice of the initial point depends on the nucleus under consideration. For example, for the nucleus Cu the initial point should be taken at a distance ~ 1 fm from the centre of the nucleus to obtain the cross sections with sufficient accuracy. The distance r 2 at which the nuclear potential effect can be ignored proved to differ little from the respective distance R + d r for the potential (1) with equivalent surface thickness. Table 2 illustrates the dependence of the results of calculation on the position of the initial and final points x z --~ k 0r 1 and x 2 ---- k 0r~ for the nucleus Cu with a certain set of parameters. Our conclusions on the m a g n i t u d e of the matching radius r 2 and the step size for solving eq. (5) are in good agreement with similar conclusions drawn b y Buck et al. 10). The results of the calculation of the cross sections for the above two methods of obtaining the initial d a t a are given in table 3. The results coincide with sufficiently good accuracy. The correctness of the calculation with the potential (1) was checked b y a h a n d test. The programme for the potential (4) was perfectly similar to t h a t

OPTICAL MODEL

83

for the potential (1), except for the difference in the right-hand parts of the equations. The calculation of the latter was also controlled by a hand test. The data obtained can be regarded as sufficiently reliable. TABLE 2 Dependence of the cross sections for Cu of x 1 and x, Xx

xs

0.871 0.871 0.871 0.871 0.871 0.4355 2.2451 3.2451 2.2451

11.871 13.871 15.871 17.871 25.871 20.4355 20.2451 20.2451 11.2451

at(b)

as(b)

at(b)

2.736 2.745 2.746 2.746 2.746 2.746 2.747 2.751 2.731

1.332 1.337 1.338 1.338 1.338 1.338 1.339 1.344 1.335

1.404 1.408 1.408 1.408 1.408 1.408 1.408 1.408 1.396

TABLE 3 Calculated cross sections for Cu Method

at(b)

as(b)

ar(b)

E x p a n s i o n in p o w e r series

2.732

1.329

1.403

Use of the solution for c o n s t a n t potential

2.733

1.329

1.404

4. C o n c l u s i o n s

1) Use of the potential with nuclear surface absorption and with taking into account the spin orbit interaction makes it possible to describe sufficiently well the experimental cross sections (the total, absorption and elastic scattering cross sections) for a wide region of nuclei with parameters independent of the mass number A. 2) Both the potential described by polynomials and the potential of the Woods-Saxon type can be used with equal success since they give equivalent results given a satisfactory approximation. 3) Our calculations with the potential and parameters of ref. 4) lead to integral cross sections differing somewhat from the respective quantities from the above investigation. In particular our total cross sections prove to be less than the experimental values (the deviation reaching ~ 6 to 8 %). The authors express their deep gratitude to Professor I. S. Shapiro for his interest in the work and discussion of the results; Prof. A. N. Tikhonov who supervised the calculations; and V. M. Martynova who did extensive work on the programme and calculations with the "Strela" computer of Moscow State University.

84

A. V. LUKYANOV gt 0,1.

References 1) R. D. Woods and D. S. Saxon, Phys. Rev. 95 (1954) 577 2) A. V. Lukyanov, Yu. V. Orlov and V. V. Turovtsev, J E T P 35 (1958) 750; Nuclear Physics 8 (1958) 325 3) G. Culler, S. Fernbach and N. Sherman, Phys. Rev. 101 (1956) 1047 4) F. Bjorklund and S. Fernbach, Phys. Rev. 109 (1958) 1295 5) P. J. W y a t t , J. G. Wills and A. E. S. Green, Phys. Rev. 119 (1960} 1031 6) I. I. Levintov, J E T P 30 (1956) 987 7) I. S. Shapiro, Uspekhi Fiz. Nauk (in print) 8) R. A. Vanetsian, A. P. Klyucharev and E. D. Fedchenko, Atomic Energy 6 (1959) 661 (in Russian) 9) Tables of Spherical Bessel Functions (Columbia University Press, New York, 1947) 10) B. Buck, R. N. Maddison and P. E. Hodgson, Phil. Mag. 5 (1960) 1181 11) H. Coon, E. R. Graves and It. H. Barshall, Phys. Rev. 88 (1952) 562 12) M. It. MacGregor, W. P. Ball and R. Booth, Phys. Rev. 108 (1957) 726 13) W. G. Cross and R. G. Jarvis, Nuclear Physics 15 (1960) 155 14) L. A. Rayburn, Phys. Rev. 116 (1959) 1571