Focusing of particles in the optical model

Focusing of particles in the optical model

Nuclear Physics 11 (1959) 5 7 4 - - 5 8 3 , ~)North-Holland Pubhshsng Co, Amsterdam Not to be reproduced by photoprmt or microfilm without written per...

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Nuclear Physics 11 (1959) 5 7 4 - - 5 8 3 , ~)North-Holland Pubhshsng Co, Amsterdam Not to be reproduced by photoprmt or microfilm without written permlssmn from the pubhsher

F O C U S I N G OF P A R T I C L E S I N T H E O P T I C A L M O D E L IAN E

McCARTHY t

School o/ Physscs, Unwerszty ot M~nnesota, M~nneapohs, M~nnesota R e c e i v e d 9 M a r c h 1959 T h e b e h a v l o u r of t h e q u i n t a l f l u x of n e u t r o n s m a t y p m a l o p t m a l m o d e l p o t e n t m l ~s e x a m i n e d as t h e e n e r g y m i n c r e a s e d f r o m 5 M e V to 30 M e V A t t h e h i g h e r e n e r g m s t h e b e h a w o u r c a n b e d e s c r i b e d q u i t e well b y g e o m e t r m a l optics u s i n g t h e classmal t r a j e c t o n e s of t h e n e u t r o n s T h e classmal f o c u s m r e p r o d u c e d q u a n t a l l y b o t h m its g e n e r a l s h a p e a n d i t s b e h a v m u r w i t h v a r y i n g e n e r g y a n d p o t e n t m l s T h e r e a r e also s t a n d i n g w a v e effects n e a r t h e f a r edge of t h e n u c l e u s w h i c h c a n n o t be u n d e r s t o o d classmally T h e relative p r o b a b i l i t y of r e a c t i o n s o c c u r r i n g m d i f f e r e n t a n g u l a r r e g m n s a n d m d i f f e r e n t radial r e g i o n s is e s t i m a t e d b y i n t e g r a t i n g t h e d i v e r g e n c e of t h e f l u x o v e r t h e s e r e g m n s

Abstract:

1. Introduction Although the optical model has so far only been used quantitatively to describe elastm scattering and reachon cross sectmns winch mvolve a knowledge of the wave functions at infinity, it is widely beheved that it also describes the motion of particles in the nucleus Elastic scattering calculations are incapable of giving an unambiguous description of the potential, b u t there are some general properties of particles xn a potential winch hold for a wide range of potentials. The quantal flux and its chvergence give a complete description of entrance channel phenomena and It is interesting to examine their behawour in some typical Potentials. In prevmus pubhcatlons 1, 2) the q u i n t a l flux of particles m the optical model has been chscussed and compared with the flux calculated an a classmal approximation. It was notmed that there was a region of very concentrated flux in the far side (1 e. the shadow side) of the nucleus winch corresponded roughly to a focus. In the part of the nucleus where the classical trajectories do not intersect, it was found that for energies of the order of 40 MeV the classmal flux vector and its chvergence, winch can be derived from the classical trajectories, was qmte similar to the corresponchng quantal quantities In tins paper it xs proposed to discuss the far side of the nucleus in more detail and, m particular, to exarmne the properties of the q u i n t a l focus Assuming that nuclear reactions are initiated b y a two-body colhslon 8) it is reasonable from the classical point of view to say that the probablhty of t Present address California

D e p a r t m e n t of P h y s i c s , U m v e r s l t V of Calffornm, L o s Angeles 24, 574

F O C U S I N G OF PARTI CLES I N T H E OPTICAL M O D E L

575

a colhslon occurring at a point is proportional to the divergence of the flux at the point This neglects coherent effects between outgoing waves arising from different points in the nucleus The relative Importance of different parts of the nucleus from the point of wew of reactions will be discussed for 30 MeV neutrons in a reasonable potential for strontium The quantal flux vector S(x) Is given by S (x) = ~

[,~(x)VV(x ) --t0(x)V,~ (x)]

(1)

The wave function ~0 is normalized to unit incident flux The classical trajectory is calculated in an equivalent real potential V', which is constructed to make the wave number k at the point x equal to the one obtained from the Schrodinger equation for a plane wave in a constant complex potential Vo+iWc(V ~ and We are the values of the real and imaginary parts of the optical potential at x) v' = -½

where E is the incident energy The optical model potential is (V+,W)/(r), where

/(r) = l/{l+exp[(r--R)/a]}

(3)

The polar coordinates of x are r, 0 and ~; the azimuth ~ does not enter the calculation since the flux IS axially symmetric about the mcldent direction (0 = 0). In section 2 we shall examme the behavlour of the focus in a fixed potential as the energy is increased, and Its behavaour for fixed energy and different values of V and W The behavlour is qualitatively the same as would be expected classically. In section 3 the focusing effects will be examined in more detail and compared with the classical effects In section 4 the divergence of the flux will be integrated over different volumes in the nucleus in order to investigate the relative probabxhty of collisions occurring in these volumes 2. Variation of the F o c u s w i t h E n e r g y and P o t e n t i a l Let us first consider the behavaoux of the flux pattern as the energy is increased. Most of the interesting features will be contained in a knowledge of the magnitude of the flux vector along the nuclear diameter parallel to the incident direction. This quantity Is plotted in fig 1 for incident neutron energies of 5, 10, 15, 20, 30 MeV and for the potential defined b y V ---- --45 MeV, W = --8 MeV, R = 5.7 × 10-13 cm, a = 0 5 × 10-13 cm for strontium 88 This wall be called potential I. It IS interesting to notice the transition from the low energy hmat to the energy range where the classical flux is a good approximation In the low

576

IAN E

McCARTHY

e n e r g y h m ] t , o n l y the s-wave flux is p r e s e n t This is s p h e n c a l l y s y m m e t r i c a n d so should b m l d u p to a p e a k at the centre of the nucleus Tins can be easily seen b y considering the w a v e functions for a c o m p l e x square well As t h e e n e r g y is increased, the p e a k a p p e a r s to m o v e t o w a r d the far side of the nucleus a n d to decrease In size I n the n e a r side of the nucleus a t t h e lower energies t h e r e are violent f l u c t u a t i o n s in t h e m a g n i t u d e of the flux which are o b v i o u s l y w a v e effects a n d in winch it IS chfficult to o b s e r v e a n y regular trend, e x c e p t t h a t t h e i r w a v e - l e n g t h s h o r t e n s w i t h increasing e n e r g y

6 • ..-..

4

r(10-t3crr0

2

0

2

4

6

~00"t3cm~

F i g 1 T h e m a g m t u d e of t h e f l u x o n t h e n u c l e a r a x i s for p a r t i c l e s i n c i d e n t f r o m l e f t t o r i g h t P o t e n t i a l 1 is u s e d , a n d t h e c u r v e s a r e l a b e l l e d a c c o r d i n g t o t h e i n c i d e n t n e u t r o n e n e r g y m M e V

At 30 MeV these f l u c t u a t i o n s h a v e b e c o m e less violent a n d the b e h a v l o u r is m o r e h k e w h a t we would e x p e c t classmally, t h a t is a falling off of flux t o w a r d s the centre a n d a large increase on the far side due to focusing of the particles in the p o t e n t i a l I t will be noticed t h a t t h e w i d t h of the p e a k (the l e n g t h of the focus) decreases w i t h increasing e n e r g y u p to 20 MeV A t 30 MeV it has increased a n d we begin to see an e l o n g a t e d focus as we would expect f r o m spherical a b e r r a t i o n in the classical sence T h e r e is no d o u b t t h a t , in the low e n e r g y h m l t , the focus is m o s t easily u n d e r s t o o d as a w a v e effect due to c o n s t r u c t i v e interference of the w a v e s reflected f r o m the nuclear surface I n s o m e of the

FOCUSING

OF

PARTICLES

IN

THE

OPTICAL

MODEL

577

potentials e x a m i n e d the reflected wave is actually stronger t h a n the incident wave at some points so t h a t the flux has zeros and reverses in direction. In p o t e n t i a l I, the I m a g i n a r y p o t e n t i a l is r a t h e r too strong to allow violent fluctuations, b u t if W is reduced to - - 2 MeV, there are two zeros on the n e a r side of the centre for 0 5 MeV neutrons. As the e n e r g y is increased, the fluctuations m o v e back, b u t in some potentials, even for wave n u m b e r s k corresponding to energies of 60 MeV or more, S has zeros on the axis between the focus and the far edge of the nucleus, in a n y case [S[ fluctuates strongly

I 6

[ 4

I 2

I 0

i 2

I 4

I 6

Fig $ The m a g m t u d e of the flux on t h e nuclear axm for 30 MeV n e u t r o n s m p o t e n t m l s I a n d I I Partmles are incident f r o m left to r i g h t

m this region a w a y from the axis In the n e x t section this region will be e x a m i n e d in more detail, from the point of view of seeing how well the focus can be d e s c n b e d b y geometrical optics. I t seems from fig. 1 t h a t at 30 MeV the flux p a t t e r n can begin to be understood b y geometrical optics. In fig 2 the flux on the axis is p l o t t e d for 30 MeV n e u t r o n s m potential I and potential I I 0.e the same p o t e n t i a l e x c e p t t h a t V = - - 5 5 MeV) Classically the stronger potential bends the trajectories more and causes t h e m to focus more strongly. This is also the case q u a n t a l l y Changing W from - - 8 MeV to - - 6 MeV does not m o v e the peak noticeably and causes an almost Imperceptible change in its width

578

IAN E McCARTHY

In this sectmn we have seen how, as the wave num ber increases, there is a gradual trans]t]on from the spherically symmetric standing wave pat t ern of the s-state flux to a p a t t e r n t h a t has properties t h a t can be described b y geometrical optics In the next sect]on we shall see t h a t some detailed features of the p a t t e r n are well reproduced b y geometrical optics.

3. Comparison of the Focal Region with Geometrical Optics Figs. 3 and 4 show the classical trajectories for 30 MeV neutrons m the potentials I and II. A t r a j e c t o r y ]s dra-~n for each value of the angular m o m e n t u m q u a n t u m num be r l. Since the daagram is a section of a three

/

P~

0

!

2

3 4

5

6

7

8

9

FIg 3 Classical trajectories of 30 M e V neutrons m potential I t

dimensmnal diagram it *s clear t h a t more flux comes from tngher values of l because the area on which the flux is incident is proportional to l. Consider the flux on the ares. In the cases illustrated the peak in the flux diagram should occur at about Z -~ 8. Tins corresponds to r ~ 4.8 × 10-13 cm in bot h cases, which agrees v e r y well with the peaks m hg. 2. In potential I the rays for l = 7 and 8 are further apart when t h e y reach the ares, so we should expect the peak for potential I m fig. 2 to be lower t han for potential II. t I n figs 3 - - 7 , t h e circular arcs m a r k e d / = 0 9 a n d / ---- 0 1 r e p r e s e n t t h e p a r t s of t h e nuclear surface w h e r e t h e f o r m factor / 1S 90 ~/0 and I0 ~/o, respectlvely, of its central value.

FOCUSING OF PARTICLES IN THE OPTICAL MODEL

|

Fig



4 Classical trajectories of 30 MeV n e u t r o n s In p o t e n t m l I I .

0

I

2

3

4

5

6

?

8

I

Fig

5 Classical tralectorles of 20 MeV n e u t r o n s m potential ]~

579

580

This is confirmed, and the although the a c c u r a c y of estimate of the ratio T h e e s t i m a t e d b y the classical

I A N ]~

McCARTHY

ratio is clearly of the right order of m a g n i t u d e , the classical calculation does not w a r r a n t an widths of the peaks are also seen to be closely calculation

02

'= 01

I

F~g 6 C o n t o u r m a p of the m a g m t u d e of t h e q u a n t a l flux m t h e far side of t h e nucleus for 30 MeV n e u t r o n s m p o t e n t i a l I The a r r o w s m d m a t e the dlrectxon of t h e flux

Fig 5 shows the classical trajectories for 20 MeV n e u t r o n s in the p o t e n t i a l I The focus on the axas again seems to be a good estimate of the p e a k in the flux In this case there appears to be a strong focusing of the classical rays a b o u t 14 ° off the axis This represents a ring of focus a r o u n d the axis This focus is not r e p r o d u c e d In the q u a n t a l calculation which is shown in fig 7 Similar, b u t not so strong, effects at 30 MeV are not r e p r o d u c e d in the q u a n t a l calculation for p o t e n t i a l I illustrated in fig. 6 The flux diagrams of figs 6 and 7 can be q u a l i t a t i v e l y u n d e r s t o o d from the classical pictures It m u s t be reallsed t h a t the q u a n t a l picture includes flux from the o t h e r side of the nucleus which has crossed the axls This appears to account for the bulge in the flux peak at the top of the diagram in b o t h figs. 6 and 7 T h e bulge at the b o t t o m of the diagram is due to flux from the side of the nucleus which is illustrated In b o t h classical cases there IS a region of v e r y small flux d e n s i t y from a b o u t 30 ° to 40 °, which is r e p r o d u c e d

FOCUSING OF PARTICLES IN THE OPTICAL MODEL

581

in the quantal pictures However, in figs 6 and 7 there are small maxama and minima in this region which have no classical analogue and must be purely standing wave effects, caused by reflections from the far edge of the nucleus which are not included in the classical approximation

f=01

Fig

7 C o n t o u r m a p of t h e m a g n i t u d e of t h e q u a n t a l f l u x m t h e far side of t h e n u c l e u s for 20 MeV n e u t r o n s in p o t e n t i a l I The a r r o w s i n d i c a t e t h e d l r e c t l o n of t h e f l u x

It should be pointed out that there are some classical effects which are not included In the classical calculation 4) The most important of these is the fact that the gradient of W contributes a term to V'

4. Divergence of the F l u x in Various Nuclear Regions The divergence of the flux of 30 MeV neutrons in potential I was integrated over different regions of the nucleus The integral of eq (4) was evaluated numerically, by integrating first with respect to r and then with respect to 0 aa

= 2zrf: dr f: dO r2smO dlv S(r, 0)

Fig 8 shows 2~rf: dr r 2 sin 0 dlv S(r, 0)

(4)

~8~

IAN E

McCARTHY

plotted against O, and m fig. 9, 2z/o dO r" sm 0 chv S(r, O) ts p l o t t e d against r

'30 MeVNEUTRONSIN F~OTEN'I"IALI'

0

20"

40e

60 ~

80 B

I00e

120e

'

14C~

160°

180°

e Fig 8 S ~ dr r" s m @ dlv S(r, 0) plotted m arbztrary umts agaxnst 0 for 30 M e V neutrons in potential I

30'MeVNEUTR()NI . . . . ~OdlvS~,O)

I

0

/

. !

.

2

.

.

3

4

t,5

, t,

6

7

, ,'~..~

8

9

IO

~(Z)-~cm)

Fig 9 [~ dO rt s m 0 dlv S(r, 0) plotted m arbitrary umt~ against r for 30 M e V neutrons m potentlal I

FOCUSING OF PARTICLES IN THE OPTICAL MODEL

583

The most interesting number obtained from fig. 8 is an estimate of the relative number of two-body collisions occurring in the focal region. About 15 °/o of the double integral in eq. (4) hes between 0 = 0 ° and 0 : 20 °. At Ingher energies, the divergence peak is not so large, since the focus occurs further out in the nuclear surface where the imaginary potential is small. Hence, it is a reasonable estimate that in medium energy reactions, not more than 20 °/o of the lmtlal reactions occur in the focus From fig. 9 it is possible to estimate the relative number of initial reactions occurring in the surface. In tins case 49 % of all imtlal reactions occur outside the 90 % density radius. In optical models in winch the imaginary potential is peaked in the surface 5), the proportion would be much higher since the divergence is proportional to W]~v[2. However, it seems that this number is fairly typical of potentials m which both V and W are of the Saxon type The author would hke to thank Mr. R. Spurrier for computing the classical trajectories and Dr. C. E. Porter for m a n y discussions. References I) 2) 3) 4) 5)

I R I D F,

E McCarthy, Nuclear Phymcs I0 (1959) 583 M Emberg, I E McCarthy and R A Spurner, Nuclear Phymcs I0 (1959) 571 E McCarthy, E V Jezak and A Krowmmga, to be pubhshed R Yenme, private commumcatlon B]orklund and S Fernbach, Phys R e v 109 (1958) 1295