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SEMICLASSICAL CALCULATION OF THE
NUCLEAR RESPONSE FUNCTION AT HIGH
MOMENTUM TRANSFER
U. Stroth, R. Hasse, P. Schuck
To cite this version:
JOURNAL
DE
PHYSIQUEColloque C6, supplCment au n06, Tome 45, juin 1984 page C6-343
S E M I C L A S S I C A L C A L C U L A T I O N OF THE NUCLEAR RESPONSE F U N C T I O N A T H I G H MOMENTUM TRANSFER
U. S t r o t h , R.W. Hasse and P. Schuck*
I n s t i t u t Laue-Langevin, 156 X , 38042 GrenobZe Cedex, France
* I n s t i t u t des Sciences NucZ&aires, 38042 GrenobZe Cedex, France
RGsum6
-
Nous c a l c u l o n s l a f o n c t i o n de l a rgponse n u c l g a i r e semiclassique- ment. 11 s ' a v h r e que l e s r g s u l t a t s obtenus pour des impulsions t r a n s f b r g e s q 2 2 fm-* s o n t en t r h s bon accord avec un c a l c u l e x a c t .A b s t r a c t
-
We c a l c u l a t e t h e n u c l e a r response f u n c t i o n i n a s e m i c l a s s i c f a s h i o n . I t i s shown, t h a t t h e o b t a i n e d r e s u l t s a r e f o r h i g h momentum t r a n s f e r s q 8 2 fm-' i n a very good agreement w i t h an e x a c t c a l c u l a t i o n .I
-
INTRODUCTIONI n e l a s t i c e l e c t r o n and proton s c a t t e r i n g £ran n u c l e i a t h i g h momentum t r a n s f e r s nece- s s i t a t e s on t h e f u l l quantum mechanical l e v e l a b i g numerical e f f o r t . Since i n t h e q u a s i - e l a s t i c peak r e g i o n s h e l l e f f e c t s a r e a b s e n t a s e m i c l a s s i c a l approach may be s u f f i c i e n t . We w i l l show i n t h i s work t h a t t h i s i s e f f e c t i v e l y t h e case f o r momen- tum t r a n s f e r s of q 2 2 fm-1.
I1 - THE THEORY
The f r e e response f u n c t i o n n:)(q,u) f o r an e x c i t a t i o n o p e r a t o r
6
can be c a l c u l a t e d from t h e p a r t i c l e h o l e Greens f u n c t i o n n(0) ( r ,,
r,, r;,
r;).
d
s h a l l b e t h e l o n g i t u d i n a l e x c i t a t i o n o p e r a t o rI n c o o r d i n a t e r e p r e s e n t a t i o n , t h e Greens f u n c t i o n a t t a i n s t h e following form:
where h i s t h e one p a r t i c l e h a m i l t o n i a n and
X
t h e Fermi enerLy. For t h e Greens f u n c t i o n we t a k e now a s e m i c l a s s i c a l approximat i o n . This approximation i s achieved by r e p l a c i n g t h e o p e r a t o r s by t h e i r c l a s s i c a l c o u n t e r p a r t s . This Thomas Fermi l i k e approximation h a s been very s u c c e s s f u l i n t h e c a s e ofs article
h o l e d e n s i t i e s/ I / .
For t h e l o n g i t u d i n a l response f u n c t i o n one o b t a i n s w i t h
k
JOURNAL
DE
PHYSIQUEt h e l o c a l Fermi Gas (F.G.) d e r i v e d p r e v i o u s l y i n a somewhat d i f f e r e n t way by Kosen- f e l d e r
121.
R:';~~,,,~
=
1
o\'a
$$%ntw)
This r e s u l t i s v a l i d f o r a r b i t r a r y p o t e n t i a l s . I f t h e p o t e n t i a l i s l o c a l , however, it c a n c e l s i n t h e denominator and e n t e r s only i n t h e arguments of t h e s t e p func- t ions
.
For a s p h e r i c a l p o t e n t i a l a l l b u t one i n t e g r a t i o n can b e performed a n a l y t i c a l l y . F o r t h e imaginary p a r t one thus o b t a i n s f o r k)>
0
:where
)
*
v )
i s t h e l o c a l Fermi momentum andct
=
& ?
$
I
The r e a l p a r t can be c a l c u l a t e d form eq. (7) by a d i s p e r s i o n r e l a t i o n
o r d i r e c t l y from eq. (6)
.
One o b t a i n sFor a s q u a r e w e l l ~ o t e n t i a l eq. (7) l e a d s d i r e c t l y t o t h e Lindhard f u n c t i o n / 3 / ; f o r a harmonic o s c i l l a t o r p o t e n t i a l
VtQ-,
Qn
u
:
R'
,
eq. (7) s t i l l can be given ana- l y t i c a l l y .111 - COMPARISON WITH MICROSCOPIC CALCULATIONS
Fig. 1
0
1
2
3
0
2
4
6
-
WW,
For two momentum t r a n s f e r s t h e response f u n c t i o n i n a harmonic o s c i l l a t i o n p o t e n t i a l (kF = 1.5 fm-l) i s compared t o a quantum mechanical c a l c u l a t i o n 141. For o r i e n t a t i o n t h e Lindhard f u n c t i o n s ( n u c l e a r m a t t e r kF = 1.36 fm-l) a r e a l s o shown.
-1
The s e m i c l a s s i c a l method reproduces f o r momentum t r a n s f e r s q > 0.6 fm t h e average v a l u e s very w e l l . For lower momentum t r a n s f e r s , t h e l o c a l Fermi Gas approximation f a i l s , because only a few eigenvalues a r e e x c i t e d and a l o c a l approximation i s n o t a b l e t o account f o r s i n g l e e i g e n s t a t e s which a r e a g l o b a l p r o p e r t y of t h e system. Energy i n t e g r a t e d q u a n t i t i e s may however s t i l l b e q u i t e a c c u r a t e a s can b e deduced from t h e f a c t t h a t t h e energy weighted sum r u l e i s e x a c t l y f u l l f i l l e d w i t h i n t h e lo- c a l F.G. approximation 121.
I n f i g . 2 t h e l o c a l F.G. response f u n c t i o n c a l c u l a t e d f o r a Woods-Saxon p o t e n t i a l
i s compared with t h e correspondent f u l l y quantum mechanical c a l c u l a t i o n 161. The r e s u l t has been averaged w i t h a Lorenzian of 3 MeV width.
OL - a1 5- local F G
-
unct Calculahon-
q : l 0 f d 1 p=2.15fm-'-
> I E U-
0 2 - + 3. 0 25 50 0 50 100 150 200Fig. 2: The s e m i c l a s s i c a l response f u n c t i o n i n a Woods-Saxon p o t e n t i a l compared w i t h an e x a c t c a l c u l a t i o n of N.van G i a i 161.
C6-346 JOURNAL
DE
PHYSIQUEaverage p o t e n t i a l i s i l l u s t r a t e d i n f i g . 3 . One f i n d s t h a t t h e n u c l e a r m a t t e r approximation (kF = 1.36 fm) i s b e t t e r f o r heavy n u c l e i t h a n f o r l i g h t e r ones. For l i g h t n u c l e i , s u r f a c e e f f e c t s a r e important and a harmonic o s c i l l a t o r ( k p _ = 1.5 fm-') response f u n c t i o n (eq. (10)) becomes more r e a l i s t i c t h a n t h e Lindhard f u n c t i o n (nu- c l e a r m a t t e r ) .
I I I F i g . 3 : The response func-
t i o n s a r e compared f o r d i f f e r e n t p o t e n t i a l s . The dashed l i n e s belong t o Woods-Saxon p o t e n t i a l s f o r two d i f f e r e n t masses. 0 10 2 0 3 0 LO
I V
-
THE DELTA RESONANCEFor i n e l a s t i c p r o t o n s c a t t e r i n g one can e x c i t e b e s i d e s p u r e nucleon p a r t i c l e - h o l e p a i r s a l s o & - h o l e p a i r s i n t h e l o n g i t u d i n a l c h a n n e l / 7 , 8 , 9 / . Thereby a nucleon can be transformed by a .rro t o a d e l t a p a r t i c l e . We a g a i n want t o s t u d y t h e d i f f e r e n c e be- tween a pure n u c l e a r m a t t e r c a l c u l a t i o n and our s e m i c l a s s i c a l approach. The t o t a l l o n g i t u d i n a l f r e e response i s given by
171.
-
Here f (q,w) and fA(q,w) a r e t h e pion b a r y o n v e r t e x f a c t o r s 1 7 1 :
N
A = 1300 P4eV and
I n analogy t o (6) one f i n d s 1 9 1 :
b a,
w i t h
hl~,?)
.%W.+
&,+
V
(R>
,
%LO, i s t h e d i f f e r e n c e between t h e Amass (m*) and, t h e nucleon mass. The d e l t a width has been n e g l e c t e d .And f o r t h e r e a l p a r t one o b t a i n s :
a
1"
~ - ~ ~ ~ ~ w ~ ~ ~ A * ~ ~ L - ~ , - ~ ~ ) ~
k~\%tqp\=
--
j d ~ $
~{A-VIRI)
dhLl
9%hP
\A+~.~,W\\\A
-
I.r.Q,WI
I n f i g . 4 t h e t o t a l response f u n c t i o n i s drawn t o g e t h e r w i t h i t s components. I n t h e imaginary p a r t ( f u l l l i n e ) t h e n u c l e a r and t h e d e l t a c o n t r i b u t i o n s a r e c l e a r l y sepa- r a t e d i n energy. The d e l t a response, however, c o n t r i b u t e s t o t h e t o t a l response f u n c t i o n through i t s r e a l p a r t (dashed d o t t e d l i n e ) even f o r lower e n e r g i e s , and modifies t h e r e a l p a r t of t h e nucleon response (dashed l i n e ) . For comparison, t h e d e l t a p a r t of t h e response f u n c t i o n i s a l s o shown f o r t h e n u c l e a r m a t t e r case ( d o t t e d l i n e ) .
n 1.5 I I I
-
Fig. 4: The t o t a l responsef u n c t i o n and i t s components (dashed: t h e nucleon p a r t dashed d o t t e d : f o r t h e A
p a r t ) . The f u l l l i n e s g i v e t h e sum of both. The calcu- l a t i o n was done i n a Woods- Saxon p o t e n t i a l f o r %a. The p o i n t s i n d i c a t e t h e re- s u l t f o r t h e n u c l e a r , m a t t e r
r? 0 2 0 0 COO c a s e of t h e d e l t a p a r t .
2 . -
I V
-
RESPONSE FUNCTION INCLUDING PARTICLE-HOLE INTERACTIONThe Greens f u n c t i o n i n c l u d i n g i n t e r a c t i o n s i s c a l c u l a t e d from t h e t r e e one by means of ( n e g l e c t i n g t h e exchange p a r t of t h e i n t e r a c t i o n ) :
One can show t h a t t o z e r o o r d e r of .ti one o b t a i n s f o r t h e r e s u o n s e f u n c t i o n :
.
..
For t h e s p e c i a l c a s e of a one exchange p o t e n t i a l (OPEP) p l u s tlidgal parameter, r e p r e s e n t i n g t h e short-range r e p u l s i o n ,
m i s t h e pion mass.
h a s been used.
C6-348 JOURNAL
DE
PHYSIQUEThe p a r t i c l e h o l e i n t e r a c t i o n causes a s o f t e n i n g of t h e response f u n c t i o n whereas t h e A c o n t r i b u t i o n causes an enhancement. We omit h e r e a d e t a i l e d comparison w i t h
t h e pure n u c l e a r m a t t e r r e s u l t because we were s o f a r unable t o e x a c t l y repro- duce t h e c a l c u l a t i o n s of r e f . / 8 / . The r e s u l t s p r e s e n t e d i n Fig. 5 should t h e r e f o r e a t t h e moment only be considered a s q u a l i t a t i v e i n d i c a t i n g t h e g e n e r a l t r e n d of t h e e f f e c c s .
I n conclusion we can say t h a t we have shown i n t h i s work t h a t t h e s e m i c l a s s i c a l approach t o t h e c a l c u l a t i o n of n u c l e a r response f u n c t i o n s works p e r f e c t l y w e l l f o r h i g h momentum t r a n s f e r s (q 2 2 fm l ) opening t h u s t h e p o s s i b i l i t y of q u i t e p r e c i s e and e a s y c a l c u l a t i o n s even i n t h e case of q u i t e s o p h i s t i c a t e d l i n e a r response theo- r i e s .
N e v e r t h e l e s s some f u r t h e r s t u d i e s and r e f i n e m e n t s should b e done i n f u t u r e :
i ) One should i n c l u d e t h e f i r s t % - c o r r e c t i o n t o improve t h e f a r t a i l r e g i o n of t h e c r o s s s e c t i o n .
i i ) A comparison of our r e s u l t (Fig. 5) i n c l u d i n g r e s i d u a l i n t e r a c t i o n w i t h an e x a c t quantum c a l c u l a t i o n should be performed, though it i s our s t r o n g b e l i e f t h a t t h e same degree of accuracy h o l d s i n t h e i n t e r a c t i n g c a s e a s i n t h e non i n t e r a c t i n g one.
REFERENCES
/1/ GHOSH G., HASSE R.W., SCHUCK P. and WINTER J . , Phys. Rev. L e t t . 50 (1983) 1250 /2/ ROSENFELDER R . , Ann. Phys. 128 (1980), 188.
/ 3 / LINDHARD J . , Dan. Mat. Fys. Medd. 28, no. 8 (1954)
/ 4 / SHLOMO S., Phys. L e t t . 118B (1982), 233
/5/ SCHUCK P . , GHOSH G. and HASSE R.W., Phys. L e t t . ll8B (1982) 237.
/ 6 / We thank D r . N. v. G I A I f o r performing t h e e x a c t c a l c u l a t i o n f o r t h e f r e e r e s - ponse f u n c t i o n f i g . 2.
/ 7 / OEST E . , TOKI H. and WEISE W . , Phys. Rep. 83(1982) 281.
