CHARGE EXCHANGE EXCITATIONS IN N≠Z NUCLEI : VLASOV AND HYDRODYNAMIC EQUATIONS

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CHARGE EXCHANGE EXCITATIONS IN N ̸ =Z NUCLEI : VLASOV AND HYDRODYNAMIC

EQUATIONS

E. Lipparini, S. Stringari

To cite this version:

E. Lipparini, S. Stringari. CHARGE EXCHANGE EXCITATIONS IN N ̸ =Z NUCLEI : VLASOV AND HYDRODYNAMIC EQUATIONS. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-27-C2-35.

�10.1051/jphyscol:1987204�. �jpa-00226469�

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JOURNAL DE PHYSIQUE

C o l l o q u e C 2 , s u p p l e m e n t au n o 6 , Tome 4 8 , j u i n 1 9 8 7

CHARGE EXCHANGE EXCITATIONS I N N z Z NUCLEI : VLASOV AND HYDRODYNAMIC EQUATIONS

E. LIPPARINI a n d S. STRINGARI

D i p a r t i m e n t o d i Fisica, U n i v e r s i t d d i T r e n t o , I-38050 P o v o , ( T r e n t o ) , I t a l y

A b s t r a c t

-

Macroscopic e q u a t i o n s of m t i o n f o r charge exchange r e a c t i o n s i n N#Z n u c l e i are d e r i v e d s t a r t i n g from the time dependent Hartree-Fock theory.

Application is m d e to t h e study o f the d i p l e g i a n t resonance i n i s o s p i n channels and p- c a p t u r e In N#Z n u c l e i .

I n t h i s work we d e v e l o p macroscopic d e l s f o r g i a n t electric i s o v e c t o r resonances e x c i t e d by o p e r a t o r s o f t h e form

+

where T - are the usual i s o s p i n o p e r a t o r s d e f i n e d by

~ + l ~ > =

^J21n> ^T

-

In>= J21p>.

These ope a t o r e n t e g i n many p h y s i c a l r e a c t i o n s involving charge exchange((p,n)

0 5 3

1

,

( n t ,a )

,

( H e , t )

. . .

^.) reactions. The m i n f e a t u r e s o f s u h e x c i t a t i o n s are as- s o c i a t e d w i t h t h e e x i s t e n c e o f a neutron e x c e s s ( (N+Z) n u c l e i )

'-'. ^mis

^is^responsi-

b l e f o r s i g n i f i c a n t energy s p l i t t i n g s between the s t a t e s t h a t are e x c i t e d i n charge exchange reactions and the ones e x c i t e d i n reactions w i t t a u t charge exchange (y,e..) A s c h m t i c s i t u a t i o n o f t h e v a r i o u s e x c i t e d i s o s p i n fragments (with their isogeome-

t r i c a l f a c t o r s ) is r e p o r t e d i n fig.1, where one starts from a nucleus ( N , Z ) i n its ground s t a t e ( T Z q ) and e x c i t e s states i n t h e same nucleus (AT7* c h a n n e l ) and i n

- -

daughter n u c l e i (AT =+1 and ATZ=-1 channels r e s p e c t i v e l y ) . Once t h e Coulomb energy is r m v e d , the fragments w i t h t h e same i s o s p i n occur a t t h e saw energy and one ob- s e r v e s the e x i s t e n c e of s p l i t t i n g s betwcen fragiwnts with d i f f e r e n t i s o s p i n ( f i g . 2 ) . The d e s c r i p t i o n of such a s p l i t t i n g is a n i n t e r e s t i n g f i e l d of t h e o r e t i c a l i n v e s t i - g a t i o n . The neutron excess is a l s o r e s p o n s i b l e f o r a n important quenching of the s t r e n g t h i n t h e aTZ=-el channel w i t h r e s p e c t t o theATZ=O and -1 cases. T h i s quenching is c l e a r l y revealed i n t h e u ' c a p t u r e e x ~ e r i m c n t a l r a t e s and o r i g i n a t e s from t h e P a u l i blocking and from dynamic c o r r e l a t i o n s .

The s t a r t i n g p o i n t o f o u r theory are t h e time dependent Hartree-Fock e q u a t i o n s (TDHF) f o r t h e one-body d e n s i t y matrix: p T T , ( r l l r 2 ) = C T r 2 h e r e as

io$:rb, .

usua1,a and T are the s p i n and i s o s p i n variables e s l n g l e particle wave

&on $. r e s p e c t i v e l y . Using t h e one-body local d e n s i t y dependent hamiltonian

f8'-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987204

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JOURNAL DE PHYSIQUE

v

^{+ +}

H = - -

2m + V ( r ) + "sym (r1p.r

-

1/2 E C ~ z (2)

* +

whereP=C. $ ( ~ , O , T ) < T ] T ] T ' > $ ~ ( ~ , U , T ' ) istheisovectorone-bodydensityand 1GTT

'

i

EC is the mdulus of the energy s h i f t between neighbcuring nuclei, one has

where

r6

%P6 are the 6 - c a p n t s of t h e isospin operator and of t h e isovector density p

.

^{I f}^{m '}^eq.(3) gives the standard TDHF equations for the neutron

( ~ = ~ ' = t l ) and proton (T=T'=-1) densities describing the nuclear e x c i ~ t i o n s in t h e AT =O channel. Conversely i f T*'

,

^eq.(3) describes the nuclear excitations i n the

AT'=+^

^channels.

Introducing the variables

& (?

^+:.) /2, =;

(g -:

⁾ ^anddeveloping the y a t i a l terms

~ ( r ~ ) - ~ ( r )

,

v ( r 2 ) p s ( r 2 ) an6 v1 (r 1)p 6

(t

1¹^{'up to} ^linearⁱⁿ^s( k c l a s s i - c a l expansion,

8

equivalently, l%$ wave length limit) one obtains the f o l l m i n <

equations :

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Taking t h e Fourier transform w i t h respect to t h e coordinate s and defining t h e di-

*

s t r i b u t i o n function

* -+

+ + -+ -+ i p * s d;

f T T l ( r , p , t ) =

2 I

^{p T T l}^(r,s,t)e (5)

(2n) 3

-+ -+ -+ +

and constructing its A - v n e n t i n isospin space f (rtp,t)=CTT _X f ( r , p , t ) <T

I

T~

I

^{T I >}

one f i n a l l y g e t s the following n a s o v type equations f o r f x (rIp,tfT

a

₊

2 ? - $ v ( ~ ) * v

^-+l f

(G *

^-+ ^-+ ^{- + +}

( Z

m r r P , p , t )

-

^Vr ^(vSF ( r ) p A ( r ) ) * V P f ( r , p , t )

-+ -+ -+ +

where f ( r , p , t ) =CT f T T ( r , p , t ) is t h e isoscalar d i s t r i b u t i o n function. By a p r c p r integration of eq.(6) in the nanentum space one can derive equations f o r the i s o v e c tor d e n s i t i e s &,current d e n s i t i e s relevant f o r charge exchange reactions

( f + - = ~ T T l f T T l < T l ~ I T 1 > ) :

+ -+ -+ -+ -+ + *

~ + ( r , t ) = j f + ( r , p , t ) d ; ^- ; J + ( r , t ) - =

J

p f + ( r , p , t ) d $ -

.

⁽⁷⁾

one finds:

where p and pl are t h e i s o s c a l a r and isovector (2-aqxment) d e n s i t i e s r e s p t i v e - l y -

?he h y d r o d m c equations a r e straightforwardly obtained by generalizing t h equilibrium approximation ( ? h a m s - F e d ) f o r t h e isoscalar density( a=3/5 (3n

i k

*

1

I p p fdp = - 6 a p 5 I 3 to the isovector case ( p 6 < < p ) : 3 ik

pppmximation (10) f o r the z - c a p n e n t (6 =z) i s irrmediately obtained applying t h e Thomas-Fermi approximation separately t o the neutron and proton k i n e t i c energy densities.

I n the follcwing w e w i l l consider eqs. (8) and (9) in the l i m i t of nuclear matter.

Inserting eq. (10) i n eq.(9) one g e t s t h e following linearized equations:

A rL

where we have introduced the volurne symnetry energy c o e f f i c i entering the semi- empirical m s formula: b = 2 / 3 ~

+

v

vol F s d ^{w i t h}^E=(5u/6m)?j3. lhis coefficient F

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C2-30 JOURNAL DE PHYSIQUE

takes contribution both from t h e k i n e t i

Ci

and potential energies. The two contribu- tions a r e k n m t o be p r a c t i c a l l y equal and yield b =50 MeV. In eqs.(11) we have a l s o introduced the quantity vol

characterizing t h e symnetry p o t e n t i a l of t h e nuclear system (V ₁

- -

^?SFP )

.

Z

Equations (11) d i f f e r f r m the ones valid i n the ATz* channello f o r the t e r m p m portional t o E and d. These tenns r e f l e c t the occurrence of isospin precession phe- nomena induced by the C o u l d and C s y m w t q p t e n t i a l s . Such a phenahnen is formally similar t o the one p u r r i n g i n an interacting spin w l a r i z e d s y s t m in the presence of a magnetic f i e l d

.

The Coulanb interaction plays here t h e r o l e of t h e magnetic f i e l d and t h e isovector interaction v i s the analogous of t h e s p i n interaction.

While magnetic systems a r e usually p o f g i z e d by an external magnetic f i e l d , the nucleus i s polarized by t h e CoulcBnb f i e l d .

In the presence of an external o s c i l l a t i n g f ' e l d ^'t e r a c f ' g with t h e uclear system

+$ -

'

^-lwr+ ^C-el') ^{where (g} ^{=g )}

through the interaction Hamiltonian Hint= (G e

+ +

G-=T.g-(r ^)it^,

,

t h e equations of notion a n t a i n an additional term. I n p a r t i c u l a r the !iYdr&narmc equations (11) become:

a

I+ +

-

N-Z

+

+iwt

%P+

2.J *

^%P+

+

2iAT pg (r)e- = 0

9 s . ( 1 3 . 2 ) 4ivy.r s e t o solutions of t h e form p + ( r , t ) = p (r)e'iwt

-la$ - f and

J + ( r , t ) = J + ( r ) e

The

q y

p l a r i z a b i l i t y r e l a t i v e t o t h e excitation operator G = E .g- (r

.

⁾

;

is gi-

l l i ven by :

a - ( a ) = 1 f g - ( r ) p - ( r ) d ;

.

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This quantity is of g r e a t i n t e r e s t because it all* t h e determination of the eigen- values of t h e nuclear Hamiltonian and of t h e strengths r e l a t i v e to the excitation o- p r a t o r . In f a c t using standard f i r s t order p e r t u h a t i o n theory one finds:

of course t h e two terms of eq. (16) cannot be sirrmltanously d i f f e r e n t f m zero f o r i+e same k-state. Ihe poles of a-(a) correspond to states

+

exci$ed by the operator

+

G

.

The dynamic p o l a r i z a b i l i t y r e l a t i v e to the operator G =E.g (r ) T is given by

1 i i a+(w) = a_(%) and its poles correspond to states excited by t h e operator G-.

W e have applied t h e above formalism t o two i n t e r e s t i n g cases:

i) t h e solutions of eqs. (13,14) w i t k i n a sphere of radius R = 1 . 1 5 ~ ~ ' ~ h, imposing the c l a s s i c a l Steinwedel-Jensen boundary condition f o r the isovector current:

-+ + +

J + ( r ) . n - = 0

,

where n is the u n i t vector orthogonal to the surface of t h e sphere;

ii) the solutions of eqs.(13,14) f o r t h e propagation of excitations of a plane wave t y p with application of the r e s u l t s to the problem of capture in N#Z nuclei.

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+ -

i) I n t h e dipole case (g ( r ) = g (r)=1/2z1 the solutions of eqs. (13.14) yield the f o l l w i n g expressions £05 the energies and strengths of t h e states excited by the d i p l e operator 1/22 z T- in the AT =+1 channels:

r ,i i i z

where 0 = qk a r e the usual eigenfrequencies of t h e Steinwedel-Jensen d e l m% q k ~ = 2 3 h

,

5.95.

. . . . . .

^One^notes^{t h a t}ⁱⁿeqs. (17.18)the e f f e c t of t h e C o u l d interaction, as treated i n the present &el, consists of a s-le s h i f t of the frequencies without ncdification of t h e dipole strength. The s h i f t i s due t o the f a c t t h a t the e x c i t a t i o n occurs in a nucleus w i t h d i f f e r e n t charge w i t h respect t o the i n i t i a l nucleus. Conversely t h e symnetry potential vs introduces dynamic effe- cts changing both t h e frequencies and t h e strength with a P c t t o t h e ones c h a r a c e r i z i n g t i e AT =O channel:

An interesting quantity t o investigate i s the isospin s p l i t t i n g AE+ between the dif f e r e n t T-isospin fragments with the same value of T

.

^One^has

^-

1 N-z

v1

~ e g l e t i n g t- i n ( (N-z)/A)

',

eq. (20) yields the well k n m r e s u l t

7 i

f o r the isospin s p l i t t i n g a l - ready e g y e d by several a-

uthors

.

I n t a b l e 1 we re- t a b l e 1 p o r t t h e predictions of eq. (20)

f o r the dipole isospin s p l i t t i g &+ AE

-

AEq

-

g s AEA (in W V ) i n several nucl e i &ether with the experimn-

48Ca 4.5

-

3 . 8

tal values £ran r e f s . (1.2.4.8).

-

The agreement i s q u i t e reasona-

b l e especially f o r t h e hE_ 6 0 ~ i 1.7 4.1 1.6 1.

s p l i t t i n g . ?he asymetzy between 9 0

t h e

aE+

and aE_ s p l i t t i n g s r e ^Zr ^3. ^3.9 ^2.6 ^2.2 f l e c t s t h e presence of

3

siqni-

f i c a n t isotensor e f f e c t ac- ^120sn ^4.7 ^5.5 ^3.7 ^3.6 counted f o r in the presen m-

0

5

d e l by t h e term i n ( d / q ) (see 208Pb 6 -2 11.2 4.3 4.5

h

eq. (20).

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C2-32 J O U R N A L D E PHYSIQUE

In t a b l e 2 we cQnpare t h e predictions of t h e Me1 f o r the quantity AE+

-

AE-=

with t h e recert experimntal r e s u l t s f o r the mnopole resonance (see r e f . 2)

.

Also in this case the agreanent is s a t i s f a c t c r y .

Another i n t e r e s t i n g e f f e c t is t h e quenching of t h e strength S

+

with r e ~ p e c t to S- (see eq. (18) )

.

?he quenching i s spectacular in nuclei with a large neutron excess and froan a microscopic pint of view arises from t h e Pauli blocking. The quenching

+

^N-Z 2

of the S strength i s t o t a l when t h e manentun ak approaches t h e value 2 -- _A _3rnF

.

This i s o b p i n e d i n practice i n the case of 208pb where t h e quenching of the dipole strength S is consequently almost t o t a l . In t a b l e 3 we caTlpare the predictions of eq. (18) f o r the dipole strength with t h e RPA predictions of ref -13

.

?he agreement is s a t i s f a c t o r y revealing t h a t the present a ~ p r m c h accounts f o r mst of the micro- scopic e f f e c t s responsible f o r the strength d i s t r i b u t i o n of t h e nuclear excitations i n t h e isospin channels.

t a b l e 2 t a b l e 3

+ -

w l e u s (AE+-AE-) ( A E + - A E ) ~ ~ ~ Nucleus S+ SwA

-

^'RPA

14-16.

ii) The

u-

capture r a t e is given by

.

2 2 2 2

In eq.(21) we have a s s m d SU4 spiq isospin set G =G

+

3G A

+

G P

-

2G P A G

a d

introduced t h e quantity e < P > (n/(Zrn muon atarcic wave function averaged over

u

il correction and M is the rel$van$ m t r i x e l m n t given by:

(22)

In eq.(22) qko is t h e energy of t h e emitted neutrino

where

yco

is t h e excitation energy i n t h e final nucleus and wu is t h e p - a t a n i c bin- ding energy. Equations (21,221 have been already investigatedBin the past usiW

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d i f f e r e n t theoretical a p p r o a c l ~ s ''-I8 mainly based on t h e closure approupition, on the use of indepndent m c l e models and of sum rules. More recently a self- consistent calculation based on the random phase approximation has been carried out i n the isospin channels and revealed t h e important r o l e of dynamic correlations.

I n the f o l l m i n g we w i l l use the hydrcdynamical model here developed t o calculate the rate(21) in N+Z nuclei. Such an approach has the merit of revealing i n a c l e a r way the phvsical inqredients characterizing t h i s reaction.

m t i o n ($2)+ shows t h a t the relevant excitation operator i n t h e p- capture is

=

9

^eiqeri^T+

.

For t h i s p r a t o r $e solutions of eqs. (13.14) yield the follc-

1 i

wing e x p r e s s i o y f o r the freque cies

Y

^(q)^of^thec o l l e c t i v e s m t e s and f o r t h e ma- t

9

trix elements s = Zk

1

^<k

1

G lo>

1

6 (w+jko) :

0 1 N-Z

q and d = -

-

V1

.

m a t i o n (25) assumes an even sinpler

5

^{" 2} ^,

form i f one neglects the term ( d / h (q) ) wlth respect t o 1, which is a good approximation i n t h e physical case considered here.and assumes7 t h a t the p t e n t i a l con- t r i b u t i o n t o h vol is equal to t h e k i n e t i c one yielding b v01- - 1/2 Vl. One then gets

+

A

S = - N-Z 2

26 ( I - - A 6 )

where 6 = RbW1/q 2

.

W a t i o y j25.26) c l e a r l y reveal t h a t t h e e f f e c t of t h e exc- e s s neutrons 1s t o reduce t h e S strength w i t h respect t o the N=Z case (Pauli blocking). The e f f e c t of the dynamic correlations is a l s o taken i n t o account i n eqs.

(25,26) v i a the s y ~ ~ t r y potential. Notice t h a t neglecting i$e e f f e c t s of such cor- r e l a t i o n s (V =O in eq. (25)) would yield an expression f o r ST containing only terms l i n e a r i n (N-Z) 1 /A :

+

_--_- ² N-Z

'uncorr. ( 1 - 2 - 6 )

6kin A kin

6

The m a t r i x element (22) k e r n s :

where t h e value of q is fixed by the solution of eqs. (23) and (24) (uko = w

+

(4))

.

0 2

N q l e c t i n g t h e small t e r m i n (d/& ) entering eq.(24) one finds:

1 N-Z

5 -

^]W!J_B

--

₂

-

_A _bvol

+ E~

q =

1

+\p ^-

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C2-34 JOURNAL DE PHYSIQUE

Within the sane approximation one g e t s t h e f o l l m i n g expression f o r t h e t o t a l r a t e :

I n the f o l l m i n g we w i l l use the known values of wlJ

,

R and A; (see f o r example r e f . (13)) and take bvol= 50 MeV. The namntum of &e neutrino r e s u l t i n g f r m eq. (291

t u r n s . o u t t o be a h s t independent on the nuclear mass ranging frcan 86 to 90 MeV.

Taking an average value of 88 &V f o r q, eq.(X)) becomes:

Equation (31) can be shown t o reproduce t h e numerical r e s u l t s o f eqs.(29,30) in an accurate way. The v a l u e q s r e d i c t e d by eq.(31) are reported i n table 4 together with the experimental valuesLu.

It is worthwile t o note t h a t

eq. (31) i n s p i t e of its sem_ Nucleus (N-Z)/A R A'

u n m r %I A'

p l i c i t y , reproduces the ex- exp

perimental values w i t h good

precision. An even b e t t e r 4 0 ~ 0 -44 .36 4.3 3.2 2.5

agreement is obtained using

a s l i g h t l y l a r g e r value of 4 8 ~ a -17 -47 .05 2.1 1.3

-

bvol ' We note t h a t eq. (31)

accounts f o r s i z e a b l e e f - %i -07 .32 .16 9.1 5.9 5.9 f e c t s introduced by t h e Pa-

u l i blocking and by dynamic "Zr .ll .175 .31 17.7 ll. 8.6 c o r r e l a t i o n s . To b e t t e r ap-

140e

p r e c i a t e t h e imprtance of .17 .075 .38 23.6 14.2 11.4 the l a t t e r e f f e c t i n t a b l e

4 we a l s o r e p o r t the values 2 0 8 ~ b .21 .029 .59 24.6 16.1 13.

f o r t h e r a t e obtained using the same value (88 MeV) f o r

t a b l e 4

the mcrnentum of the n e u t r i - _f

no, b u t ignoring the e f f e c t s of dynamic c o r r e l a t i o n s (see eq.(27) f o r S ) . One can see t h a t the e f f e c t of the dynamic c o r r e l a t i o n s is t o 1- t h e s t r e n g t h by-40%.

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.

Wiley and Sons, New York, N.Y.,1978.

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