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Giant-resonance damping: semimicroscopic description

Dedicated to the 100-th anniversary of the birth of Academician Arkady Benedictovich Migdal


Giant-resonance damping: semimicroscopic description

Giant-resonance damping: semimicroscopic description
(metho ds, results, p ersp ectives)

M.G. Urin
National Research Nuclear University MEPhI, Moscow, Russia

Collab orators:

M.L. Gorelik, S.Yu. Igashov, S.E. Muraviev, V.A. Ro din, V.S. Rykovanov, I.V. Safonov, B.A. Tulup ov, and G.A. Chekomazov


Giant-resonance damping: semimicroscopic description Outline
Outline Intro duction General remarks Theoretical approaches Metho ds (schematically) Standard elements of the cRPA Nonstandard elements of the cRPA Phenomenological treatment of the spreading eect Practical realizations Implementations of the approach Previous results New results Persp ectives (p-h optical mo del) Basic equations Ways to implement The GR energy shift due to the spreading eect Summary


Giant-resonance damping: semimicroscopic description Introduction General remarks

1. General remarks
Being an universal phenomenon, GRs corresp ond to collective (p-h)-typ e excitations of low multip olarity (to sp ecic do or-way states (DWS)). Dierent combinations of the DWS quantum numb ers (

J , L, S , T , T3

) lead to a wide variaty of GRs.

A lot of observables (a collection is given by Harakeh and Van der Woude, Oxford, 2001) need for an unied theoretical description. In particular, it's concerned with GR damping. GR main relaxation mo des:
(i) distribution of the prop er p-h strength (Landau damping) is the result of the nucleus shell structure



(ii) DWS coupling to s-p continuum leads to direct particle decay and related phenomena (iii) DWS coupling to many-quasiparticle congurations (mqp c) leads to the spreading eect


Giant-resonance damping: semimicroscopic description Introduction Theoretical approaches
Very schematically:

= + + .
Actually, an interplay of main relaxation mo des, which is changed with increasing of excitation energy, takes place in the GR phenomenon. GR main prop erties:
(i) gross prop erties (strength distribution, transition density) are the integral characteristics; (ii) direct-decay prop erties (partial direct-nucleon-decay branching ratios and related phenomena) are the dierential characteristics.

2. Theoretical approaches RPA-based microscopic and semimicroscopic approaches are the most advanced in description of GR prop erties (a mean eld and p-h interaction are the input quantities).


Giant-resonance damping: semimicroscopic description Introduction Theoretical approaches

Landau damping + s.p. continuum



continuum-RPA

(cRPA). Shlomo and Bertsh, NPA'75.

Ç

No description of the spreading eect and partial particle decays.

cRPA+DWS coupling with a limited numb er of 2p-2h states. Kamerdziev et al., Phys. Rep'04.

Ç

No termalization of DWS that leads to interaction of dierent DWS via 2p-2h states in a contradiction with the statistical assumption.

Ç

No description of partial particle decays.

cRPA + a phenomenological treatment of the spreading eect (in spirit of an optical mo del)



outlined b elow

semimicroscopic description of GR damping. U., NPA'08 (a brief review), PAN'11 (in press).

Ç Ç

A statistical assumption is supp osed to b e valid. A metho d to describ e direct particle decays is develop ed and widely implemented.


Giant-resonance damping: semimicroscopic description Methods (schematically) Standard elements of the cRPA

1. Standard elements of the cRPA
The (lo cal) p.-h. Green function sp ectral expansion:

A(x , x1 ; )

taken in

co ordinate representation is the basic quantity and satises the

A(x , x1 ; ) =
s
Here,

(x1 )s (x ) (x )s (x1 ) s -s - s + i 0 + s - i 0 ^ ^ s (x ) = s |+ (x )(x )|0

.

s = Es - E

0 and

are the

excitation energy and transition density, resp ectively. the s.p. creation op erator. In the continuum region ( energy)

^ + (x )

is

> BN , BN

nucleon separation

the energy-dep endent transition density.

1 - ImA(x = x1 ; ) = |(x , )|2 ,

(x , ) = s (x )|

=s is


Giant-resonance damping: semimicroscopic description Methods (schematically) Standard elements of the cRPA
The strength function

SV0 ( ) =

s

^ |Vs 0 |2 ( - s )

for a lo cal

s.p. external eld (prob e op erator)

^ V=

V0 (xa ) = A:

^ ^ ^^ + (x )V0 (x )(x )dx V0

is

a determined by

1 + SV0 ( ) = - Im V0 [[AV0 ] ,
integration).

([...] means prop er

+ SV0 ( ) = | V0 ( ) |2

at

> BN

.

The Bethe-Goldstone integral equation for within the RPA the p-h interaction which leads to GR formation (

A(x , x1 ; ) contains F (x , x1 ) F (x ) (x - x1 ), |s |d DWS w.f.):

ARPA ( ) = A0 ( ) + [A0 ( )FARPA ( )] ,
where

A0 (x , x1 ; )

is the free lo cal p-h propagator.


Giant-resonance damping: semimicroscopic description Methods (schematically) Standard elements of the cRPA

Alternative formulation of the cRPA in terms of an eective eld (eective prob e op erator)

V (x , )

dened as

[V0 ARPA ( )] = [V ( )A0 ( )]
equation:

and satisfying the integral

V ( ) = V0 + F [A0 ( )V ( )],

so that

1 + SV0 ( ) = - Im V0 [A0 ( )V ( )]

and

1 1/2 (x , ) = - ImV (x , )/F (x )SV0 ( ).
This formulation is used within the Migdal's FFST.


Giant-resonance damping: semimicroscopic description Methods (schematically) Standard elements of the cRPA

Free p-h propagator Green functions

A0 (x , x1 ; )

is determined by a mean eld ers

via the b ound-state wave functions

Å (x ), single-particle g (x , x1 ; = Á ), and occupation numb

nÅ

with taking the full single-particle basis into account:

A0 (x , x1 ; )

=
Å

nÅ { (x )Å (x1 )g (x , x1 ; Å + ) Å (x1 )Å (x )g (x1 , x ; Å - )}; Å

+

(H (x )-Å )Å (x ) = 0; H (x )a

(H (x )-)g (x , x1 ; ) = - (x -x1 );

single-particle Hamiltonian.


Giant-resonance damping: semimicroscopic description Methods (schematically) Nonstandard elements of the cRPA

2. Nonstandard elements of the cRPA
Direct-nucleon-decay probabilities can b e describ ed with the use of an alternative expression for

SV ( )

at

> Bn |M

:

1 SV0 ( ) = - Im V + ( ) [A0 ( )V ( )] =
a kind of the optical theorem.

V0 ,Å

( )|2

Å

(+) =Å + (x )Å (x ) (+) b eing the free decay-channel wave function and >0 (x ) b eing the continuum-state wave function.

MV

0

,Å

= nÅ

1/2

Å,0 ( )V ( )

(+)

with



(+) Å,0

=

MV

0

,Å is the amplitude of a direct+semidirect (DSD)

nucleon-escap e reaction induced by the external eld

V0 (x ).


Giant-resonance damping: semimicroscopic description Methods (schematically) Nonstandard elements of the cRPA
Partial branching ratio for direct decay from the excitation-energy interval one-hole state

= 2

1 with p opulation of

Å-

1

of the pro duct nucleus is determined as:

bÅ ( ) =
( )
Unitary condition

|MV

0

,Å

( )|2 d /
( )

SV0 ( )d .
.

bÅ ( ) = 1
Å

is valid indep endently of

Alternative description of GR particle decay within the cRPA The scattering amplitude Denitions

(+) Å,0 result in equations:

[( )

(x , x1 ; ) is the [A( )F ] = [A0 ( )( )] and (+) ] = [F Å ( )]

basic quantity.

( ) = F + [FA0 ( )( )] and (+) (+) (+) Å ( ) = Å,0 + [A0 ( )F Å ( )]
(+) Å

( )the

eective decay-cannel w.f.


Giant-resonance damping: semimicroscopic description Methods (schematically) Nonstandard elements of the cRPA

Consequences:
(i) DSD-reaction amplitude:

M

V0 ,Å

( ) = [

(+) Å

( )V0 ];

(ii) S-matrix for nucleon-nucleus scattering: pt SÅ Å ( ) = SÅoÅ Å Å - 2 i Å Å ( ), (-) (+) Å Å ( ) = [Å ,0 F Å ( )]; (iii) unitary conditions: |SÅ Å ( )|2 = 1, or
Å

-

1

ImS

pot ÅÅ ÅÅ

( ) =
Å

|

ÅÅ

( )|

2

;

(iv) partial branching ratios:

bÅ ( ) =
( )

|

ÅÅ

( )|2 d
( ) Å

|

ÅÅ

( )|2 d

.


Giant-resonance damping: semimicroscopic description Methods (schematically) Nonstandard elements of the cRPA
Pole approximation for the main cRPA quantities is valid for low-energy (subbarrier) GRs at their maximum. DWS resonances are nonoverlapp ed (Rd

i ( ) ( - d + )- 2d

1

):

ARPA (x , x1 ; ) =
d

(x )d (x1 )Rd ( ); d

1 SV0 ( ) - Im Sd Rd ( ); = d 1/2 V (x , ) Sd vd (x )Rd ( ); =
d

1 1/2 Sd ( ,Å )1/2 Rd ( ) ( )| = d 2 d (x , x1 ; ) vd (x )vd (x1 )Rd ( ); = |M
V0 ,Å d

;

1 Å Å ( ) = 2


d

d ,Å

d

1/2 ,Å

Rd ( )

.


Giant-resonance damping: semimicroscopic description Methods (schematically) Nonstandard elements of the cRPA

Here,

d (x )

and

vd (x ) = F (x )d (x )

are, resp ectively, the DWS

transition density and transition p otential;

+ Sd = |[V0 d ]|2

the DWS strength;



(+) 2 d ,Å = 2 nÅ |[Å,0 vd ]| the DWS partial width for direct nucleon decay;

= d bÅ =

Å

d

,Å the total (escap e) width;

d

Sd ( ,Å / )/ d d

d

Sd

the partial branching ratio.

All these values can b e evaluated within the cRPA. For high-energy GRs (DWS resonances are overlapp ed) only

SV0 ( ), (x , ), Å Å ( ),

and

bÅ ( )

can b e evaluated.


Giant-resonance damping: semimicroscopic description Methods (schematically) Phenomenological treatment of the spreading eect
3. Phenomenological treatment of the spreading eect
In the p ole approximation the spreading eect on the energy-averaged GR characteristics can b e rst taken into account in terms of the DWS spreading width under reasonable statistical assumption:

^ ^ 2 d |H |m m|H |d m = dd . d
Here



m is the mqps density,

^ H

is a residual interaction.

Thus:

i i ? Rd ( ) Rd ( ) = ( - d + + ), d 2 2
d is the mean DWS spreading width. As a ? ? result, for the energy-averaged quantities SF ( ), MV ( ) we
where

=

get a sup erimp osition of the generally overlapp ed DWS resonances. The width calculations of



is adjusted to repro duce in

? SV ( )

the exp erimental total width. Then the

direct-decay prop erties (in particular, without use of free parameters.

? bÅ ( )

) are describ ed


Giant-resonance damping: semimicroscopic description Methods (schematically) Phenomenological treatment of the spreading eect
An alternative way for using the cRPA as a base for taking the spreading eect into account consists in the substitution

i + I (r , ) 2
directly in the cRPA equations for energy-averaged quantities

?? V, M

V . It means that the

quantities (Green function, wave function

?(+) =

Å

+

-dependent single-particle g (x , x1 ; Å Á ), continuum-state ? (x )) are determined by the

single-particle p otential having an imaginary part:

i H (x ) - (Å Á Á I (r , )) g (x , x1 ; Å Á ) = - (x - x1 ), ? 2 i ?(+) H (x ) - (Å + + I (r , )) Å + (x ) = 0. 2
This way allows us to take spreading eect into account in the same manner for low- and high-energy GRs. Actually, we use an eective



-dep endent single-particle optical-mo del p otential.


Giant-resonance damping: semimicroscopic description Methods (schematically) Practical realizations

4. Practical realizations
Phenomenological Landau-Migdal forces are taken as the p-h interaction (

C = 300

MeV fm ):

3

F (x1 , x2 ) = C {(f (r1 ) + f 1 2 ) + (g + g 1 2 )1 2 } (r1 - r2 ).
The dimensionless intensities are taken as follows:
(i) the isoscalar strength f (r ) from the translation-invariance - conditions (the 1 spurious state has the zero energy and exhausts almost the total isoscalar dip ole EWSR); (ii) the isovector strength from the isospin-selfconsistent description of the mean-eld isovector part; (iii) the spin-isovector strength (iv) the spin-isoscalar strength

g g

from the exp erimental

Gamow-Teller resonance (GTR) energy;

0

wasn't used in

implementations of the semimicroscopic approach


Giant-resonance damping: semimicroscopic description Methods (schematically) Practical realizations

A partially self-consistent phenomenological mean eld contains isoscalar, isovector, and Coulomb parts:

U (x ) = U0 (x ) + U1 (x ) + UC (x ),

with

U0 (x ) = U0 (r ) + U

SO

(r ),

U0 (r ) = -U0 fWS (r , R , a), R = r0 A1/3 , USO (r ) = USO U1 (x ) = 1 v (r ) 2
(3)

1 dfWS ls, r dr

,

UC (x ) =

1 (1 - 2 (r )

(3)

)UC (r ).

The symmetry p otential neutron-excess density

v (r ) = 2Cf n
(-)

( -)

and the mean

Coulomb eld are calculated selfconsistently via the

n

= nn - n

p

and proton density

resp ectively. Only ve phenomenological parameters (

U0

,

np f,

,

USO , r0 , a

) are used and adjusted to repro duce the

exp erimental single-quasiparticle sp ectra in doubly-closed-shell nuclei. These parameters determine also the values

f

ex

and

f

in

in the radial dep endence of the isoscalar p-h strength

f (r ).


Giant-resonance damping: semimicroscopic description Methods (schematically) Practical realizations
The spreading parameter (the imaginary part of an eective optical-mo del p otential) is parameterized similarly to some versions of the single-particle optical mo del:

I (r , ) = I (r )I ( ), Ivol (r ) = fWS (r , R , a),
or

where

I

surf

(r ) = -4a

dfWS . dr

Using the volume absorption we revealed the saturation-like energy dep endence of interval:

I (r , )

from description of the

exp erimental total width for dierent GRs from a wide energy

I

vol

( ) = vol vol 0.125

( - )2 , 1 + ( - )2 /B 2
MeV

where

the intensity

-1

, the gap parameter

=3

MeV, the saturation parameter charge-exchange GRs the nal nucleus

B=7

MeV. In description of

is replaced Ex = - Q .

by the excitation energy of

No adjusted parameters in description of the GR direct decays!


Giant-resonance damping: semimicroscopic description Implementations of the approach Previous results

1. Previous results (referred and partially describ ed in U., NPA'08, PAN'11) are related to direct-nucleon-decay prop erties of various isoscalar (T

= 0)

and isovector (T

= 1)

GRs. the results are mainly

concerned with the GRs studied exp erimentally.
(i) ISGMR (partial proton widths), ISGDR (partial direct-proton(neutron)-decay branching ratios); (ii) charge-exchange ( IVGSDR

T3 = -1

(-)

and IVGSMR

) GRs: GTR (partial proton widths), (-) (partial direct-proton-decay

branching ratios). IAR (partial proton widths have b een analysed without taking the isospin-forbidden spreading eect into account); (iii) the isovector

T3 = 0

GRs: photo-absorption, DSD

photo-neutron and inverse reactions accompanied by excitation of the IVGDR and IVGQR, IVGDR2the overtone of the IVGDR (partial neutron and proton branching ratios).


Giant-resonance damping: semimicroscopic description Implementations of the approach Previous results

In the main, satisfactory description of the available exp erimental data has b een obtained. Some results are under revision due to the necessity to satisfy the exp erimental conditions known later and/or to improve the metho ds.


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

2. New results
2.1 Coulomb description of the main relaxation parameters of the IAR and IVSGMR U., PAN'10.
The IAR prop erties are closely related to the isospin SU(2)-symmetry in nuclei. This symmetry is weakly broken mainly by the mean Coulomb eld

(-)

(IAR overtone). Gorelik, Rykovanov,

UC (r )

. To avoid spurious

contributions, we describ e the main relaxation parameters of (-) IAR (and also of IVGMR ) in terms of UC .


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

Starting relationship

^^ [H , T

(-)

^ (-) ] = UC

^ (T

(-)

=
a



(- ) a

,

^ (- ) UC =
a

U (ra )

(-) a

),

which is valid within the RPA under the afore-mentioned (-) self-consistency condition v (r ) = 2F n (r ), allows to get the basic equation:

S F ( ) =
Here,

S C ( ) . | - C |2

(-)

is the Fermi strength function corresp onding to (-) -) ; SC ( ) is the Coulomb strength function (-) (-) corresp onding to VC ,0 (x ) = VC (r ) , where VC (r ) = UC (r ) - C is the variable part of UC .

V

F ,0

SF ( ) (x ) = (


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

Being also valid for the energy-averaged quantities, the basic equation can b e parametrized as (exp eriment!):

1 SA A ? S F ( ) = 2 ( - A )2 + 1 4
with

2 A

A b eing, resp ectively, the IAR Fermi strength, energy, and total width.

S

A

(N - Z ), A

, and

? A = + A

From comparison of the ab ove equations we get:

C = -
A and

i A , 2

and

A = 2 S

-1 ? (-) A SC

( = A ).

The nonlinear equation for



A (!) can b e solved provided the

A values are found from cRPA description of SF ( ). 2 Very schematically: A = MA M ( = A ), where M ( ) is (-) the IVSGMR total width, MA is the amplitude of the (-) Coulomb mixing T> -IAR and T< -IVGMR .

S


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

Within the approach (Section 2) the IAR partial width for -1 direct proton decay into the neutron one-hole state Å of the pro duct nucleus is determined by the reaction amplitude

? M ?

VC

(-)

,Å

( )

:

A,Å

-? = 2 SA 1 |M

VC

(-)

,Å

( = A )|2 ,

? M

VC

(-)

,Å

1 ?(+) ? = nÅ/2 [Å,0 V

(-) C

].

Finally, the calculated spreading width

= A - A
Å

?

A,Å

(tends to zero without absorption) and the reduced partial proton widths

?

A,Å

? = SÅ

A,Å

P Å(

exp

)/P Å( = Å + A ) Å
-1

(SÅ the sp ectroscopic factor of the

state,

PÅ ()



p otential-barrier p enetrability for protons) can b e compared with the corresp onding exp erimental data (Table 2).


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

The Coulomb strength function

? (-) SVC ( )

exhibits an almost
(-)

single-level resonance corresp onding to the IVGMR

(Fig. 1). Only one sp ecic parameter S (taken to describ e (-) the exp erimental IVGMR total width) is used for () description of the IAR and IVGMR main relaxation mo des. The contribution of the spreading eect in formation of the (-) high-energy IVGMR is relatively low (Fig. 1). It leads to the large (ab out 0.5) total direct-proton-decay branching ratio for this GR.


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

Figure 1


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

Table 1. The calculated and exp erimental IVGMR
(-) M , exp (-) (-) M , exp

()

parameters
(-)

Nucleus
90 Zr 140 Ce 208 Pb



M





(-) M

S
0.06 0.11 0.09 -1 (MeV )

btot

34.6Á2.9 35.4Á3.5 37.2Á3.5 (MeV)

36.6 37.8 39.7 (MeV)

18.9Á4.1 16.6Á4.2 15.0Á6.0 (MeV)

18.8 18.6 15.8 (MeV)

73 41 51 (%)

Nucleus
90 Zr 140 Ce 208 Pb

M

(+) , exp

M

(+)



(+) M , exp



(+) M

22.0Á2.0 19.9Á2.4 12.2Á2.8 (MeV)

19.8 17.3 14.4 (MeV)

15.0Á2.1 8.5Á4.5 11.6Á7.1 (MeV)

8.0 8.0 3.4 (MeV)


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

Table 2. The calculated and exp erimental IAR-decay parameters



,

207

Pb

S

, exp



exp

+ A
11.12 10.35 9.89 7.26 (MeV)



A, , exp



A,



A, exp



A

3p1/2 2f5/2 3p3/2 2f7/2

1.0 0.98 1.0 0.7

11.46 10.91 10.59 9.15 (MeV)

51.9Á1.6 26.4Á2 64.7Á3.4 4.2Á0.6 (keV)

51.5 17.7 64.0 3.1 (keV) (keV) (keV) 78Á8 71

Nucl.
91


2d5/2 2f7/2 2g9/2

S

, exp



exp


4.20 9.55 14.06 (MeV)



A, , exp

A

,



A, exp



A

Zr 141 Ce 209 Pb

0.89 0.80 0.78

4.67 9.68 14.83 (MeV)

4.0

Á0.5

0.93 11.9 20.3 (keV)

20Á2 51Á4 75Á7 (keV)

21.6 47.7 84.5 (keV)

11Á1 22.7Á0.6 (keV)


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

2.2 Particle decay of the IVGSMR (phys), '09.

()

. Safonov, U., Bull. RAS.

In accordance with the p-h structure, the GTR and (-) IVGSMR are the spin-ip partners of the IAR and (-) IVGMR , resp ectively. In connection with the rather unexp ected exp erimental data (Zegers et al., PLB'03), we (i) slightly revised previous calculations of the partial direct-proton-decay branching ratios (-) for the IVGSMR (Tables 3,4), (ii) evaluated the partial (+) direct-neutron-decay branching ratios for the IVGSMR (Table 4). Results: (i) a contradiction with the exp erimental data of Zegers et al. concerned with the partial (not total)
(-) direct-proton-decay branching ratios for the IVGSMR in 208 Bi still remains; (ii) found in calculations a large total (+) direct-neutron-decay branching ratios for the IVGSMR can

b e apparently observed (Harakeh, private communication).


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

Table 3. Calculated partial branching ratios (in %) for the direct proton (-) 208 decay of the IVGSMR in Bi. The results are given for the decays from the excitation-energy range

Ex

31

50 MeV. The exp erimental data are

taken from Zegers et al., PLB'03

Å

-1

3p1/2
1.44

2f

5/2

3p3

/2

1i13/2
16.37

2f

7/2

1h9/2
4.33

1h11/2
8.85

theo bÅ r exp bÅ

3.93

3.22

6.97

13 Á 5 3s
1/2

22 Á 8 bÅ
54.62

Å

-1

2d3/2
1.77

2d5

/2

1g

7/2

1g

9/2

theo bÅ r exp bÅ

1.16

3.55

2.02

4.01

17 Á 8

52 Á 12


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

Table 4. Calculated partial branching ratios (in %) for the direct neutron (+) 208 decay of the IVGSMR in Tl. The results are for the decays from the excitation-energy range

E

x

9

14 MeV.

Å

-1

3s1

/2

2d3/2
2.66

1h11/2
32.82

2d5/2
4.76

1g7

/2

1g9

/2

2p1/2
0.14

2p3/2
0.13

bÅ
46.06

bÅ

1.49

3.26

0.80


Giant-resonance damping: semimicroscopic description Implementations of the approach New results
2.3 Photoabsorption and DSD neutron radiative capture cross sections. Tulup ov, U., EMIN-2009 (Moscow), ICNP-2010. An unied semimicroscopic description of these quantities in the IVGDR region is p ossible . Because the p-h interaction isovector strength

f

is xed, we add the isovector (separable)

momentum-dep endent forces

F
section

m -d

(x1 , x2 ) =

1 k (1 2 )(p1 p2 ) mA

to repro duce in calculations of the photoabsorption cross

a ( )

the exp erimental IVGDR energy.

16 3 e 2 1 (3) S (3) ( ), V0 (x ) = - (3) rY1 . 3 c V0 2 int int In such a case a = (a )TRK (1 + k ). The absorption exp intensity is adjusted to describ e the a ( ) shap e line (Fig. The partial DSD (n )-reaction cross sections, calculated a ( ) =
agreement with the corresp onding exp erimental data.

2).

without the use of free parameters (Fig. 3), are in satisfactory


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

Figure 2


Giant-resonance damping: semimicroscopic description Implementations of the approach New results

Figure 3


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) Basic equations

1. P-h optical mo del for phenomenological description of the spreading eect on (p-h)-typ e excitations at arbitrary (but high enough) excitation energies. U., PAN, '10; arXiv: 1005.2349v2 [nucl-th], PAN'11 (in press).
i + 2I

The shortcomings of the

metho d. It is (i) physically

clear but not-well-grounded enough; (ii) valid in the p ole approximation, i.e. only in the GR energy region (in implementations far low- and/or high-energy tails are the subject of interest). An analogy with formulation of the single-particle optical mo del has b een used in formulation of the p-h optical mo del. Below we consider the basic equations and ways for practical realization of the mo del.


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) Basic equations

The basic quantity is the p-h nonlo cal Green function

A(x , x ; x1 , x1 ; ),

whose sp ectral expansion is similar to the

afore-shown expansion for the lo cal Green function

A(x , x1 ; ) = A(x = x ; x1 = x1 ; ): (x , x )s (x1 , x1 ) (x1 , x1 )s (x , x ) s -s - s + i 0 + s - i 0
b eing the transition matrix allows us to realize the

A(x , x ; x1 , x1 ; ) =
s
with

density. This sp ecic feature of

^ s (x , x ) = s |+ (x )(x )|0 A

statistical assumption in description of the spreading eect. The strength function for a (nonlo cal) external eld

1 + SV0 ( ) = - Im[V0 A( )V0 ].

V0 (x , x )

:


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) Basic equations

Sp ecic quantities
(i) the free p-h nonlo cal Green function mean eld:
(0) s s = - , s (0) s

A0

is determined by a

= (x ) (x ), [

(0) s

, s ] = ss

(0)

(!)

;

(ii) the RPA nonlo cal Green function (nonlo cal) p-h interaction

ARPA is F (x , x ; x1 , x1 ):

determined also by a

ARPA ( ) = A0 ( ) + [A0 ( )F ARPA ( )],
within the discrete RPA (dRPA):

s d ,
lo cal p-h interaction:

s d , A

[ , d ] d



dd

;

(iii) the lo cal p-h Green function

(Sect. 2.1) is determined by a

F (x , x ; x1 , x1 ) F (x , x1 ) (x - x ) (x1 - x1 ).


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) Basic equations
An additional p-h interaction



( x , x ; x1 , x1 ; )

takes place is a sharp

due to DWS coupling to mqps. Having high-density p oles corresp onding to virtual excitations of mp qs,



? ?

function of

( ) =



and, therefore, only the ( + iJ ) (J -1 ) can be m

energy-averaged quantity reasonably parametrized:

(x , x , x1 , x1 ; ) = {-i W (x , x ; ) + P (x , x )} (x -x1 ) (x -x1 ).
-functions here are due to a large

The

( pF )

momentum

transfer at decay DWS to mqps. P-h optical-mo del basic equations

? A( ) = ARPA ( ) + [ARPA ( )

? A( ) = A0 ( ) + [A0 ( )(F 1 +? ? SV0 ( ) = - Im[V0 A( )V0 ] W ( ). P ( )
is determined by

()A()] ?? + ( ))A( ??

or

)],

can b e formally solved, using a phenomenological value of

W ( )

via a prop er disp ersive

relationship (Sect. 4.2).


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) Basic equations

The statistical assumption can b e simply realized, considering the dRPA+p ole approximation as an example. This assumption is valid under the conditions:
(i)

?

is nearly constant within the nucleus volume, i.e.

-i W (x , x ; ) + P (x , x ; ) -i W ( ) + P ( ),
(ii) dRPA transition matrix densities

d (x , x )

are orthonormalized.

In such a case, the solution of the B.-G. eq. is:

? A( ) = ARPA ( + i W ( ) - P ( )),
and the strength function is a sup erimp osition of DWS resonances:

1 ? SV0 ( ) = - Im

+ |[V0 d ]|2 ( - d + i W ( ) - P ( ))-1 . d


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) Ways to implement
The alternative presentation of the basic eqs. (corresp onds to the mo del of interacting and damping quasiparticles):

? ? ? ? A( ) = A0 ( ) + A0 ( )F A( ) , ? A0 ( ) = A0 ( ) + A0 ( )
The solution of the last eq.:

where

?

? ( )A0 ( ) .

? A0 (x , x ; x1 , x1 ; ) =


? (x ) (x ) (x1 ) (x1 )A ( ),

where

? A ( ) = W
,

n - n - - + (n - n )(i W ( ) - P ( ))W


,
,

(x ) (x )W (x , x ) (x ) (x )dxdx s (x , x ) = s (x , x ) . ? (fWS )ÅÅ fÅ , W = f f .
(0) (0)

statistical assumption (!),

W (x , x ) fWS (r )fWS (r ),

,


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) Ways to implement
After coming to the lo cal limit, i.e.

? ? A0 ( ) A0 ( ),

? ? A( ) A( ),

F F,

SV0 ( ) SV0 ( ),

we get the mo died dRPA eqs. (with taking the spreading eect into account) which are valid at an arbitrary (but high enough) excitation energy. Two-level mo del

1 > 2 ,

R

1,2

(r ),

j1 1

j2 =

1 2 -

1, L 1

.

1 L SV0 ( ) = - ImSd
DWS energy,

- d +

i 2 d

( )

i + d - 2 ( ) d

strength,

spreading width:

d = 1 - 2 + F [

22 R1 R2 ],

L Sd

=

L [R1 V0 R2 ]2 ,

( ) = 2W ( )f1 f2 d


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) Ways to implement

From dRPA to cRPA



(x ) (x1 ) - Å - + i W ( )fÅ f

g (x , x1 ; Å + ), ?

{h(x ) - (Å + - i W ( )fÅ fWS (r ))} g (x , x1 ; Å + ) = - (x -x1 ). ? Å (x1 ) (x ) Å - Å - + i W ( )f fÅ

g (x1 , x ; - ), ?

Å

{h(x1 ) - ( - + i W ( )f f

WS

(r1 ))} g (x1 , x ; - ) = - (x1 -x ). ? Ç ir ?

g ? ?

reg


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) Ways to implement
The cRPA basic quantity into account:

? A0 (x , x1 ; )

that takes the full p-h basis

? A0 (x , x1 ; ) =
Å

nÅ (x )Å (x1 )g (x , x1 ; Å + ) + ? Å n (x1 ) (x )g (x1 , x ; - ) + ?


+ +
Å

nÅ n (x )Å (x1 ) (x1 ) (x ) Ç Å 2i W ( )f fÅ . 22 ( - Å + )2 + W 2 ( )fÅ f

Ç

Def.:

? ? ? ? ? ? [A( )V0 ] = [A0 ( )V ( )] V ( ) = V0 + [F A0 ( )V ( )];

1 1 +? +? ? ? SV0 ( ) = - Im[V0 A( )V0 ] = - Im[V0 A0 ( )V ( )];


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) Ways to implement
Nucleon-nucleus resonance scattering and DSD-reactions Def.:[A(F

+



)] = [A0 G ], G ( )

nonlo cal scattering amplitude;



(Á) Å,0

( ) =

(Á) =Å + (-) ,0

(x )Å (x1 )nonlo
(+)

cal free decay-channel w.f.;

GÅ

,Å

( ) = [Å
(+) Å,0

G Å,0 ]

resonance-scattering matrix elements.

Def.:[G

] = [(F +
(+)



(+) )Å ],



(+) Å

( )eective

d.-ch. w.f.

? Basic eqs.:Å

?(+) ? ?(+) ? ?(+) ? = Å,0 +[A0 F Å ] (+) = 0 [1-A0 F ]-1 ,

(+) ?(+) ? ?(+) Å,0 = Å,0 +[A0 Å,0 ],

?(+) ?(+) Å,0 = =Å
WS

+

(x )Å (x1 )mean
Å

free w.f.

{h(x ) - (Å + + i W ( )fÅ f

?(+) (r ))}=

+

(x ) = 0.


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) Ways to implement

? MV

0

,Å

?(+) ?(+) ? ( ) = [Å , V0 ] = [Å,0 V ].

Lo cal-limit basic relationships:

? ? ? V ( ) = V0 + [F A0 ( )V ( )],

1 +? ? ? ? SV0 ( ) = - Im[V0 A0 ( )V ( )], MV
Consequences:

0

,Å

?(+) ? ( ) = [Å,0 V ( )].

(i) a semimicroscopic mo del for description of the main relaxation mo des of high-energy (p-h)-typ e excitations is prop osed and formulated for practical implementations; (ii) a conrmation is obtained for the
i + 2 I method of evaluation of the GR main relaxation parameters.


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) The GR energy shift due to the spreading eect

2. The GR energy shift due to the spreading eect. Tulup ov, U., PAN, '09. The analytical prop erties of b e presented in the form:



A and are similar ( ) = ( )- (0)

and the latter can with



(x , x ; x1 , x1 ; ) =
m

vm (x , x )vm (x1 , x1 ) vm (x1 , x1 )vm (x , x ) - - m + i 0 + m - i 0

.

Here and



vm (x , x ) is the ( = 0) = 0.

transition matrix p otential for a mqps

|m

,

Supp osing the energy-averaged quantities can b e factorized, i.e.

? Im ( )

and

Re ( ) ?

(x , x ; x1 , x1 ; ) = W (x , x )(-i W ( ) + P ( )) (x - x1 ) (x - x1 ), one can get the disp ersive relationship for P ( ):

?


Giant-resonance damping: semimicroscopic description Perspectives (p-h optical model) The GR energy shift due to the spreading eect

P ( ) =

2 P .V .



W ( )
0

2 d , ( 2 - 2 )
taken in the simplest

which formally takes the full basis of mp qs into account. For the saturation-like energy dep endence of form:

W ( )

W ( ) = 1 2 (1 + 2 /B 2 ) 2 P ( ) =

-1 ones get:

B 22 ln . 2 + 2 B B
i + 2 I ( )
metho d (Sect. 2.4). In

The analytical expression can b e also obtained for the dep endence

W ( ) = 1 I ( ) 2

used within the

such a case the metho d is transformed in the ab out 1.52.0 MeV for high-energy GRs.

i + 2 I ( ) - P ( )



one. This transformation leads to the GR energy shift,which is


Giant-resonance damping: semimicroscopic description Summary

A semimicroscopic approach to description of giant-resonance damping is presented. The basic p oints of the approach are the continuum-RPA and the phenomenological treatment of the spreading eect directly in the cRPA equations in terms of an eective s.p. optical-mo del p otential. The main advantage of the approach is the p ossibility to describ e direct-decay prop erties and related phenomena for various GRs. The approach is realized in many implementations with the use of the Landau-Migdal particle-hole interaction and phenomenological partially selfconsistent mean eld.


Giant-resonance damping: semimicroscopic description Summary

An attempt is undertaken to take phenomenologically the spreading eect into account within a RPA-based particle-hole optical mo del for arbitrary (but high enough) excitation energies. The mo del is formulated for practical implementations to describ e the main relaxation parameters of high-energy particle-hole-typ e excitations.


Giant-resonance damping: semimicroscopic description Summary

Many thanks for attention!