One-particle transfer fragmentation in the 40Ca region

Volume 99B, number 2

PHYSICS LETTERS

12 February 1981

ONE-PARTICLE TRANSFER FRAGMENTATION IN THE 40Ca REGION G.G. DUSSEL

1,

R.P.J. PERAZZO 1 and S.L. REICH

Departamentode F{sica,Comisi6nNationalde EnergihAtdrniea, 1429BuenosAires,Argentina Received l0 December 1979

The one-body strength distribution is analyzed theoretically by calculating the residues of a renormalized one-body Green's function. The results provide a good description of the experimentally known features.

The odd neighbours of a doubly magic core have spectra bearing properties that in general depart from a pure single-particle pattern. This can be attributed to the admixture of fermionic and collective degrees of freedom. The Nuclear Field Theory [l] (NFT)provides a perturbative framework to study the excitation modes of a (normal) Fermi system. It is based upon three essential ingredients: (i) the introduction of elementary modes of excitation, both bosonic (collective) and fermionic; (ii) the use of a dimensionless expansion parameter, 1/~2, ~2 being the effective degeneracy of the valence levels; (iii) a precise prescription to evaluate the coupling of the different degrees of freedom in terms of a given two-body residual hamiltonian. In the present note we show that a consistent use of the first two ingredients already provides a reasonable understanding of the fragmentation of the one particle transfer strength. The spectroscopic factors for one nucleon transfer are the residues of a one-body Green's function G. If for G we use the (bare) Hartree-Fock (HF) propagator G O, all the strength is concentrated in only one state. The fragmentation into many levels of the same angular momentum and parity occurs when the HF particles couple to other (composite) states. This process gives rise to a renormalized propagator G, defined by the equation

G(jj'; t)= Go(J, t)fjj, - ~ f f dr dr' Go(J, "r) (1)

x 5~(/',/; T ' - O G 0 ( / ; t - r). In this calculation ~7represents the coupling of HF particles to a one-fermion plus one-boson ( l f l b ) state through a diagram in which a fermion line emits and subsequently absorbs a boson line [2]. This corresponds to the lowest order (1 ] ~ ) correction of a single particle state. The solution of eq. (1) with this kernel has been extensively discussed within schematic [2] and realistic situations [3]. The positive (negative) energy poles of G correspond to the eigenenergies of the states In) (IN)) of the A + 1 (A - 1) system, A being the number of particles of the doubly closed core. In the present note we use eq. (1) to describe the fragmentation of the one-nucleon transfer strength in the 4°Ca region. The choice of 4°Ca as a closed core is due to the abundant experimental evidence [4-6] showing in its odd neighbours sets of levels of the same ]'~ exhausting most of the one-particle sum rule. The input data to eq. (1) are the (bare) single particle energies ej, the particle-phonon coupling vertex functions, and the choice of the boson space. The ej's are obtained from the experimental excitation energies Ei (E N) of the A = 41 (39) nuclei, using the Baranger prescription [7] : 1

1 Fellow of the Consejo Nacional de Investigaciones Cientfficas y T~cnicas.

,

l

=

n

r(nlc,+10>12 , e +

0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company

N

1(Nasa0>126 N.

(2) 71

Volume 99B, number 2

PHYSICS LETTERS

The stripping (pick-up) spectroscopic factors I(n[ cf[ 0)l 2 (I (NI c/I 0>l 2) were also taken from experiment. The e]'s obtained with eq. (2) include all processes that determine the short time behaviour [5] of the one-body Green's function. HF contributions are included buL more generally, are contained all corrections implying diagrams in which the fermion line is linked in only one point to an otherwise unlinked set of interacting fermion and/or boson lines. These corrections leave unchanged the feature of the HF propagator concentrating all the strength in one state. We only considered the levels in which the sum rule

1=

i<.l%+lo>t 2 + n

i

c/o>l 2

Table 1 Collective and single-particle data used in the calculation.

E (MeV)

1-

5.90

EB__.___(ER;O~.)B(Eh, 0 --+ h)exp Sclass(h, r = 0) (e 2 fm 2k) -

0.05 a) 90 b)

2{

3.90

0.035

2~

5.63

0.005

2~

6.91

0.05

2~

(17)

~100 c)

8 b) 70 b) _

31 32

3.73 6.29

0.06 0.34

1.7 X 104 b) 0.2 X 104 c)

4+

6.50

0.05

0.26 X 103 d)

lpl/2 3.3 e) lP3/2 1.7 e) 0fT/2 - 0 . 4 0 e) 0d3/2 - 6 . 6 e) lsl/2 - 9 . 3 e) 0ds/2 - 13.4 e)

a) Extracted from r m [5]. b) Ref. [5]. c) Theoretical RPA value, d) Assumed transition rate. e) From refs. [5,6] and Baranger's prescription [71 (see the text).

72

the differences among the various DWBA analyses involved, and the fact that (3) has been imposed. One of the interesting features of the 40Ca region is the appearance of several weakly populated positive (negative) parity states above (below) the Fermi surface. These states have a significant contribution to the e/'s and are usually ignored by experimentalists in calculating weighted averages of measured single-particle energies. The boson space is defined with the low lying collective states of 4°Ca. The coupling of the ith nucleon to the nth mode of a ?vmultipole collective oscillation is given by:

(3)

N

was exhausted at least to 60%. Whenever it was possible, the (small) differences between neutron and proton single-particle data were averaged out to absorb Coulomb effects. Bare energies are given in table 1. The empirical spectroscopic factors thus obtained have an error that can be estimated to be ~30% lumping together the experimental error estimated in refs. [5,6],

k~

12 February 1981

Hcoup = - K x ~r.arn) " /.t l k./*(g2.)a! t A#

(4)

The intensity K~, is given by [8] : K~,= [47r/(2~ + 1)]

Mw2/A(r2X-2),

(5) (rS>=

:~

.

occ. orbits m

The fermion-boson vertex functions are related to the B(EX) value through: B(EX, 0 -+ X) = e2(2X + 1)2A2(X)/K 2.

(6)

The unknown B(E4) value was obtained by assuming that the 4 + state exhausts the same percentage of the classical sum rule as the 25. Pairing phonons and high lying isovector modes and positive parity particlehole excitations across two major oscillator shells are not relevant to the fragmentation within the experimentally explored region. When introduced in the calculation they produce negligible changes in the spectroscopic factors. The consistency of the "empirical" single-particle and collective data, was checked by inserting the value of K~, into an RPA calculation of the 3 - modes. The experimental energies and B(E3) values are correctly reproduced. The positive parity modes appear to involve large static quadrupole distortions and are predominantly of 2 p - 2 h and 4 p - 4 h nature. No RPA description of them was thus attempted. The coupling to the negative parity modes of 4°Ca ("backward diagrams" [3] ) causes a transfer of onebody strength across the Fermi level. The strong fragmentation of particle (hole)levels above (below)the Fermi level is instead due to the coupling to the 2 + and 4 + states of 4°Ca ("forward diagrams" [3] ). The

Volume 99B, number 2

PHYSICS LETTERS

d 3/2

E'

s 1/2

E

[M.V]

.

.

_f ii

f7/2 ® ~~

f7/2®37

"

f 7/2 ~3T

5-'~ f 7/2 ® 37

.i,- -''1

P 3/2 @ 3~"

O-

O-

-5-

-5.

f7/2®37

" '- ~

O-

.

=

5-

f 7/2 ® 3T

.

[MeV]

i~,

5- " - " " P 3/2®3i-

d 5/2

E

[MeV] f 7/2@32

I- .

12 February 1981

£ d 312

.

//d,/2 22

C S 1/2

I.-, . . . . . . . . . b

-10-

-10"

d 3/2 ® 2~"

I--,

d 3/2 ® 2

1 ti ,/2.2

I- . . . . . . I

~ - d 312® 2; .,,.../S 1/2 ® 2~

~d

"-'--d 3/2 ® 2~ -15" - . - - s 1/2® 2~

I.,.,,,

I.

,

h.,,,,

.I

-

Sl/2®21

-

s/2®2T

,

d

I.- . . . . . . .

, si/2®2~

-15-

~,- S I/2 @ 2+3

p,, 4

-

.01

V2 ® 2;

Imll,J

I.

I

hl..t

.I

d5/2 @2~

= l.-.ut

I

.01

- d 5/2@2~

~--

I.

hm..,

.I

.01

Fig. 1. Theoretical (dashed bars) and experimental [5,6] (full bars) spectroscopic factors for one-nucleon pick-up reactions on a 4°Ca target. The strengths (length of the bars) are plotted on a logarithmic scale. Spectroscopic factors < I% axe not drawn. Theoretical results of ~1% are indicated by stars. The active configurations within the energy range that is plotted are displayed at the right of each axis.

(small) amount of strength that in all cases is transferred across the Fermi level is properly accounted for by this calculation. The increasing fragmentation of the d3/2, s1/2, d5/2 levels is also properly described. The d5/2 case [6] is a consequence of the large density of 1fl b states in the neighbourhood of the (bare) singleparticle level t l . This is a remarkable example in which adding up an infinite series, as implied in the solution of eq. (1), is more adequate than perturbative treatment that is impaired by vanishing by small energy denominators. t-1 A similar situation is found with thejl5/2 in the 2°spb region [3,9,10].

The particle and hole states share the same property of showing a greater fragmentation farther away from the Fermi surface. The present model is however unable to match this feature for the Pl/2 and the less dramatic situation of the P3/2" This is basically because there are not enough configurations available in the proper energy range. The present calculation has no adjustable parameters and can nevertheless reproduce the essential characteristics of the one-body strength distribution. The theoretical interpretation hinges more on the available configuration space and the relative location of singleparticle and 1fl b states rather than on the residual two-body interaction or on the actual value of the par73

Volume 99B, number 2

PHYSICS LETTERS E

f7/2

12 February 1981

p3/2

E

E

[M~V]

[M'V] I

/2®4T / pl/ • 10 1 # P3/2~2 3 • -~/P3/2 ® q

pi/2

[rlev]

/ . P 1/2@2~ 10-

Pl/2 ® 2~ ~ P 3 / 2 ® 2~

10-

/

. p3/2®2;

• -- I"~'/P 3/2 ® 2~ ~-//f 7/2 ~ 2; • ~-~ 7/2 ~ t,~ 5 ~ ' P 3 / 2 ®2~

--f

~"" ~'-f

I

-10.~•

•

d5/2 ®3~

.I

.01

m/2

O-

-10-

. d 3/2@3T

~d

_..jd 3/2 ®I-

3/2 ® I-

-15 OI

~

I.

- c

I- . . . . . . . . .

__~d 3/2 ®3~

• ~ ~SI/2@3;

~.~,

I

"£ P 3/2

'm-~l,

jd3/2®3}

Sl/2 ®3~

2t

\f 7/2 ~ 2%

----4

~ d 3/2 ®3~"

.

f 7/2 ®

o_1_

~-

£ 'f 7/2

-15"

P3/2 @ 2T

5-

I

, ~ ~]~.

-10~

. p3/2 ®2~ , f7/2 @4;

7/2 ® 2~

I-N 0

@ 21

7/2 @ 2~

~,,H,

I.

J

~'IIII

.I

. sl/2®F --~- d 5/2 ®3~

I

~ S I / 2 @1d 5/2 ® 3( --

.01

h,.III

I.

I

h,ul,

.I

,

.01

Fig. 2. The same as fig. 1 but for one-nucleon stripping reactions on a 4°Ca target.

t i c l e - p h o n o n vertex functions. The procedure used in this calculation is also able to describe, within a weak coupling framework, strong fragmentations that are hard to explain within first-order perturbation theory. The situation prevalent in 40Ca may, however, be considered in the limit of applicability of the NFT procedures due to the occurrence of static deformations. All these results suggest a margin for practical use of the NFT prescriptions that is larger than the one foreseeable from formal arguments.

References [ 1 ] D.R. Bes, R.A. Broglia, G.G. Dussel, R. Liotta and B.R. Mottelson, Phys. Lett. 52B (1974) 253;

74

[2] [3] [4] [5] [6] [7] [8] [9] [10]

D.R. Bes, R.A. Broglia, G.G. Dussel, R. Liotta and R.PJ. Perazzo, Nucl. Phys. A260 (1976) 77. D.R. Bes, G.G. Dussel, R.P.J. Perazzo and H. Sofia, Nucl. Phys. A293 (1977) 350. R.P.J. Perazzo, S.L. Reich and H. Sofia, to be published. C.M. Lederer and V. Shirley, Table of isotopes, 7th Ed. (Wiley, 1978). P.M. Endt and C. van der Leun, Nucl. Phys. A214 (1973) 1, and references cited therein. P. Doll, G.J. Wagner, K.T. Knopfle and G. Mairle, Nucl. Phys. A263 (1976) 210. M. Baranger, Nucl. Phys. A149 (1970) 225. A. Bohr and B. Mottelson, Nuclear structure, Vol. II (Benjamin, Reading, MA, 1975) Ch. VI. S.L. Reich, H. Sof{a and D.R. Bes, Nucl. Phys. A233 (1974) 105. I. Hamamoto and P. Siemens, Nucl. Phys. A269 (1976) 199.