Spectroscopy of mixed-parity configurations (pseudo-nuclei)

Nuclear Physics A120 (1968) 273--284; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from tho publisher

S P E C T R O S C O P Y O F MIXED-PARITY CONFIGURATIONS (PSEUDO-NUCLEI) RAJ K. G U P T A and L. E. H. T R A I N O R

Department of Physics, University of Toronto, Toronto 5, Canada Received 6 May 1968 Abstract: A pseudo-nuclear model of mixed-parity configurations lp½ and ld~ has been analysed for both degenerate and non-degenerate cases. It is found that concealed configuration mixing can occur provided non-degeneracy is allowed in the model. A comparison with earlier pseudonuclear models suggests that the contribution of core excitation to low-lying energy levels is more pronounced when only two-particle-two-hole excitations occur than when one-particleone-hole excitations also make significant contributions. In principle, a systematic study o f inelastic electron scattering is shown to be a sensitive method for detecting the configuration interactions in both even- and odd-parity configurations.

1. Introduction The shell model was introduced less than two decades ago. In the earlier years, it achieved many notable successes in the low-energy spectroscopy of nuclei near closed shells. In recent years, however, improved experimental techniques, e.g., the development of high voltage Van de Graaff and solid state counters, have revolutionized nuclear spectroscopy and greatly increased the demand for more sophisticated theoretical models and better wave functions. The old shell-model calculations, which were based on a single active shell and which treated the nuclear core as inert, were clearly shown to be unsatisfactory by the appearance of low-lying negative-parity states, particularly in doubly even nuclei. In more recent years, very sophisticated shell-model programs and particle-hole theories have been developed in order to treat the nuclear structure problem on a more realistic basis. Notwithstanding these new developments, an interest in the old-fashioned shellmodel calculations persists. In part, it is motivated by the general complexity of the problem and the desire to push the simple models as far as possible; in part, it is also motivated by the desire to better understand the successes and limitations of the older and simpler calculations with a view toward effective treatment of their more complex counterparts. More specifically, one can ask to what extent the successes of the older treatments were more apparent than real. Perhaps, the answers have also some bearing on the conclusions which are drawn from the successes and failures of the more recent developments. In view of these general considerations, Cohen, Lawson and Soper ~) (hereafter referred to as CLS) decided to critically test the shell model by replacing actual nuclei by pseudo-nuclear models, i.e. models capable of exact solution. Specifically, 273

274

R. K . G U P T A A N D L. E. H . T R A I N O R

they chose neutrons filling two degenerate subshells isolated from all other singleparticle states; the properties of these model nuclei were compared with the results of shell-model calculations using a truncated configuration space. In their first investigation, they considered "pseudonium or Ps isotopes" (pseudo-nuclei consisting of neutrons only filling degenerate ld~ and lf~ single-particle states). Later Lawson and Soper 2) extended their study to both neutrons and protons filling degenerate lp~r and ld~ single-particle levels. (The latter study, hereafter referred to as LS, has the disadvantage that it does not possess even a loose correspondence with any region of actual nuclei.) In both cases, it was found that the low-lying levels of appropriate parity have spins consistent with the pure shell-model configurations ((f~)~-4 in one case, (d~r)N-2 in the other). In addition many other properties, such as binding energies relative to closed shells, many B(E2) values, M1 transitions and single-nucleon spectroscopic factors, were shown to be remarkably insensitive to the large amount of configuration mixing (induced by residual forces between nucleons in degenerate levels). They interpreted these results to indicate that large configuration mixing (of the order of 50 ~o or more) can be concealed by an effective two-body potential chosen to optimize the shell-model calculation. Recently, the present authors 3) extended the method of CLS to 2s~r and ld~ states (so-called pseudoneon or Pn isotopes) in both degenerate and non-degenerate models. In contrast to the results obtained in the CLS and LS models, the calculations for the s--d shell showed that the configuration mixing cannot be concealed. When the configuration mixing is large (degenerate st and dt), the level ordering given by the shell model is incorrect (even the first excited state is assigned the wrong angular momentum!). If, on the other hand, the s~ level is shifted downward relative to the d~ level so as to achieve the correct level ordering, the configuration mixing is reduced to negligible proportions. Furthermore, our study showed that the Coulomb excitation form factors for inelastic electron scattering are very sensitive tests of configuration mixing, particularly in the neighbourhood of the pure configuration diffraction minima. This sensitivity (to configuration mixing) of the widths and presumably also of the heights of the diffraction peaks led us to invent the term "configuration interaction spikes" for their description. The purpose of the present paper is to investigate whether or not this contrast between the results of Cohen, Lawson and Soper and our own has its origin in the fact that they used levels of opposite parity differing in orbital angular momentum by one unit, whereas we used even-parity levels differing in orbital angular momentum by two units. To achieve this purpose with a model somewhat relevant to actual nuclear species and bearing a close correspondence with our previous work, we consider neutrons only filling lp~r and ld~ subshells (pseudo-oxygen or Px isotopes) in both degenerate and non-degenerate circumstances. The model itself is discussed briefly in sect. 2. Our calculations on Px isotopes (both shell model and exact) are presented in sect. 3, while sect. 4 is concerned with the results on Coulomb excitation form factors. Our conclusions are summarized in sect. 5.

MIXED-PARITY CONFIGURATIONS

275

2. The model We consider an exactly soluble model of only neutrons in the single-particle levels lp½ and ld~ assumed isolated from all other single-particle levels. The residual neutron-neutron interaction is taken to be the usual spin-dependent potential Vnn ---- [¼Vs(1--al" er2)+¼Vt(3+a," a2)] exp ( - r 2 / r ~ ) ,

(2.1)

where V, and Vt are the singlet-even and triplet-odd strengths and r0 the range of the Gaussian potential. With this residual two-body interaction, the matrix elements are then evaluated *) using the single-particle harmonic oscillator wave functions ~k oc exp

-

,

(2.2)

where r~ is the radial extension parameter. In actual calculations, the two radial parameters ro and r~ occur ortly in the ratio 2 = ro/r ~ which is thert referred to as the range parameter. The number N of neutrons contained in the two levels runs from two to eight. Since N = 2 fills the p-shell completely, we shall refer to our model nuclei as pseudo-oxygen isotopes 16px to 22Px. We first solve this model exactly and compare the resulting "pseudo-experimental" data or "exact" data with the data obtained from the usual shell-model approximation in which the p~ core is considered inert. The three adjustable parameters coming into the calculations are the strengths V~ and Vt of the potential and the range parameter 2. We have performed our calculations in two different possible ways. First, keeping the strengths Vs and Vt of the potential fixed, we have obtained spectra for several values of the range parameter 2. This enables us to determine a value 2erf (effective range) which gives the best fit between pseudo-experiment and shell theory. As a variant of this procedure, we also try fixing the value of 2 at 0.6 and varying the potential strengths in order to find the effective values of V~ and Vt which give the best fit. Only central forces are considered [this point was discussed in our earlier paper a)]. The Hamiltonian incorporating the different choices of the parameters of the potential is then diagonalized for the various values of spin J and for all neutron numbers N between 2 and 8. The calculations are carried out on the IBM7094 using a shell-model program developed by the authors. 3. The spectra The pseudo-experimental data for all even- and odd-parity states in the pseudonuclei 16px to 22px were first obtained by considering the two levels lp~ and ld~ of

our model to be degenerate. The calculations were carried out for many choices of the potential parameters; for the first method in which the range parameter 2 is considered variable, we have calculated the spectra for three choices of the potential strengths (i) Vs = - 3 0 MeV, Vt = - 1 0 MeV, (ii) Vs = - 2 0 MeV, Vt = - 8 MeV,

276

R. K. GUPTA AND L. E. H. TRAINOR

and (iii) Vs = - 3 0 MeV (Serber force); for the second method in which the range parameter is fixed at 2 = 0.6, we have varied Vt from - 2 0 to 0 MeV keeping Vs fixed at - 3 0 MeV and then varied V~ from - 4 0 to - 1 5 MeV for fixed Vt = - 8 MeV. We find from our calculations that the ground states of all the even- and oddN nuclei have spin and parities consistent with the pure shell-model configuration (d~) g-2, viz. J = 0 + for even-N nuclei and J = }+ for odd-N nuclei as shown in table 1. This result is the same as obtained for all the earlier models. However, TABLE 1 G r o u n d s t a t e s o f P x i s o t o p e s f o r A = 0.6, V8 = - - 3 0 M e V a n d Vt = - - 1 0 M e V

N

Configuration w.r.t. x6Px

Spin-parity J~

Percentage content of configuration (p.l.)S(d~)N-s

Percentage probability of the excitation of core (p~)~ 62.5

2

(p~r) ~

0+

37.5

3

d~

~+

42.0

58.0

4

(dl.)z

0+

49.7

50.3

5 6 7

(dt)a (d~)4 (d~p

~+ 0+ t+

57.3 61.0 100

42.7 39.0 0

8

(d~) s

0+

100

0

as to the order of spins (for N > 3) for low-lying levels of appropriate (i.e. positive) parity, we find that a different situation occurs. Whereas lSpx gives (see fig. 3) the level order 0 +, 2 +, 4 + . . . . independent of the choice of force parameters, which matches the order that would be obtained from the pure shell-model configuration (d~) 2, the isotopes 19px and 2°px show a significant dependence on the choice of parameters of the two-body force. Fig. 1 shows the level spectrum for 19px calculated as a function of the range parameter 2 using the potential strengths Vs = - 30 MeV and Vt = - 1 0 MeV. We notice that the order of positive-parity states for (say) 2 = 0.6 is ~+, ½+, ~+ and ~+, which goes over to the order ~+, ½+, ~+ and ~r+ for 2 ~> 0.85 (the latter one matches the order ~+, ~+, ~+ obtained from the shell-model configuration (d~)3). In view of this result, we have also studied a non-degenerate "exact model" of the Px isotopes. Separating the two levels by a small energy gap A = 0.5 MeV ensures the right order for correspondence with the shell-model results as shown in fig. 2. As regards the configuration mixing, the ground state now becomes 78.4 % pure for 2 = 0.6 (Vs = - 3 0 MeV, Vt = - 1 0 MeV) compared to a purity of only 57.3 for the degenerate case. We emphasize again at this point that whereas the degenerate CLS and LS models gave the spins of the low-lying states of appropriate parity consistent with shell-model configurations, the degenerate Pn model did not.

MIXED-PARITY

277

CONFIGURATIONS

_ ENERGY (in MeV) 9/2-

V s =-30

--

V t =-I0 MeV

/ 5/2512+ {d~) 5 912+ (p~)2 (d.~)3 \ 7•2-

m

3/2--

312+ (P½}2(d~)3 I --

1/2--

~

l

I

05

o.7

I X

I

I

I s12 + (,.,12(':'..m)3

0.9

,.,

z

z

Fig. 1. Energy s p e c t r u m o f zaPx for t h e exact (degenerate) m o d e l as a f u n c t i o n o f the r a n g e p a r a m e t e r with 1/8 = - - 3 0 M e V a n d Vt = - - 1 0 MeV. T h e configurations s h o w n for the positive-parity states are those expected f r o m t h e shell-model p o i n t o f view. T h e negative-parity states arise f r o m t h e configuration (p~.)(d~r)4. ENERGY (in MeV)

Vs:-30

F

,,,:-,o ""

I-

Z~: o.s

~

,~~.. ~.~

'

/

~

o'6

5/2+'d~) 5

,/2

'

o'.~'2+(3"('{;'

Fig. 2. Energy spectrum of zgPx for the exact (non-degenerate) model as a function of the range p a r a m e t e r 2 with 1/8 = - - 3 0 MoV, V, = - - 1 0 M e V a n d level separation d = 0.5 MeV. T h e configurations s h o w n for t h e positive-parity states are t h o s e expected f r o m the shell-model p o i n t o f view. T h e negative parity states arise f r o m t h e configuration (p½)(d½)%

278

R . K . G L r F r A A N D L. E. H . T R A I N O R

T a b l e 1 also lists the p e r c e n t a g e c o n t e n t o f v a r i o u s c o n f i g u r a t i o n s i n the g r o u n d states o f the p s e u d o - n u c l e i o f o u r m o d e l c a l c u l a t e d for the choice o f p a r a m e t e r s 2 = 0.6, V~ = - 3 0 M e V a n d Vt = - 1 0 M e V . F o r N = 4, for e x a m p l e , the g r o u n d state is o n l y 49.7 ~ p u r e (p~)2(d~)2, i.e. ~O.r=o+ = 0.705(p½)2(d~) 2 + . . . . T h i s r e s u l t c o u l d be i n t e r p r e t e d to m e a n t h a t the r e m a i n i n g 50.3 ~o is the p r o b a b i l i t y o f core excitation. A m e a s u r e o f the softness o f the core is t h u s s h o w n in the last c o l u m n o f table 1. T h e core e x c i t a t i o n s o f positive p a r i t y i n the p r e s e n t m o d e l ( a n d also i n the C L S a n d LS m o d e l s ) c o r r e s p o n d e n t i r e l y to t w o - p a r t i c l e - t w o - h o l e exc i t a t i o n s i n c o n t r a s t to o u r p r e v i o u s m o d e l o f p s e u d o n e o n i s o t o p e s (s½, d~ subshells), w h e r e o n e - p a r t i c l e - o n e - h o l e c o n t r i b u t i o n s are also significant. T h e i m p o r t a n t role o f p a r t i c l e - h o l e c o n f i g u r a t i o n s i n l o w - l y i n g states o f real n u c l e i is, o f course, well k n o w n s). TABLE 2 Percentage content of the states in tSPx (degenerate model) as a function of three parameters of the two-body force Case Vs = - - 3 0 MeV Vt = --10 MeV

J~ 0 + g.s. 2 +' 4 +'

Configuration / 2 (p½)~(d~)2 (d~r)~ (p½)2(d~r)2 (d~r)4 (p½)2(d~r)~ (d~r)4 /--Vt

= 0.6 Vs = --30 MeV

0 + g.s. 2 +, 4+ ,

(p½)2(d~)Z (dl.)4 (p~r)2(d~)2 (d.l.)4 (p~_)2(d~t)~ (cl~)4 / --Vs

2 = 0.6 Vt = --8 MeV

0 + g.s. 2 +' 4 +'

(p~r)*(d~t)2 (d~) 4 (pat)2(dt) 2 (d~_)4 (p~)2(d~)2 (d~r)4

0.5

0.6

0.7

0.8

0.9

48.5 51.5 46.9 53.1 46.9 53.1

49.7 50.3 49.5 50.5 49.5 50.5

51.5 48.5 52.9 47.1 52.9 47.1

53.5 46.5 57.0 43.0 57.0 43.0

55.8 44.2 61.4 38.6 61.4 38.6

20

15

10

5

0

45.1 54.9 40.4 59.6 40.4 59.6

47.6 52.4 45.2 54.8 45.2 54.8

49.7 50.3 49.5 50.5 49.5 50.5

51.6 48.4 53.2 46.8 53.2 46.8

53.2 46.8 56.4 43.6 56.4 43.6

40

30

25

20

15

51.3 48.7 52.5 47.5 52.5 47.5

50.5 49.5 51.0 49.0 51.0 49.0

49.9 50.1 49.8 50.2 49.8 50.2

48.9 51.1 47.8 52.2 47.8 52.2

47.1 52.9 44.3 55.7 44.3 55.7

The prime denotes the first excited state. Fig. 3 shows t h e s p e c t r u m for the d e g e n e r a t e exact m o d e l o f 18Px c a l c u l a t e d as a f u n c t i o n o f the r a n g e p a r a m e t e r 2 u s i n g p o t e n t i a l s t r e n g t h s Vs = - 3 0 M e V a n d

5-

2- (2) 2+

I-

(d~)4

.3-

2; (I)

(d~) 4

4+

(pm)Z(ds)z

2+

(pt)~' (riD..) z z

,,I

0.5

I

I

o;, X

I

o.9

0 +

2

{p )z ( d ) : '

½

-~

Fig. 3. Energy s p e c t r u m o f zsPx for t h e exact (degenerate) m o d e l as a f u n c t i o n o f the r a n g e p a r a m e t e r A with V~ - - - - 3 0 M e V a n d V, = - - 1 0 MeV. T h e configuration s h o w n for the positive-parity states are t h o s e expected f r o m the shell-model p o i n t o f view. T h e negative-parity states arise f r o m the configuration (p~r)(d~r) 3. O f t h e two negative-parity 2 - states, o n e labelled 2-(1) arises f r o m the configuration (p~r)~r(d~r)~S a n d the other labelled 2-(2) arises f r o m the configuration (p~r)~r(dt)~ 3. "EX~,CT" (DEGENERATE) "EXACT" (DEGENERATE)

3 4÷ 5" -i-÷

>~ ._~

2

-

-

..

"2"(I) O* 3-

"SHELL MODEL" (d5/2)2

-4"-

-

~ 2 " , _

-

_ _

4."

/~:R)

"SHELL MODEL"

(d5/2)2

z.÷CO)

4÷

<0_

_~.~. - -

4"

_ 2÷

2÷

2.

Z O

¢.,.) X b..I

o

- - o "

-

px 18

(o)

-

o÷

~

o

pxI8 '

'

~--(b)

o÷ - - , (c)

'

Fig. 4. C o m p a r i s o n o f the exact (degenerate) s p e c t r u m o f lsPx with shell m o d e l spectra. (a) V8 = - - 3 0 MeV, Vt = - - 8 M e V for b o t h calculations, ;~ = 0.5 (exact) c o r r e s p o n d to ~ r t ~ 0.64 (shell). (b) Vt = - - 8 MeV, ;[ = 0.6 for b o t h calculations, Vs = - - 2 0 M e V (exact) corresponds to (V~)er t --29.5 (shell). (c) VB = - - 2 0 MeV, Vt = - - 8 M e V a n d ;t = 0.6 (exact) c o r r e s p o n d to effective values VB = - - 3 0 MeV, Vt = - - 1 0 M e V a n d ;[ ~ 0.59 (shell).

280

R. K. GUPTA AND L. E. H. TRAINOR

Vt = - 10 MeV. This pseudo-experimental data was also obtained for various other choices of V, and Vt. As mentioned above, the order of positive-parity states matches the order 0 ÷, 2 +, 4 + obtained from the pure shell-model configuration (d~) 2 and is independent of the choice of potential parameters. Actually, even the structure of these states does not change with the triplet-odd strength Vt of the potential. As to the amount of configuration mixing, table 2 shows that for any choice of the potential strengths and range considered here, the three states reveal large configuration mixing. To look for the possibility that the large configuration interaction responsible for this situation can be concealed in the effective two-body potential, we compare in fig. 4 the shell-model spectrum (taking lp~ as closed core) with the exact data. A comparison in terms of the "effective range" (V~ = - 3 0 MeV and Vt = - 8 MeV for both exact and shell-model calculations) is shown in fig. 4(a) and that in terms of "effective strengths" (2 = 0.6 and Vt = - 8 for both models) in fig. 4(b). Fig. 4(c) gives a comparison for the best fit between the exact model and shell-model spectra when effective values for both the potential strengths and the range are used in the shell-model calculations. Thus, we see that the core-excited configurations can be concealed in the present model, insofar as spectra are concerned, with any one of several choices of potential parameters. This result agrees, in principle, with that of Cohen, Lawson and Soper but differs from our s-d (Pn isotopes) results where concealed configuration mixing could not easily occur. 4. Nuclear form factors

In this section, we consider inelastic scattering of electrons from pseudo-nuclei as a tool for measuring the relative purity of wave functions in exact and shell-model approximations. We compare the transition probabilities as functions of the momentum transfer q. The reduced transition probability or so-called nuclear form factor for an electron in momentum state Ki to scatter into Kf by exciting the nucleus from initial state li) to excited state If) is (in first Born approximation) given by

n(A, q, JfJi)

- 1 2Ji + 1

Il 2,

(4.1)

where the operator M(2, q) represents the interaction field and 2 the multipolarity of the transition. Thus, for a known form of the multipole operator M(2, q) (spherical tensor), the transition probability can be calculated 6) as a function of momentum transfer q using the appropriate wave functions for initial and final states. We have calculated only the quadrupole (2 = 2) transition probability B(C2, q, JeJi) for Coulomb excitation using the isotope tSpx as an example. The configurations involved are shown in fig.I 5 for both the exact and shell-model cases - a t , a2 etc. being the eigenvector components for states in the exact model. The transition probabilities B(C2, q, 20) and B(C2, q, 42) for the two allowed transitions 0 + ~ 2 +

281

MIXED-PARITY CONFIGURATIONS

and 2 + -+ 4 + are then given by the following expressions: exact model B(C2, q, 20) = I((ba(p½)2(ds)2+ b2(d~)4), Jr --- 2+JIM( C2, q)[[ (al(p~)2(d~) 2+a2(d~)4), Ji = 0+>12 _

2 (e'r2) 2 (al bl +a2 b2)2(7-2x)2e -2~, 7 7z

(4.2)

B(C2, q, 42) =~l((cl(p½)2(d~)Z+c2(dl)4), Jf=4+llM(C2,q)ll (bdp~)2(d~)2+b2(d~)'), Ji = 2+5] 2 =

243 (e'r~) 2 (C 2 b 2 _ c , 3430

bi)2(7-2x)2e-2X;

(4.3)

shell model B(C2, q, 20) = I((d~) 2, Jf = 2+11M(C2, q)ll(d{) 2, Ji = 0+>l 2 2 (e'r~) 2 (7_2x)2e_2X ' 7

(4.4)

B(C2, q, 42) = ~1 L ( (d~ ) 2, J f - - 4 + I I M ( C 2,q)ll(d,) 2, Ji = 2+)12 243 (e'r2) 2 (7_2x)2e_2C 3430 7r

(4.5)

SHELL MODEL CONFIGURATION

EXACT MODEL CONFIGURATION

(d_5)2 z

4 +-

CI (Pl)2(ds_)2+C2(ds)4 ~" z 2

(ds_)2 2

2+

bl (p±)2(d5)2 ~b(d5)4 2 2 2

Id_s

o+

2

)z

ojp_, )z ¢d_5)z + PX 18

2

2

o2~d5_)4 2

Fig. 5. Shell-model configurations for laPx. The mixed configurations for the corresponding levels in the exact model are also indicated.

In these expressions, e' is the effective charge of the neutrons, and the momentum transfer dependence is contained in the definition

x = ¼q2r2,

(4.6)

282

R. K. GUPTA AND L. E. H. TRAINOR

with r I the radial extension parameter of the harmonic oscillator wave functions. Since the aim of the present study is to compare the results of two model calculations rather than to obtain absolute values, we have not included the small corrections due to the finite size of the nucleons and the so-called Darwin term in the Coulomb multipole operator. To compare the shell-model wave functions with the exact wave functions in ~8px, we take the ratios R(C2, q, 20) and R(C2, q, 42) of the exact-to-shell-model transition probabilities. The two ratios are [from eqs. (4.2), (4.4) and (4.3), (4.5)] R(C2, q, 20) = (a~ bl +a2 b2) 2,

(4.7)

R(C2, q, 42) = (c2 b 2 - C l bl) 2.

(4.8)

It is interesting to note that these ratios are independent of the electron momentum transfer q. This result is in sharp contrast to that for the Pn isotopes where the corresponding ratios show prominent structure as functions of q. We shall refer to these ratios as the "real photon" transition ratios, since at q = 0 the reduced transition probability for the inelastic electron scattering process becomes equal to the real photon transition probability. However, to determine the effect of concealed configurations on these real photon transition ratios, we study their dependences on the parameters of a two-body potential. Since the shell-model configurations are pure, the corresponding reduced transition probabilities are independent of the parameters of the two-body potential; hence the 2-dependence (or alternatively the dependence on the strengths VS and Vt of the potential) of the transition ratios is an indication of the configuration mixing in the exact model states. The calculations are carried out for several choices of the two-body potential parameters (see sect. 2). For the case 0 + ~ 2 +, one has the particularly simple result R(C2, q, 20) ~ 1,

(4.9)

which is approximately independent of the choice of the potential strengths and range parameter and hence, independent of the amount of configuration mixing. On the other hand, the ratio R(C2, q, 42) shows an interesting dependence on the parameters of the two-body potential. Table 3 gives the ratio R(C2, q, 42) as a function of 2, Vs and Vt. We have also included in this table the results for the non-degenerate model with level separation A = 0.5 MeV. The amount of configuration interaction is clearly reflected in this ratio since it never reaches the value unity. This happens because of the structure of eq. (4.8). The ratio R(C2, q, 42) can approach unity only if, either the eigenvectors cl and bl or c2 and b 2 are negligible; in either case the exact model states would become pure shell-model states. In other words, the 0 + ~ 2 + transition probability is completely insensitive to configuration mixing, while the 2 + ~ 4 + transition probability depends upon it in an entirely essential way. Translating this result into a practical consideration, one could be misled by a measurement

MIXED-PARITY CONFIGURATIONS

283

of a single transition probability into believing in pure sheU-model configurations, but a systematic survey of all transition probabilities would dramatically reveal any underlying strong configuration interaction. TABLE 3

Ratio R(C2, q, 42) of exact (degenerate, A = 0 and non-degenerate with A = 0.5 MeV) to shellmodel transition probabilities in 18Px Case Vm= --30 MeV, Vt = --10 MeV

d (MeV) / 2

0.5

0.6

0 0.5

0.004 0.472

0.000 0.365

0.003 0.330

0.020 0.339

20

15

10

5

0.037 0.325

0.009 0.347

0.000 0.365

0.004 0.381

0.017 0.394

40

30

25

20

15

0.002 0.263

0.000 0.371

0.000 0.455

0.002 0.566

0.013 0.708

\ -vt Vs = --30 MeV, 3. = 0.6

0 0.5 \ --V8

Vt = --8 MeV, 3. = 0.6

0 0.5

0.7

0.8

0.9 0.052 0.371 0

5. Conclusions

Summarizing our present work on the p~ and d~ subshells, we find that insofar as spectra are concerned, concealed configuration mixing can occur provided one allows non-degeneracy in the model. In comparison with earlier pseudo-nuclear models, one could possibly interpret this result to mean that it is much easier to conceal configuration mixing and core excitation in shell-model studies where such effects involve two-particle-two-hole excitation (mixed parity configurations) than it is for studies where one-particle-one-hole excitations also make substantial contributions. It really comes to the same thing, however, since in the case of mixed parity configurations one-particle-one-hole excitations give rise to negative-parity states, which are in themselves direct and tangible evidence of the importance of core excitations. Stated in another way, the one-particle-one-hole excitations directly reveal configuration mixing and core excitation, either through the appearance of negative-parity states or alternatively through the direct failure of the shell-model calculations to give a satisfactory spectrum ordering for the positive-parity states. While these conclusions may be somewhat academic since they are drawn from models involving only two subshells, they probably suggest a lesson for any studies using a truncated configuration space. The present study and our previous work on pseudoneon isotopes both seem to show that systematic studies of electromagnetic transition probabilities, and more particularly of inelastic electron scattering results, would, in practice, dramatically reveal large amounts of configuration mixing. Again, one can make the same comment as for spectra, that the role of one-particle-one-hole excitations is more revealing

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t h a n t h a t o f t w o - p a r t i c l e - t w o - h o l e excitations (no d o u b t exceptions exist); w h e n m i x e d - p a r i t y states occur, one o b t a i n s s t r o n g E l , M 2 , etc. t r a n s i t i o n s a n d certain discrepancies in the E2 systematics; when only e v e n - p a r i t y states occur (e.g. in P n isotopes), E2 t r a n s i t i o n p r o b a b i l i t i e s for inelastic electron scattering show a significant c h a n g e in b e h a v i o u r as functions o f m o m e n t transfer q with increasing a m o u n t o f c o n f i g u r a t i o n interaction. W e are i n d e b t e d to the N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a for financial s u p p o r t .

References 1) 2) 3) 4)

S. Cohen, R. D. Lawson and J. M. Soper, Phys. Lett. 21 (1966) 306 R. D. Lawson and J. M. Soper, Proc. Int. Conf. on nuclear physics Gatlinburg, Tenn. (1966) p. 511 L. E. H. Trainor and R. K. Gupta, Nucl. Phys. AI08 (1968) 257 S. K. Shah and S. P. Pandya, Nucl. Phys. 38 (1962) 420; R. K. Gupta and P. C. Sood, Phys. Rev. 152 (1966) 917 5) D. J. Hughes and A. B. Volkov, Phys. Lett. 23 (1966) 113 and the earlier references given therein 6) R. S. Willey, Nucl. Phys. 40 (1963) 529