Semiempirical calculations of shell model states in 212Rn

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Semiempirical calculations of shell model states in 212Rn

T. Lönnroth

To cite this version:

T. Lönnroth. Semiempirical calculations of shell model states in 212Rn. Journal de Physique Lettres,

Edp sciences, 1980, 41 (8), pp.185-187. �10.1051/jphyslet:01980004108018500�. �jpa-00231755�

L-185

Semiempirical

calculations

of shell

model

states

in

212Rn

T. Lönnroth

_(*)

Department of _{Physics, University} of Jyväskylä, Jyväskylä, Finland and Research Institute of Physics, Stockholm, Sweden

(Re~u le 13 decembre _1979, revise le 11 _{fevrier, accepte} le 29 fevrier 1980)

Résumé. 2014 En utilisant

_l’algèbre

standard des moments _angulaires et les énergies d’interaction

_empiriques,

c’est-à-dire les _{énergies d’interaction à 2 corps extraites des niveaux d’énergie expérimentaux} connus

du 210Po,

on étudie les _propriétés des états du 212Rn. Les _énergies d’excitation, les _périodes et moments _magnétiques sont donnés. L’effet de la troncation dans

_l’espace

des protons est examiné. On trouve que les données sont bien vérifiées par

l’expérience.

Abstract. 2014

Using standard

_angular

momentum

_algebra

and

_empirical

interaction

_energies,

i.e. two-nucleon interaction _energies extracted from

_{experimentally}

determined levels

_{in 210Po,}

_properties of valence nucleon states

in 212Rn are calculated. Excitation _energies and some half-lives and _magnetic moments are _given. The effect of the truncation of the

_{configuration}

_{space is discussed. Comparison} with

_{experimentally}

determined

_{properties gives}

good agreement. J. _Physique - LETTRES 41 (1980) L-185 - L-187 15 AVRIL 1980, Classification Physics Abstracts 21.60

1. Introduction. - The shell model

predictions

of nuclear structure have met with the _greatest success

for nuclei in the

_regions

where N or Z =

20,

28, 50,

(64),

82 or 126. Here the states can well be

_depicted

as few-nucleon

configuration plus

a closed core. Such core nuclei are

e.g. 160,

40Ca

and

208Pb.

The closure mentioned above is a relative

_concept

and retains some

_meaning

_only

for cases where the _energy of the

first excited state of a core-nucleus is well above the

single

nucleon orbital

_spacing.

Such is the case for

208 Pb

where its first excited state, the collective

octupole

vibration

_{(3 -)}

is located at 2.6 MeV whereas

e.g. the first proton orbitals in

209Bi

are situated at

0.9 and 1.6

MeV,

respectively.

Thus

209Bi

can be

depicted

as _proton orbitals outside a closed core.

The

_problem

of 209

_particles

is thus reduced to

208 +

1,

where the 208 first ones

_represent

a zero-energy, Jft =

0 +,

_{vacuum state.}

When _{e.g. protons} are added we _get the N = 126 iso-tones

209Bi,

2lopo,

211At, 212 Pn,

etc. In all these nuclei it has been

_{experimentally}

verified

_[1]

that the

lower-lying

states are formed from valence nucleons.

In order to

_study

the structure of _many

_proton

states and to check the limitations of

_{configurational}

purity,

we have undertaken a calculation of valence

proton states in

212Rn.

The

_lower-lying

states in this nucleus are thus

_supposed

to be made _up from

four-(*) Now at the University of Jyvaskyla.

proton

configurations.

As is seen in the

_discussion,

this

_assumption

is well verified. It is also seen that the _yrast states _agree with

_experiments

to an

asto-nishingly

high degree.

2. Calculational

_procedure.

- For the

proton orbi-tals outside the Z = 82 shell

_closure,

the level

order-ing

is established

_[2].

We retain the three first _proton

orbitals,

partly

to reduce the

_{configuration}

_space and

partly

because we are

_mostly

interested in _yrast states

_(an

_yrast level is that level of a

given

J that has

the lowest excitation

_energy).

We thus calculate

four-proton

configurations

of the

_following

_{types :}

_h~~2,

h 3 9/2

f7/2

and

_{h 9/2 3}

_il3/2-

_{Configurations}

of _e.g. the

type

_h~/2

f7/2

ii 3/2

would meet with the

_problem

that the _two-proton interaction

_f7/2

_i13/2

is not

_empirically

known.

The calculational

_procedure

_closely

follows that introduced

_by

I. Talmi for

_light

nuclei

_[3]

and

_applied

to the lead

_{region by}

J.

_Blomqvist

_[4].

We have a Hamiltonian of the form

L-186 JOURNAL DE _{PHYSIQUE -} LETTRES

where i,

j,

k run over available _proton orbitals. The

terms

_Ho

are the bare interaction

energies

whereas the

7~

are

energies resulting

from the

polarization

of the

208 Pb

core.

Taking

the terms

_pair-wise,

_e.g.

we

_get

energies

which allow for core-correlations.

These are

_represented

_by

the

empirical single-nucleon

energies

(proton energies

in

_209Bi)

and

_empirical

two-nucleon

energies (two-proton energies

in 2 1 lpo).

The

_three-body

effective

energies

are shown to be small

[5],

and we

tacitly

assume the

_higher-order

correlations to be even smaller. Our Hamiltonian thus

reduces to the

_expression

Since the calculation is based on the

_assumption

that

the

_protons

move in the _average field of

208 Pb,

this core thus

_represents

the vacuum and its _energy is set

to zero. We have the

_following

relation

where

_Sp(209Bi)

is the _proton

_separation

_energy in

209Bi

_{and Bji}

is the excitation _energy of orbital

_ji

in

209Bi

[2].

The _{two-proton interaction energy is} extracted from

210po

_(Ref.

_[lb])

_using

the

_expression

We thus _get the excitation _energy of levels

_{in 212Rn by}

adding

all interaction

_energies

dj

to the zero-order

proton

energies

E(ji,

209Bi)

and

_subtracting

the

_ground

state _energy

of 212Rn

relative to

20 8Pb,

viz.

The

_ground

state _energy

of 212 Rn

can be calculated as

We use the

_proton

separation

energies,

since

_they

are found in tables

_[6].

The reduction of the

_hg/2

_{configurations}

to

h9/2-pairs gives

coefficients of fractional

_parentage

(cfp)

and the

_{recoupling gives 6j-symbols.}

The

_cfp’s

are

given

in the

_Appendix

of ref.

_[3]

and

_6j-symbols

have been calculated for each case.

Using

the wavefunctions calculated above we also

calculate some allowed transition

_{probabilities,}

i.e.

for E2 and E3

_multipoles.

The reduced transition

probability

can be written as

_[3]

where

_Ci

and

_Cj

are

geometrical

factors

_arising

from

the

_decoupling

of the

_{four-proton configurations}

and the radial matrix

_elements

r~ )

are

_integrated

from

the numerical tables of reference

_[7].

_

The

_g-factors

of some states with measurable half-lives were also

_calculated,

_mostly

with the aid of

Lande’s formula. 3. Results. -

Using

the method outlined in

sec-tion 2 the

_following

levels were calculated

_(v

indicates

seniority

within a

multiplet) :

4

h~/2

h9/2f?/2

3 .

h9/2 113/2

The

_low-spin

levels of all

_{configurations}

_{h9~2 ~}

were

not

calculated,

since

_they

are not

_likely

to be

_populated

in a

(HI,

xn)

reaction,

and we wanted to

_give

an

explicit comparison

with

_experiments.

On the other

hand,

with the levels

above,

one can make a check of

the

_linearity

of the mean excitation _energy of a

multi-Table I. - Calculated excitation

L-187 SEMIEMPIRICAL CALCULATIONS IN 212 Rn

plet

as a function of

seniority

_quantum number.

The results are

_given

in the table

above,

and in the level

scheme in

_figure

1 we

_present

a

comparison

of our

calculation with

_{experimentally}

determined levels.

As is seen in the

figure,

the _agreement between

theoretical and calculations and

_experimental

results

Fig. 1. - Theoretical level scheme of 212Rn. The

experimental-ly [8] found levels are _given on the left.

is

_strikingly

_good,

_except for

_low-spin

levels. This

discrepancy

is

_readily

accounted for as a _consequence

of the truncation of the

_{configuration}

_space.

4. Conclusion. - We find that the rather

simple-minded calculation

_presented

above

_{gives remarkably}

good

results.

_However,

the most

_striking

disagree-ment, i.e. the

large discrepancies

in excitation

_energies

for

_low-spin

_states, can well be accounted for if one

includes

_{configuration mixing.}

This amounts to

enlarging

the

_{configuration}

_space

_beyond

_1h9~2,

2f7/2

and

_lil

_3/2. Thus _e.g.

_the

_{I 0 + > in 210pO,}

is

accord-ing

to

_{Kuo-Herling [9]}

Even if one truncates as

above,

one is faced with the

problem

that the

_empirical

excitation

_energies

of the

states

_{(f ~2)0+}

and

_(1~3/2)0~

are not known.

In contrast to the

_low-spin

states, the

high-spin

configurations

of

[9]

are not mixed and this is used as

the basic _argument for

_doing

calculations

_assuming

pure

configurations.

As is also seen in

figure

1,

there is

_agreement

between

theory

and

_experiment

for the

_quoted

half-lives and g-factors. In these the

_{configuration mixing}

enters

linearly

and thus

_larger

deviations

_might

be

_expected.

Acknowledgment.

- The author is much

indebted

to J.

_Blomqvist

for his kind

_supervision

at the

begin-ning

of the work and for _many

_subsequent

_stimulating

discussions. Thanks are due to I.

_Bergstrom,

the head of the Research Institute of

_Physics,

for kind

hospi-tality.

Financial

_support

_by

The Nordiska Accelerator-radet and

_Magnus

Ehmroots Stiffelse are

acknow-ledged.

References

[1] Nuclear Data Sheets 22 ₍₁₉₇₇₎ 545 ; FANT, B., Phys. Scr. 4 (1972) 175 ;

BERGSTRÖM, I., FANT, B., HERRLANDER, C. J., WIKSTRÖM, K., BLOMQVIST, J., Phys. Scr. 1 (1970) 243.

[2] SPETH, J., WERNER, E., WILD, W., Phys. Rep. 33C (1977) 127.

[3] DE SHALIT, A., TALMI, I., Nuclear Shell Theory (Academic

Press, New York and London) 1963.

[4] BERGSTRÖM, L, BLOMQVIST, J., FANT, B., HERRLANDER, C. J., LINDÉN, C. G., WIKSTRÖM, K., Proceedings of the XIIth International Winter Meeting on Nuclear Physics, Villars,

Switzerland, 1974.

[5] LÖNNROTH, T., BLOMQVIST, J., BERGSTRÖM, I., FANT, B., Phys.

Scr. 19 (1979) 233.

[6] Atomic Data and Nuclear Data Tables 19 ₍₁₉₇₇₎ 215.

[7] BLOMQVIST, J., WAHLBORN, S., Ark. _{Fys. 16 (1960)} 545.

[8] AFI Annual Report 1976, 3.3.21 ; HORN, D., HÄUSSER, O.,

FAESTERMANN, T., MCDONALD, A. B., ALEXANDER, T. K., BEENE, J. R., HERRLANDER, C. J., Phys. Rev. Lett. 39 (1977)

389.

[9] Kuo, T. T. _S., HERLING, G. H., NRL Memorandum _Report