Calculation of δ 〈r2〉 for mössbauer transitions using coriolis mixed wave functions.: Application to 153Eu

[ 1.E.4 [ Nuclear Physics A142 (1970) 273--290; (~) North-Holland Publishing Co., Amsterdam I t

Not to be reproduced by photoprint or microfilm without written permission from the publisher

C A L C U L A T I O N O F ~ ( r 2) F O R M O S S B A U E R T R A N S I T I O N S

U S I N G C O R I O L I S M I X E D W A V E F U N C T I O N S .

Application to ~SSEu

JOHN HESS Department of Physics, The Hebrew University, Jerusalem, Israel t

Received 29 July 1969

Abstract: An energy fit to the positive parity levels of XSaEu was carried out. From the energy fit, Coriolis mixed wave functions were obtained and compared with the extensive experimental data available for B(E2) and B(M1) reduced transition probabilities, magnetic moments, branching ratios and isomer shifts. In general, good agreement was obtained. However, there is a striking exception for the J(r 2) (difference in mean square radius) associated with the 103.2 keV transition. For this case the theoretical estimate of c$(r2)xo3.2 was six times larger than the experimental value. A simple quantum mechanical formulation for ~(r2) is given and it is shown that the very large 103.2 keV ~(r2) can be calculated almost as well using a classical formulation. Thus the origin of such a large discrepancy is unclear.

1. Introduction

The fo rmula for the i somer shift o f a M6ssbaue r abso rp t ion line

AE = ~nZe2[Da-Ob]b(r 2) erg,

where Z is the a tomic number , e the e lect ron charge, D a and D b are relativist ic elec-

t ron densities a t the nucleus in the absorbers a and b and 6 ( r 2) is the difference in

the mean square charge rad ius between the nuclear states associa ted with the M6ss-

bauer t rans i t ion , seems to be well es tabl ished 1).

The quan t i ty 6(r 2) has always been o f interest in nuclear s t ructure; however, since

independen t de te rmina t ions of the chemical fac tor [ D a - Db] have no t been considered

to be comple te ly reliable, de te rmina t ions o f f i ( r 2) f rom M.E. measurements have

been cons idered to be s imilar ly unrel iable . Thus a large d isagreement by factors o f 5

o r 10 between exper imenta l and theoret ica l de te rmina t ions of cS(r 2) has not been

r ega rded too seriously.

Recent ly , measurements o f 6(r 2) in deformed nuclei have been made by measur ing

energy shifts of muonic X-rays 2). Re- in te rpre ta t ions of these measurements and also o f the M6ssbaue r measurements have b rough t a b o u t agreement between the two

methods o f de te rmining 6 ( r 2) resul t ing p re sumab ly in dependab le values for this

quant i ty . The d iscrepancy between theory and exper iment still remains.

t Supported in part by the Israel Academy of Sciences and Humanities.

273

274 J. HESS

In this paper we are mainly interested in a favourable case for theoretically esti- mating 6(r 2) in a deformed nucleus. It is the isomer shift of the 103.2 keV transition in ~ 53Eu. The points which we consider to be favourable are:

(i) The value of 6(r 2) associated with the 103.2 keV shift is 50 to 100 times larger than shifts associated with transitions between rotational states. Dealing with a large effect has advantages from both an experimental and a theoretical point of view.

(ii) The phenomenological parameters obtained from a fit to the band mixing model, which is applied here, may be tested against the extensive data (other than for 6(r2)) available for positive parity states of ~ S3Eu.

The main point we wish to make is that while the results for 6(r 2) may be inaccurate to 30 or 40 %, it is unlikely that they should be wrong by a factor of 5 or 10. Thus the results may eventually be of use in bringing about agreement between theory and experiment.

2. Quantum mechanical formulation for 6(r2). Justification of classical form for case 2 of 6(r ),03.2

The mean square radius of the nuclear charge distribution ( r 2) is given by the expectation value of the operator (1/z)~Z=~ r/Z. I where r• = x~ +y~ + z~ is a function of the coordinates of the ,th proton, Z is the number of protons and I is a unit operator which operates in the space of the complete set of nuclear state vectors.

In order to make contact with experimental measurements and classical expressions we want the result ( r 2) to be expressed in terms of matrix elements of multipole operators in the proton variables. To this end we first expand the unit operator I into a complete set of multipole operators Xff ) which transforms under rotation of the coordinate system like spherical harmonics t. The operator whose expectation value we wish to evaluate then takes the form

1 z 1 z 0o ,=+~

Since we are interested in the electric charge distribution we take the projection of the above operator onto the operator subspace corresponding to proton spatial coordinates. We then specialize to an effective operator reef made up of constants a~ and operators Q~), which have the same transformation properties, under rotation, as the Xff ), but operate only in the subspace corresponding to proton angular momen- tum variables. The constants a~ will appear in the expectation value (r 2) and will represent the contributions to (r 2) from summations and integrations over the other variables. The operator r~f then has the form

i u=+A

ref f = a~ ~ ( - _ ~ ~---u~-u • 2=0 /~= -A

TO within a constant factor the Xu(;-) are the operator equivalents introduced by Stevens2~). For a logical development o f the orthogonali ty and completeness propert ies o f these operators (of which use is made here) see Schwinger 22).

153Eu MOSSBAUER TRANSITIONS 275

The matrix elements of the Q~a) are proportional to multipoles of electric charge. I f we normalize the Q~2) so that in the classical limit

Q~) ~ V 16z

" ~ 2 2 + 1 raY2""

we may establish a correspondence between the expectation value /,= +.g

<r~2e> = a2< ~ ( - 1~2-,O(2) O(a)x, , _ , , . , _ , , . , , , (1) 2=0 #= - 4

and the classical expression for (r 2) of Fradkin ~) for a deformed axially symmetric nucleus,

( r 2 ) = 3 2 3 3-Ro + ~ R 2 f12. (2)

= 3 2 = 5/12R2oZ 2, From the correspondence we obtain the values ao ~ R o , a2 a 1 = a a = 0. The details of establishing the correspondence and some additional justification for neglecting E1 and E3 contributions are given in the appendix.

Substituting the values of a 2 into eq. (1) and dropping the subscript eft, we have for the expectation value (r 2) the quantum mechanical expression

5 /z=+2 ( r 2) = 3 2 _ _ 3-Ro+ leR2Z2 < ~ ~ (-1)UQ_t,O,), (3)

/a= - 2

where in the absence of a superscript on Q, it is understood that 2 = 2. The t~(r 2> associated with the M6ssbauer transition b ~ a is then given by

~ r 2 > __ 5 t*=+2 ~ = + 2 12Ro Z2 [(bl ,=~-2 ( - 1 ) "Q- ,Qu lb ) - ( a l u=Z-z(- 1)'Q_uQula)]. (4)

We now introduce formally the Coriolis mixed wave functions whose use, for the case of 153Eu ' we will try to justify. The wave functions have the form

IIM> = a ( K ) I I M K > + u ( K + 1)IIMK+ 1>, (5)

where ~(K) and c~(K+ 1) are coefficients to be determined by fitting the theory to the experimental energy spectrum and ]IMK> are symmetrized rotational wave functions [ref. 3) ]

]/2I + I I M K ) = r - i ~ z (Dtu~lK) + ( - 1)r-KD~_~¢I-K)). (6)

The matrix elements of eq. (3) are then # = + 2 / t =+2

<al ~ ( -1) 'Q,Q_~Ia> = ~ ~ (aI(-1)~'Q_~,II'M'K'><I'M'K'IQ,la>. (7) t~ = - - 2 I ' M " IL = - - 2

K'=. . ' , ' } ,~

It is assumed that the 153Eu spectra under consideration are the K ~ and K -

2 7 6 J. HESS

rotational bands only and that contributions from other states may be neglected, at least as far as the nuclear data under discussion is concerned. An additional K = rotational band based upon a fl-vibration could easily be included in the formulation at the cost of introducing extra parameters. It will be seen however that for the 103.2 keV 6(r 2) this would introduce a negligible correction.

I f we now substitute for la) in (7) a wave function of the form (5) we have an expression in matrix elements of the form (I'M'K'IQulIMK). Since we are dealing with a spherical invariant ( r2) , it is possible to eliminate angular variables which are measured in the laboratory by the substitution 4)

(I'M'K'[Q~,,_~tlMK)

= (_ I)M,_K,X/2--~+ lx/2~7~i ( I'M, ,2 I ) ( I'K, ,2 IK)

x (K']QK,-K]K), (8)

, , , ( 2 ) which expresses the matrix element II M K IQM,_M[IMK) evaluated in laboratory coordinates in terms of the matrix element (K'lQ(r2)_tclK) evaluated in intrinsic nuclear coordinates. In general a second term involving the matrix element (K'[Qw +K[K) also appears; but, in our case K' + K = ~ + ~ = 4, so it vanishes.

Substituting (5) and (8) into (7) and applying orthogonality relations of the 3j- symbols yields

/ * = + 2

(al }", ( - 1 ) " Q _ u Q u J a ) = ~ ~(K)2I(K'IQw-KIK)I 2, (9) , u = - - 2 KK' =~-, {

where the coefficients ~(K) are defined in the theory to be real. Following refs. 7-9), we introduce the notation

Q(K'K) = <K'IQw-KIK),

which for the case of K' = K is abbreviated by QK- I f furthermore, we introduce the notation: B K = Q2K+Q2(K , K+ 1) where K-values can only be ~ or 2 s then (9) takes the form

c~(K)2I( K'[QK'-KIK)[ 2 = ~ c~(K)EBr, KK" K

for 3(r 2) we then have, substituting into (4),

b(r2 ) _ 5 [ E cq(K)2BK- Z %(K)2BK] 12R~ Z 2 K K

5 12R g Z 2 ~ BK(%(K)E -- eb(K)2),

where a refers to the excited state ]a) and b refers to the ground state ]b).

153Eu M/JSSBAUER TRANSITIONS 2 7 7

For the specific case of two rotational bands this result may be further simplified since the coefficients e(K) are normalized according to the relation

Z (Ky = 1, K=k,~

then

t~(r2) _ 5 12R2 Z2 (B~ -- Bk)(O~a(~) 2 -- ~xb({) 2)

5 - - 2 2 5 2 12R~ Z 2 (Q~- Q~)(~a(~) - ~b({)2),

6(r2 ) _ 5 2 12R 2 Z 2 ( Q I _ Q2)(C~b(~2)2 _ =,({)2). (10)

This result was used in a previous paper 5) to calculate the ratio of the 83.4 keV 6(r 2) to the 103.2 keV 6(rZ).

We note that for pure, unperturbed rotational bands the result for the interband 103.2 keV transition goes over to the classical expression

6 ( • 2 ) 1 0 3 . 2 - - 5 2 2 12Ro 2 Z 2 (Q~-Q~), (11)

which can be obtained by substituting the approximate classical expression [ref. 1)] Qo = 32R2fl/x/~ into eq. (2). On the other hand, 6(r2)aa.4, which is a measure of the amount by which the K = { rotational band is perturbed, is zero.

In this paper we use eq. (10) to calculate ~ ( r 2 ) absolutely by taking phenomeno- logival values of Q~ and Q2 from experimental data for eight B(E2) values of E2 transitions between states of the K -- -~ and K = ~ bands of 153Eu. In applying E2 data from transitions to the evaluation of static E2 moments the question of core polarization effects arises, in particular, whether or not in a transition the core polarization follows the particle motion adiabatically. It is known that for the neighbouring doubly even nuclei 152Sm and 154Gd ' which are very similar to each other, that the core deformation varies strongly as one goes up in energy (500-600 keV) [ref. 6)], so it is probable that deviations from adiabatic behaviour occur. On the other hand we deal with low-energy rotational levels which from the experimental evidence of 6 ( r 2 ) and the energy fit of sect. 3 are seen to be weakly perturbed. The evidence from a comparison of the ground state quadrupole moment obtained from E2 transition data and optical measurements (see sect. 4) shows that our values of Q~ and Q~ could be off by 10 to 20 ~ due to non-adiabaticity.

3. Wave functions and energy fit

The band-mixing theory of Kerman 7) treats the Coriolis term

_ h 2 "-~c = ' (I+ j_ + I _ j+),

21o

278 s. hESS

which appears in the Hamiltonian of a rigid top plus odd particle, exactly. Other inter actions which could perturb the pure rotational states, based on intrinsic states of the odd particle, are neglected. By treating the strength of the Coriolis term and the parameters describing the pure rotational bands as effective parameters, Kerman achieved a remarkably accurate fit to the energy level scheme (7 excited levels fitted to better than 300 eV) of 183W. Introduction of additional parameters yielded good fits to the abundant data for transitions. More recent investigations have shown that the introduction of an additional parameter corresponding to K = ___ 2 band-mixing effects 8) or the introduction of other parameters 9) does not greatly change Kerman 's original solution 7).

~+481.1

~*325.1

o~

~* 193.2 --

7+ 83.37

~+ 0.00

t t l K =-~ band

K=-}bond

½+ 269.5

7285t'

to 97./, 3 keY 5-tevet

Fig. 1. Positive parity rotational bands of ~S3Eu. The data is taken from refs. ts. z6.27).

The energy level scheme of 153Eu (fig. 1) with which we are concerned is well suited to a band-mixing calculation. The two positive parity rotation bands which are coupl- ed by the Coriolis interaction are more closely spaced in energy than for the case of 183W, while additional intrinsic states of positive parity are absent. On the other hand the N = 90 core of 153Eu places it in an unstable region between spherical and stably deformed nuclei so that strong perturbations of the rotation bands from rotation- vibration interaction might be expected. This has already been mentioned in connec- tion with changes in core polarization associated with non-adiabatic effects.

Following the notations of refs. 7- 9), the energy fit was carried out as follows. The energies of the two unperturbed rotation bands in keV are: E{(I) = E~°)+ E~)I(I+ 1) for the ground state K = { rotation band and E~(I) = 103.179+E~t)[I(I+ 1)-~-~] for the K = ~- rotation band.

l$3Eu MOSSBAUER TRANSITIONS 279

The Coriolis or rotation-particle (RPC) interaction mixes only states of the same I so that the case of 153Eu differs from the case of lS3W in that the ground state is

mixed, whereas the excited K = k intrinsic state is not. This requires using the I = ground state energy (zero) as input data with about the same error as is assigned to the 83.37 keV, I = ~ level.

The interaction parameter A is defined by the equation

H c = <I, K + II,ZfcllK> = -Ax/(I- K)(I+ K+ 1),

where IIK> is the symmetrized rotational wave function of eq. (6). For 15aEu ' K = { and K + 1 = s. There are then four parameters, E~ (°), E~ t) and E~ 1) and A with which to fit the energies. Since there is no K = ½ band with a decoupling parameter, there is one less parameter than for the case of ls3W.

Energies of the Coriolis mixed states are given by

EH, L(I) = ½ [E~(I) + E~(I)] + ½AE4'I + (2Hc/AE) 2, (12)

where A E = E d I ) .

The mixed wave mixed functions are

The coefficients

are given by

where

IIM>L,n = ~L,n(~)ll, K = ~}>+~L,n(~})]I, K = ~>.

aL({), ell({), eL({) and ~n({)

= =

~/1 + ( R + x/1 +R2) '

~H(~) = - - ~ L ( ~ ) = - 1

x / l + ( - R + ~ / 1 +R2) '

AE R= 2Hc

As described in ref. 9), fits were carried out by minimizing the symmetric quadratic form

17 = Z2-~-~ 2,

where

Ei-- Ti 12

~ 2 = ~ ( ~ - 1 ) \ad (13)

280 J. HESS

where El is an experimental energy with error __+ a~ and T~ is the theoretical energy

calculated f rom eq. (12). The calculations were carried out on the Hebrew Universi ty

CDC 6600 computer .

The energy fit obta ined and the resulting wave funct ions are summarized in table 1.

The parameters which have the best fit were

EG o) = - 106.870 keV,

E(, 1) = 12.739 keV,

E(1) ~_ = 13.020 keV,

A~ = 12.407 keY.

TABLE 1

Energies and amplitudes of positive parity states of lS3Eu

K I Experimental Energy from ~(~) e(~) energy (keV) fit (keV)

~ 0 0.0217 0 . 9 8 6 7 --0.1627 ~ 83.37 (3) 83.28 0.9715 --0.2371 ~ 193.2 (3) 191.00 0.9561 --0.2939 ~ 325.1 (3) 323.46 0.9417 --0.3366 ~a 481.1 (3) 480.88 0.9286 --0.3712 ~ 654.9 (5) 663.44 0.9170 --0.3989 ~ 103.179(3) 0 1 ~ 172.854(3) 172.854 0.1627 0.9867 ~ 269.5 (3) 269.9 0.2371 0.9715 ~ 394.02 0.2929 0.9561

") Energies for the K = ~r band of the 1 = ~ state and up are taken from reL 15). b) The remaining energies with the exception of the 269.9 keV state are averages given in ref. 26). 0 Data for and evidence of the existence of the 269.9 keV level is taken from ref. 27).

The fit obtained here for 153Eu is somewhat less remarkable than that obtained for

1saw, especially for the case of the K = 2, I = @ level which is placed at 663.5 keV,

8.6 keV higher than the experimentally measured 654.9 keV. This displacement is

p robab ly due to the neglected ro ta t ion-vibra t ion interaction. On the other hand there

is one less parameter available here and also for the case of ~83W, energies only up to

554 keV were considered.

4. Determination of the E2 intrinsic matrix elements Q(KK') from experimental B(E2) values. Calculation of 5(r2~

For Coriolis mixed ro ta t ion bands the B(E2) reduced t ransi t ion probabi l i ty is given

by [cf. eqs. (7)-(9)1


2 I i B(E2, I i ~ If) = l @ ( 2 1 f + l ) [ ~ ei(Ki)c~f(Kf)(-1)h +rf (IKf K , - Ke- Ki]

L-Kt, Kf=2 -,

× Q ( K f K 3 e 2 b 2, (14)

where the transition is between the initial state

I/i M i > = ~i(~)l/i n i K = 3> + ~i(~)[/iMi K = ~>

and the final state

I l f m f ) = ~f(~)l/rMrK = {)+o~e({)llrMeK = ~>.

As for the energies, the symmetric form eq. (13) was minimized using eq. (14) with the c~(K) of table 1 to calculate the theoretical B(E2) values. The experimental data for both B(E2) and B(M1) values are summarized in table 2. In this table we have listed all the measurements with errors which went into calculating each B(E2) and B(M1).

The Q(KK') values which have the best fit were

Q(5) = 6.502,

Q(3) = 4.779,

Q(3, ~) = -0 .892.

For comparison the experimental and fit B(E2) values are given in table 3. Substitution of Q(~), Q(~) and amplitudes ~(K) from table 1 into eq. (10) gives the

following results for the 83.4 and 103.2 keV 6(r2);

6(r2)83.4 = --0.0147 fm 2,

t ~ ( r 2 ~ l O 3 . 2 = -0 .483 fm 2.

The constant 5/12R2Z 2 in eq. (10) was evaluated as 0.0255 fm 2 for Z = 63 and Ro = 1.2 × (153) ~ = 6.42 fm. In sect. 6 we will discuss and compare these results with experiment.

As a measure of how accurate our assumption of adiabaticity is, we want to compare the quadrupole moment of the I = 5 ground state obtained from the calculation with the result obtained from spectroscopic data. We have in general for Coriolis mixed 153Eu states

Qs = ( IM = I[2r2p2(cos O)IIM = I )

= ( 2 I + 1 ) - I 0 KK~,~x(K)o~(K,)(_I),+K, I K' K - K' JK' Q(K'K),

which on substituting for the Q(KK') and the ground state ~(K) coefficients yields

Q~ = 2.361 b.

282 J. HESS

TABLE 2

Relative 7 intensity normalized to 106 units for the 103.2 keV 7 and 2 (branching ratio) measurements

Transition Lifetime Relative gamma Electron con- B(E2) B(M1) energy in t~ of initial intensity nor- version coetii- (e 2 • b 2) ( e ~ / 2 M c ) z

keV state malized to 106 Multipolarity cient l+C~rJ) (nsec) units for the and/or other

103.2 keV 7;t conversion measurements f) electron in-

tensity data

103.179 3.80(2) a) 100 × 10 ¢ 98.3 ~ M1 2.76(20) d) 0.0076(40) 0.0033(5) 1.7(8) ~E2 ')

19.82 3.80(2) ~) E2 2286(200) ~) 0.0043(20) L/KIoa.z = 0.014 d)

172.854 0.14(4) b) 2450(500) c) 63 ~ M1 3300(400) d) 37 ~ E2 ~) 1.10(20) c) 0.0024(8) 0.9(3) ×

10-4 69.676 0.14(4) b) 192(12) × 103 98.1 ~o M1 5.99 d) 0.75 (40) 0.13 (5)

1.9(8)~o E2 g) 89.48 0.14(4) b) 4300(600) d) 98 ~o M1 6.3 d) 0.0051(30) 0.0014(5)

2(I) ~o E2 ' ) 193.2 E2 0.42 (3) k) 109.8 )~ = 0.385(8) e) 32 ~ E2 0.871 (60) 0.0156(8)

68 ~ M1 8) 83.37 36 ~ E2 1.71 (7) k) 0.0148(8)

64 ~ M1 g) 75.43 0.14(4) b) 65(8) × 103 d) E1 h)

a) Ref. 28). b) Ref. 29). c) Ref. zo) ~) Ref. 31). e) Ref. 16). r) Relative gamma intensities are taken from the compilation of ref. no). s) Ref. 32). ~) Gamma intensity of the 75.43 keV transition is needed to calculate some of the other gamma intensities. ~) Appendix 5 of ref. 5) and ref. 3a). J) Experimental data from compilation ofref, zo). 19.82 keV 1 + ~ r is from theoretical tables ofref. 33). k) Ref. 34).

TABLE 3

Comparison of B(E2) values obtained from experiment and from theoretical fit

Transition energy (keV) 83.37 103.179 172.854 69.676 89.48 109.80 19.82 193.2

B(E2) exp 1.71(7) 0.0076(40) 0.0024(8) 0.75(40) 0.0051(30) 0.871(60) 0.0043(20) 0.42(3)

B(E2) theor. 1.50 0.00004 0.0026 0.80 0.0055 1.283 0.00025 0.40

O p t i c a l Qs m e a s u r e m e n t s g ive t h e r e s u l t s ~°)

Os Ref.

2.92(20) Muller et al. 23) +2.42(20) Krebs and Winkler 24) +2.94(23) Winkler 25)

15 3Eu MOSSBAUER TRANSITIONS 2 8 3

One o f the opt ical measurements agrees with the result deduced f rom B(E2) t rans i t ion

d a t a while two do no t and suggest tha t the value for Q(~) could be out by 20 ~o. This

20 ~ can be t aken as a measure o f the uncer ta in ty , for 1 5 3 E u ' in t roduced by assuming

ad iaba t i c i ty and tak ing the Q(KK') f rom E2 t rans i t ion da ta to be the same as the

Q(KK') appea r ing in expressions for the static moments .

5. Determination of M1 intrinsic matrix elements G(KK') from experimental B(M1) values. Comparison with static M1 moment data *. Branching ratios

This sect ion is in tended as a fur ther check on the calcula ted wave funct ions and

on the consis tency o f the da ta of table 2.

Ana logous to eq. (14) for B(E2) t ransi t ions , we have for B(M1) t ransi t ions,

3 ( 2 I f + 1) B(M1, I i --~ If) =

x [ ~ ~,(Ki)~f(KO(-l)"+<f ( If ] I, )G(K,,K,)]2(eIi ]z. r,, rf=k. -~ Kf Ki -- Kf - Ki \2Mc /

The same fit as was carr ied out with the E2 t rans i t ion da t a o f table 2 was carr ied out

for the M1 t rans i t ion data . The G(KK') ob ta ined were:

G(~) = 0.459,

G(~) = 0.983,

a ( L ~) = - o . o o l -~ o.

The compar i son between the fit and exper imenta l values is given in table 4.

TABLE 4

Comparison of B(MI) values obtained from experimental and from theoretical fit

Transition energy (keV) 83.37 103.179 172.854 69.676 89.48 109.8

B(M1) exp. 0.0148(8) 0.0033(5) 0.9(3) × 10 -4 0.13(5) 0.0014(5) 0.0156(8) B(M1) theor. 0.0123 0.0025 0.8 × 10 -4 0.06 0.0035 0.0188

In table 5, we compare o-factors o f static M1 momen t s calcula ted f rom the above

G(KK') values with exper imenta l data . F o r Coriol is mixed wave funct ions the mag-

netic momen t s are given in nuclear Bohr magne tons by:

# = I E ~ (K)~(K ' ) K,K'=~,~

1 gR(K)r~KK, 1 = gI. " I ( I + 1 )

t The analysis given here of magnetic data has previously been carried out by Atzmony 11) and Richter, Henning and Kienle 12). These results differ somewhat from the results given here due to different input data.

284 J. HESS

To calculate magnetic dipole moments one needs an additional constant, gR(K), the collective gyromagnet ic ratio, which cannot be obtained f rom transition data. For

the g round state K = ~ band we use the theoretical result ~3) gR(~) = 0.48 and for the K = 3 band we take, for lack of information, as a reasonable guess gR(~z) = gR(~)-

TABLE 5

Magnetic dipole moments

Band State energy Spin #(exp) g(exp) Experimental g(theory) refs.

K = ~ ground state ~ -}-1.53 (2) 0.612 (8) 35) a) 1.5292(8) 0.6117(4) a6) a) 0.6126

K = ~ 83.4 keV ~ 1.88 (4) 0.54 (1) a7) 0.550 1.80 (8) 0.51 (2) 12)

K = 23 103.2 keV .~ -}-2.04 (8) 1.36 (6) as) ~) 0.87 3~) a.b)

a) Taken from the compilation of Shirley lo). b) Shirley, recalculation based on data given in other references.

For M1 moments it is to be expected that the transverse magnetic polarization associated with transitions should differ f rom the longitudinal polarization associated with static moments 14). in theoretical calculations of the matrix elements G(KK') this is expressed by introducing two effective spin gyromagnet ic ratios g~o f and e f f gsl " Although in general g~o ff # g~( f, the possibility that g~o ff = g~ f is not ruled out and

one may hope that if one magnetic momen t o f a rotat ion band is given correctly by the procedure used here, then another magnetic momen t o f the same band should

also be given correctly. This appears to be the case for the K = ~ band. On the other hand for the K = 3 it appears that g~f is definitely not equal to g~f. Part o f the large

discrepancy between the theoretical and experimental result for the K = ~ 103.2 keV level may also be due to uncertainty in gR(3).

TABLE 6

For 2 = cascade/crossover

Initial level 2(theor.) 2(exp.)

Seaman et al. ~5) de Boer ag) a) Boehm et al. 4o) a)

325.1 keV I = ~ 0.198 0.151(9) 0.42(20) 0.141(11)

481.1 keV I = ~ 0.131 0.088(21) 0.93(60)

a) Denotes data taken from the compilation of ref. as).

In addit ion to the x 53Eu data o f table 2 which were used to calculate experimental B(MI~ and B(E2) values, two more branching ratios for the K = ~- band have been measured. In table 6 we compare the theoretical results with the experimental data.


6. Discussion. Comparisan of results for ~(r) 3 with experiment

In table 7 the theoretical values for ¢~(r2) which were calculated in sect. 4 are compared with experimental data.

TABLE 7

Theoretical and experimental values for 6(r 2) with factor 2 included (see text)

Transition energy (~(r 2) theor. 6(r 2) exp. Exp. refs. (keV) (fm 2) (fro 2)

103.2 --0.483 --0.085 (13) 41) --0.080 (25) 42)

83.4 --0.0147 --0.00085(60) 2) --0.0021 ( 8 ) 12)

In the past it has been assumed that the chemical factor in the isomer shift expression (eq. 1) can, for solids containing Eu ions, be estimated from spectroscopic data on gaseous ions. Recently, a spectroscopic measurement has been made in solid Ca(Eu)F2 [ref. 16)], from which it was concluded that this assumption is wrong by a

factor of 2.06, or within the experimental accuracy 2 *. It is concluded that previous determinations of the chemical factor for europium compounds were too small by a factor of two and the resulting 6(r 2) were too large by a factor of two. In table 7 all the data for 6(r 2) includes the factor of 2 of ref. 16).

Referring to table 7 it seems that the procedure used here gives a poor estimate of the 83.4 keV shift. The disagreement is probably due mostly to rotation-vibration interaction, which has been neglected and is expected to give a positive contribution of about the same size as the calculated Coriolis interaction contribution. The experimental data for the 83.4 keV transition has however an interesting consequence in that it makes it possible to estimate the possible rotation-vibration contribution to the 103.2 keV shift. This is due to the negative sign of 6(r2)s3.4 which shows that for this case the Coriolis contribution is larger than the positive rotation-vibration interaction contribution and hence for the 40 to 100 times larger 6(rZ)lO3.Z, may be neglected entirely.

While one does not expect agreement between the calculated and measured 6(r 2) for the 83.4 keV transition, the large disagreement for the 103.2 keV 6(r 2) is surprising. To illustrate this point we note that for pure rotational bands the quantum mechanical expression for 6(r2)1o3.2, can be written, [cf. eq. (10)]

-- 5 t(g)2 ,O2~ /5~2 __ 5 ¢Q2_Q2~ 0.9735, ~(F2~103.2 12R2Z2 ~ , ~ - - - ~ / g.s.t~/ 12R2Z2 t ~ 7/

* From the published work [ref. 16)] it seems that a theoretical calculation of the lattice zero point vibration energy is included in this determination, the accuracy of which might be a cause of some concern. On the other hand, it is the only direct determination in a solid and the result brings about agreement between experimental determinations of 6(r 2) by M6ssbauer effect and muonic X-ray measurements [ref. 2)l.

286 s. HESS

which differs by less than 3 ~ from the expression for a classical charge distribution [eq. (11)],

5 Q2 Q2 6 ( r2 )1o3 .2 - 12R2Z~( ~ - ~).

This is, in part, a consequence of the fact that the 103.2 keV I = ~ state is unaffected by the Coriolis interaction. I f we consider the rotational levels to be unperturbed then the squares of the intrinsic quadrupole moments can be obtained directly from the usual expression for pure rotational bands 17),

B(E2, I i ~ I 0 = (2If+ 1) 0

by using only experimental B(E2) values for the 83.4 keV and 69.7 keV transitions which yield (see table 2)

Q~ = 48.14_+ 0.20,

Q~ = 22.0 _+11.0 .

The 4 % experimental error in Q~ can then be neglected next to the 50 % experimental error in Q~ so that ifeq. (11) is written as,

~<r2>1o3.2 = - 12Ro2Z 2 ~ \ ~ / ]

the entire experimental error appears in the ratio (Q~/Q~)2 and can be attributed to Q$ only. We have then the approximately correct theoretical result,

6<rZ> 1 o3.2 = - (0.67 _+ 0.30) fm 2.

The purpose of the preceding approximate calculation was to support the assertion that while for the theoretical result, ~<r2>1o3.2 = -0 .483 fm 2 one might expect a 50 % discrepancy with experiment, a 600 % discrepancy is surprising.

To summarize the arguments: (i) 6 ( r 2 > 1 o 3.2 may be accurately calculated with the classical expression eq. (11 ),

a fact justified by the quantum mechanical formulation of sect. 2. (ii) The negative sign of 6(r2>s 3.4 shows that perturbations from Coriolis inter-

action have a larger effect on 6(r2>sa.4 than perturbation from R-V interaction. Since for c5(r2>1o3.2 Coriolis perturbations are less than 3 %, the same is true for R-V perturbations in this case.

(iii) The agreement to within 20 % of values of the ground state quadrupole moment obtained from optical measurements and from B(E2) transition data gives a measure of the uncertainty introduced by non-adiabaticity. That this uncertainty is probably less than 20 % is also suggested by the experimental evidence for the small- ness of R-V perturbations.

15 3Eu MOSSBAUER TRANSITIONS 287

In conclusion it seems possible that some ambiguities still remain in experimental determinations of ~(r2).

The author would like to thank Prof. S. Ofer, Dr. I. Nowik, Dr. J. Burde, Dr. R. Bauminger and Dr. I. Unna for help and discussions.

Appendix

We wish to establish a correspondence in the limit h or h ~ 0 between the expectation value ( r~f) [eq. (1)]

2 f~-~t3(x) O(x)b (reff> = a~( X (---J ~ - u ~ u - ) .=0 p= --),

and the classical expression [eq. (2)]

3 R2-2,

~a=+2f 1)~_a~a in the ~ variables used by where f12 is the spherical invariant /__.~=-2~- A. Bohr to describe the nuclear surface 3). The e~, (for charge distribution) are given classically by tT)

_ 1 fvr2y2~dar" %

A corresponding "quantum mechanical" expression suggested by the above expression is

4 __1 (air 2 y2~]a)" ~x~, = -gn Z R 2

Using this expression for % it is possible to establish the correspondence for the 2 = 2 term. Considering expectation values we have,

/ ~ = + 2

a2 E ~, (a l ( -1 )"Q-v la ' ) (a ' lQ~ , la ) , ~=-2 a"

i t = + 2

a22¢ -7~ E (alr2y2vla)(alr2y2.la) *" #= -2

In this step the space in which the Qv operates goes over from a space of dimension 2 I+ 1, where I is the total proton spin (finite number of quantum mechanical degrees of freedom) to coordinate space (infinite quantum mechanical degrees of freedom). Furthermore, off-diagonal elements of the matrix (a lQ, la') are neglected, a step which we will show is justified for t 5 3Eu" The effective vanishing of these off-diagonal matrix elements for 2 = 1 and/ t = 3 implies that at = a3 = 0, since for states of a definite parity static E1 and E3 moments (diagonal matrix elements) vanish. If we

288 J. HESS

now go over to a completely classical expression using the substitution

we have the correspondence

JL=+2

a2(alp=C_2(-l~Q-,Q,la> -+ %,4Z2P2a2.

Comparing this with the corresponding B-term in the classical expression [eq. (2)]

gives the value for a,,

5 a2=-.

I2R; Z2

Similarly a,, = &Rf from the correspondence

(alQ’“‘Q(o’la) --+ 4.

We now examine on the quantum mechanical level the neglecting of off-diagonal

terms of the matrix (alQy)la’) for the case of 1 53E~, which leads to the conclusion that a, = 0. In the classical expression [eq. (2)] the term linear in j3 vanishes due to

the orthogonality of the spherical harmonics used to describe the nuclear surface

[see ref. ‘)I. These spherical harmonics correspond, in the classical limit, to the dia-

gonal elements of the matrix (alQF)la’). Without the neglect of off-diagonal matrix

elements - the classical expression can be approximately regarded as an expression

in the parameter rip=+?,

9 p=C_i+lJPQ$Ql”‘h

which takes the form

In sect. 2 it was shown that for rotational wave functions the terms may be written

as the sum over squares of intrinsic matrix elements,

(K]Q’“‘]K’) = Q’“‘(KK’),

p= +1

a=C_n((-lYQ?Q:l)) =~,l(KlQ’i’lK’)12, then

The intrinsic matrix elements Q(“)(KK’) have been investigated for ’ 53E~ and there

exist theoretical estimates supported by experimental work for these quantities. In


terms of Nilsson mat r ix elements 18) the Q(Z)(KK') are given by Q(Z)(KK') = (6p + ( - 1 )zZ/A~)(h/Mo~o)~Z2GE~ where (6p + ( - 1 )~Z/A ~) is a factor which describes

the effective charge, x/h/Mo~ o is a mean nuclear rad ius and GE~ is the Ni lsson mat r ix e lement of the ope ra to r E~.

F o r the electric d ipole matr ix elements M a l m s k o g et al. 19) give theoret ical values

for 153Eu ' cor rec ted for pa i r ing interact ion. Inser t ing app rop r i a t e factors o f (1 - Z / A )

and (h/Mo~o) gives the intr insic mat r ix elements

Q(1)(~ 2 ~) = 0.0097 fm 2,

Q(1)(2~ ~) = 0.00073 fm z.

Evidence f rom exper imenta l B(E1) values show that these mat r ix elements should

be even smal ler so that Q(1)(~, ) ) ~ 0.01 serves well as an upper bound . Tak ing for

1SaEu, Z R 2 = 0.22 b, the d ipole te rm in eq. (A.1) is then app rox ima te ly

Q(1)Qo) ~ (0.01)2 × 10-26 ZR 2 - 2 x 1 0 - 2 3 = 0 . 5 x 1 0 -7.

The cor respond ing es t imate for the E2 term, using the phenomenolog ica l values for Q(2) o f sect. 4 is

Q(2)Q(2) ~ 40 x 10 -4s - 0.08.

(ZRo2) 2 -- 5 × 1 0 -46

Thus neglecting the E1 te rm next to the E2 te rm is justified. F o r the E3 te rm the only

i n fo rma t ion avai lable is f rom an exper imenta l B(E3) value for collective t ransi t ions o f 152Sm [ref. 2o)], which gives the es t imate

Q(3) Q(3) _ 0.3 e 2 • b 3

and a co r respond ing te rm in eq. (A.1) o f

Q(3)Q(3) _ 3 0 x 10 -74 - 3 x 1 0 -5.

(ZRo2) 3 10 -68

Thus the E3 te rm and also p r o b a b l y higher mul t ipole cont r ibu t ions can be safely neglected.

References

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290 J. HESS

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Documents

Calculation of δ 〈r2〉 for mössbauer transitions using coriolis mixed wave functions.: Application to 153Eu