Isomer-shift analogue in neutron resonances

Nuclear Physics @ North-Holland

A362 (1981) 18-44 Publishing Company

ISOMER-SHIm

ANALOGUE

A. MEISTER*,

D. PABST,

IN NEUTRON

L.B.

PIKELNER

and

RESONANCES K. SEIDEL*

Laboratory of Neutron Physics, JINR, 141980 Dubna, USSR Received 18 July 1980 (Revised 3 November 1980) Abstract: For the first time, the recently predicted chemical shift of neutron resonances, to be regarded as an analogue to the Mijssbauer isomer shift, has been experimentally observed studying the 6.67 eV shifts were determined by a chi-square fitting technique from resonance of 238U. The experimental the time-of-flight transmission spectra of metallic uranium and four uranium compounds measured at the Dubna IBR-30 pulsed reactor. A computational method has been applied to estimate, and compensate for, the influence of the crystal-lattice vibrations on the experimental values thus obtained. The electron density differences at the nucleus have been calculated for the various sample pairs using available data on chemical X-ray shifts in uranium compounds, on Mijssbauer isomer shifts in isovalent neptunium compounds and on free-ion electron densities. The resonance shift results lead to the conclusion that the mean-square charge radius of 238U diminishes by 1.7’;: fm* upon capturing the resonance neutron.

E

NUCLEAR REACTIONS shifts. 238U deduced

238U(n, y), E = 6.67 eV; measured chemical resonance energy nuclear mean-square charge radius change. Natural target.

1. Introduction The

well-known

Mijssbauer

isomer

shift ‘) measures

the

minute

difference

between the excitation energies observed for the same isomeric state of a given nuclide in two (chemically) different samples. This energy difference is caused by the hyperfine interaction between the nucleus and its surrounding electron cloud. The same hyperfine interaction must be expected to affect the position of the neutron resonance levels of the nucleus in quite a similar way as first suggested in 1973 by Ignatovich eta/. *). The location of the resonance, as observed, e.g., in a transmission experiment, should therefore somewhat depend on the chemical environment in which the nuclide under consideration is embedded. As may be anticipated from the Mossbauer spectroscopy analogue [cf. eq. (26)], the resonance energy shift between two samples turns out to be roughly proportional to the product of the electron density difference Ape(O) at the nucleus found between the two samples, and the change A(rE) in the nuclear mean-square charge radius due to the capture of the neutron. * Present Mommsenstr.

address: 13.

Technische

Universitlt

Dresden, 18

Sektion

Physik,

DDR-8027

Dresden,

A. Meister et al. / Isomer-shift analogue

It is obvious

from the preceding

effect would have a twofold may be expected to provide

that the experimental

detection

19

of this hyperfine

physical significance. Firstly, measurements information about the shape of the nucleus

of this kind at relatively

high excitation energies since the neutron binding energy released upon the formation of the compound-nucleus state is transferred to the nucleus. Generally, little is known about the shape of the nucleus in this excitation energy range and cannot be obtained by other means. Secondly, such data should very nicely supplement the information on electronic structure problems drawn from Mossbauer experiments. A crude assessment reveals that, under favorable circumstances, the expected chemical shift of a neutron resonance should be of the order of 10e4 eV. This has to be compared to the widths of low-energy neutron resonances which typically lie near 0.1 eV. Nevertheless, as far as statistics is concerned, even such small level shifts can reliably and rather accurately be measured in transmission experiments employing the time-of-flight method and a chi-square fitting technique. This was demonstrated in experiments aimed at measuring the magnetic moments of compound-nucleus states 3). However, the experimental shift AE,, thus determined will be, in the present case, a superposition of the wanted chemical shift AEo and a falsifying displacement AEev due to the influence of the crystal-lattice vibrations on the shape of the resonance cross-section curve:

AE,, = AEo + AEev .

(1)

The additivity of AEo and AEE, follows from the fact that AEo implies a simple translation only; this translation having been performed, the remaining small cross-section difference found for the two respective samples will produce the displacement AEtv in the course of the chi-square fit. Thus, the perturbing vibrational shift AEev must be properly taken into account, which per se is a difficult problem since AEp, proves to be of the same order of magnitude as AEo. In the present paper, we report on work in which it was possible to overcome this problem and to determine, for the first time, the chemical shift of a neutron resonance. The experiment was carried out at the Dubna IBR-30 pulsed reactor using various compounds,

as well as the metal,

of natural

uranium

and measuring

precisely

their

transmission spectra in the vicinity of the 6.67 eV resonance of 238U. This resonance offers the advantages of being well isolated and pronounced. In addition, uranium distinguishes itself by possessing several oxidation states. This ensures relatively large electron density differences between properly chosen chemical compounds. Another favorable feature of uranium lies in the fact that its nucleus is strongly deformed. There is reason to suppose that the nuclear deformation becomes smaller as the excitation energy increases; thus, the mean-square radius change could also be expected to assume an acceptable value. In the following, we shall first describe our experimental approach and the technique used to determine AE,,. After that, a method will be outlined which permits the determination of AEf,. Then electron density differences for the various

20

A. Meister et al. / Isomer-shift analogue

sample pairs used in our measurements will be derived. This will be followed by the deduction of the mean-square radius change from the chemical shift values. Finally, the result thus obtained will be compared with available theoretical findings. 2. Experimental

setup and data handling

The time-of-flight transmission spectra of the samples were measured with the pulsed reactor operating in booster mode “). A TPA-i minicomputer was employed on-line in the experiment 5, (cf. fig. 1). A set of three samples was always used, the samples being consecutively placed into the neutron beam for about five minutes. After this period, the spectrum accumulated was checked for any instability in reactor power and for electronic drifts using the accumulated number of counts of a BF3 monitor, the sum of the counts within a suitable channel range of the transmission spectrum and the accumulated number of reactor pulses. If certain requirements were met, the spectrum was then transferred to the respective section of the computer storage and added to the previously measured spectra; otherwise, the run was repeated. This approach ensures equal experimental conditions for all samples and instabilities largely cancel each other.

Fig. 1. Experimental setup. I, II and III, uranium samples and their pertinent accumulated transmission curves, respectively. T, timer; Cl and C2, counters; A and AC, sample-changing control modules; VK-5, time-coder; CC, CAMAC crate controller; D, neutron detector; R, terbium reference sample.

The neutron detector used comprises a glass scintillator enriched in 6Li and a FEU-49 photomultiplier. The over-all counting-pulse path with direct access to the computer buffer storage is operable up to rates of 5 x lo5 pulses/s guaranteeing the high statistical accuracy needed. A ls9Tb sample was permanently put in the beam. The resonance dips of this sample permitted an unbiased evaluation of the shifts between the measured spectra. A typical experimental curve obtained for UOS and accumulated together with the curves of the other two samples (not shown) within a ten hour period, is presented in fig. 2.

A. Meister et al. / Isomer-shift analogue

N

HO61

209

‘, \'<,

I 245

667

IL.4111

335

21

233 u

eV

n.

ev

n .15'Tb

1.0 -

Fig. 2. Experimental

time-of-flight spectrum obtained with a channel width of 2.0 JLS.t is the channel number, N the counts per channel.

The half-width of the 6.67 eV resonance dip equals about fifteen channels whereas the dip shift looked for is of the order of several hundredths of a channel. It is determined by the following chi-square fit. Let the pulse numbers of the measured time-spectra 1,2 in channel t be denoted by Nr(t), Nz(t). Transmission spectrum 2 is then adjusted to spectrum 1 by varying A, At and Nr, in the expression (24

and minimizing ‘2 [Nl(t)

“=,?t,N

1

-Ni (t>12

(t)+Nz(t)

(2b)

’

for a given channel range (tr, tz). The parameters A and Nn take into account any minor differences in the dip depths and backgrounds remaining in the transmission spectra after properly equalizing the samples with respect to thickness, potential scattering, beam position, etc. Fortunately, whilst the parameters A and Nn are somewhat correlated with one another, At was found to be uncorrelated to the two other parameters. The displacement At together with the flight path L gives the experimental shift AEex of the resonance energy E. via the relation AE,, = -2.77

x lo-*E;'*At/L

,

(3)

with AE,, and E. in eV, At in l.~.sand L in m. The fitting procedure described above was applied not only to the 6.67 eV uranium resonance but also to the lJ9Tb resonances at 3.35, 11.1 and 24.5 eV. The total measuring time spread over several measuring periods amounted to some 800 h. Altogether, five chemically different samples prepared from natural uranium were used. Three samples out of the set of five were employed during each measuring period. It may be added that some measurements were performed at elevated sample

22

A. Meister et al. / Isomer-shift analogue

temperature three samples

or with increased

sample

used in any measuring

For the purpose shift are presented The 238U resonance 11.14 eV.

of illustration,

thickness. period

However,

agreed

to better

some typical results obtained

in fig. 3 and table 1. The errors quoted shifts were corrected

n

the thicknesses

of the

than 1% . for the experimental

in table 1 are statistical.

for the diplacement

of the ls9Tb dips at

“‘Tb

14 t

-r

t

10 8 6

0 -10-6-2

2 6 10

At -2

2

6 10 12

At

Fig. 3. Histograms of experimental shifts for the 6.67 eV resonance of *% and the 24.5 eV resonance of 15’Tb. These results refer to the UOz -UF4 sample pair. The shifts are defined as At = r(UO,)-t(UFJ where @JO,) and t(UF,) are the respective time channel resonance positions with UOI, or UF4 placed in the beam. They are given in units of a thousandth of one channel width. The fit ranges comprised 41 channels with the uranium and 21 channels with the terbium resonances. Experimental conditions: sample thickness, 0.552 x 10” nuclei per cm*; sample temperature, 374 K; flight path, 64 m; channel width, 2.0 ps.

The results of table 1 demonstrate that the experimental shifts depend rather strongly on the width of the channel fit range. This phenomenon can be traced back to the influence of the crystal lattice vibrations in the samples as will be shown next.

3. Evaluation

of the crystal-lattice

vibration effect

In principle, the crystal-lattice vibration shift AEev is deduced in a two-step procedure 6S7). First, the neutron transmission curves of each sample pair are calculated as accurately as possible assuming experimental conditions which correspond precisely to those of the measurement. In these calculations, the effect of the crystal-lattice vibrations is described within the framework of a model, the parameters of which may be determined from the experimental data. However, the resonance energies of both samples of a given pair are put exactly equal, i.e., the chemical shift AEo is assumed to be zero. After that, the same fitting as described by

A. Meister et al. / Isomer-shift

23

analogue

TABLE 1 Experimental shifts of the 6.67 eV resonance

14

of 238U “)

10

6

8

1-2 (A)b) uo3-unet

U03-UFd.HaO

-16.4*2.7 4.3k2.0

5.0*3.1 8.9jz2.3

-9.9+2.8

17

40

15.6zt4.7 2.1k3.1

9.2*2.6

11

7

10

(B)‘) uo3-umt

UF.+.HsO-U02

-11.3zt2.9 -10.1 f 1.5

-3.3*3.0

lO.lzt3.3 0.7* 1.7

27

30

3.9zt3.6 5.2*1.9

20

18.8i6.1 14.3zt3.1

8

12

(qb) uo3-unet

-69.Ok2.0

UOZ(NCMY~H~O-U,,, UOz(N03)y6Ha-UOs

-69.9k2.5 -1.5+2.5

-69.7*1.9 -70.8k2.6 -0.8 * 2.3

-65.3i2.5 -66.1+2.3 -2.4k2.3

-41.1*2.1 -42.6~2.5 -2.8+2.5

-19.6+2.7 -21.8*3.0 -3.4*3.5

“) Time channel shifts defined as At = tl - t2 where tl and tz are the resonance positions of the samples 1,2. The shifts are given in units of a thousandth of one channel width. b, The labels A, B and C refer to the following experimental conditions: (A) Sample thickness, 0.552 x lo*’ nuclei per cm*; sample temperature, about 293 K; flight path, 55 m; channel width, 2.0 es. (B) Sample thickness, 0.552 x lO*r nuclei per cm*; sample temperature, 374 K; flight path, 64 m; channel width, 2.0 IJ.S. (C) Sample thickness, 1.19 x lo*’ nuclei per cm*; sample path, 60 m; channel width, 2.5 t.~s.

temperature,

about 293 K; flight

eqs. (2) for the experimental spectra is applied to these calculated transmission curves. The resulting displacement At will then directly yield LIE,,. To be able to calculate the neutron transmission spectrum in the vicinity of the resonance under investigation, one must know the cross section u as a function of the neutron kinetic energy E, in the laboratory system. Generally, this function is given by the convolution integral u(E,) =

I

S(E’)B(E,

- E’) dE’ ,

where B(E,) denotes the Breit-Wigner relation and S(E’) the transfer function describing the exchange of kinetic energy between the incident neutron and the sample crystal lattice. Let the crystal be isotropic. The transfer function can then be expressed in terms of the normal-mode frequency spectrum p”(hv). For uranium

24

A. Meister et al. / Isomer-shift analogue

compounds of the type UXI with the uranium mass Mv being much greater than the ligand mass M,, the normal-mode frequency spectrum can be expected to be similar to that of an infinite chain of coupled oscillators in which large masses alternate with small ones 8V9).To a good approximation, it can be described by pu(hv) = a1 S(hv - hVl) + a2 S(hv - hY2) . The well-separated

(5)

normal frequencies v1 and YZare interconnected l&2=

In addition, there is a normalization

by the relation

(MJM”)1’2.

(64

condition, namely al+az=l.

(6b)

The frequency spectrum pu(hv) to be used in the present case is weighted by the squared moduli of the polarization vectors describing the vibration of the uranium atoms. This is so because only those modes come into play which are excited by the impact of the neutrons on the uranium nuclei9-l’). It thus differs from the total frequency spectrum as measured in inelastic neutron scattering or specific heat experiments where all modes may be excited. Using the weighted frequency spectrum (5), the integral in eq. (4) can be split up to yield

+S2r(Ss,B(En+~hv1-

77~2)+S4~B(En+~~hvl+77hv2)1).

(7)

The quantities Si, and S, are statistical weight factors associated with the excitation and de-excitation of p phonons of frequency vi and n phonons of frequency ~2. They are given by &)“(b:)“E @:)“(~;)“-P w!(w-p)! ’ w!(w-p)! ’ s2p=

sl,=

W=p

s3p=

W=p

(~:)“&)“-p

f

W=p

f

w!(w-p)!

sgp=

’

E W=p

(8)

&)“(b:)“-P w!(w-p)!

*

The factors b: (i = 1,2) are obtained from Bose-Einstein statistics which relates them to the recoil energy E,/(A + 1) transferred to the nucleus, and to the phonon energies hVi : 6:

=E,[(A

br =E,[(A

The quantity D(E,) expression %%)

= exp {-&/(A

+ l)hvi]-‘ai[l + l)hvi]-‘a,[exp

-exp (-hvJkT)]-’ (hv,/kT)

in eq. (7) is the Debye-Waller

- 11-l .

,

(94

(gb)

factor calculable from the

+ l)[(allh vl coth (hv1/2kT) + (uJhv2)

coth (hv2/2kT)]}. (10)

25

A. Meister et al. / Isomer-shift analogue

The Breit-Wigner formula including interference be written as B(E)

=

and potential scattering terms may

(~~‘/2m,)~,/E~‘*~~,/E”*+~~/E~‘* + WWWd’*R(E -Eo)}+~~~z (E-EO)*+$*

,

(11) where r, is the neutron width at resonance energy EC,,f, the radiation width, f the sum of r, and r,, R the potential scattering radius, and m, the neutron mass. For the present calculations, the standard BNL-325 cross-section atlas values of EO = 6.67 eV, r, = 1.52 meV, and r = 27.5 meV have been used. For metallic uranium and U308 the weighted frequency spectrum parameters listed in table 2 have been found by fitting the pertinent uranium resonance patterns TABLE

2

Parameters of weighted frequency spectra Sample

umet 7 U308

9,

uoz

lo)

UFz,.HzO uo3

UOz(N0&6HzO

al

hvI (eV)

1.00 0.90* 0.05 0.93

0.011 0.013 l 0.003

0.012 0.008~0.001 0.53*0.06 0.82~kO.04 0.014*0.001 0.82ztO.04 0.014*0.001

a2

hvz (ev)

0 0.10*0.05

0 0.052 f 0.008

0.07 0.47ztO.06 0.18iO.04 0.18kO.04

0.046 0.028~0.003 0.054*0.004 0.054*0.004

to experimental data measured at 4 K, 77 K and 293 K [ref. ‘)I. For the purpose of clarifying the concepts involved, these parameter values have been used to compute the cross-section curves of those two materials. The results are displayed in fig. 4. Here, the left-hand top part which is for the metal clearly demonstrates how the total cross section is made up of individual constituents. The heavy solid line within the curve family represents the recoilless component while the respective curves on the left and right are related to the de-excitation and excitation of allowed lattice vibration modes. The figure also reveals that substructures in the weighted frequency spectrum finer than the width of the Breit-Wigner term are smoothed out in the cross section curve. This explains why the use of eq. (5) yields a good description of the resonance pattern although it contains only simple S-functions. The recoilless component is also separately shown in the right-hand top part of fig. 4 pertaining to U30s. The graphs marked with U and X represent the cross section constituents due to the excitation and de-excitation of phonons of frequencies v1 and v2. The curve labeled U + X stands for the simultaneous excitation and de-excitation of both hvl and hv2 phonons. It may be mentioned that the curves U, X and U + X are typical not only of II308 but of all the compounds used. As far as the wanted lattice vibration shift AEev is concerned the energydependent difference between the cross sections of any sample pair is particularly

A. Meister et al. / Isomer-shift analogue

26

A6

7~

6.60

6.70

c

loo : 6 U,O,-6ilUl

-100:

T=300K

61max

y , ,,,, ,,,,, , , 111~11111111111111111111 650

6.60

6,770

6.80

E&VI

Fig. 4. Comparison of cross-section curves for the 6.67 eV resonance of 23sU at 300 K. Top left: metallic uranium.

Top right: U30s.

Bottom:

difference

of U,,,

and U30s

cross sections.

important because this difference must be expected to affect the minimum chi-square value eventually obtained in fitting one. transmission spectrum to another. For metallic uranium and U308, the cross-section difference is shown in the bottom part of fig. 4. Similar curves should occur with all the sample pairs used since the locations of the peaks and dips observed are largely determined by the total resonance width lY The effect on the chi-square value, and hence on A&, arises mainly from the asymmetrical shape of this graph which reflects itself in the transmission spectra. This phenomenon is illustrated in fig. 5 showing the results of calculations in which typical experimental conditions had been assumed (flight path, 63.6 m; channel width, 2.0 t.~s;sample thickness, 0.552 x 10zl nuclei per cm*; temperature, 300 K; background, 2.8 x 10’ counts per channel). Within drawing accuracy, a distinction is hardly discernible in the resultant neutron transmission spectra of metallic uranium and U308. However, closer examination reveals differences of the order of one percent. The shape of the pertinent difference curve depends on the relative positions of the two transmission spectra as shown on the insert. The lattice vibration

A. Meister et al. / Isomer-shift analogue

1

/

I

I

870

a80

890

I

900

27

I

t

Fig. 5. Comparison between the resonance transmission dips of metallic uranium and UjOs. Top: calculated time-of-flight transmission curves. Insert: difference curves obtained prior to, and after, the fitting described by eqs. (2).

shift d&, is directly obtained from that value of At at which ,y2 reaches its minimum. In the present case, At amounts to 0.026 channels corresponding to about 400 FeV. In computations aimed at simulating actual measured transmission spectra, allowance had to be made for some experimental peculiarities not mentioned so far. These include dead-time losses, the resolution function of the time-of-flight spectrometer 4), and, as far as necessary, the grain-size distribution in powdery samples. The relative difference between the experimental and calculated transmission spectra was then generally of the order of one per cent only. This suffices for investigating the influence which each of the parameters entering the calculations exerts on the computed transmission curve. Since the crystal-lattice shift A&, is eventually determined by the difference between the transmission spectra of the samples compared, it was decided to check the agreement between the calculated and the experimental difference curves also by a chi-square test. This can be done only after the fit procedure described in sect. 2, has already been applied; otherwise, the chemical shift A&, contained in the experimental data would affect the experimental differences. After properly adjusting any free parameters, the test yielded chi-square values always approximately

28

A. Meister et al. / Isomer-shift analogue

equal to unity. This result suggests that there are no major systematic errors arising from the crystal lattice model employed. The free parameters just mentioned include those quantities related to the weighted frequency spectrum given by eq. (5). Calculations performed for a very large set of parameters have demonstrated that the transmission difference curve of any sample pair is hardly affected by any reasonable change in hvr or 01 provided the mean energy per degree of freedom (E)~ of the uranium atom vibrations is kept constant. Moreover, any alteration in (E)~ leads mainly to a change in the magnitudes of the peaks and dips observed but only to a slight change in their relative positions as evident from the top of fig. 6 for the equivalent cases of the U,,, - UO3 and U,,t- U02(N0& - 6H20 sample pairs. This fact provides a clue for finding sufficiently accurate values of a 1. The mean energy (E)~ is connected with the uranium weighted frequency spectrum p&v) by the expression +)I, = t\OWhvpb(hv) coth

d(hv) .

(hv/ZkT)

(12)

. I

2

I

,

-

UO,-

I

UO,[NO,I,

’ 6H2

0 -1

-2

I’

l-1 650

.

l I

660

I

I

670

680

t

J

Fig. 6. Comparison between experimental values and calcutated transmission difference curves. Solid line, (e)“/kT= 1.08 for U03 and U02(NO&6HzO; dashed curve, (E)JkT= 1.038 for both compounds. Experimental conditions: channel width, 2.5 es; sample thickness, 1.19 x 10zl nuclei per cm’; samples kept at room temperature.

A. Meister et al. / Isomer-shift analogue

29

This relation allows one to obtain al if hvl and (E)” are given. The parameter hvi may be found from available data on the specific heat cv. For compounds of the type UXl, d cv =-_(3(1+ dT

l)J%)“XJ

7

(13)

where L is Avogadro’s number, and (E) ox! is the energy per atomic degree of freedom averaged over the whole UXI lattice. The value of (&)ux, can be deduced from the expression m hvp (hv) coth (/W/~/CT) d(hv) , (E)“X, = $ (14) I0 which differs from eq. (12) only in that po(hv) has been replaced by the total spectrum p (hv) of the normal vibration modes of the whole lattice. Taking advantage of the fact that Mu >>Mx, it can be shown that, to a first approximation,

p&v) = &S(hv-hvl)+&

S(hv - hvz) .

The values of hvI and, considering eq. (6), hvz are chosen so as to reach via eqs. (15), (14) and (13), agreement with available experimental specific heat data. The values of ai and hzq thus found are then checked and, if necessary, changed applying the above-mentioned least-squares procedure aimed at getting optimum agreement between the experimental and the calculated transmission difference curves. This amounts to precisely determining the difference in the mean energies (E)” of the two samples compared. It would therefore be expedient for each sample pair to include at least one compound for which (e)u is accurately known from other findings. This is the case’with metallic uranium and uranium dioxide. For the latter, the data given in refs. l”,ll) yield according to eq. (12) (E)JkT = 1.035 at a temperature of 300 K whereas (&)“/kT = 1.015 for the metal 9). The metallic state represents a particularly favorable case in so far as p”(hv) coincides with p(hv). Hence, only one parameter is unknown which can be fixed from specific heat data. Apart from the UOz(N03)2*6H20 - UOj and U03 - UF4.H20 combinations, all sample pairs employed did contain either metallic uranium or uranium dioxide. Therefore, the optimum lattice parameters of the former could eventually be fixed by cross comparison. It may be added that the crystal lattices of uranyl nitrate and uranium trioxide are known to be quite similar “). This is in concordance with the fact that the transmission spectra of these two compounds do not reveal any major differences as evident from fig. 6. The values of the weighted-frequency spectrum parameters which turned out to be optimal, are listed in table 2. These data have been used to check, among other things, how the lattice-vibration shift AEeV depends on the channel range of the fitting procedure described by eqs. (2). Since the chemical shift AEo is sure to be

A. Meister et al. / Isomer-shift analogue

30

independent of any fit range, the lattice vibration shift AEey must be expected to run parallel with the experimental displacement AE,,. This proves indeed to be true as apparent

from fig. 7. The curves

dependence feature

in the AE,,

observed

of the experimental

shown

there

also demonstrate

that the fit-range

data of table 1, has to be regarded

approach

as an inherent

employed.

2000

_---__AEIV -

___----

,/*/’ 1500

11.94020

___------

___--1

~1000

5 w

a

500

0

I

-5ooc

I

I IO

,

/

I

I

I 20

/

I

I

I

I

I

I

I

I

t2-

1,

30

Fig. 7. Variation of experimental and lattice vibration shifts with fit range. Dashed lines enclose error range of calculated AEev curves. Closed circles, U,,, -U03 sample pair with 1.91 x 10” nuclei per cm2; sample pair with 1.19 x 10zl nuclei per cm’; open circles, U,,,-UO, triangles, Umet- U02(N03)2.6H20 sample pair with 0.552 x lO*l nuclei per cm*.

For all sample pairs used, table 3 summarizes the final results for AEo. To illustrate the order of magnitude, range-averaged values of AE,, and AEev are also listed. The errors indicated are standard deviations and include the statistical errors of AE,, as well as the estimated errors of all those quantities entering the calculation of the transmission curves of each sample pair. The size of AEev and, in accordance with eq. (l), AE,, turns out to be thickness-dependent as must be expected in a transmission experiment from the asymmetrical shape of the cross-section differences of the respective sample pairs (cf. fig. 4). The table also contains data on the electron density differences Ap,(O) which are required for interpreting the AEo results in terms of the related nuclear mean-square radius change A(rz). The unambiguous separation of the chemical shift into the electronic factor Ape(O) and the multiplicand A(rz) is equivalent to the solution of what is known in Mijssbauer spectroscopy as the

“1

Urn,* “)

0.221t 1.01 0.22 f 1.01 0.22 It 1.01 0.75 * 0.27 0.75 kO.27 6.55 It 2.95 6.55 f 2.95 6.55 ~~2.95 6.17 it 2.95 8.65i1.93 8.65 f 1.93 9.40* 1.96

(1O26cme3)

Ape@)“i

bI AdO) = P,(% - p,(O)z. ‘) A.!?,,= Eo, - E,,2 etc.

0.552 1.190 1.190 0.552 1.000 1.000

0.440 0.410 1.190 0.552 1.000 0.552

Sample thickness ( 10zl nuclei cm-‘)

“) Sample temperature was 374 K, otherwise room temperature.

UOz(N0&6Hz0-U,,, U03-UF4.H20 U03-UF4*H20 uoyuo2

U034Jmet

uo,-

uo3-um?t

UOz(N0&6Hz0-U03 UOz(N0&~6H20-U03 UOz(N0&.6Hz0-UOs UFz,.H20-U02 UF,,*Hz0-U02

TABLE

3

-_(EhJ2

kT

0 0 0 0.015*0.014 0.011 * 0.002 0.065 it 0.010 0.065 f 0.010 0.065rtO.011 0.065~0.011 0.040* 0.005 0.046 rt 0.007 0.057 zt 0.006

at T=293K

(Ekll

3

-74&87 +108+56 i-44*51 -38125 +266icSO +261t48 -48+44 i- 1069 f 39 +1097*50 -9Ort36 + 1331t88 +348*80

be%

AL

Results for the chemical shift of the 6.67 eV resonance of 238U

-2i90 +24*16 +15*12 -69*11 +266*46 +214*86 +145*89 + 1525 i205 + 1552d~205 +25oAz 112 i-579* 121 +7931106

beV)

A&v ‘1

-72*126 +g4*59 +29*53 +31+27 Ozt68 -188i99 -193199 -456*209 -455*210 -340*117 -446il50 -445* 132

MV)

AEo ‘1

w

: 2 g2

9 ‘:

5

P : ;;

3

‘,?

32

A. Meister et al. / Isomer-shift analogue

isomer shift calibration problem. The independent evaluation of Ape(O) necessary for this purpose will be dealt with next. 4. Determination

of electron density differences

The problem may be tackled within the framework of a generalized free-ion model with the occupation numbers P,r of the nl valence electrons not restricted to integral values. This model may be used to connect experimental chemical X-ray shifts with the desired electron density differences 13*14).Relativistic Hartree-Fock-Slater (HFS) calculations for free ions demonstrate the X-ray shifts involving K, L and it4 electrons to be approximately linear functions of the occupation numbers P,,. Let SE;;’ be the shift caused by completely removing one nl valence electron. The shift observed between two hypothetical chemical compounds 1,2 with a difference of SP,l= x in the occupation numbers of their nl electrons may then be assumed to be AE$

= (E$)I - (E;)z

=x SEC.

(16)

If two or more valence electrons are involved, their contributions add yielding the total shift AE,

= 1 bP,[ 6E:. nl

are supposed to

(17)

In the present case, the sum applies to the 5f, 6d and 7s valence electrons of the uranium atom. The aforementioned HFS calculations also show that analogous relations hold for the electron density difference between the two compounds, viz., &:‘(O) = (G’(O))1 - (p:‘(O))2 = x &C’(O) , 4,(O)

= C @,I c%,“‘(O) . nl

(184 Wb)

For uranium, chemical K,r ahd L,r X-ray shifts have been measured “-l’). As is evident from eqs. (16) and (17), these data may be used to evaluate 6P,l provided 6E$ is known. The latter quantity may be obtained from orbital energy eigenvalues as compiled for diverse appropriate valence electron configurations by Band et al. “). For any two ionic configurations 1, 2, 6E$ = (~1, - en,rj,)l- (e,lj - w,,)z

,

(1%

where enli and E,‘~,~, are the electron energy eigenvalues of the nlj and n’l’j’ orbitals involved in the X-ray transition under consideration. The 6P,, values thus found may then be used to obtain Ape(O) via eq. (18b). The Sp,“‘(O) data required may be calculated from the data given in 18) for the quantity _ N,,l = ha

l-a

hi

(&‘(r)P2’)

)

(20)

A. Mister

et al. / Isomer-shift

0

1 occupation

Fig. 8. Theoretical

number

33

analogue

2 change

x

uranium Kal X-ray shifts plotted versus valence electron occupancy that ordinate scales in top and bottom figure parts differ).

where (Y is the fine structure contribute to the total electron

constant. Only the s- and p-electrons density ~~(0) of any ionic configuration.

change.

(Note

with j = $ Hence,

Some numerical results for the K,r and L,i X-ray shifts and the electron density differences as functions of the 5f, 6d and 7s electron occupation numbers are shown in figs. 8,9 and 10. As for sign conventions, configuration 1 corresponds to the origin and configuration 2 to some value of x > 0. It is obvious from these plots that the linear relationships (16) and (18a) are indeed reasonable approximations. Moreover, it turns out that the contribution of the 5f electrons to the chemical X-ray shifts by far outweighs the influence of the 6d and 7s electrons. On the other hand, fig. 10 demonstrates distinctly that the different valence electrons affect dp,(O) to comparable degrees. Temporarily, the chemical compounds in question may now be grouped according to their respective oxidation numbers. The X-ray shifts observed between any two such groups are largely independent of the ligands attached to the uranium atoms. This is obvious from the experimental results 15-r7) compiled in table 4. Therefore, the same valence electron configuration may be ascribed to all the compounds of

A. Meister et al. / Isomer-shift analogue

34

1000

750

2

500

1 ‘2 .q

250

s % B i

0

J -25

-50 0

Z

1

occupatron

number

change

x

Fig. 9. Calculated uranium L,, X-ray shifts as a function of the valence electron occupation change. (Note that ordinate scales in top and bottom figure parts differ.)

number

L

5 aJ

G -40 0

1 occupation

Fig. 10. Theoretical

electron

density differences

2 number

change

resulting from changes 7s shells.

x in the occupancy

of the 5f, 6d and

35

A. Meister et al. / Isomer-shift analogue 4

TABLE

Experimental

Uranium states l-2

and theoretical

Exp. L,I shift 15,16)

Samples l-2

(mev)

UBr3-U02 UF3-UF4 UC13-U02

+ 190* 15 +201*16 +260+50

X-ray

shifts “)

Theor. La1 shift

0

&I,,-uoz

-11*14

-31

uo*-u*o5

+80*50

+80

+119*13 +140*12

+ 178

differences

Ex (sample

Theor. Kal shift

(meV)

WV)

+ 265

-3*1.5 -4zt20

shifts are the energy

Exp. K, I shift “)

(meV)

UF4-U02 UBr4-U02

U0AJQJNW~ uo*-uo3 U02-UOzS04,3H20 U02-U03HP04.4H20 “) The X-ray

uranium

l)-_Ex (sample

+230+20 +160+30 +190*30

+194

2).

such a group. There are strong arguments in favor of a (5f)2(6d)2(7s)2 configuration for the metallic state of uranium 16*19*20). This may be used as a starting point in fixing appropriate SP,,, values which are found by calculating the X-ray shifts to be expected with the aid of eq. (17), and comparing the results with the experimental values. The accuracy of the values thus obtained may be further improved by considering available Mossbauer isomer shift data ‘) for isovalent neptunium samples. Apart from one more non-bonding 5d electron, the neptunium atoms in such compounds may be supposed to have the same electron occupation numbers as the respective uranium ions. This, however, does not hold for the metallic state since the difference between the 5f occupation numbers of the neptunium and uranium metals is known to be about two 20). The electron density difference ratios of the various neptunium sample pairs calculable from eq. (18b) after extracting Spenr(0) values from the tables of Band et&. 18),must exactly match the corresponding isomer shift ratios. This condition finally leads to the configurations listed in table 5. The theoretical uranium X-ray shifts resulting from these occupation numbers are included in table 4 whereas in fig. 11 the electron density differences obtained are plotted versus the oxidation numbers. As to the occupation numbers of table 5, not too much value should be placed on the magnitude of the P,,l figures themselves since the decisive factor for. the related X-ray shifts and &,(O) values are rather the differences in the valence electron occupancies. It seems worthwhile mentioning at this point that the chemically induced decay rate changes found for the 75 eV transition of 235mU[ref. 2’)] should be helpful in checking the conclusions reached so far. The fractional decay constant variation of

A. Meister et al. / Isomer-shift analogue TABLE

Valence Actinide

electron

configurations

state

5

of neptunium

and uranium

Neptunium

states

Uranium

5f3 66d1.47s2 ’

metallic 3+ 4+

5f3.756d0.77s0

5f3

5f2.856d0

5+

6+

5

056d0.7,s0.5 57s0

5

5f* 606d0 ‘7s’ 5

a 6

10

12 14 16 0

1

2

Uranium

Fig. 11. Electron

density

difference number

this transition 22.23 as 1

in which the total interval

3 oxldallon

4

5

6

number

as a function of the uranium oxidation number. is meant to designate the metallic state.)

conversion

coefficient

(The zero oxidation

CY>>1, may be written

where the sum runs over all shells involved in the internal conversion. The experimental values of Ah/h are of the order of some percent. For the free uranium atom,

A. Meister et al. / Isomer-shift analogue

37

the partial internal conversion coefficients CQ have recently been computed 14’24’25). These data show that three terms representing the internal conversion in the 6~112, 6~312and 6d3,* subshells, have to be taken into account in calculating the sum of eq. (22). Since the 6d electron belongs to the uranium valence electrons, the AA/A values should therefore yield direct information as to the 6d occupation numbers. The free-ion model does explain the order of the effect observed. However, closer examination reveals that two important phenomena are unjustifiably neglected in such an approach. The first is connected with the 6p electrons. The chemical bonding leads not only to a change in the normalization constants of the 6p wave functions producing a non-zero A&, but the ligand molecular orbitals may also contain some admixture of 6p character as demonstrated by Gubanov et al. 19) for the cases of the oxygen 2s molecular orbitals in UO and U02. The resulting change in the 6p occupation numbers is likely to be of the order of a few percent only. However, since the relative internal conversion coefficient of the whole 6p shell is about 97%, the pertinent ACY~~/LYvalue will also amount to some percent. Thus, this term must not be neglected as done in the free-ion model. The second circumstance to be considered is of a similar nature: The bonding may cause the uranium 7p orbital to be partially populated. Data on the internal conversion coefficient of a 7p electron have not yet been published. However, to a crude approximation 23), -3 ffdan

,

(23)

whence (~7~

Aa7,

=

=

0.63a+,

0.63~~6~SP,p .

(244 Wb)

Obviously, the term Aa,,/a must also not be neglected in eq. (22) even if SPT~equals only a few tenths. For comparison, it may be pointed out that Gubanov et al. 19) found P7p = 0.1 in UO. Interestingly, their results also show that theoretical Pnr values used for discussing the Ah/A data should be based on relativistic calculations. With respect to Ape(O), the general conclusion from the preceding discussion is that the present state of the theoretical interpretation of the AA/A results does not allow to extract more refined information on the valence electron occupation number than is contained in table 5. However, one more refinement on the Ape(O) values can be achieved by postulating the thus far calculated mean electron density difference between the Np6+ and Np3+ compounds to be true. This allows the members of each neptunium oxidation state group to be separately placed along the electron density scale since the errors of the experimental isomer shifts are rather small. The resulting electron density difference may be transferred to the respective uranium compounds multiplying the neptunium values by a correction factor of 0.9. This factor is based on a comparison of the electron density differences obtained by HFS calculations for isovalent

38

A. Meister et al. 1 Isomer-shift analogue

neptunium and uranium ions r8). This approach yielded the final results for Ape(O) listed in table 3. The errors given were estimated using the relation

deduced from eq. (18b). The A(SP,J values were fixed so as to cover for any combination of oxidation state groups the experimental errors and major deviations in the uranium X-ray and the neptunium isomer shifts. The relative errors of the &E’(O) values were assumed to range from 15 to 20%. Interestingly, the Ape(O) value obtained for the Np6+ - Np3+ combination may be used to calculate the mean-square radius change for the 60 keV Mdssbauer transition of 237Np. The isomer shift AEIs ,is 76.Ok3.8 mm/s [ref. ‘)I and the electron density difference amounts to (20.9 f 3.4) x 1O26cmd3. One then obtains from the well-known formula ‘) A&s = (e2/6~0)ZAp,(0)

A(&

(26)

a value of - (25.7h4.1) x lop3 fm* for A(r:) which is in good agreement with the data of Kalvius and Shenoy 26) as well as the results of Makarov et al. *‘). 5. Resonance

shift and nuclear mean-square

radius change

The mathematical description of the hyperfine interaction between the nucleus and its surrounding electron cloud is well established I***).Obviously, the analogy between the Mossbauer isomer and the chemical neutron resonance shifts suggests that a relationship similar to eq. (26) must hold for the latter, viz., AEo = (e*/6~~)2Ap,(O)

A(ri)

.

(27)

In this formula, the nuclear mean-square radius change A(r:) of the Mossbauer -y-ray transition has simply been replaced by the radius change A(ri) due to the capture of the resonance neutron. Eq. (27) may indeed be regarded as expressing the desired connection between the chemical resonance shift measured and the related nuclear mean-square charge radius change. However, the deduction of this expression is based on the assumption that the electron density is constant over the whole region of the nucleus. This condition is not met with as heavy an element as uranium as evident from fig. 12, which illustrates the radial dependence of the angularaveraged proton and electron densities at the 238U nucleus. These results are based on a three parameter Fermi proton distribution of the type ~~(8, rP) = P~o{I + exp (4 ln 3(r,[l-P

Cl

- W/W1 ,

(28)

with /3 = 0.253, t = 1.46 fm and R = 7.15 fm [ref. *“)I. The electron density was computed by means of the Dirac-Fock program of Desclaux 30).

A. Meister et al. / Isomer-shift analogue

0123456789

39

r PI

Fig. 12. *Angular-averaged proton (p) and election (e) density distributions at the 238U nucleus. At nucleus center, p,(O) = 56.9 x 1O36protons per cm3 and p,(O) = 41.4 x 103’ electrons per cm3.

The question of how the inconstancy of the elecctron density in the vicinity of the nucleus affects the resonance shift, can be answered by considering some conclusions which are deducible from the more rigorous expression lS31)

The symbol Ap,(r,) denotes the difference between the proton densities of the excited compound-nucleus and the target nucleus ground states. The above relation can be simplified taking advantage of the fact that the electron density difference is proportional to the electron density itself throughout a rather wide radial range: AP&~)/P&~) = &(O)/P,(O)

(30)

= const .

In this region, the chemical binding produces only changes in the normalization constants of the electron wave functions but does not affect their shapes. This behaviour is demonstrated in fig. 13, where the relative Dirac-Fock electron density difference for the 5f26d27s2 and 5f26d’7s2 uranium valence electron configurations is plotted versus the radial distance. Resolving eq. (30) for Ape(re) and inserting into eq. (29) yields AE

=

0

_4me2 AdO> -

co

-

P,(O)

IW &Jr&~{ r,=O

I,“=, ($-$)p,(r,)r? c

dr,) dr, .

(31)

The fractional alteration of the electron density pe(re) due to the chemical bonding is smaller than lop3 . The integral factor in eq. (31) may therefore be considered to be independent of the valence electron configurations in question. Hence, to a very high degree of accuracy, the neutron resonance shift AEo is, in any case, directly

A. Meister et al. / Isomer-shift analogue

40

2

-3

Fig. 13. Relative

electron density difference for the 5f26d27s2 and 5f26d17s2 uranium tions plotted versus the radial distance to nucleus center.

electron

configura-

proportional to the electron density difference Ape(O). In this respect eq. (31) agrees perfectly with the physical meaning of eq. (27). Contrarily, though the value of the integral factor of eq. (31) is obviously determined by the shape of the nucleus, it is not immediately clear how to extract the desired change in the nuclear mean-square radius as pe(re) is radius-dependent. It is convenient to circumvent this difficulty by substituting Kpe(0) for p&J with the correction factor K chosen such as to retain the value of the double integral. This gives A&, = (e2/6~o)ZKAp,(0)

A(ri)

.

(32)

Fig. 12 suggests that K should differ but little from unity. Numerical calculations necessary for the exact determination of K require some assumption to be made on the proton distribution. If a distribution of the type described by eq. (28) is supposed to hold for both the ground and the excited states, then K turns out to be 0.90 f 0.03 for 238U. The experimental A& values (cf. table 3) are plotted in fig. 14 versus Ape(O). The horizontal bars near the abscissa indicate the estimated Ape(O) errors of the respective sample pairs. The solid straight line is the result of a chi-square fit through the experimental points and the origin. The error range of the slope of this straight line for a confidence level of 95% is enclosed by the two dashed lines. Making use of eq. (32), these data yield A(rE) = -(1.7)?&$

fm*,

where the errors indicated again correspond to a 95% confidence level. From eq. (28), the mean-square charge radius of the 238Uground state is found to be 34 fm*, so A(ri)/(rg) equals -0.05, or -5%.

A. Meister et al. / Isomer-shift analogue

41

./’

II A U02(N(

200

I

’ 0

2

,

f L

I

I

I

I

I

a

6

10 Ape(O)

Fig. 14. Chemical

shift values for the 6.67 eV neutron resonance eiectron density differences.

of 238U plotted

[102'cm3]

versus the respective

6. Conclusions It is easy to verify that higher-order hyperfine effects cannot produce the chemical resonance shifts reported in this paper. As for the electric quadrupole interaction, the ground-state spin of 238U equals zero; hence, J = fin the compound state of 239U reached on capturing an s-neutron. Consequently, there is no quadrupole splitting at all in either state. The same is true of the magnetic hyperfine splitting of the 238U state does undergo magneting splitting. ground state whereas the 239U compound The latter, however, is of the order of a few FeV at most and, moreover, symmetrical around

the original

state

such that the effects

of both sublevels

on the chemical

resonance shift largely cancel each other. Obviously, the resultant value of A(ri) given above must properly be regarded as a combination of an isotopic radius alteration and a radius change caused by the excitation of the nucleus. The isotope effect on (rf) can be assessed using experimental data on the optical isotope shift in uranium and plutonium 32). Such an evaluation proves that the ground-state mean-square radius of 238U tends to increase by only about 0.05 fm2 per neutron added. Therefore, the isotope effect may be neglected in the present discussion. On the whole, it is thus reasonable to conclude that the experimentally observed resonance shift is connected with a decrease of the nuclear mean-square radius. This reduction must be due to the excitation of the nucleus to an energy of 4.8 MeV released on binding the resonance neutron. An intuitively appealing and, at the same time, unambiguous interpretation of the mean-square radius diminution is not possible since it may be brought about by

42

A. Meister et al. / Isomer-shift analogue

several factors. These include a decrease of the radial parameter R of the proton distribution, a reduction of the nuclear deformation, or an alteration in the type of the proton distribution. In fig. 15, the relative change of the mean-square radius is plotted versus the deformation parameter. These data refer to a proton distribution as described by eq. (28). In the pertinent calculations, the radial parameter R and the diffuseness t were kept constant whereas the normalization constant pPOwas adjusted so as to satisfy the condition 2 =

I

p(r,) drp .

(33)

Obviously, the experimentally observed A+:) value may be reached by the nucleus assuming a more spherical shape upon the formation of the compound state. Such an explanation may perhaps be the most plausible since shell effects disappear in sufficiently highly excited nuclei 33).

80

-

70

a

30-

0

Fig. 15. Relative

change

010

020

of the 238U mean-square charge parameter.

030 radius

p as a function

of the deformation

A rigorous interpretation of A+:) would, of course, require the calculation of the wave function of the nuclear compound state. This problem has not yet been solved. However, statistically averaged values of compound state properties such as the changes of the mean-square radius and of the quadrupole moment which are produced by an excitation to energy E*, can be assessed within the framework of the statistical model introducing an appropriate nuclear temperature T. Such a method has been applied by Bunatian 34*35).This author computed mean-square radius changes taking consistently into account the strong nucleon interaction in the particle-hole channel and the pairing interaction. For an excitation energy E” near the neutron binding energy, he found the temperature T to be somewhere between 0.7 and 1 MeV yielding A(rE)=-- 0.2 fm*. This value agrees in sign with our

A. Meister et al. / rso~er-shift anaiogue

43

experimental result but is considerably smaller. However, an accurate computation has been done for spherical nuclei only 35); calculations for deformed nuclei might prove helpful to clarify to what extent the discrepancy is due to the influence of the nuclear deformation. In addition, it must be borne in mind that such calculations are bound to yield mean values averaged over many compound states because they are based on a statistical approach. Therefore, more experimental data for several resonances of one nucleus or for comparable resonances of some neighbouring nuclei are required before a more detailed comparison between the theoretical and experimental findings can be made. The authors wish to express their gratitude to their many colleagues without the help and encouragement of whom this work could not have been accomplished. Thanks are especially due to Prof. I.M. Frank, Dr. V.I. Luschikov and Prof. D. Seeliger for their continual interest and support. We have enjoyed a great many illuminating discussions with Dr. G.G. Bunatian, Dr. L. Cser and Dr. Yu.M. Ostanevich. The help and assistance of Dr. S. Szalai and V.A. Vagov who developed the on-line processor system employed, is gratefully appreciated. We are thankful to E. Arndt, Dr. E. Hartmann, Dr. M. Nagel and Dr. G. Zschornack for computing some uranium electron wave functions, to Dr. R. Hahn, S. Meister and Dr. H. Bruchertseifer for the preparation and chemical analysis of the samples used, to Profs. V.A. Khalkin and R. Dreyer for fruitful discussions and for providing sample preparation facilities. The assistance of M. Boettge, Dr. P. Eckstein, S. Eckstein, Dr. Yu.D. Mareev, Dr. W. Pilz and R. Tschammer who helped in diverse phases of experimentation, data taking and analysis, is also gratefully acknowledged.

References 1) Miissbauer isomer shifts, ed. G.K. Shenoy and F.E. Wagner (North-Holland, Amsterdam, 1978) 2) V.K. Ignatovich, Yu.M. Ostanevich and L. Cser, JINR Report P4-7296 (1973) 3) V.P. Alfimenkov, L. Lason, Yu.D. Mareev, O.N. Ovchinnikov, L.B. Pikelner and E.I. Sbarapov, Nucl. Phys. A267 (1976) 172 4) V.V. Golikov, Zh.A. Kozlov, L.K. Kulkin, L.B. Pikelner, V.T. Rudenko and E.I. Sharapov, JINR Report 3-5736 (1971) 5) G.G. Akopyan, V.A. Vagov, K. Seidel, A. Meister, D. Pabst, L.B. Pikelner and S. &alai, JINR Report P3-11740 (1978) 6) K. Seidel, A. Meister, D. Pabst and L.B. Pikelner, JINR Report P3-80-135 (1980) 7) K. Seidel, A. Meister, D. Pabst and L.B. Pikelner, JINR Report P3-11741 (1978) 8) J.M. Ziman, Principles of the theory of solids (Cambridge University Press, London, 1972) 9) H.E. Jackson and J.E. Lynn, Phys. Rev. 127 (1962) 461 10) G.M. Borgonovi, D.H. Houston, J.U. Koppel and E.L. Sfaggie, Phys. Rev. Cl (1970) 2054 11) A.T.D. Butland, Ann. Nucl. Sci. Eng. l(1974) 575 12) S.L. Ruby,G.M. Kalvius, B.D.DunIap,G.K.Shenoy,D. Cohen,M.B. Br0dskyandD.J. Lam,Phys. Rev. 184 (1969) 374 13) K. Alder, G. Baur and U. Raff, Helv. Phys. Acta 45 (1972) 765 14) K. Seidel, A. Meister and D. Pabst, JINR Report P3-11742 (1978)

44

A. Meister et al. / Isomer-shift analogue

15) V.M. Vdovenko, L.L. Makarov, LG. Suglobova, V.A. Volkov and N.P. Chibisov, Dokl. Akad. Nauk SSSR 202 (1974) 868 16) L.L. Makarov, LG. Suglobova, RI. Karaziya, Yu.M. Zaitsev, Yu. F. Batrakov and N.P. Chibisov, Vestn. LGU 16 (1975) 87 17) P.L. Lee, F. Boehm and P. Vogel, Phys. Lett. 63A (1977) 251 18) I.M. Band and M.B. Trzhaskovskaya, LIYaF Report 92 (1974) 19) V.A. Gubanov, A. Rosen and D.E. Ellis, J. Inorg. Nucl. Chem. 41(1979) 975 20) H.L. Skriver, O.K. Andersen and B. Johansson, Phys. Rev. Lett. 41 (1978) 42 21) M. Neve de Mtvergnies and P. Del Marmol, Phys. Lett. 49B (1974) 428 22) I.M. Band, L.A. Sliv and M.B. Trzhaskovskaya, Nucl. Phys. Al56 (1970) 170 23) H.C. Pauli, K. Alder and R.M. Steffen, in The electromagnetic interaction in nuclear spectroscopy, ed. W.D. Hamilton (North-Holland, Amsterdam, 1975) p. 341 24) D.P. Grechukhin and A.A. Soldatov, Yad. Fiz. 23 (1976) 273 25) D. Hinneburg, M. Nagel and G. Brunner, Z. Phys. A291 (1979) 113 26) G.M. Kalvius and G.K. Shenoy, Atomic Data and Nuclear Data Tables 14 (1974) 639 27) L.L. Makarov, B.F. Myasoedov, Yu.P. Novikov, Yu.F. Batrakov, RI. Karaziya, A.N. Mosevich and V.B. Gliva, Zh. Neorg. Khimii 24 (1979) 1014 28) Hyperfine interactions, ed. R.B. Frankel and A.J. Freeman (Academic Press, New York, 1967) 29) S.A.De Wit, G. Backenstoss, C. Daum, J.C. Sens and H.L. Acker, Nucl. Phys. 87 (1967) 657 30) J.P. Desclaux, Comp. Phys. Corn. 9 (1975) 31 31) B. Fricke and J.T. Waber, Phys. Rev. B5 (1972) 3445 32) K. Heilig and A. Steudel, Atomic Data and Nuclear Data Tables 14 (1974) 613 33) U. Mosel, P.G. Zint and K.H. Passler, Nucl. Phys. A236 (1974) 252 34) G.G. Bunatian, Yad. Fiz. 26 (1977) 979 35) G.G. Bunatian, Yad. Fiz. 29 (1979) 10