Electric field gradients in pyrochlore compounds

ELECTRIC FIELD GRADIENTS IN PYROCHLORE COMPOUNDS Y. CALAGE Institut de Chimie, UniversitC d’Oran, Algkrie

J. PANNIERS Laboratoire

de Chimie hair&ale

D, Universit6

de Rennes, Faculti: des Sciences, 35031Rennes, Cedex, France

(Received 6 Augusi 1976;accepted IS October 1976) Abstract-The

lattice contribution to the electric field gradient was calculated for pyrochlore compounds as a function of the distortion parameter x(48f). It is shown that this simple model accounts well for the experimental MSssbauer quadrupole splitting of various compounds containing Fe’+, Fe*‘, Sn’+,Sn*+,Sb” or Gd” ions. The use of MBssbauer data to determine the structural parameter x(48j) is outlined. 1. INTRODUCTMIN

The calculation of electric field gradierfts (EFG) has been carried out for a lot of crystal structures, especially spinels [I, 21, ~o~ndum[~i l] and garnets& 121. These investigations generally used a point charge model (including or not including multipolar contributions) and were restricted to a few compounds. The usefuiness of such calculations is thus limited by the precision of the ionic coordinates, the nuclear quadrupole moments and the Sternheimer antishielding factors; so, they were used to rarely improve crystal structure determinations [ 131. In this paper, we use a point charge model (monopolar and dipolar contributions) to calcutate the EFG as a function of the only structural parameter x(48f) of the pyrochlore. Using previously determined distortion parameters, these results allow the ~alcuIation of quadrupole splittings which are in a fair agreement with the experimental ones.

each

peak. Each intensity was measured at least three times and the average of the three measurements was used for the structure determination. The intensity of a group of equivalent reffections was taken to be I=SCx,uxLPxF* where SC is the scale factor, p the multiplicity factor, LP the Lorentz-polarization factor (taking into account the polarization on the monochromator) and F the structure factor. The scattering factors for cations were taken from Refs. [20,21]; for O*-, we used Tokonami’s value[22]. The least-square refinement of the data was performed by the full-matrix program Maryse[l9], specially written for powders, which minimizes the function Ew(& - I,)2. Several cycles of refinement of the scale factor, positional parameter ~(48~) and individual isotropic temperature factors led to the results of Table 1. 2.2 A theoretical detqmination by minimization of lattice

2 !5TRUCTURFsDETJBMINA’TTONS In order to check the accuracy of point charge model calculations, we have first to determine the x(48f) parameter of some pyrochlore compounds containing a Mijssbauer nucleus. The two series Ln,Sn,O,[14] and LnzFeSbO,[lS] are. of particular interest and we used two different ways of obtaining w(48ff:

energy

The method proceeds in two steps: -first, to define a lattice model, i.e. a mathematical function giving the Iattice energy vs structural parameters: the Born-Land6 model[23,24] and the Born-Mayer model [2_5,26]are the most widely used. -second, to determine the value of the structural parameters which minimize the lattice energy; the pioneering work in this area was done by Busing who 2.1 A st~et~re re~nement from X-ray border doto developped a generalized ieast-squares computer proThe compounds were prepared by solid state reaction gram to study the structure of alkaline-earth as described by Brisse and Knop[ 14,151. Carefully grinchlorides[25]. ded samples were packed in a tray and the intensity of Due to the high symmetry of the pyrochlore structure each peak was collected with a CGR Theta 60 diffrac(Fd3m-0:) we were obliged to use a very simplified tometer using Cu-Ka radiation, a quartz monochromator model which represents the lattice energy of a pyrochand a scintillation detector; the scanning speed was O.OS”/mn. Background was determined before and after lore compound Ln,M,O, as the sum of only two terms: -a coulombic term E, = -M(x)/ao where aa is the length of the cubic unit-cell and M(x) the Madelung tPresent address: E.I. duPont de Nemours & Co., Experimental Station C&D 356/327,Wilmington, Delaware, DE 19898, constant relative to that length. The values of M(x) were U.S.A. previously reported[26,27]. 711

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CALAGE and

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PANNETIER

Table 1. Structure refinements from X-rays powderdata

Compound

no(IV

Nib NHKLb ~(4Sf)“~

B(8b)“

B(16c)d B(l6#

B(48f)”

1.12(6) 0.33(7)

1.92 U41 2.44 3.84 This work O.lO(57) 3.89 6.08 This work

La2Sn20, La2Sn20,

10.7005(10) 16 10.7005(10) 43

16 72

0.3284(1) 3.65@O) 0.3252(19) -1.30(60)

Gd2Sn207 ErZSnZO, Gd2Ti,0,

10.4600(8) 38 10.3539(9) 39 10.1852(9) 26

63 64 26

Gd2Ti207 Y2Ti207

10.1852(9)

0.3320(36) 3.40(4.69) 0.55(11) 0.18(9) 0.3345(14) -1.03 (59) 0.02(4) 0.54(5) 0.3237(34) -1.57 (1.15) 0.58(52) 0.74(23)

1.20(6) 0.31(6)

R

Rw

1.07(4) 0.71(41)

0.63(30) 1.55 2.61 Thiswork 2.84(1.26) 1.82 WI

40

64

0.3220(21) -0.19 (64)

O&6(11) 0.60(6)

0.70(40)

Y1Ti207 Y2Ti207

10.0949(5) 26 10.08%(14) 75 10.08%(14) 63

26 75 39

0.54(15) 0.54(7) 0.63(25) 0.49(11) 0.29(4) 0.28(8)

0.83(41) 1.15 0.29(12) 6.6 0.48(23) 3.72 4.9

Sm2FeSb0,

10.3389(15) 18

31

Y2FeSb07 Cd2SbZ0, A82SbZOL

10.2122(11) 34 10.2700(20) 35 10.2490(20) 35

58 59 57

0.3299(10) 2.06(120) 0.3288(5) 0.00(10) 0.3281(13) -0.44 (45) 0.3246(36) -0.40 (180) 0.3256(18) -1.43 (69) 0.3205(55) 3.5(532) 0.3171(21) -

0.02(49)

Ref.

1.09(48) -0.22 (92)

4.12 6.41 This work Ml 3.18 4.17

1171 Thiswork Thiswork

0.37(9) 0.92(11) -0.27(36) 2.86 5.52 This work 0.86(19) 1.00(22) 1,14(100) 6.93 7.95 This work 0.10 2.25(11) 1.67(63) 3.82 6.30 This work

“Unit-cell parameters of the rare-earth compounds are taken from Refs. [14-161. bNI = number of intensities; NHKL = number of diffracting planes used for the structure refinement. ‘The origin used is Bo[18]. “The estimated standard deviations given in parenthesis apply to the last significant place.

-a repulsive term which takes into account only the repulsion between nearest neighbours:

+4BL.exp IRLn~K’aol where B is a softness parameter and R a radius characteristic of the repulsion between a given cation and oxygen. We gave a fixed value of 0.300 to the parameter p; Kg, K2 and K, are parameters depending only on the value of x(48f)[27] and relating the cation-anion distances to the cell length. In a first stage, we tried to use for the repulsive energy the form proposed by Gilbert[28,29] with the soft sphere parameters calculated by Buttefield[30]; however, these parameters greatly overestimate the repulsive energy and sttucture therefore unusable for they are determination[31]. The values of the parameters RLn and Ru were taken as the sum of the ionic radius of oxygen and cation[32,33]. The model parameters BLn and BM were determined from known structures (five Ln2Sn207, five LnZTi207 (Table 1) and three LnzZr207[34]) by a least-squares adjustment procedure similar to that used by Yuen[31]. Due to the imprecision of the experimental parameters x(48f), we determined only one BL” for all the lanthanide cations and one for each M4’ cation (T?‘, Sn’+, Zr4+). The model parameters derived in this way were then used to calculate crystal structures, that is, the model parameters were held fixed at their previously determined values, the unit-cell length at their experimental values while the x(48f) parameters were allowed to vary so as to minimize the lattice energy. The results of the calculation are given on Fig. 1 for the three series of compounds as a function of the radius of the rareearth cation. As we used a very simplified model and did not allow the unit-cell parameter to vary together with x(48f), we cannot consider that method as a true minimization of the lattice energy; however, it seems to be a good way to study the behaviour of simple structures .along a series of compounds [23,24].

Fig. 1. Curves x(48f)=f(rLn)+) calculated by lattice energy minimization.

3. CALCULATMlNOF THE FLECTRIC FIELD GRAD-

We consider the lattice EFG as the sum of the monopole and dipole contribution: the calculation of the monopolar term is straightforward but the dipole one requires previous knowledge of the electrostatic field EO and ionic polarizability a: EFG = EFG(,,,,, + KaEo. A complete calculation has, in fact, to consider the effect of induced dipoles. However, for the pyrochlore

structure, we found this correction to be very small and we neglected it. 3.1 Monopolar contribution The EFG,mono,is described by a symmetric secondrank traceless tensor whose components are given by:

where V is the electrostatic potential, xik and xik the coordinates of the kth atom, Sii the Kroneker index and ek the charge on kth atom. This tensor can be di-

Electric field gradients

in pyrochlorecompounds

agonalized in a principal axis system: using the (111) $$I6 ~11 direction of cubic unit-cell as z-axis, we can specify the 750. EFG in the pyrochlore structure by only one parameter V,,, the usual asymmetry parameter 7 = (V,, - V,,)VL’ 500 being zero. The value of V,, for a given site (16~ or 16d) of the structure is the sum of the four contributions arising from the sites 8b, 16c, 16d and 48f. V,,(16c) = f: e,V:,(16c) i=l z-5(Cd2N\0,)

V,,(16d) = f: eiV:z(Md). i=l

The first three contributions (i = 86, 16c, 16d) are of course x(48f) independant. Contributions Vi,(16~) and g,(16d) were evaluated by a direct summation method; however, instead of the usual spherical boundary, we took advantage of the better convergence of a cubic boundary method[35]; we performed the calculation for different values of the limit (one to six unit-cells) and the results were extrapolated analytically to infinity by using a Neville plot [36]. The summation for V,, was performed using a unit-cell length of 1 A; for a compound with a unit-cell length ao, the electric field gradient Vh is then given by Vi, = Vzz/ao3. The results of the calculations are given in Table 2.

Fig. 2. Calculated EFG(16c) vs x(48f).

410

Table 2. Values of the EFG on sites 16~ and 16d Contribution from site: 8b 16c 16d

4w

x = 0.3000 0.3050 0.3100 0.3125 0.3160 0.3200 0.3250 0.3300 0.3350 0.3400 0.3450

i

I

300

VL(l6c) -53.9738 +115.6836 -115.6836

V:,( 16d) +393.8327 -115.6836 +115.6836

t375.6157 +288.3537 +206.8797 +168.3947 + 117.0847 +62.1478 -0.9714 -57.9984 - 109.0923 - 154.4833 - 194.4503

-363.6615 -366.7345 -369.7015 -371.1205 - 373.0085 -374.9855 -377.1025 -378.7105 -379.6664 -379.8094

From this table, it is easy to calculate the EFG,,,.,) on sites 16~ and 16d for any pyrochlore compound. Results for AZq’B*‘3--q”0,, A’BB’O,, A2’B:‘06 and A’BB’F, compounds are presented on Fig. 2, 3. These results show that EFG,,,.,, is generally positive on (16~) sites, but negative on (16d) ones. Further, V,,(16c) is strongly x(48f) dependent, whereas V,,(16d) is nearly constant. We must also emphasize the following point: the value x(48f) = 0.3125, which corresponds to a regular coordination of the B cation by the nearest anions, is by no means a particular point on the curves V,: = f(x(48f)); in particular, V,,(16c) is not zero for this value of x(48!), as erronneously stated by Knop [ 151and Loebenstein [37]. 3.2 Electrostatic field Due to the lattice symmetry, the electrostatic field is zero on each site except on the anion site (48f) (mm

1-ZS(RD

X(48,1

,310

,320

,330

Fig. 3. Calculated EFG(16d) vs x(48!).

symmetry); these 48 positions split into three equal groups corresponding to the three possible orientations of the field EO along one of the axes (MO), (010) or (001) of the unit-cell. It is well known that the calculation of Eo by direct summation methods gives very poor convergence (l), so we used the Bertaut method[38]; the formula recently proposed by Weenk aqd Harwig[39] gave in fact a poor convergence (see Appendix). Calculations were carried out on a CII 10070computer as a function of x(48f) and the results are presented on Fig. 4 (for a given compound, the values of Fig. 4 must be divided by 02 in order to obtain E,, in e.A-’ units). 3.3 Dipole contribution This contribution is given by EFG~di~ol.) = KaEo where aEo represents the dipole moment on a (48f) site; K is in fact a second rank-tensor with elements: Kii =

when me is the i component of the dipole moment on the kth atom and the other letters have the same meaning as in the expression for Vh. As for the calculation of EFG 0nono,rthe tensor K, can be specified by only one parameter K; this parameter was evaluated by direct summation using cubic boundary; a cube of edge six a0

Y. CALAGE and J. PANNETIER

714

moment of the $‘Fe (14.4 KeV) state and “/- the Sternheimer anti-shielding factor. The experimental values of AEq [15,40] are compared with our calculations on Fig. 6 which shows a plot of AEq (exp) vs 4e2Vz&o-3.The values of x(48f) used to calculate V,, are from this work (Chap. 2) and from Refs. 141,421.The slope of the curve AEq (exp) = f($e2V,aa-3) is the product Qp.(l - r-). The usual values of these factors: QR = 0.28 barn [7] (1 - r-) = 10.14[43] give a good agreement between calculated and measured values. Thus, the monopole approximation accounts fairly well for the evolution of the Mbssbauer quadrupole splitting with the distortion parameter x(48f).

Fig. 4. Electrostaticfield E vsx(48f). gave a good convergence. Results are presented on Fig. 5; again, for a compound with unit-cell ao, the values of Fig. 5 have to be divided by a,‘.

.jil .-10

.20

$d$_

.30

Fig. 6. Fe” compounds:A&,,,, vs &?VJad. 4.2 Pyrochhiores containing Fe” The totai q~d~~le splitting in ferrous com~~ds given by:

is

where

Fig. 5. Dipole contributions X(16c) and K(16d) vs x(48f).

4. DI!XUSSION

In a first stage, we do not consider the dipolar concaution. Indeed, it is d&u& to assign a vaiue to the polarizabitity of the anions and, moreover, this contribution is genera&y smaller than the monopolar one. 4.1 Pyrochlore containing Fe3’ The ferric ion Fe’+ is spherically symmetric (38) and gives no valence contribution to the EFG. Thus, the quadrupole splitting measured by Mbssbauer spectroscopy is given by:

where aa is the unit-cell length, QF~ the quadrupole

we used (1 - r-) = 11.97[44]. The valence contribution AEqtvde.,., arises from the charge distribution of the non-sphered 3d valence-electrons of the ferrous ion (3d6, ‘D); this con~but~n is known to be more important than the lattice one, but d&u& to evaluate; moreover, A&~~,,~., usually shows a considerable temperature dependance [45]. Five pyrochlore compounds containing Fe*’ ions were studied by Miissbauer spectroscopy[40]; unfortunately, the sign of AEq is not known and only two structures were determined[42]. We therefore assumed a constant value x(48f)=O.319 for the five compounds. For this value of the distortion parameter, the octahedron (peF$) is trigonally distorted by compression along the (111) axis as is the octahedron (Fe@C&JN in iron &&Ii&e FeSiFs, 6&0[45]. We can therefore assume that A& is also negative for those pyrochiore comes: then AEq~“~=“~~ and A~q~~,ti~~~ are of opposite sign. The values

715

Electric field gradients in pyrochlorecompounds of AEqt,,,,c,,,,,calculated with that assumption are given in Table 3. Those calculated values are nearly constant and compared well with the value -3.67 mm/s observed for FeSiK 6Hz0 which is considered as one of the most ionic ferrous compounds[64]. It must be noted that the lattice contribution is much more important in pyrochlore than in other structures1461 and that AI&~-~ shows a linear dependence on AEqt,atWttas suggested by Ingalls 1451.

4.4 Pyrochlores containing Sd’ Only two pyrochlore compounds containing stannous ions are known: Sn&lb& and Sn2TaDJ491: although these compounds contain both cation and anion vacancies and are not truly cubic[50], their distortion is certainly very small and we considered them as cubic pyrochlores Sn22%42J’07.They exhibit a very large quadrupole splitting[SO] which cannot be explained by the co~elation diagrams of Lees and Flinn[!ill. For s~nnous compounds, the quadrupole splitting is given by:

Table 3. Fe*’ pyrochlores: experimentaland cahdated @XidrUpole splitting IAE,B”I

AE:*

Compound

1%

VJad

AE,-’

(calc)t

WJ1t

Wc)t

RbRCrF6

10.274 10.300 10.321 10.407 10.476

0.1475 0.1464 0.1455 0.1419 0.1391

to.741 +0.735 +0.731 +0.713 +0.699

2.641 2.570 2.556 2.412 2.319

-3.388 -3.305 -3.287 -3.143 -3.018

TlFeCrF, RbFeVFs CsFeCrFa CsPeVF,

Walues in mm/s.

4.3 Pyrochlore containing Sn” The Miissbauer quadrupole splitting for pyrochlores Ln&rlCb is given by the expression: AEq = ; e23

Qs.(l - y-).

The quadrupole moment of tin nucleus was hrst calculated by Boyle [47] to be Qs~= -0.08 t 0.04 barn and then by Nick&z 14glwho gave Qsn = +0.065 * 0.005 barn, but the Ste~eimer ~tis~eld~g factor was not previously determined for Sri”. Using our structural determinations, calculations of the EFG and the experimental Miissbauer data from I_oebenstein[37], we obtain the results presented on Fig. 7: the fitting of experimental values gives Qsn(l - r_) = 2.0 barn. Thus, we can propose for the Sternheimer factor rJSn4’) the values + = -24. (Q = -0.08 barn) or ?/- = -30.0 (Q = -+0.065barn).

where the tirst term arises from the unbalanced pz electrons. It was previously proved by Donaldson[52] that the lattice contribution to the quadrupole splitting is generally negligible for stannous compounds and of opposite sign to the valence contribution. However, in the pyrochlore structure, we obtained very large negative values for the EFG (lattice) in 16d sites: that means the valence and lattice con~ibution to the total q~d~~le splitting are of the same sign. Then, if we add to the calculated lattice contribution the valence con~ibution interpolated from a correlation diagram[Sl], we obtain a total quadrupole splitting in good agreement with the experimental value (Table 4). So, we can suppose that some of the discrepancies observed in the correlation diagrams are caused by the neglect of the lattice contribution to the total EFG. 4.5 Pyrochlores containing Sb’+ only two pyrochlore compounds containing Sb” were studied by Miissbauer spectroscopy: Cs2Sb20, and &Sb~OsjS4, 551. The accepted value of the quad~~le moment Q of “‘Sb is -0.28 bam[56], but, as far as we know, the Stemheimer antishielding factor has not been previously reported. Using our structural data of Table 1 and the experimental vahres of AEq for these two compounds, we obtained: Cd&KW AEqc.xp,= -5.0 mm/s (55)(1 - y&& = 61.0? 28. AgBbzOa: AEq
bA~mm,,J

The ~ce~nty on (1 - y_), calculated from the estimated standard deviation of x(48f), is of course very large and it would be of interest to check this value by other 12’SbMiissbauer experiments, for instance with the Ln*FeSb(x series.

.80, .7O Slope c?,,(l- a, I q 2.0 barns .60 SO’ 40,

0

.lO

Ln,.Sn,O,

Compounds

.20

.30

.40

Fig. 7. Sn” compounds: AEqt.xP,YS ~ezVz,/ao3.

JPCSVol. 38No. 7-E

4.6 Rare-earth pyrochlores In the rare-earth compounds, the EFG usually arises from the 4f electrons and that contribution cannot be easily evaluated. However, the electron configuration of Gd3’ (4f’) has a spherical symmetry and cannot produce a valence-electron ~n~bution to the EFG; scaly, the Eu3’ (4f”>co~ation has a ground state of ‘F. which has a zero total angular moments [64]. Accordingly, for

716

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and

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PAMETiER

Table 4. Sn*‘ pyrochlares:experimentaland cafculatedqnadrnpolosplitting AE,,C*C (mm/s) Compound

a0WI

Sn2PWI? Sn,Ta&

10.588 10.565

x(48j) 0.314(53) 0.312(SO)

sz::,

VJa,’

Valence

Lattice

Total

AE,‘“’ (mm/s)

+1.73 l-1.64

-0.328 -0.333

1.32 1.43

a.59 0,62

1.91 2.05

1.87 1.83

Table 5. Gdf’ and Eu-” pyrochlores:experimentaland e&n&ted quadrnpolesplitting Compound

@of&

x@ff

Transition

10.1852

0.3220

‘$“Gd(87KeV)

10.4600

0.3320

lJJGd(87KeV)

10.23b

0.328”

“‘Gd (87KeV)

lOSO

0.3333

“‘Cd (87KeV)

IO.1962

o.3nd

““En (103KeV)

(I- r-) 81.00[59] 61.871621 81.00[59] 61.87[62] 81.00[59] 61.8?[62] Sl.OO[S9] f&g?]621 62.38[62]

AR*‘*‘.(mm/s!" AE,‘“’(mm/s) -9.25 -7.07 -8.16 -6.23 -8.72 -666 -8.02 -6.12 -10.41

(Ref.)

-11.37t0.17 -10.392 0.20 -7.86

IS’;1;1 [Sal WI

-8.08

WI

-7.5OirO.13

PI

- 10.3

]6i]

“Calcukitedwitlt Q(Gd)= LS9b[St] and Q(Ru)= 2.95[St]. “From Ref. [63]. ‘V&e estimatedby lattice energy rnia~i~t~n. “From Ref. fS7]. Gd3* and Ed+ pyrochlores, the quadrupole splittiug is given by: Vzz(16d)

The results of the calculation are in Table 5. The agreement with exigent data is generally good, except for G&Ti& The reasons for that desagreement are not very clear, but it must be noted that the greatest diflerence between x(4@) X-rays and x(4Sf) theor. (Fig. I) was also observed for that compound. 4.7 DQvolarcontribution to EFG (lattice) So far, we have not taken into account the dipolar contribution to EFG
I$$ 0.919(15) 0.759(15) 0.424(37) 0.736(37) 0.718(37)

cr=o 0.955 0.808 0.445 0.700 0.800

d =0.8 a!= 1.5 a = 2.4 A’ A’ Bi’ 0.850 0.731 0.409 O&S? 0.758

tQp. = 0.28barn (t-r_) = 10.14@e”). Qs. = -O&8bani (I- -+I = 25,OO (Sir”‘).

0.760 0.W 0.666 0.644 0.378 0.337 0,621 OS74 0.722 0.676

compounds, as a function of cu(O’-); it can be seen that the addition of dipolar contribution does not really improve the agreement between experimental and calculated values and we can assume that a(O*-) is certainly not greater than 0.8 A”. Moreover, it is significant that, on Fig. 6, points corresponding to Sm2FeSb07 and YzF&b% are on the same straight line as those correspo~mg to the fluorides AMzFe which must give a negIiSible dipohu term, The higher values of the electrostatic Seld & in compounds A2Z”BpQr must give rise to greater dipolar contributions; however, the lack of precise experimental data for such compounds does not allow us to give any conclusion about the value of a@). Finally, we must note that the value of K(16d) is so small that the dipolar contribution to EFG(16d) is practically zero. 5. CoNCLtMrNS Because of its high cubic symmetry, the pyrochlore strature forms a suitable system for testing the applicability of a point charge model to the calculation of 4uadrupole splittings. This rather crude model can hardly be applied to a sinSle compound because of the uncertainties on Q and (1- y-), but, when working with a complete series of iso-structural compounds (and the rare-earth pyrochlores’ are a good example of such a series) one can use a few previously determined structures to calculate the factor (1 - y-) and then, conversely use the Miissbauer data to calculate or improve structures. Of course, the physical meaning of Q and (1- y-) obtained in this way is not clear and they must be considered only as pro~r~on~ity factors which contain, in fact, ~n~~ut~ons from higher m~ti~lar terms and from covaIerrey effects. Fortunately, for pyrochlore compounds A&G,: (a) the dipo]ar ~ou~bution is

Electric field gradients in pyrocblore compounds

generally small (see Section 4.7); and (b) the distance B-O does not vary along a series; for instance the variation of the distance Sn’+-O is about 2% from L&n&, to Lu&O,, so the covalency effects must be nearly constant (as is, in fact, the isomer shit). Table 7 presents a few x(48!) values calculated from Miissbauer data, using QF&1 - y-1 = 0.28 X 10.14 barn.

Table 7. Calculation of x(48f) from Miissbauer data

Compound SmZFeSb& EuzFeSbO, GdzFeSb07 Tb2FeSbol Y2FeSlh

Er*FeSbO, B&FeNbO, CsNiFeF, CsMnFeFs TIMgFeFs

AEd” (mm/s)

(Ref.)

0.759 0.790 0.801 0.848 0.919 0.925 0.536 0.519 0.583 0.488

X(48!) caIc.

x@f) X-rays

[l51 [15] [l51 1151 rlsj u51

0.3238 0.3246(36) 0.3243 0.3244 0.3247 0.3252 0.3256(18) 0.3252

&I [40] [401

“o:z 0.3175(10) 0.3188 0.318 0.3164

(Ref.)

t

t 1411 1421

Q’bis work. SF. Varre&-Private communication.

Acknowledgements-We

are grateful to Dr. F. Varret for stimulatingdiscussionsand to Dr. E. F. Bertaut for pointing out our

attention to the convergence of fhe potential formula derived by J. W. Weenk[39].

717

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718

Y.

CALAGE

and J. P ANN~~ER

Weenk and Hanvig[39]. However, the convergence of their formula proved to be much poorer than the convergence of the Bertaut series. As an example, we give in the following table the electrostatic field and potential at the 48j site of the pyrochlore Az’+Bz~‘O,calculated for x(48j) = 0.3200 with both formulas. (We used a uniforme charge distribution: j3 = 27rhRis the limit of the summation in the reciprocal space.)

j3/* E (Bertaut) I

2 3 4

5.4069 6.9533 6.9519 6.9505

E (Weenkf

1.8411 6.8869 7.1068 6.9508

V(Bertaut)

V (Weenk)

19.8149 19.0960 19.0888 19.0877

23.0054 19.1978 19.1739 19.1202