A single-crystal neutron diffraction study of the distribution and thermal motion of silver ions in alpha- and beta- Ag3SI

Solid State lonics 18 & 19 (1986) 1150-1162 North-Holland, Amsterdam

1150

A SINGLE-CRYSTAL NEUTRON DIFFRACTION STUDY OF THE DISTRIBUTION AND THERMALMOTION OF SILVER IONS IN ALPHA- AND BETA- Ag3SI J.-J. DIDISHEIM*+, R. K. MCMULLAN~ and B. J. WUENSCH* *Department of Materials Science and Engineering, Massachusetts I n s t i t u t e of Technology, Cambridge, Massachusetts 02139, U.S.A. ~Chemistry Department, Brookhaven National Laboratory, Upton, Long Island, New York 11973, U.S.A. +

Single-crystal neutron d i f f r a c t i o n data have been used to examine the d i s t r i b u t i o n of mobile Ag ions in anion-ordered B-Ag3SI (space group Pm3m) and anion-disordered a-Ag3Sl (space group Im3m) at 8 temperatures in the overall range 23 ° to 475°C. The Ag density in the B-phase is confined to a c l u s t e r of four tetrahedral sites grouped at the centers of the cell faces, closely spaced as a r e s u l t of the d i f f e r e n t distances between Ag+ and the two species of anion. The i n d i v i d u a l sites are not resolved in Fourier syntheses because of large thermal displacements w i t h i n a f l a t potential w e l l , but the refinements permit r e j e c t i o n of occupancy of the octahedral s i t e . The Ag+ d i s t r i b u t i o n in ~-Ag3SI displays a 2-coordinated s i t e on the body diagonal of the cell and density delocalized in bands which display f i n e structure not present in ~-Agl or B-Ag2S. This feature may be q u a l i t a t i v e l y interpreted as an average over positional disorder of the e q u i l i b r i u m position w i t h i n the tetrahedral i n t e r s t i c e , the location depending on the short-range configurat i o n of anions about the s i t e . Models employing p a r t i a l Ag+ occupancy of the 2-coordinated posit i o n plus e i t h e r octahedral or tetrahedral s i t e occupancy involve the same number of s t r u c t u r a l parameters and refined to comparable l e v e l s . Tetrahedral s i t e occupancy is judged to be somewhat more l i k e l y on the basis of agreement f o r the higher temperature data sets and physical c r i t e r i a .

I. INTRODUCTION

studies 7.

The compound Ag3SI, described by Reuter and Hardel I - 3 5 is intermediate to the f a s t - i o n

tion energy for Ag transport, Ag2S was found to

In spite of having a higher activa-

contain delocalized bands of Ag density along

conductors Agl and Ag2S. Single-crystal

, Fig. lb.

n e u t r o n - d i f f r a c t i o n analysis of the high-

that the d i s t r i b u t i o n represented v i b r a t i o n a l

I t was accordingly u n l i k e l y

temperature bcc structures of the l a t t e r two

disorder, and was instead interpreted as an

phases 4'5 revealed very d i f f e r e n t p r o b a b i l i t y

average of positional disorder 5. The transport

d i s t r i b u t i o n s for the mobile Ag ions.

mechanism was viewed as a correlated motion.

Silver

ions undergoing strongly anharmonic thermal v i b r a t i o n occupied the tetrahedral i n t e r stices in ~-Agl, Fig. la.

Scattering density

The d i f f e r e n t behaviors of ~-Agl and B-Ag2S was ascribed 5 to d i f f e r e n t bonding characteri s t i c s between Ag and the anion (as shown in

which bridged neighboring tetrahedral sites

the ordered room-temperature structures) as

suggested a d i f f u s i v e hop through the shared

well as the concentration of Ag ions r e l a t i v e

face between the sites as a transport mechan-

to the number of available tetrahedral s i t e s : each Ag ion in ~-Agl has available a jump for

ism. This i n t e r p r e t a t i o n was supported by a molecular dynamics c a l c u l a t i o n 6 and correspon-

which both a nearest and second-nearest neigh-

dence between the s t r u c t u r a l r e s u l t s , the

boring s i t e are vacant, while those in Ag2S

a c t i v a t i o n energy for Ag migration as well as

do not. The stoichiometry AgI.5X possessed by Ag3SI separates these two compositional

the results of i n f r a r e d and Raman scattering

+Present address: I n s t i t u t de Cristallographie, UniversitedeLausanne,CH-lOl5 Lausanne, Switzerland. 0 167-2738/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Z-J. Didisheim et al. / A single-crystal neutron diffraction study

1151

ranges of behavior and is accordingly of great

> T > -I16°C) the anions in the phase designa-

interest in an understanding of the transport

ted as B order in a CsCl-type array, space

properties of this family of cation-disor-

group Pm3m; the Ag ions remain disordered

dered conductors having body-centered cubic

among interstices.

At yet lower temperature

(-ll6°C > T) the material was recently found8'9

anion arrays. Three structural modifications of Ag3SI

to undergo a further transformation to the y

are known as a function of temperature. The

form in which the Ag ions order in a subset

high temperature ~ phase (T > 240°C) has anions

of the available interstices in a structure

disordered in a s t a t i s t i c a l bcc array, space

which l i k e l y has space group R3 but remains

group Im3m, with cations distributed among

metrically cubic.

interstices 3.

With lowered temperature (240°

Subsequent to the original description of Reuter and Hardel3, the structures of the y and B phases were examinedwith combined x-ray and neutron powder d i f f r a c t i o nl O ' l l .

Single-

crystal x-ray data with refinements which incorporated provision for anharmonic thermal vibration have been used to examine the details of the Ag probability distribution in all three phases12-14. Noneof the structural models remain without some ambiguity. The Ag-S bond length is much smaller than for Ag-I.

The equilibrium position for tetrahe-

dral coordination in the anion-ordered B-phase a

thus shifts from the ~ 0 location of a bcc array to x~O (x ~ 0.4).

Neighboringtetra-

hedral sites are too close to be resolved in Fourier syntheses. The Ag density was found to consist 12 of a f l a t , squarish maximumin (lO0) which is centered at the location of the octahedral site. The question is:

(See Fig. 3, below.)

Does this represent partial

occupancy of closely-spaced tetrahedral sites, f u l l occupation of the octahedral site at ½½0 by a Ag ion undergoing marked anharmonic thermal vibration, or a coorDination of occupancies? Parenthaler et al} 2 found that models based upon either tetrahedral or octahedral site occupancy described FIGURE l Comparison of partial Fourier syntheses of the scattering density for Ag in (lO0) for ~-Agl and ~-Ag2S. (a) ~-A~I at 300°C. Contour intervals 0.0024 lO-IZ cm/AJ (Ref. 4). (b) ~-AgpS at 325~C. Contourintervals 0.0048 l~-12 cm/A3 (Ref. 5).

to equal satisfaction diffraction data obtained at room temperature. They state 12 that the data "do not allow a decision as to which of the two models is physically more meaningful," but preferred the tetrahedral

Z-J. Didisheim et al. / A single-crystal neutron diffraction study

1152

model on chemical grounds.*

exception.)

Similarly, the Ag distribution in the ~-phase of Ag3SIl3 was found to qualita-

Systematic dependenceof the

temperature-factor coefficients upon temperature provides a valuable basis for

t i v e l y resemble that observed for B-Ag2S

distinguishing time-averaged harmonic or

at temperatures just above the phase trans-

anharmonic thermal vibration from positional

formation--that is, delocalized bands

disorder, and also a basis for assessing the

of density with weak local maxima at the

physical consistency of a model. The present

locations of the tetrahedral and octahedral

single-crystal neutron-diffraction analyses

interstices.

were accordingly performed at four different

Perenthaler and Schulz13

proposed a model in which both positions

temperatures within the s t a b i l i t y ranges of

were occupied, but between 0.46 and 0.167

both ~- and B-Ag3SI.

Ag ion could be assigned to the octahedral position (the remaining ions being consigned to tetrahedral sites) without change in agreement between observed and calculated structure factors.

The preferred model con-

2. EXPERIMENTAL Diffraction measurements were conducted on a four-circle diffractometer at the High Flux Beam Reactor at Brookhaven National Labora-

tained 0.37(2) and 0.055(6) Ag ions in an

tory using a neutron beamof 1.05099(6)~

octahedral and tetrahedral site, respectively.

wavelength monochromatedby reflection from

The present single-crystal neutron diffraction analyses were undertaken with the hope of resolving these ambiguities.

As there is

(002) of a Be single crystal.

Melt-grown

single crystals of Ag3SI were kindly provided for study by H. U. Beyeler of the Brown

no decrease in scattering power of the ions

Boveri Research Center, Baden, Switzerland.

with increasing scattering angle as with

The specimen selected for study was 2.1 mm

x-rays, and as absorption by the sample is

in diameter and 6.8 mm in length.

negligible, neutron diffraction generally

drical approximation to the shape was used

provides a greater number of observable high-

in subsequent correction of the intensity

A cylin-

angle intensities for such highly-disordered

data for absorption by the sample,(uI = 0.951

structures.

cm- l , ~l R = O.lO0). The crystal was mounted

Resolution of the density maps

should thus be improved. Moreover, Fourier

on a vanadiumpin held in the copper base of

synthesis of the nuclear scattering density

a controlled vacuum furnace which was heated

provides the probability distribution of a

resistively.

point nucleus, rather than a convolution of

to ±l°C over periods of several days.

this probability with the distribution of

Temperaturesremained stable

Data were collected at 8 temperatues

electron density on an atom as in the case of

(four for the B-phase and four for ~)

x-rays.

order of increasing temperature. The crystal

This further improves a b i l i t y to

in

resolve closely-spaced sites separated by less

was recentered, and the l a t t i c e constant and

than an atomic diameter.

an orientation matrix determined by a least-

Previous structural

examinations of the individual phases of

squares f i t to measurement of 20 reflections

Ag3SI had been restricted to a single tempera-

with 30° < 29 < 50° prior to measurement of

ture.

the room temperature data set and after each

(The B-phase, examinedat both -123°

and 22°C by Perenthaler et al.12, is an

incremental change in temperature. Remeasure-

*In a subsequent note14 i t is stated that the Ag distribution is better described by tetrahedral site occupancy at low temperature (-123°C) and octahedral occupancy at room temperature.

J.-J. Didisheim et al. / A single-crystal neutron diffraction study

l 153

Table I. Lattice Constants and Data Set Statistics for m- and B-Ag3SI as a Function of Temperature (Ntot = total number of intensities measured, Nav= total number of averaged independent reflections used in the refinements, Nav>O = number of independent reflections with IFLI ~ o(F~), Rint(1) = internal agreement of intensities, including unobserved reflections.) B-Ag3SI Pm3m

m-Ag3Sl Im3m 323°C

380°C

442°C

475°C

4.934(2)

4.946(2)

4.966(I)

4.970(I)

209

142

107

104

96

72

72

37

31

28

26

60

55

57

20

21

19

18

3.55%

2.82%

1.91%

3.31%

2.84%

2.88%

3.47%

0.78

0.78

O. 78

0.78

0.77

0.75

0.67

23°C

95°C

168°C

232°C

a(X)

4.892(I)

4.900(I)

4.904(I)

4.912(I)

Ntot

428

213

209

Nav

72

72

Nav> o

61 3.74%

Rint(1)

(Sine/~)max 0.78

ment of the l a t t i c e constant at room tempera-

where n is the number of equivalent r e f l e c -

ture a f t e r completion of experiments extend-

tions.

ing to 475°C provided the same value, w i t h i n

between equivalent r e f l e c t i o n s was evaluated

standard deviations as that obtained p r i o r to

as

the heating cycle.

Integrated i n t e n s i t i e s

were recorded with ~/20 scans to sin@/~ <

An overall i n t e r n a l agreement factor

N

N

Rint(1) = S r i t z l i i :I "=I

0.79A - l for the B phase, a l i m i t which was found to include a l l observable r e f l e c t i o n s .

where N is the t o t a l number of independent

This l i m i t was gradually reduced in the

r e f l e c t i o n s and

temperature range of the m phase as increased thermal v i b r a t i o n weakened the i n t e n s i t i e s at higher s c a t t e r i n g angles.

_i N r i = ni Z j=l

lii - lij I

Four sets of

symmetry-equi val ent r e f l e c t i o n s were recorded

is the i n t e r n a l agreement f a c t o r f o r an

at each temperature f o r the m-phase and three

i n d i v i d u a l set of equivalent r e f l e c t i o n s

for the B-phase (except f o r the room-tempera-

with average i n t e n s i t y l i "

ture data f o r which s i x sets were recorded.)

relevant to the data sets obtained at each

The variance due to counting s t a t i s t i c s f o r

temperature are summarized in Table I .

an i n d i v i d u a l r e f l e c t i o n was calculated as

"unobservable" r e f l e c t i o n s , defined as those

Statistics The

Oc2(I) = T + 5B where T and B are, respec-

IF21% ~(F2), were included in a l l subsequent

t i v e l y , the t o t a l counts and background

c r y s t a l l o g r a p h i c calculations as measured,

counts in the scan.

except that negative IF21 were redefined as

Symmetry-equivalent

r e f l e c t i o n s were averaged and the variance

having zero magnitude.

of the combined i n t e n s i t i e s was taken as 2 ( i ) = ~c 2 + ~p2 where ap 2, the variance

3. REFINEMENTSOF THE STRUCTURES

due to population s t a t i s t i c s , Op2 = ( n -

is given by

n ( I _ l j )2 I) -I Z j=l

All refinements were performed with the f u l l - m a t r i x least squares program UPALS14 which minimized the function

ZwIF~- F~I,

1154

.L-J. Didisheim et al. / A single-crystal neutron diffraction study o

where Fo and Fc are observed and calculated

t i c e constant at 300°C, 4.928A, which is

structure factors, r e s p e c ti v e l y , and the

provided by Fig. 2, is s i g n i f i c a n t l y smaller

weights, w, assigned to the observations were W -I

=

c~2(Fo2) + 0.01 Fo2, where c~2(Fo2) is

than values of 4.99(I) and 4.994(8)A found by Reuter and Hardel 3 and Perenthaler and

the variance o f the averaged squares of the

Schulz 13, respectively.

observed structure factors as defined above.

5. RESULTS FOR B-Ag3SI

Values of 0.597, 0.2847 and 0.528 lO-12cm

The structure o r i g i n a l l y proposed by Reuter

were employed as the coherent scattering

and Hardel 3 was employed as the i n i t i a l

lengths f o r Ag, S and I, respectively.

for the anion-ordered B phase:

Description of anharmonic thermal v i b r a t i o n

Pm3m, with

model

space group

I located in position 1 a m3m 000,

of the atoms, when employed, used the Gram-

S in 1 b m3m ½½½, and 0.25 Ag in 12 h mm x½0

Charlier expression f o r expansion of the tem-

with x % 0.4.

Refinement of this model

perature f a c t o r 15 in terms up to fourth order.

(seven parameters upon employing anisotropic

The level of refinement was assessed by means

harmonic temperature-factor

of the standard residual, R, weighted residual,

converged to a residual R(F2) of 8% f o r the

Rw, and goodness of f i t ,

S, which are defined

as

room-temperature data.

coefficients)

A difference Fourier

synthesis showed as the largest anomalies a R(F2) =

ZiFo2 -

Fc21/ZFo2

negative peak at 000, a p o s i t i v e peak at ½½½, and a broad p o s i t i v e maximum centered at

2 4-½ Rw(F2) = {EWlFo2 - Fc21 /ZwFo }

0½0. The anomalies at anion sites c l e a r l y suggested p a r t i a l disorder of S and I , a par-

s

=

{SWiFo2 -

Fc212/N-

P}½

t i a l quenching of the anion array of the I0

where N and P are the number of r e f l e c t i o n s

phase previously noted by Hoshino et al.

and the number of parameters in the model,

The anomaly at 0½0 represents Ag+ occupancy

respectively.

of the corresponding tetrahedral sites in the

4. TEMPERATUREDEPENDENCEOF THE LATTICE CONSTANT The l a t t i c e constants obtained fo r both

"antiphase" structure.

An order parameter n

(n ~ 1 i f no I is present at S s i t e s , n ~ 0 i f h a l f of the I ions are found at S posi-

~- and •-Ag3SI are presented in Table 1 and

tions) was accordingly included in the r e f i n e -

plotted as a function of temperature in Fig. 2.

ment as well as an "antiphase" tetrahedral

The v a r i a t i o n is well described by a l i n e a r dependence which corresponds to l i n e a r thermal

5.00

f

I

i

I

expansion c o e f f i c i e n t s of 1.86 and 5.05 10-5 K-I f o r the ~- and ~ phases, respectively. Extrapolations of the l i n e a r segments of

4.95 o

Fig. 2 i n t e rs e c t at 236°C, in good agreement

4.90

with the ~-B t r a n s i t i o n temperature of 235°C reported by Reuter and Hardel 2 and close to the temperature of 246°C at which an anomaly in the s p e c i f i c heat was reported I0

Our

l a t t i c e constant determination at room temperature is in good agreement with values obtained in other i n v e s t i g a t i o n s .

The l a t -

236 ° C .J

=

4.85 0

I I00

J

I 200

I

I

t

300

TEMPERATURE

I 400

I 500

('C)

FIGURE 2 Lattice constants f o r ~- and ~-Ag3SI as a function of temperature.

Z-J. Didisheim et aL / A single-crystal neutron diffraction study

Ag ion. Refinement of the I/S r a t i o showed no deviation from unity and was constrained

1155

the a l t e r n a t i v e models which were considered w i l l be subsequently published.) Introduction of the octahedral s i t e as an equilibrium position for Ag ions did not, in p a r t i c u l a r ,

to this value in the f i n a l cycles of refinement. The occupancy of the Ag sites in the structure and in the antiphase structure

improve the agreement between Fo and Fc

were refined independently; the t o t a l Ag con-

despite the introduction of a greater number

tent of the cell corresponded, within stan-

of parameters.

dard deviations, to stoichiometric Ag3SI, Table 2.

ing data, unlike the single-crystal x-ray

The present neutron scatter-

results, thus permit rejection of the octa-

A number of a l t e r n a t i v e models for the

hedral i n t e r s t i c e as an equilibrium position for Ag+ ions, a feature which, at elevated

structure were considered, including the introduction of anharmonic temperature-

temperatures, is shared by the related fast-

factor coefficients to fourth order and/or

ion conductors ~-Agl and B-Ag2S.

f u l l or p a r t i a l occupancy of the octahedral i n t e r s t i c e . No other reasonable refinement

synthesis of the scattering densities at

could be obtained.

levels which pass through the I - ion and

Figure 3 presents sections of Fourier

(A f u l l discussion of

Table 2. Site Occupancies, Atomic Coordinates and Anisotropic Temperature-Factor Coefficients for B-Ag3SI (Estimated Standard Deviations in Parentheses) Atom

Parameter

I 1 a m3m 000

23°C

95°C

168oc

232oc

B(X2)

3.43(7)

4.00(7)

4.24(7)

4.52(8)

B(X)

3.2(I)

3.4(I)

3.9(I)

4.0(I)

0.400(2) 0.13(I)

0.407(4) 0.13(2)

0.406(5) 0.14(I)

0.411(9) 0.15(2)

0.085(5)

0.II(I)

0.106(7)

0.12(I)

0.028(2)

0.028(3}

0.032(2)

0.035(2)

0.77(I)

0.83(I)

0.79(I)

0.85(I)

S i l v e r content~in Ag6SI

3.2(3)

3.1(4)

3.1(2)

3.1(3)

Scale factor

0.184(7)

0.184(7)

0.185(7)

0.188(8)

BII=822=833 823=B13=B12=0 S 1 b m3m ~½ BII=822=833 823=B13=B12=0 Ag 12 h mm x½0 823=~13=B12=0

Anion order parameter

x

BII 822 833 n

Residual*

R(F2)

0.0480

0.0586

0.0387

0.0468

Weighted residual*

Rw(F2 )

0.0546

0.0496

0.0468

0.0510

1.181

1.283

1.176

1.490

Goodness of f i t *

S

*Includes "unobservable" structure factors F21< a(F2)

J.-J. Didisheim et al. / A Jingle-crystal neutron diffraction study

1156

•

I

i h

d b

F

I

Ioi d

FIGURE 3 Sections of Fourier syntheses of the scattering density in anion-ordered ~-Ag3SI at 23°C (top) and 232°C^(bottom), at levels passing through I- ions at the cRrner~ ~f the cell ( l e f t ) and through S~- at the center ( r i g h t ) . Contour i n t e r v a l 0.08 I0 -~c cm A- ° , zero contour omitted. The strongest negative feature, -0.06 10-12 , f a l l s w i t h i n the contour i n t e r v a l and does not appear (a) p(xyO) 23°C, (b) p(xy½) 23°C, (c) p(xyO) 232°C, (d) p(xy½) 232°C. S2- ion, respectively, at 23°C and 232°C.

ments of the ions.

The resolution of the maps17 may be e s t i -

The parameters for the most s a t i s f a c t o r y

mated as 0.36 ( s i n S / ~ )x~~ = 0.46~ ~ a/lO,

descriptions of the structure of B-Ag3SI at

which is smaller than the 0.69X separation

the four temperatures examined are presented

between neighboring Ag+ sites.

in Table 2, along with residuals (including

The four

tetrahedral sites in a cluster are not

unobserved structure factors) and goodness-

resolved, however, and merge into a single

of-fit.

broad region of scattering density which

and positional parameter f or the room-temper-

The temperature-factor c o e f f i c i e n t s

tends to become more i s o t r o p i c with increas-

ature structure are in very good agreement

ing temperature.

with the x-ray results f o r the tetrahedral

The results are quite sim-

i l a r to those obtained by Perenthaler et

model reported by Perenthaler et al. 12.

a l . 12 despite the fact that the present syn-

p l o t of mean-squared thermal displacements

A

theses employ a number of observable struc-

as a function of temperature shows a satis-

ture factors which is up to 25% larger and

factory f i t

that the p r o b a b i l i t y d i s t r i b u t i o n is free of

r e l a t i o n s f o r the present data extrapolate

to a l i n e a r r e l a t i o n ; the

convolution with the d i s t r i b u t i o n of elec-

to good agreement with the x-ray results

trons on the Ag+ ion.

obtained at -123°C 12.

The overlap instead

arises from the large mean-squared displace-

Further extrapola-

t i o n to 0 K provides large residual mean-

J.-J. Didisheirn et al. / A single-crystal neutron diffraction study

1157

squared displacements for all atoms:

U~I(Ag ) = 0.121~ 2, U~2(Ag) = 0.065~ 2, UT33(Ag) = 0.022~2, U-'~(S) = 0.028~2, and U~(1) = 0.026~2.

The atomic displacements

thus contain positional disorder which is not

::::-i......... i />----J

"""--,i..ii........i/-:

accounted for by time-averaged thermal vibration.

I t is l i k e l y that the potential for the

Ag+ ions is v i r t u a l l y constant within the square area delineated by the cluster of four tetrahedral sites in (lO0) and that the tetrahedral positions represent very shallow potential minima. This interpretation is consistent with measurements of r e f l e c t i v i t y in the farinfrared 18,19. 6. RESULTSFOR ~-Ag3SI Alpha-Ag3SI s t a t i s t i c a l l y has the same bcc anion array in space group Im3m as ~-Agl and + B-Ag2S. The highly disordered Ag distribu-

FIGURE 4 Section p(xyO) of a Fourier synthesis of the scattering density in anion-disordered ~-Aq3SI at 475°C. Contour intervals at 0.017 lO-12 cm A-3. Positive and negative contours are shown as f u l l and dashed lines, respectively; zero contour omitted.

tion causes the signs of the structure factors to be determined by the r e l a t i v e l y immobile

e i t h e r m-Agl or B-Ag2S.

anion arrangement.*

not previously found in this family of struc-

This situation permits

A second new feature

a p r i o r i Fourier synthesis of the distribu-

tures is a well-defined local maximum in posi-

tion of Ag+ scattering density.

presents the section p(xyO) for ~-Ag3SI at

tion 8 c 3m ~ , midway between the pair of anions along the body diagonal §. This density

475°C. The Ag scattering density is delocal-

is displayed in the partial Fourier sections

Figure 4

ized in weak bands which closely

PAg(Xyk) of Figs. 5b and 5d.

resemble the result found by Perenthaler and

cates a clear trend towards greater delocali-

Schulz13.

The distribution is more remin-

iscent of that in B-Ag2S, Fig. Ib, than that for ~-AgI. The Ag d i s t r i b u t i o n was examined in more detail by means of partial Fourier syntheses in which the dominant scattering density of the anions was subtracted.

Figures 5a and

Figure 5 indi-

zation of the Ag density with increasing temperature, contrary to the behavior found for B-Ag3SI. The pronounced maximum at the octahedral interstice at lower temperature is greatly diminished at 475°C. I t seems unlikely that the complex d i s t r i bution of Figs. 5a and 5c represents a time-

5c present the partial synthesis PAg(XyO) at 323° and 475°C. The maps reveal a maximum

of the Ag+ potential.

at the octahedral site ~.0 and a fine struc-

mediate to ~-Agl and B-Ag2S in terms of cation/

ture of local maxima which was not found for

anion r a t i o , but the anion arrangement is iso-

averaged distribution which provides a measure Alpha-Ag3SI is inter-

*This was confirmed in refinement of the f i n a l models. §Lin~arly-coordinated. Ag occurs in several s u l f i d e structures 20 but with a bond distance of 2.44A which is considerably longer than the 2.14A separation here found for m-Ag3Sl. The coordination resembles that in Cu20.

Z-J. Didisheim et al. I A single-crystal neutron diffraction study

1158

i

....... (

.4,

....:::-,:'

i~.-:::" '.bJ,)/J~t~--,

b . . , - ~ ~ . ~ , ~ ,---,:<.1 .~

i "-"

<<->-,,-I

"-~---"

"-.-'Th

'---._~~ ~ r', I f / l /

/-.,~

d%' J r, ! I

"SSk'k I

a

..... ...... .,-~ <" i+:-:.~{((,~

'~~

,.... c~//#lu~,,~,',,J o

:'i~':.'~.--::"

!:,,~

..... .,," ~\ ~ ) 1 '--........

,

'X\.

..........~k~_j<. ..... -" c"-"~ "\,"~2 ,,,

~i

-I ,tl "-I

L,....I:..........~/'-",~(~- /) .......... \./ ii,, X'

..:[-... . . . . . . . . . . . . . . . . ~ i - f . L .

..../

,.?--

.

......,. : .......... , '"~

"~

....@/"

If(("~'q~}

," ,"~"~

-

i\.

d

FIGURE 5 Sections of p a r t i a l F o u r i e r syntheses o f the Ag s c a t t e r i n g d e n s i t y in a n i o n - d i s o r d e r e d ~-Ag3SI a t 323°C ( t o p ) and 475°C ( b o t t o m ) . Contour i n t e r v a l s a t 0.0075 10 -12 cm ~ - 3 zero contour o m i t t e d (a) PAg(XyO) 323°C, (b) PAg(xyk) 323°C, (c) PAg(XyO) 475°C, (d) PAg(Xy~) 475°C.

Table 3. Approximate Coordinates f o r a S i l v e r lon i n the Tetrahedral I n t e r s t i c e o f a Body-Centered Array o f Anions w i t h D i f f e r e n t NearestNeighbor C o n f i g u r a t i o n s o f S2- and I - Anions + Location o f Ag s i t e in Anions forming Distinct Im3m and approximate the t e t r a h e d r o n configurations coordinates 41 4S

1 1

12 d ZI2m ~0½

31 + IS

4

48 j m Oyz y = 0.212 z = 0.462

I I + 3S

4

48 j m Oyz y = 0.293 z = 0.543

21 + 2S

6 ('2

24 x 48 x

g = i =

mm x0½ 0.332 2 ~x½-x 0.040

J. -J. Didisheim et aL / A single-crystal neutron diffraction study

1159

structural to the binary endmembers only i f

be in proportion to the sum of the ionic r a d i i .

a long-range average of the structure is con-

All but one of the equipoints (48i) provide

sidered.

locations which occur in the section p(xyO).

The difference between Ag-S and Ag-I

i n t e r i o n i c distances makes i t reasonable to

Figure 6 compares the calculated e q u i l i b r i u m

expect that the e q u i l i b r i u m p o s i t i o n of an

positions with a portion of the p a r t i a l s i l v e r

Ag+ ion residing in an i n t e r s t i c e w i l l depend

density map, PAg(XyO) at 323°C, shown on an

on the short-range anion environment about

expanded scale.

the s i t e .

l o c a t i o n and shape of the local maxima is

Sixteen d i s t i n c t anion configura-

The correspondence with the

tions are possible at the vertices of each

striking.

tetrahedron and these provide 15 d i f f e r e n t

note that the greatest density is located at

e q u i l i b r i u m locations for an Ag ion residing

the 24g p o s i t i o n which corresponds to the

w i t h i n the tetrahedron (the regular environ-

environment of 21 + 2S which occurs in the

I t is of p a r t i c u l a r i n t e r e s t to

ment of e i t h e r 41- or 4S2- is assumed to lead

B-phase.

to occupancy o f the ideal center o f the t e t r a -

s i t e ½~0 which is predicted as the e q u i l i -

hedron ~½0) which are described by the 5

brium p o s i t i o n f or a tetrahedron formed

equipoints l i s t e d in Table 3.

e n t i r e l y of e i t h e r S or I anions.

The coordinates

Minimum density occurs at the 12d

The d i f -

f o r these positions were computed by requiring

fering occupancies of these positions may be

that the distances to the anion species at the

due to a tendency f o r Ag+ ions to preferen-

corners of the c e l l or body-centered position

tially

occupy sites coordinated by 21 and 2S

as they migrate through the structure and to ¼00

avoid i n t e r s t i c e s formed by other anion configurations.

A l t e r n a t i v e l y , the anions might

not be d i s t r i b u t e d completely at random; short-range interactions may favor the occurence o f local anion configurations consisting of 2S and 21 as in the B-phase.

The change

in Ag d i s t r i b u t i o n with temperature might, in this case, p a r t i a l l y r e f l e c t a decrease o f such short-range order with increasing temperature. A s i m i l a r analysis might be made of the d i s t r i b u t i o n of e q u i l i b r i u m sites w i t h i n the octahedral i n t e r s t i c e .

As 67 configurations

are possible this seemed u n j u s t i f i e d given the r e s o l u t i o n o f the density maps. The small number of observable i n t e n s i t i e s - between 18 and 21--creates problems in any FIGURE 6 Expanded portion of a p a r t i a l Fourier synthesis of the Ag scattering density pAa(XyO) in1~ ~-Aq3SI at 323° . Contour i n t e r v a l D.O05 I0 -to cm A-3. Superposed are the calculated e q u i l i brium positions o f Ag+ ions with d i f f e r e n t short-range anion configurations. 12d(41 or 4S) • ; 24g(21 + 2S) + ; 4 8 j ( I I + 3S) • ; 48j(31 + IS) • .

attempt to perform a refinement o f the structure.

The observed scattering density may

c l e a r l y be modeled to an a r b i t r a r i l y good agreement i f enough atomic positions or higherorder anharmonic temperature-factor c o e f f i cients are introduced, yet the preceding dis-

Z-J. Did~heim etaL / A sm#e~rys~lneutron diffraction smdy

1160

Table 4. Atomic Coordinates and Temperature Factors as a Function of Temperature for the Most Satisfactory Model for m-Ag3Sl Temperature Atom (S,I) 2 a m3m 000

Parameter BII

323°C

380°C

442°C

475°C

0.079(I)

0.083(2)

0.087(2)

0.093(2)

n

2.5(2)

2.2(2)

1.9(2)

2.2(5)

~i 1=822=833 823=BI 3=BI 2=0 Ag(1) 24 g mm x0½ 823:B 13:BI 2=0

Ag(2) 8 c ~m kk~ BII=822=833 623=~13=B12

x

0.344(3)

0.348(6)

0.36(I)

0.35(I)

BII

0.21(2)

0.24(5)

0.30(9)

0.27(6)

822

0.20(2)

0.19(2)

0.21(3)

0.23(3)

833

0.063(4)

0.063(6)

0.058(9)

0,07(I)

n

0.6(2)

1.0(3)

1.6(3)

1.4(5)

0.27(8) -0.12(4)

0.4(I) -0.17(5)

0.5(I) -0.23(5)

0.4(2) -0.17(7)

0.19(I)

0.19(2)

0.19(I)

0.19(2)

R(F2)

0.030

0.029

0.020

0.018

Rw(F2)

0,043

0.045

0.033

0.042

S

1.14

1.32

1.03

0.97

BII B12

Scale factor Residual* Weighted residual* Goodness of f i t *

*Includes "unobservable" structure factors, F2<~(F 2)

t r u l y involves an average of disorder over

perature and equal in magnitude to the value found for the B-phase; (d) refined Ag s i t e

d i s t i n c t closely-spaced sites which are not

occupancies which agreed with a stoichiometry

equally populated.

of 3 Ag+ per c e l l .

A large number of models for the structure

Only two simple models were found to meet these c r i t e r i a . In the f i r s t , Ag ions with

cussion suggests that the Ag d i s t r i b u t i o n

at 323°C were tested (the results w i l l be f u l l y discussed elsewhere) which involved occupancy of various sites with or without anharmonic temperature factors. Several c r i t e r i a were used to evaluate the physical reasonableness of these models: (a) the residuals R(F2) and Rw(F2); nice i f small, but suspect i f below the measures of i n t e r nal agreement; (b) a goodness-of-fit, S, close to unity--not easily achieved in view of the small number of degrees of freedom, N-P; (c) a scale factor independent of tem-

anisotropic harmonic temperature-factor coefficients were assigned to the displaced tetrahedral position 24g shown by Fig. 6 to be the location which is p r i n c i p a l l y occupied. The second model placed Ag in the octahedral s i t e and included anharmonic temperaturefactor coefficients to fourth order. Both models included p a r t i a l occupancy of a second type of Ag s i t e at position 8 c k½½. Two of the four independent fourth-order anharmonic coefficients permitted by symmetry in the

J.-J. Didisheim et al. / A single-crystal neu tron diffraction study

1161

second model (all third-order terms are iden-

disorder rather than a time average of dynamic

t i c a l l y zero), D2222 = D3333 and Dll22 = Dll33, refined to small, nonsignificant values and

disorder which might provide a measure of the

were set equal to zero.

Both models then

Ag potential.

Satisfactory models for the

structure placed a partial Ag+ in a linearly-

required lO parameters; both refined to the

coordinated position plus either an anhar-

same residual (including unobserved F2

monically-vibrating ion in the octahedral

o(F2)) of 3.0% and displayed similar d i f f e r -

site or a harmonic Ag+ ion in a tetrahedral

ence maps. There accordingly was no basis

site at the equilibrium position correspond-

for discrimination between the models on the

ing to a local environment of 21 + 2S as in

basis of a test of s t a t i s t i c a l significance

the B-phase. Both models require the same

or the above c r i t e r i a of physical reasonable-

number of parameters for their description

ness.

and refined to similar residuals.

Similar ambiguity was experiencedin

This ambi-

refinements performed with the data sets

guity was e a r l i e r encountered in single-

which had been obtained at higher tempera-

crystal x-ray analyses13, and was not resolved

tures.

The octahedral model was s l i g h t l y

more satisfactory at 380°C, but the tetra-

in the present neutron study despite the fact that the probability distribution is not

hedral model was clearly "better" at the two

further broadenedby convolution with the

highest temperatures. The refined parameters

distribution of electrons on the Ag ion.

and residuals obtained for the l a t t e r model

tetrahedral model seemed, on the basis of

are presented as a function of temperature

physical c r i t e r i a , to be a s l i g h t l y more

in Table 4.

preferable representation of the structure.

The

The distribution of mobile ions in Ag3SI 7. CONCLUSIONS Fourier syntheses of the Ag+ scattering

has characteristics similar to those in both ~-Agl and ~-Ag2S but not quite in the manner

density in anion-ordered ~-Ag3SI failed to

anticipated.

resolve occupancy of discreet octahedral or

~-Agl in that Ag ions residing in tetrahedral

tetrahedral sites due to large thermal vibra-

interstices l i k e l y hop between neighboring

Anion-ordered B-Ag3SI resembles

tion amplitudes of the ions which occupy

sites.

these closely-spaced positions, but refine-

ever, that individual sites are not resolved,

ments permitted rejection of f u l l or partial

and residual mean-squareddisplacements at

occupancy of the octahedral interstice as an

0 K indicate positional delocalization which

equilibrium position. Anion-disordered ~-Ag3SI contains a Ag+

The potential minimum is so f l a t , how-

is not accounted for by thermal vibration. The Ag ions in anion-disordered B-Ag3SI l i k e l y

distribution which displays a local maximum

(but not irrefutably) occupy tetrahedral

at a linearly-coordinated position at kkk

interstices.

plus delocalized scattering density in

probability distribution represents an aver-

In commonwith B-Ag2S, the Ag+

bands. The bands contain a fine structure

age of positional disorder over distinct

of local maximawhich may be explained in

closely-spaced equilibrium sites within the

terms of dependenceof the equilibrium posi-

tetrahedral cavity.

However,the disorder

tion within the tetrahedral void on the short-

arises from the short-range configurations

range configuration of anion species about

of two different anion species about the

the site.

interstice rather than from an irregular dis-

The probability distribution thus

primarily represents an average of positional

tribution of anion-cation bond distances

1162

J.-J. Didisheim et aL

/ A single-crystal neutron diffraction study

between the same pair of species as in

neutron d i f f r a c t i o n , in: Fast lon Transport in Solids, eds. P. Vashista, J. N. Mundy and G. K. Shenoy (Elsevier North Holland, N.Y., 1979) pp. 217-220.

Ag2S. Alpha-Ag3Sl is unique in displaying p a r t i a l occupancy of a l i n e a r l y - c o o r d i n a t e d s i t e on the body-diagonal of the c e l l .

8, S. Hoshino, T. Sakuma and Y. F u j i i ,

J.

Phys. Soc. Japan 45 (1978) 705.

ACKNOWLEDGEMENTS One of us (J.-J. D.) is pleased to acknowledge the support of a fellowship received from the Swiss National Science Foundation.

9. A. Magistris, G. Chiodelli and A. S c h i r a l d i , Z. Physik. Chemie 112 (1978) 251.

The neutron d i f f r a c t i o n measurements were

I0. S. Hoshino, T. Sakuma and Y. F u j i i , J. Phys. Soc. Japan 47 (1979) 1252.

carried out at the High Flux Beam Reactor

II.

at Brookhaven National Laboratory, operated under contract AE-AC-O2-CHO016 with the U.S. Department of Energy.

We are indebted to

Joseph Henriques of BNL for assistance with the diffractometer measurements and operat i o n of the vacuum furnace. REFERENCES I. B. Reuter and K. Hardel, Naturwiss. 48 (1961) 161. 2. B. Reuter and K. Hardel, Z. Anorg. A l l g . Chem. 340 (1965) 158. 3. B. Reuter and K. Hardel, Z. Anorg. A l l g . Chem. 340 (1965) 168. 4. R. J. Cava, F. Reidinger and B. J. Wuensch, Sol. State Comm. 24 (1977) 411.

T. Sakuma and S. Hoshino, J. Phys. Soc. Japan 48 (1980) 1036.

12. E. Perenthaler, H. Schulz and H. U. Beyeler, Acta Cryst. B37 (1981) 1017. 13. E. Perenthaler and H. Schulz, Solid State lonics 2 (1981) 43. 14. E. Perenthaler, H. Schulz and H. U. Beyeler, Solid State lonics 5 (1981) 493. 15. J.-O. Lundgren, UPALS, a Full Matrix Least-Squares Refinement Program, I n s t i tute of Chemistry, Upala, Sweden (1979). 16. C. K. Johnson and H. A. Levy, Thermal motion analysis using Bragg d i f f r a c t i o n data, i n : International Tables for X-ray Crystallography, Vol. IV, eds. J. A. Ibers and W. C. Hamilton (Kynoch Press, Birmingham, 1974) pp. 311-336. 17. R. W. James, Acta Cryst. 1 (1948) 132.

5. R. J. Cava, F. Reidinger and B. J. Wuensch, J. Sol. State Chem. 31 (1980) 69. 6. P. Vashista and A. Rahman, Phys. Rev. Lett. 40 (1978) 1337. 7. R. J. Cava, F. Reidinger and B. J. Wuensch, Conductivity mechanisms in the superionic phases of Agl and Ag2S as determined by

18. B. Gras and K. Funke, Solid State lonics 2 (1981) 341. 19. P. BrUesch, H. U. Beyeler and S. Str~ssler, Phys. Rev. 825 (1982) 541. 20. W. Nowacki, Schweiz Min. Petrogr. M i t t . 49 (1969) 109.