Non-stoichiometry in uranium sesquinitride

J. Phys. Chem. Solids,

1973, Vol.

34, pp. 1611-1626 9

Pergamon

Press.

Printed

in Great

Britain

NON-STOICHIOMETRY IN U R A N I U M SESQUINITRIDE T. FUJINO and H. TAGAWA Japan Atomic Energy Research Institute, Tokai-mura, Ibaraki, Japan

(Received 18 July 1972) A b s t r a c t - - A statistical model is proposed for analyzing thermodynamic properties of non-stoichiometric uranium sesquinitride. It consists of the combination of the site exclusion mechanism and cluster formation reactions. The model was applied to the three ionic possibilities of U ~ U ~ + N f -, U4+US+N33- and UzJ+N3 z-. By fitting the theoretical equations to the experimentally o b s e r v e d data, 4+ 6+ 3it was found that the set U 32U, zU3 /3 -- - 2,, 1 : 2 cluster gave the most satisfactory results. From 9 . / . . ' /3 = 2, the upper hmR o f non-stolchlometry is expected to be N / U = 1.75. The result that the l : 2 clusters predominantly exist in UzN~+~ suggests the existence of UNv7~ phase. T h e energy required for nitrogen atom for occupying the lattice site of the crystal, E, was calculated to be - 120.5 kcal/ mole, and the interaction energy to form clusters, E~, was - I 1.07 kcal/mole.

1. INTRODUCTION

to the several rare-earth sesquioxides, uranium sesquinitride crystallizes in two different crystal systems. High temperature form /~-U2N3 which is stable at higher temperatures than 800~ is hexagonal with the La203 type structure [ 1,2]. Its composition range is presumably in the hypo-stoichiometric side and very narrow[3-5]. Another sesquinitride o~-U2N3, which is more familiar than the /~-form, is cubic with the /3-Mn203 type structure [6]. The crystal of this form is stable at room temperature, and has rather wide non-stoichiometric range of composition. The type of the non-stoichiometry has been reported to be nitrogen excess, U2Nj+~, and according to the X-ray and neutron diffraction analyses the excess nitrogen atoms are in 16(c) position of the space group Ia3, while regular nitrogen atoms in 48(e) position[7, 8]. Statistical mechanical method is one of the most powerful tools for investigating thermodynamic quantities of non-stoichiometric compounds in relation to the atomic scale behaviour of defects or clusters. Katsura and Sano[9] have analyzed the thermodynamic data of U2Nj+x with a statistical model that attractive and repulsive forces are limited SIMILAR

between the most neighbouring atoms in U2N~+~.. They assumed that the interaction energies between these atoms are not varied but the numbers of uranium-uranium, uranium-nitrogen and nitrogen-nitrogen pairs are varied with x of U.,N:~+j., and calculated the partial molar free energy with BraggWilliams approximation. Although this kind of treatment is successful for explaining the cooperative phenomena in metallic alloy systems, it would give not always good results for ionic compounds where the nonstoichiometry causes the change of cation valencies, i.e. change of the interaction energies. The bonding character of U.,N3+x is not definitely defined yet, but it is generally accepted that the compound is partially ionic and partially covalent[10]. In the present paper, then, we deal with the problem with attention to the ionic behaviour of this compound. The statistical model we are presenting here may be regarded as an ionic counterpart of the model of Katsura and Sano [9]. In recent works of Atlas on ionic fluorite type crystals of CeO2_~ and UO2+x[I 1, 12], he proposed a method where the local fluctuation of anion and reduced (or oxidized)

1611

1612

T. FUJINO

and H. TAGAWA

cation concentrations is expressed in terms of the transformed distribution of normal concentration of such cations and decreased (or increased) concentration of neighbouring anions around an anion defect. However, the method involves the procedure to determine, a priori, the coulombic energy states by classical dielectric theory in calculating repulsive energies between anions and those between cations, so that it cannot be applied to the U2N3+x system where ionicity is likely to be incomplete. Present method is developed on the treatment of Anderson[13]. The model used here is characterized by the following points. First, the arrangement of excess nitrogen anions in the anion sites is considered to follow site exclusion mechanism originally given by Speiser and Spretnak[14]. The exclusion value /3' was treated as a constant. Secondly, as the results of interaction between (relatively) positive and negative defects, they are considered to form clusters. The formation reaction of clusters is analyzed with the quasi-chemical method [15] by which we evaluated the interaction energy without Bragg-Williams approximation. Three ionic models of I'-'I 43/2+ I"'sI 61/2 + Nx ~3 J 3-[16] , U4+US+N83- [ 17] and U2~+N32- [ 18] have been proposed concerning the bonding nature of U2N3. In the present stage, however, no evidences are known which conclusively support one of these models. This is presumably due to difficulties in discerning the ionic species in solid crystals. It is one of the aims of this paper to discuss these ionic models in relation with the thermodynamic properties. Regarding to experimental pressurecomposition-temperature data on the nonstoichiometric U~N3+~ phase, there have been considerable discrepancies in both values of equilibrium pressure and its dependence on composition. Thus, in the present paper, we carefully performed the pressure measurements with the aim to contribute to the thermodynamic assessment of the U2N3+~ system.

2. C O N S T R U C T I O N

OF THE

MODEL

2.1 Anions in U2N3+x

Non-stoichiometric a-uranium sesquinitride is known to have N / U ratios over 1.5 by chemical analysis. The measurement of density shows that the value increases as the N / U ratio increases[19]. From these results, the type of non-stoichiometry in this phase is concluded to be nitrogen excess U2N3+x rather than uranium deficient Uz-xN3. X-ray and neutron diffraction analyses also give strong support for this type. According to the diffraction techniques, the peak intensities for U2N3+~ are consistently expressed by assigning excess nitrogen atoms to 16(c) position and regular nitrogen atoms' to 48(e) position of the space group Ia3 [7]. These anions interact to each other. Consider first the interaction of the excess nitrogen atom with the regular nitrogen atom. If E is defined as the energy required for a neutral nitrogen atom in gas phase to occupy one of the 16(c) positions in the U~N3+x crystal, the above interaction energy is included inseparably in E. In this paper, the value of E is assumed to be constant irrespective of composition and temperature. Next, the interaction between the excess nitrogen atoms should be considered. Perhaps, it will be most natural to consider the mechanism for this interaction energy that the number of sites assigned to a defect is varied with varying interaction force between defects [20]. That is to say, the value of/3' in the site exclusion concept proposed by Speiser and Spretnak[14] varies with defect concentration and temperature, where/3' is the number of sites around a defect which cannot be occupied by the other defects on account of repulsive interaction between them. Conversely, the apparent/3' value can be obtained by means of statistical mechanics if energy levels are defined as a function of/3'. In the work of Atlas [11, 12], he carried out this calculation by expressing the interaction energies as a function of distance between defects with the dielectric theory on the assumption that the interaction is exclusively coulombic. How-

NON-STOICHIOMETRY

IN

ever, for crystals where non-coulombic interactions are mixed as in the present case, this treatment cannot be applied because the energy levels are given by no explicit means, i.e. the value offl' cannot be determined. In this situation, the method of Hagemark [21] seems to be useful. He assumed in UO2+~ that 12 nearest neighbour interstitial sites around an excess oxygen atom are not able to be occupied by the other oxygen ions, and found that the theoretical (~o~ and So~ values agree well with experimental data in rather wide range of x in UO2+~. The method of Hagemark is equivalent to the treatment that /3' of Speiser and Spretnak is a constant, i.e. not a function of composition. Although this method has been applied only to the ionic UO2+~ crystals, it will also be applied to crystals of which bonding nature is partly covalent. One theoretical justification for this consideration may be given by (1) in covalent crystals the interaction force is short range character and far more rapidly decreases with distance than in ionic crystals; (2)/3' excluded sites will reasonably exist under the condition that the repulsive force between an excess anion and most neighbouring anions is larger than the force to push the neighbouring anions toward the excess anion as the results of mutual repulsion between a great number of excess anions. Thus, in the present UsN3+~ system, 13' is assumed to be independent of composition and temperature in the experimental range studied. The complex which consists of an excess nitrogen ion and /3' neighbouring vacant 16(c) sites is assumed to be distributed randomly in the 16(c) position. 2.2 Cations in U2Na+x As for the valence model of uranium sesquinitride, (I) U4+U6+N (II) 312 1/2 3 a-rt 16], U4§ [17] and (III) U2a+Na2- [1 8] have been proposed. Experimental evidence on the magnetism shows that U2N3 is paramagnetic and the magnetic moment decreases with x of U2N3+x on the line toward zero at UN2 [ 16]. Because U 6+ is diamagnetic, the valence

J P C S Vol. 34, N o 1 0 - B

URANIUM

SESQUINITRIDE

1613

state of uranium is reasonably + 6 in UN2. However, it is not yet clear whether U 5§ or U 6§ exists in U2Na§ Possibility II is proposed by Trzebiatowski [ 17] on the basis that the spin only magnetic moment for ll4+ll 6+--3/2--1/2 N~3- is considerably greater than the experimental value. However, the [T4+[T6+~ 3model will not necessarily be discarded because the theoretical magnetic moment for this model will be diminished by considering ligand field perturbation. Possibility III is proposed by Nasu et al. [18] for/3-U2N3. It is somewhat unfamiliar that the valency of nitrogen ions is --2 in this model, but according to Didchenko et a1.[22], - 2 valency is seen in the mononitrides of rare-earth elements and then its possibility may exist also for uranium sesquinitride. Aside from the problem which the true model is in this system, the introduction of excess nitrogen into the crystal causes some uranium ions oxidized by charge neutrality condition. This oxidation-reduction reaction will be described for each model as below:

(I)

X ,.-~ 3/2 ~..., 1/2a "q3

+ N 3§ a- r-n 3(1-x)/2U 6(1+3x)/2

= U4+

(II)

X

U4+US+N33-[] + ~N~ TT4+

TTs+

~3-~

,,.a l _ 3 x ~ J 1 + 3 x ~ 9 3 + x ~ . . J l - x ,

(lid

(1)

(2)

U+a+N3'~-W] + 2N+ = IT3+ TT,+~2- vn 9,,.~ 2(1 --X) ,.-~ 2 x 9 - 3+xt.--J l - X ,

(3)

where [] is the vacant 16(c) lattice sites for excess nitrogen ions. We refer the models represented by equations (1), (2) and (3) to 4-6 model, 4-5 model and 3-4 model, respectively. These models may be regarded as an extension of each of possibility I, 11 and 111 to non-stoichiometric range. For possibility 111, 3-5 and 3-6 models are not considered here because of far more than linear rapid increase of the ionization potential of uranium from + 3 to +6[23]. The ionic species in the solid will,

1614

T. F U J I N O and H. T A G A W A

of course, not so simply be defined; crystallographic configuration and entropy effect etc. may also play an important role. As one possible mechanism of the 4 - 6 model, the entropy effect can be considered to cause a disproportionation reaction, 2 U 5 + = U 4+ + U 6+, in the solids.

lattice is no more random after the formation of the clusters. Consider first the 1:2 cluster. Denoting [excess N 3 - + f l ' vacant anion sites] by [excess N 3-] for brevity, the reaction is expressed as 2 U 6+ + excess N 3- = U 6+ - N 3- - U 6+,

2.3 4 - 6 model We give full statistical description first for the 4 - 6 model. Let the number o f lattice sites for uranium ions be 2M, then the number of vacant sites available for being occupied by excess nitrogen ions is M for stoichiometric U2N,~ (equation(l)). If the n u m b e r of the excess nitrogen ions in the U2N3+~ crystal is m, the relation m = M x holds. In this non-stoichiometric crystal, the number of vacant anion sites decreases to M - r e . Because the crystallographic results indicate that all uranium ions are on the same 24(d) position[7], it will be reasonable to assume at first that U 4+ and U 6+ ions are distributed randomly on this position. These ions are present with 3 : 1 ratio in the stoichiometric U2N3. As the crystal accomodates the excess nitrogen ions, as shown in equation (1), the numbers of the U 4+ and U 6+ ions are varied to ] ( M - m ) and 8 9 respectively. Because the average valency of the uranium ions is ~ ( 3 + x ) in the range of 0 ~< x < 1, the U 6+ ions behave as positive defects in this crystal. T h e s e ions have an attractive coulomb interaction with excess nitrogen anions, so that the mechanism that these cations and anions form clusters can be considered. Now that the excess nitrogen ion has already made up a complex by enveloping fl' vacant anion sites, the cluster formed with the complex and with one of the most neighbouring U 6+ ions will be expressed as [excess N 3 - + / 3 ' vacant anion sites]U 6+ which we refer to 1:1 cluster. If an excess nitrogen anion bonds with two U 6+ ions, the cluster is [excess N 3 - + / 3 ' vacant anion sites] U26+ which we refer to 1 : 2 cluster. It is noteworthy that the distribution of U 6+ in the cation sub-

(4)

where the right side of this equation shows the 1:2 cluster. If 2YM cluster is formed in U2Na+x, since the numbers o f U 6+ and excess nitrogen ions are 8 9 and m--2YM, respectively, the equilibrium constant K of this reaction is Y

K = [88 l + 3x) _ 2y]2( 89

Y),

(5)

where x is the value in U2N3+x. This equilibrium constant depends on the temperature as

K = e~Sme-amRr = Ae-Ec/~T

(6)

which is given by the relation AG = - - R T I n K . In equation (6), A is a constant and Ec is formation energy of a cluster. F r o m equation (5), /33 2 +gx 7 +l+l/K)y 4 p - (sx+ 1)r +tNx --lx(3x+

1) 2 = 0.

(7)

Therefore, the concentration o f the cluster, Y, is given by equation (7) by knowing K. The number of ways of arranging each of the 1:2 clusters, the excess N 3- ions which do not form 1:2 clusters, the U 4+ and free U 6+ ions o v e r the lattice sites is represented by

f~(m) = x [89

fl"(M/fl)! 62YMm! (M/fl--m) !m!" ( m - - 2 Y M ) ! ( 2 Y M ) !

(2M--4m) ! + 3m) -- 4YM] ! [ 89 1 l m ) + 4YM] !.

(s) T h e first term gives the n u m b e r o f ways o f arranging m excess N a- ions over M/fl avail-

NON-STOICHIOMETRY

IN U R A N I U M

able anion sites, where /3 = f l ' + 1 [14]. T h e second term is the number of ways of selecting 2 Y M excess N 3- ions which form the 1:2 clusters out of m excess N ~- ions. Because there are four nearest neighbour uranium sites around an excess N 3- ion, the number of ways that two of four sites are occupied by U 6+ ions is 4C2 = 6. T h e factor 62YM is necessary because there are 2 Y M such clusters. If the structure parameter of U2N3+~ is taken into account, the coordination o f uranium around an excess N 3- is not exactly tetrahedral, but the approximation above will be permitted in the present treatment where the other factors are not very accurate. The third term is the number of ways of arranging the remaining ,~,-(M+3m)-4YM U s+ ions o v e r 2 M - 4 m cation sites which are not neighbouring to the excess N 3- ions. The vibrational mode of the crystal varies as it accommodates the excess nitrogen ions. In Einstein model of the lattice specific heat, each atom in the crystal is supposed to be an independent oscillator. Although this model agrees to lesser extent with experimental values at low temperatures than D e b y e model, it is used in usual statistical discussion provided that the temperature of thermodynamic interest is higher than the characteristic temperature where the discrepancy between these two models is small. By means of the Einstein model, the change of the characteristic temperature by impurity atoms can be estimated to be of the order of Y power of the change of lattice constant[12] where Y is Griineisen constant. Then, the characteristic temperature can be regarded not significantly be changed by 0.04 A [19] change of the lattice constant with non-stoichiometry in the present

SESQUINITRIDE

1615

crystals; the effect of excess nitrogen ions on the whole vibrational modes can be treated simply as an increase of the number of oscillators in the Einstein model. T h e semi-grand partition function for the non-stoichiometric U2N3+j. crystal is represented by M e-O 12T =---_,= ~ h't2(m) 1 - e -~ l~

m=O

where h is the absolute activity of nitrogen atoms, ~ ( r n ) is the number of configurations given by equation (8), and O is the Einstein characteristic temperature. The third term in the sum shows the vibrational partition function of the excess nitrogen atoms. The last term is concerned with the energy required in the process that the m excess nitrogen atoms occupy the 16(c) position in the crystal and form 2 Y M 1:2 clusters. Because E is safely described by the maximum term in the sum of equation (9), by the condition OlnE

Ore* the equation ln h-f Oln~)(rn*)

Orn*

I e-~ + 31n 1 ~ / ~ 1

E + 2MEc dY]dm* kT = 0

]

(10)

is obtained. In this equation, m* is the value of m which gives maximum contribution to E. By rewriting m* again as m, and differentiating l ) ( m ) , we get

1 3 [E--ZEc] ( 1 - - ~ x ) (2Y)Z({--J~-zx+4Y)2Z+11/2Z+l 1 ~. 2Z+3[21-1 [ 1 e -~ [6z(Z--4x)4(x--ZY) (-~+zx--4Y) ' ] ~e_-~/r] e x p [ ~ - -],

h= [

(11)

where

Z=_2MdY

[ 88189188

~mm= -- [88 1 + 3x) -- 2Y] 2+ 4( 89

-2Y] Y) [88 1 + 3x) -- 2Y] + l/K"

(12)

1616

T, F U J I N O and H. T A G A W A

one. This factor takes the form exp [-mB/2] in equation (9), which can be regarded as a correction term for the configurational entropy. For the case of the 1 : 1 cluster, the reaction of cluster formation is

On the other hand, the absolute activity of nitrogen atoms in the gas phase in equilibrium with N~ molecules is expressed as [24] h =

MN

= F |.

(P.f.)u

'1.4 "Na

q112 [

k(P.f.)x~" e~/krJ

'

(13)

where MN, M.~2 are the number of nitrogen atoms and molecules in the gas phase, respectively, (p.f.) is the partition function for microcanonical ensemble, and D is the dissociation energy of a nitrogen molecule into two nitrogen atoms. Since (P.f.)N~ can be expressed as the product of translational, vibrational and rotational partition functions [24], equation (13) becomes )k -~- e -D/2kT

[ ~41rrn~kT13'2kT

XLL

#

j

e -~

U6+ + e x c e s s N a- = U 6 + - N 3-,

where U 6 + - N a- means the 1:1 cluster in this model. The equilibrium constant is

K= ( 89188

,

O(m) •

[ 89

=

~m( M/fl) ! 42rUm ! (MI B --m)!m! ( m - 2YM) r(2YM) ! ( 2 M - 4m) ! 3m) - 2 Y M ] r[ 89 l lm) + 2YM] r (19)

The first term of the right of this equation is the same with that in equation (8). The second term is the number of ways of choosing 2YM excess nitrogen atoms which form 1:1 clusters out of m excess nitrogen atoms. The factor 4 zYMcomes from the fact that the hum-

~. [6z(2-4x)4(x--2Y)Z+l( 89 . . . . . . k .( 1 -.# x ). (2Y)Z(j-42-x+4Y) 2z+11/2 J

In PN, = z m ,

+C(T) + (2E+D) -2ZEc kT + B, where

9 rf4rrmNkTq31~.~ e- Ov/2T ~r]

,,-

1

-

-

(18)

T 1-,, ~

1-e-% rO,.J

where mN is the mass of nitrogen atom, p is the pressure of N~ gas and | | are the vibrational and rotational characteristic temperatures of the gaseous nitrogen molecules, respectively. In equilibrium state, the absolute activity of the nitrogen atoms in the gaseous phase is equal to that in the crystal phase. Then, from equations (1 1) and (14) the logarithm of the N2 pressure is

#

Y

provided that the number of the I : 1 clusters formed is 2YM. The number of configurations is given as

(14)

c(r)=,nLt-

(17)

e- ~

The value B in equation (15) is only one arbitrary parameter introduced in order to adjust theoretical pressure with experimental

(15)I

ber of ways that an U 6§ ion occupies one of four most neighbouring cation sites around each excess nitrogen atom is four. The third term is the number of ways that the 89 + 3m) --2YM U 6+ ions which do not form clusters occupy 2 M - 4 m cation sites randomly, where 2 M - 4 m is the number of cation sites not neighbouring to the excess nitrogen ions.

(16) '

[ e-e/2r ] L1

NON-STOICH1OMETRY IN URANIUM SESQU1N1TRIDE

Similarly as before, the logarithm of the nitrogen pressure is given as lnp,~,~ = 21n[ 4z(2-4x)4(x-2Y)z+~[89 ~ ~

1617

U 5+ + e x c e s s N 3- -- U 5 + - N 3-,

(27)

] - 6 1 n [ 1 - ~e -~ //JF ]

~

+ C ( T ) 4 ( 2 E + D ) --2ZEc ~-B, kT

(20)

where Ec is the formation energy of the 1 : 1 cluster, and

Z=--

20

{~(l+x)2+ 89

2.4 4-5 model Consider first the reaction to form the 1 : 2 clusters. For this model, the expressions become 2 U 5+ +

excess N a-

.

(21)

Y

K = ( 89189

(28)

f~(m) is given by changing the third term of equation (19) to

= U 5+ - N a - - U 5+,

(2M - 4m) !

(22)

(M+3m-2YM)!(M--Tm+2YM)!'

Y

K = (89 Y) [89 l + 3x) _ 2y] 2.

(23)

The number of configurations, f l ( m ) , is given by changing the third term of equation (8) for the 4-6 model to

(29)

and In p.v~ is given by changing the first term of equation (20) to , F4z(2 -- 4x)4(x--2Y)Z+l(1 + 3x-- 2Y) z+a-] (30)

(2M - 4m) !

where

(M + 3 m - 4YM) !(M-- 7m + 4YM) !" (24) The logarithm of the nitrogen pressure is represented by changing the first term of equation (15) to 2Y) z+l ( 1 + 3 x - - 4Y) 2z+~1 (1--fix) (2Y)Z(1 - - 7 x + 4 Y ) zz+7 (25)

. F6 z ( 2 - - 4 x ) 4 ( x - -

mL

J"

___1[, l+2x+4/K

1

{88 + 2x)2+ (1 + 4 x ) / K + l/K2} 1/2 " (31) 2.5 3-4 model For the case of the 1 : 2 cluster, the reaction is written as 2U4++excess N 2- = U 4 + - N 2 - - U

4+,

(32)

where

Z=--

[ 89189189 [89( 1 + 3x) - 2 Y] 2+ 4 (89 - Y) [89( 1 + 3x) -- 2 Y] + 1/K"

Consider next for the reaction of 1:1 cluster. In this case,

and

K =

y ( 89 Y) ( x - 2Y) 2"

(26)

(33)

1618

T. F U J I N O and H. T A G A W A

T h e n u m b e r of configurations and the logarithm of the nitrogen pressure are given by replacing the third term of equation (8) to ( 2 M - - 4m) ! (2m -- 4YM) !(2M -- 6m + 4YM) !

(34)

and the first t e r m of equation (15) to , [-6z(2 -- 4x)4(x -- 2Y) z+' (2x -- 4Y) 2z+2] (35) respectively, but Z=

3( 89 Y)2 3( 89 y ) 2 + 1/4K"

(36)

F o r the case of the l : l cluster, U 4+ + e x c e s s

N z-

=

U 4 + - - N 2-,

(37)

and K =

(89

Y Y) ( x - Y)"

(38)

H e r e , f l ( m ) is given by changing the third t e r m of equation (19) to ( 2 M - 4m) ! (2m -- 2YM) ! ( 2 M - - 6m + 2YM) !'

(39)

and the logarithm of the pressure is o b t a i n e d by replacing the first term of equation (20) to , [4 z (2 -- 4x) 4 (x -- 2Y) z+' (2x -- 2Y) z+2]

'nL

J' (40)

where

___113

lx+3/K ] (88 + 3x/K + 1/K2),/2j. (41)

3. E X P E R I M E N T A L

3:1 Apparatus T h e experimental apparatus is the s a m e as described in the earlier paper[25]. It consists of a Cahn R H - t y p e automatic electrobalance, a Kanthal resistance furnace, a pressure

m e a s u r e m e n t system, hydrogen and nitrogen supply system and v a c u u m pumps. T h e balance was adjusted to have a maxim u m weight change of 500 mg and a sensitivity of 0.01 mg. T h e dimension of quartz crucible was 25 mm in height and 18 m m in outer diameter. T h e temperature of a specimen was measured by a P t / P t + 13% Rh thermocouple mounted close to the crucible inside the reaction tube, by which the t e m p e r a t u r e was automatically regulated within + l~ T h e over-all system was able to be e v a c u a t e d till 2 • 10 -6 m m H g or below. Its gas leak was very small; after standing for 1 week without pumping, the pressure was still as low as 1 0 - 4 m m H g . F o r the m e a s u r e m e n t s of the nitrogen pressure, oil and mercury manometers were used. Nitrogen and hydrogen gas used are commercial ones with purities higher than 99.99 per cent. Nitrogen was purified by passing o v e r a c o p p e r gauze heated at 520~ with subsequent passage through a liquid nitrogen trap. H y d r o g e n was purified by passing through palladium asbestos heated at 300~ and through a liquid nitrogen trap.

3.2 Procedure Experiments were carried out by varying the nitrogen pressure at a fixed t e m p e r a t u r e between 500 and 1000~ U r a n i u m nitride was prepared in situ in the crucible. U r a n i u m turnings of about 1 g were washed in carbon tetrachloride, pickled in cold dilute nitric acid to r e m o v e thin oxide films, and washed in distilled water and aceton. T h e bright metal was transferred into the crucible which was suspended from one end of the balance arm. T h e reaction system was e v a c u a t e d to 5 • 10 -6 m m H g . H y d r o g e n gas was introduced into the system up to about 150 m m H g . Then, the reaction was conducted at 200-250~ The metal was converted to uranium hydride. Nitrogen gas was added up to the total pressure of 3 5 0 - 4 0 0 m m H g , and the temperature was raised till about 400~ T h e hydride

N O N - S T O I C H I O M E T R Y IN U R A N I U M S E S Q U I N I T R I D E

readily converted to the nitride having the composition n e a r U N 1 . 7. After the mixture of hydrogen and nitrogen gas was evacuated, nitrogen was again introduced into the system to the pressure of 50 mmHg. T e m p e r a t u r e was raised further and then kept at a fixed value between 500 and 1000~ T h e attainment of the equilibrium was judged with a weight recording chart. T h e composition of the sesquinitride was determined as the weight gain of the uranium metal. Because temperature was fixed in a run, the measurements consisted of reading of the weight change at various equilibrium pressures. In the course of experiments, hysteresis effect was examined by putting the nitrogen pressure back to its initial value. The results showed that the effect was negligibly small under the experimental conditions. All experimental data were corrected for buoyancy. T h e data below 1 0 m m H g were also corrected for the thermomolecular flow because its effect becomes not negligible in this range[26-28]. This was performed by measuring the weight of nichrome wires as a dummy sample with other experimental conditions unchanged. 4, DETAILS OF COMPUTATION

Computations were carried out on a F A C O M 2 3 0 - 6 0 computer. T h e experimental values of (~x~, /~N2 and SN.~ were calculated as follows: As will be shown in the later section of this paper, the isotherms of lnpN~ VSX can be regarded to lie on straight lines over fairly wide composition range. Thus, the fitting to linear equation was performed in this range by the least square method for each of 600, 700, 800, 900 and 1000~ isotherms. After the relation between ln pN~ and x is obtained for several temperatures, the relation between lnp.v~ and 1 I T can be examined for a fixed x value. According to the results of computation, the latter relation was regarded to be linear. This means 1

lnpx~ = r + s - ~ ,

1619

where r and s are parameters determined by the least square calculations. Partial molar thermodynamic quantities are obtained by

/4~ = R d In Px2 = R s , 9

d(l/T)

g~T2 _ Flx.~ -- G.~,.~ _ R r . 9

(42)

T

F r o m these equations it is seen that/4.v~, S x~ are independent of temperature so far as lnpN~ is a linear function of 1/T. T h e fitting of the statistical mechanical results of (15), (20), (25), (30), (35) and (40) to the observed In Px2 vs x curves is performed as follows: The unknown coefficients in these equations are A, B, E and E,., where d is the value in equation (6). K, Y and Z are variables which are dependent on composition and temperature. Since equations (15), (20), (25), (30), (35) and (40) are not linear with respect to the unknown coefficients, they were expanded in T a y l o r series around the approximate values of the coefficients, and the least square calculations were carried out by means of successive approximation method. Because these equations contain logarithmic terms, it occurs for some statistical models that all antilogarithms obtained from each of the input data cannot remain positive as the coefficients converge. In this case, we understood that the model was not 'correct'. In computing the successive approximation series, the factors less than 1 (minimum 1/1000) were multiplied to the correction terms in order that the series rapidly converges without excessive corrections. T h e values of characteristic temperatures of N2 gas used were | = 3340 and | 2.86~ As the value of | D e b y e temperature for U N (324~ was substituted since the value for U2N3 has not been reported. S. RESULTS AND DISCUSSION

Figure 1 shows the reported data of the equilibrium pressure together with ours,

1620

T. F U J I N O and H. T A G A W A 1000

lO00 x+

500

+ A

~x A

~

0 x Z~ Z~

~x

/~

A

o

II

oo

z~

~+x

,s,,,

A

lltil

x

9

+ x

•

ooo*c

:,o o

ii!: /? //

Z 10

/)7 "b o 5 o o * c

x

e

E E

i~ ~o

A

1OO

5O

looo'c 9o0'c 7oo'c 8oo*ceoo~c

500

lo

g 700"C

Z"

/

~

/

o

| o

1

I

I

|

I~,

I

I

1-5 1'55 1.6 1.65 1,7 N/U 0 0.1 0"2 0'3 0.4 x Fig. 1. Comparison of the consistency between our data and previously reported data. • Lapat and Holden[28]; + Bugl and Bauer[29]; A Mfiller and Ragoss[31]; (Z) Present work.

where as an example only 1000 and 700~ data are described. It is seen from this figure that our values consist with those of Bugl and Bauer[29] excepting the values for larger x at 700~ but the agreement with those of Miiller and Ragoss[31] is not good. The discrepancy of the data seems to be larger at 700~ but the scatter of the observed points is small within the data of the same author, which suggests that the errors are of systematic nature. As for the origin of these discrepancies, the oxidation of nitride sample during an experiment can be considered. A part of sample being oxidized, the apparent N / U value increases. Another origin may be traced to the non-equilibrium of nitrogen pressure between the sample and the gas phase because the equilibrium is not easily attained at low temperatures. For example, about 1 day is necessary for equilibrium at 500~ On the other hand, long period heating would cause the oxidation of the sample by small amounts of oxygen in the nitrogen gas.

1.5

1.s5

0

0.1

1.'6 0.2

1.65

117 %

0.3

0.4

x

Cornposifion

Fig. 2. Composition dependence of lnp,vr 0 Observed point; ...... 3-4 model with input data Region 1 and 3; ....... 4 - 6 model with input data Region 1 and 3; - 4 - 6 model with input data Region 1.

In Fig. 2, all values we measured are given with open circles. These data indicate that the dependence of lnpN2 on x is almost linear in the pressure range over 10mmHg and that the slope of these isotherms becomes steeper in the series from 1000 to 500~ The experimental data for 1000, 900 and 800~ will be most reliable because at these temperatures the equilibrium is easily attained. However, the effect of thermomolecular flow becomes significant in the pressure range less than 10 mmHg, and therefore, the accuracy of the data in this range is somewhat lower even after it was corrected. We refer these PN= => I0 mmHg and ps, < l0 mmHg regions to Region I and 2, respectively. Since pressure data for x > 0.33 are corresponding to the lower temperatures of 700, 600 and 500~ these data may be much less reliable. We refer this region to Region 3. As input data for the computations of statistical mechanical equations, we used

NON-STOICHIOMETRY

1N URANIUM

first 65 observed points of Region I. The results showed that the slope of the curve of In pu2 vs x was dominantly influenced by the configurational term. For the case of the I : 1 cluster, computed slopes were extremely large and far from fitting to the experimental values for all of the 4-6, 4 - 5 and 3 - 4 models. However, if 1:2 cluster was supposed, both of the 4 - 6 and 3-4 models were correct for/3 = 1 and 2 (/3 = 3: not), although the 4 - 5 model was not correct for all of/3 = 1,2 and 3. With the input data of Region 1, no great discrepancies were seen between the InpN~VSX curves of these 4 - 6 and 3-4 models which were calculated with the A, B, E and Ec values obtained by the least square calculations. The effect of /3 appears in the range of larger x values; for /3 = 1, the slope of the lnpu2 V S X curve becomes fairly smaller than that for /3 = 2 in the range x > 0.33 and it inclines to separate from the observed data. With 100 input data of Region 1 and 3, we again carried out the least square computations for the 4 - 6 and the 3-4 models. The result was that both of the models are correct for/3 = 2 but not correct for/3 = 1. This means that if the observed data in Region 3 are, in spite of their low accuracies, still valid for discriminating the value of/3, they show that /3 = 2 is the case in the present system. The curves in Fig. 2 are obtained from the least square values of A, B, E and Ec, which are tabulated in Table 1. Dashed curves are the results for the 3-4 model with the input data of Region 1 and 3, and dash-dot curves are for the 4 - 6 model with the same input data. Comparing these curves, it is found that the

SESQUINITRIDE

1621

dashed curves rather well coincide with the experimental points in the range of larger x values but the deviation from the experimental data is very large in Region 2 (1
Table 1. Computed results of unknown coefficients L i n e in Fig. 2

Model

Input data (Region)

A

2E + D

U 3+ - U 4+

1 and 3

0.005128

land3 1

B

k

- 21.32

2257 • 10

0.1500

-1.74

1836

0.7181

- 1-59

k - 1035 • 10

/3=2 U 4 + - U 6+ /3=2 U 4+ - U 6+ /3=2

-8094

--1010• -5569

10

1622

T. F U J I N O and H. T A G A W A

below 10 m m H g well follow the experimental curvatures in this region although Region 2 data are not used as input for computing out solid lines. Moreover, according to the computed results, the curves of In PN~ are expected to show sigmoidal variations with x provided that the x value covers the wide range of composition, which is consistent with the results of Naoumidis[32]. He measured the equilibrium nitrogen pressures up to 100 atm by means of autoclave techniques and found that all curves show sigmoidal dependencies on the composition x. The meaning of/3 = 2 is that any one excess nitrogen atom has one more site which is not to be occupied by the other excess nitrogen atoms. As its consequence, the limit of the non-stoichiometry must be N / U = 1-75 (x = 0-5) from equations (8) or (15). This is actually fulfilled in the real crystals; crystals with N / U > 1.75 have not been formed under the usual experimental conditions. In addition to this, the plot of lattice constant against composition shows a discontinuity near N / U = 1.75119]. T h e s e facts are self-consistent with the results of fitting to the experimental In PN~ that/3 is 2. The rapid increase of the experimental equilibrium pressure in the vicinity of N / U = 1.75 may be caused by the stronger repulsive interactions between the neighbouring excess nitrogen atoms when /3 = 1, to which rearrangement is necessarily undergone at N/U = 1.75. Statistical mechanical values of partial thermodynamic quantities for the 4 - 6 , / 3 = 2 model are given from equation (15) as the following equations: G v~ = R T

2In

These (~N, /4x2 and SN~ values were calculated by using A, B, E and Ec values on the lowest line of Table 1. T h e y are presented as solid curves in Figs. 3, 4 and 5. Figure 3

800~

15 -~ \ ~

900

lc

i~ t 5

o 1.5

1'55

1.6

1.65

0

0'I

0.2

0.3

N/u x

Fig. 3. Variation o f - G,v~ with composition. 9 Experimental value; Theoretical line.

shows the variation of GN2 with composition for various temperatures. It is seen in this figure that the calculated values are very well in agreement with the experimental values at 1100 and 1000~ At 900 and 800~ curves and experimental points consist with each other in the range of larger x values, but the departures appear as x decreases. Figures 4 and 5 represent /~Uzand SN,, respectively. As is seen in equation (42), experimental values of/~U2 and SN~ are not functions

6Z ( 2 - - 4 x ) 4 ( x - - 2 Y ) Z + I (.2t + ~3 x - - 4 Y ) -(i~---~x) ~

2Z+3/2

6In(1 ~-O-Y]e-~ ~

+ C ( T ) + B ] + 2E + D--2ZEc, J

fflu = 2E + D -- 2ZE~, F 1 6z(2--4x)4(x--2Y)Z+'( 89 n i-2 +C(T)+B].

2z+a/z

// e-O/2T \

61n~,-1 ~ ) (43~

NON-STOICH1OMETRY

IN U R A N I U M

60 9

50 E

40 O

o

~

z

IT

9 __9

3o 20

10 O

f

i

i

1.5

1-55

1-6

1.65

1.7 N/U

i

0

0.1

0.2

0.5

0.4

x

Fig. 4. Variation o f - - / 1 N , with composition. 9 Experimental value; C) Experimental value of Bugl and Bauer [29]; Theoretical line for 1000~

40 O

9

/ 1

/

g7

. ~

- ~ 0

9

2C

",,,2

%%%.

I

J

~"

i

1"5

1"55

1'6

1'65

1.7

N/

0

0"1

0"2

0"3

0"4

x

Fig. 5. Variation of partial molar entropy of nitrogen. 9 Experimental value; C) Experimental value of Bugl

and Bauer[29]; Lines indicated are the results of theoretical calculation for 1000~ of temperature in this range. On the other hand, theoretical values are dependent on it; the theoretical curves in Figs. 4 and 5 are for 1000~ It is seen from Fig. 4 that experimental--/tN~ value decreases fairly greatly with increasing x value, which is in contrast to the dependence of the theoretical curve where the decrease is much smaller. The

SESQUINITRIDE

1623

latter dependence will be expected if E does not vary with composition and if its absolute value is fairly larger than the interaction energy of defects. In Fig. 5, there are greater discrepancies between experimental and theoretical Sx~ values. In this figure, the experimental --SN~ value decreases with x, but the theoretical value increases with x. It is not yet clear what the reason of these discrepancies is. If we trace the reason in the experimental values, one possibility is that in the process of the least square calculations to plot lnp;v~ vs I/T we used together with the data of Region I those of Region 3 which are considered to have lower accuracies. The effect of Region 3 is especially large in the range of larger x values. An alternative reasoning will be traced in the theoretical values, i.e. the statistical model above is not applicable to the present system. However, the experimental values o f - S o ~ increase with x in Th,U1_,O2+x[33], Mg,U1_,O,~+.,.[341, UO2+;[12, 35, 36] and Ce,,O:~+x[l 1,371 systems for example, and these facts are consistent with each of statistical mechanical results. It can be considered that in these cases the sum of the vibrational entropy of the solid of the added anions and the gaseous entropy which occurs on removing one mole of anions from the gas phase does not vary greatly with composition and that the change of total So~ is mainly caused by the configurational entropy. This is also seen in the theoretical values of the present system. As shown in Fig. 5, the negative vibrational and gaseous entropy, -- S,v~(vib. + gas), remains almost unchanged with x, while the negative configurational entropy, - Sx~ (conf.), remarkably increases with x. It is sure that the value of -So~ decreases in Pr203+~ system in the transient range to form intermediate ordered phase, but this range is rather narrow between O / P r = 1.78 and 1.83138]. Therefore, if it is true in the present case that--,~.~ decrease monotonically in the wide composition range from x = 0.1 to 0.4, this phase will be concluded to have unusual configurations and/or

1624

T. F U J I N O

a n d H. T A G A W A

vibrational modes of nitrogen atoms. For this problem, first of all, more precise data on L.~N2are required. Figure 6 represents the variation of (~U~,

-~Na(/'"vib.+gos) 40 ~J"-"

.........

~.~

30

2O o E

10

10

gNu(conf.) _

0 400

600

800

Temp.,

0 1000 1200 *C

Composition : x= 0.2 Fig. 6. D e p e n d e n c e of partial m o l a r q u a n t i t i e s on temperature.

HN, and ~-~N2with temperature. A slight decrease o f - - H N, at higher temperatures is ascribed to the interaction energy term. Variation o f - ~-~U2is also small. In the preceding discussion, we regarded that UzNa+z is completely ionic. In the case where the covalency is added to the bonding nature, its effect will be recognized as the decrease of apparent charges in the statistical mechanical treatments because electrons concerning to the covalent bonding are not distinguishable between the atoms. If the degrees of covalency of U 4+, U 6+ and N 3- ions are expressed by l-y, 1-8 and l-E, respectively, the apparent charges of these ions will be U (4~+, U (~)+ and N (3')- for the 4-6 model. By the condition of charge neutrality, the numbers of U ~4~+and U (68)+become 12~--3T(3 + x ) M 6e -- 48

and 37(3 + x ) -88_M ' 6e -- 48 respectively. The number of configurations given in equation (8) should be modified in this case by the numbers of U ~4~)+and U ~ + , and thus SN, having the form of equation (43) varies with the degree of covalancy. It will be able to estimate how much the crystal is covalent on the accurate experimental data, but here we do not treat this problem quantitatively. In the present paper, we have considered the formation of 1 : 1 and 1:2 clusters. However, it is possible that such other clusters having compositions I :3 and 1:4 are formed because the uranium atoms coordinate around an excess nitrogen atom nearly tetrahedrally. A finding which seems to be related to this problem is that in Fel_xO system the structure and size of the cluster is independent of the composition although it is not yet clear how Roth cluster[39] and Koch-Cohen cluster[40] exist in the three ranges of Fex_xO [41]. These two clusters are different in the structure but the composition is close to that of Fe304 which is the next phase occurs in the extension of the non-stoichiometry of the Fel_xO. This means that the non-stoichiometry of the wiistite is composed of random and atomic scale dispersion of the clusters whose composition is close to that of next ocurring FeaO4 in the matrix of stoichiometric FeO[41]. It is surely not ascertained that this kind of mechanism holds for the present system, but direct application reveals that the 1:4 cluster requires UN2 as the next compound; this cluster would be denied because UN2 is not known. On the other hand, if next compound is UN~.Ts (= 88 the 1 : 2 clusters are expected to be formed predominantly. The equilibrium constant of the cluster formation reactions are shown in Fig. 7 for the 4-6,/3 = 2 model. Although the difference of the two curves in this figure is not small due to the great sensitivity to the input data, the

NON-STOICHIOMETRY 5x[o'

IN U R A N I U M S E S Q U I N I T R I D E

\

Y

.Z 103

'-' 5 x l o 2

S = I,.,tJ

lo' 5xlO 400

i

i

600

I

I

800

I

I

1000

Temp.,

Fig. 7. Equilibrium constant of the reaction forming 1 : 2 clusters in the 4-6 model (equation (5)). - . . . . Input data, Region 1 and 3; Input data, Region I.

equilibrium constant is high enough to regard that almost all clusters formed are 1:2 type with remaining free U n+ less than several per cent. This fact supports the existence of UNI.r~ phase, which is consistent with our former conclusion that/3 is 2 because it means that the limiting composition of the U2Na+x phase is N / U = 1.75. Let us discuss the energy values in Table 1. Because D = 2 2 5 k c a l / m o l e [ 4 2 ] , we have E = - 120.5 kcal/mole with the use of Region 1 data in the table. The interaction energy obtained is Ec=-ll.O7kcal/mole; E is negatively by far larger than E~. In the present statistical model, we assumed that the excess nitrogen atoms occupy vacant 16(c) sites with the stabilizing energy E at first and then, U 6+ ions coordinate around excess nitrogen ions with the interaction energy E~. F o r these mechanisms it is necessary that E is negatively larger than E~, which is shown to be fulfilled in this case. 6. CONCLUSIONS The characteristics of the present statistical model are (1) the site exclusion mechanism originally proposed by Speiser and Spretnak

1625

was taken; ( 2 ) t h e value offl' was considered not to vary with composition and temperature; (3) anionic and cationic defects were considered to form clusters. This model was applied to the three ionic possibilities of uranium sesquinitride, namely ~ r4+! T6+N.T3U4+US+N3 3- and Uz3+N32-. By examining the fitting of theoretical equation for each of these possibilities to the observed data, best fit was obtained for the i~.--~Ta+l~6--~ 3- model with 3/2 ~J 1/2z "3 fl = 2 and 1 : 2 cluster. Although discrepancies were found in partial molar entropy values, the above set was successful for explaining the partial molar free energies of nitrogen for nonstoichiometric U2N3+~. The result o f / 3 = 2 means that the highest composition of the UzN3+x phase is N / U = 1.75, which is supported by several experimental facts. T h e 1:2 cluster suggests the existence of U N , 7~ phase, which is consistent with the result that /3 = 2. The energies calculated are E = - 120.5 and Ee = -- 11.07 kcal/mole. Present model can be modified for partial covalency in the crystals at least in principle. The refinement of the model along this line would be hopeful for better fit between experimental and theoretical values of thermodynamic quantities. Acknowledgernents-The authors are grateful to Dr. T. Ishimori and Dr, K. Ueno for a number of helpful suggestions.

REFERENCES 1. M A L L E T T M. W . and G E R D S A . F., J. electrochem. 102,292 (1955). 2. V A U G H A N D. A . , J Metals 8, 78 (1956). 3. BENZ R. and B O W M A N M. G.,J. Am. chem. Soc. 88,264 (1966). 4. SASA Y. and A T O D A T., J. Am. ceram. Soc. 53, 102(1970). 5. STOCKER H. J. and N A O U M I D I S A., Ber. Deut. K eram. G es. 43,724 (1966). 6. R U N D L E T. E., B A E N Z 1 N G E R N. C., WILSON A. S. and M c D O N A L D R. A.,J. Am. chem. Soc. 70, 99 (1948). 7. TOBISCH J. and HASE W., Phys. Status Solidi 21, KI1 (1967). 8. MASAKI N. and T A G A W A H., to be published. 9. K A T S U R A M. and SANO T.,J, nucl. Sci. Technol. 4,283 (1967). 10. C O R D F U N K E E. H. P., in The ChemisttT of Uranium, p. 195. Elsevier, Amsterdam (1969).

1626

T. F U J I N O and H. T A G A W A

11. ATLAS L. M.,J. Phys. Chem. Solids 29, 91 (1968). 12. ATLAS L. M., J. Phys. Chem. Solids 29, 1349 (1968). 13. A N D E R S O N J. S., Proc. R. Soc. Lond. A185, 69 (1946). 14. SPEISER R. and S P R E T N A K J. W., Trans. Am. Soc. Metals 47,493 (1955). 15. A N D E R S O N J. S. and B A R R A C L O U G H C. G., Trans. Faraday Soc. 59, 1572 (1963). 16. S E G A L N. and SEBBA F., PEL- 165 (1967). 17. T R Z E B I A T O W S K I W. and TROC R., Bull. ,4cad. Polon. Sci. 12,681 (1964). 18. NASU S., TAMAK1 M., T A G A W A H. and KIKUCH I T., Phys. Status Solidi 9, 317 (1972). 19. T A G A W A H., to be published. 20. REES A. L. G., Trans. Faraday Soc. 50, 335 (1954). 21. H A G E M A R K K. and BROL! M., J. inorg, nucl. Chem. 28, 2837 (1966). 22. D I D C H E N K O R. and G O R T S E M A F. P., J. Phys. Chem. Solids 24, 863 (1963). 23. C H I L D S B. G., CRMet-788 (1958). 24. R U S H B R O O K E G. S., Introduction to Statistical Mechanics, Chapter 6, Oxford, London (1949). Japanese Translation by KUBO S. and KINOSH ITA T., Hakusuisha, Tokyo (1955). 25. T A G A W A H.,J. nucl. Mater. 41,313 (1971). 26. C Z A N D E R N A A. W., in Vacuum Microbalance Techniques (Edited by M. J. Katz), Vol. 1, p. 129. Plenum Press, New York ( 1961). 27. T H O M A S J. M. and POULIS J. A., Vacuum Microbalance Techniques (Edited by K. H. Behrndt), Vol. 3, p. 15. Plenum Press, New York (1963). 28. LAPAT P. E. and H O L D E N R. B., in Compounds o f Interest in Nuclear Reactor Technology (Edited

29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

39. 40. 41. 42.

by J. T. Waber, P. Chiotti and W. N. Miner), Vol. 10, p. 225. Met. Soc. AIME, Edws. Bros., Ann Arbor (1964). B U G L J. and BAUER A. A., ibid. Vol. 10, p. 215 (1964). S C A R B R O U G H J. O., D A V I S H. L., F U L K E R SON W. and B E T T E R T O N J. O., Jr., Phys. Rev. 176, 666 (1968). MIJLLER F. and R A G O S S H., in Thermodynamics o f Nuclear Materials, 1967 p. 257. IAEA, Vienna (1968). N A O U M I D I S A., JiJl-472-RW (1967). A R O N S O N S. and C L A Y T O N J. C., J. chem. Phys. 32, 749 (1960). F U J I N O T. and N A I T O K., J. inorg, nucL Chem. 32, 627 (1970). ROBERTS L. E. J. and W A L T E R A. J., J. inorg. nucl. Chem. 22, 213 (1961). K I U K K O L A K., ,4cta Chem. Scand. 16, 327 (1962). B E V A N D. J. M. and K O R D I S J., J. inorg, nucl. Chem. 26, 1509 (1964). J E N K I N S M. S., T U R C O T T E R. P. and E Y R I N G L., in The Chemistry o f Extended Defects in Nonmetallic Solids (Edited by L. Eyring and M. O'Keefe), p. 36. North-Holland, Amsterdam (1970). ROTH W. L., ,4 cta crystallogr. 13, 140 (1960). KOCH F. and C O H E N J. B.,,4cta crystallogr. B25, 275 (1969). A N D E R S O N J. S., in Modern Aspects o f Solid State Chemistry (Edited by C. N. R. Rao), p. 75. Plenum Press, New York (1970). ,4merican Institute o f Physics Handbook (Edited by D. E. Gray), p. 7-140. McGraw-Hill, New York (1957).