THOMAS R. P. GIBB,JR.
1096
Crystallographic Requirements and Configurational Entropy in Body-Centered Cubic Hydrides
by Thomas R. P. Gibb, Jr. Contribution No. d l 1 from Department of chemistry, Tufts University, itfedford, Massachusetts (Received November 1.8, 1863)
The number of ways w in which foreign atoms or ions may be arranged in a b.c.c. host lattice is enumerated as a function of composition for structural units containing two host atoms. The result is shown to be simply related to w for a mole of substance, which leads directly to the configurational entropy. The relatively large contribution of configurational entropy to the free energy of b.c.c. metallic hydrides renders accurate calculation of w particularly important for this class of compounds. It is shown that conventional (combinatorial) calculation of w leads to large errors when site occupation is restricted, e.g., in the case where occupation of a given site renders the nearest sites unavailable. The value of w is obtained by actual enumeration for restricted occupancy of octahedral and of tetrahedral sites by foreign atoms X and the results are presented as graphs of w us. n where n is the atomic ratio defined by MX,. A simple method of calculating w for restricted site occupancy is derived heuristically and shown to give excellent results for the cases studied.
Introduction The number of ways in which interstitial atoms may be arranged in the lattice of a nonstoichiometric crystal may influence or, geometrical factors being equal, even determine the changes in structure and magnetic properties resulting from the insertion of these atoms. The number w of such arrangements controls the configurational or statistical entropy S, through the wellknown relation s, = k In w,where k is Boltzmann’s constant (3.296 X cal./deg.). For highly nonstoichiometric solids where the number of arrangements per mole may be of the order of loz4,and particularly if the heat of formation of the solid is low, the T S term of the general expression for free energy may be nearly comparable to the heat of formation or enthalpy term a t room temperature or above. This is the case with some of the metallic hydrides, q., of group V metals and palladium. This article will deal only with the number of ways in which hydrogen atoms, for example, may be arranged in a b.c.c. metal lattice such as that of vanadium. The treatment is designed primarily to bridge the gap between purely geometrical considerations, such as radius ratio, atomic radii, etc.,l and the bases for The Journal of Physical Chemistry
statistical-mechanical treatment, 2--5 including orderdisorder One of the difficulties in setting up a statistical-mechanical treatment is always the choice of the array or cluster to be used. In some cases this is straightforward as in Snoek’s classical work on martensite (cj. ref. 9), where the carbon atoms are free to occupy any of the octahedral sites in the unstressed b.c.c. iron. I n other cases, occupancy of one
(1) T. R. P. Gibb, Jr., Advances in Chemistry Series, No. 39, R. F. Gould, Ed., American Chemical Society, Washington, D. C., 1963, p. 99. (2) S. L. H. Martin and A. L. G. Rees, Trans. Faraday SOC.,50, 343 (1954). (3) G. G. Libowitz, “Advances in Chemistry Series,” No. 39, R. F. Gould, Ed., American Chemical Society, Washington, D. C., 1963. (4) J. E. Mayer and M . G. Mayer, “Statistical Mechanics,” John Wiley and Sons, Inc.. New York, N. Y . , 1946. (6) L. Kaufman, ASD-61-445 (1961); Trans. A I M E , 224, 1006 (1962). (6) J. M. Honig, J . Chem. Educ., 38, 638 (1961). (7) F. E. J. K. Aretz, Physica, 26, 967, 981 (1960); 28, 736 (1962). (8) T. Muto and Y. Takagi, “Solid State Physics,” Vol. I , F. Seitz and D. Turnbull, Ed., Academic Press, Inc., New York, N. Y., 1955, p. 193. (9) J. D. Fast, “Entropy,” McGraw-Hill Book Co., New York, N. Y., 1962, p. 123.
CRYSTALLOGRAPHIC REQUIREMENTS IN BODY-CENTERED CUBICHYDRIDEH
site by a rather large interstitial atom may prevent occupation of some neighboring sites. The nature of such restrictions is sometimes subtJle1s3and especially so when progressive anisotropic deformations occur.
Interstitial Configurations in a B.c.c. Lattice If all of the octahedral sites of a b.c.c. metal are equivalent, as they must be in an undistorted crystal (Fig. l), then the number of ways w of arranging hydrogens
I
I
I
A
n i n MHn
Figure 1. Octahedral sites (black dots) and tetrahedral sites (triangles) in b.c.c. lattice. Tetrahedral sites are shown only on three faces.
w=
( 3 N )! ( B ~ N ) ! [-( Ie
B
=
-3N[LJ In 0
Figure 2. Natural log of the number of ways of arranging interstitial atomo H in one unit cell of an undistorted b.c.c. metal: curve A, equivalent octahedral sites; B, octahedral sites, mutual exclusion of face center and edge center sites; C, nonsuperposable arrangements for B ; D, equivalent tetrahedral sites; E, one tetrahedral site per face; F, tetrahedral sites, exclusion of three nearest sites. Curves are guide lines between discrete points.
p~]!
where N is Avogadro’s number, and 0 is the occupied fraction of the 3N available sites. The number of octahedral sites per unit cell is six, as is evident from Fig. 1 , and the unit cell contains two metal atoms, hence 3N sites per gram-atom of metal. This formula simplifies via Eltirling’sapproximation to In w
curve A of Fig. 2 , but with the ordinate increased by an (see below). appropriate factor of the order of If the three octahedral sites per metal atom are not equivalent (but are equally available), then the number of arrangements is given by multiplying together three combinatorial relations of the type shown above with the necessary changes in total sites and fractions occupied. Such methods may be used to evaluate the change in entropy resulting from redistribution of interstitial atoms as in a martensitic transformation, or from stress due to distortion of the crystal s t r u ~ t u r e . ~ A similar formula may be set up for occupancy of tetrahedral sites where the total number of sites per unit cell is twelve.
In w
I
in these sites in a gram-atom of metal is given by simple combinatorial formula
1097
+ (1 - 8) In (1 - e ) ]
where the nuinber of sites 3N is outside the brackets. As long as 3N is large enough so that Stirling’s approximation is applicable, In w is directly proportional $0 the total number of sites a t a constant value for the occupancy. I[f the composition of the interstitial compound were represented by MX, or MH,, then 0 = n / 3 . A graph of ln w us. n is similar in shape to
If the size of the interstitial atom is such that occupancy of a given site prevents or inhibits occupation of certain neighboring sites, the simple combinatorial method used above is no longer applicable, and in most cases the number of arrangements cannot be calculated in this way. To understand the relation between purely crystallographic considerations and the calculation of the number of arrangements, it is necessary to investigate what happens in a simple unit cell or in a minimum rtsseinbly of such cells, where the evident relationship between sites helps to visualize what will happen in a very large assembly. In Fig. 2 , curve A shows the natural log of the number of arrangements possible in a single unit such as Fig. 1 plotted us. the stoichiometric proportion of interstitial atoms Volume 68,Number 6
M a y , la64
THOMAS R. P. GIBB,JR.
1098
represented as n in equivalent octahedral sites. This recapitulates the previous combinatorial calculation but now in a much smaller unit. It is surprising that the number of arrangements is so large (maximum is 48,620) for a single unit cell. If the interstitial atoms are found not to occupy the edge-center positions (octahedral sites) 0, 0; 0,O; l,l/~ 0;, l/2,1,0 (Fig. 1)when the face center site 0 is occupied (or vice versa), which is a not uncommon situation in a b.c.c. matrix, then one may not use a simple combinatorial calculation to get the number of possible arrangements for a given fractional occupation. The reason for this is that the availability of sites is no longer a simple function of the fraction filled. The size of the unit shown in Fig. 1 is small enough to permit an actual enumeration of the number of possible arrangements as a function of occupancy.1° The results of the enumeration are plotted as curve B of Fig. 2. I n this curve, the total number of arrangements is counted as a function of the atomic ratio n. In curve C the number of these arrangements not superposable by rotation about any cube axis is shown. This would be of interest if only an isolated “cubic molecule” were under consideration, but if the cube is a portion of a large assembly, then it is rigidly fixed as part of this assembly and arrangements which would be equivalent in a molecule are no longer so. If the tetrahedral sites of a b.c.c. metal are the only sites available for occupancy, and if they are all equivalent (Fig. l), then the number of arrangements may also be calculated by a simple combinatorial formula. The result is shown as curve D in Fig. 2. If one makes an arbitrary restriction that only one tetrahedral site per face may be occupied, curve E results. If one makes the more meaningful restriction that occupation of a tetrahedral site prevents occupation of the three nearest tetrahedral sites, curve F is obtained. (This restriction may be stated in the form r = a o f i / 8 , where r is the radius of the interstitial atom, a0 is the cube edge, and the radius of the matrix atom or cation is such that contact occurs between interstitial atoms.a If in Fig. 1 site 0, is occupied, for example, sites l / 2 , 0, 3 / 4 ; l / 2 , 0, ’ / 4 ; and 1, l/4, ‘/2 are excluded.)
Discussion The relation of the number of arrangements in a single unit cell to the number of arrangements in a large number of unit cells, say a mole of solid or (6.02/ 2) X unit cells, is not obvious, although it is easily demonstrated when a simple combinatorial formula is applicable. Thus for a gram-atom of solid containing three equivalent octahedral sites per atom, the abovementioned combinatorial formula and its logarithmic The Journal of Physical Chemistry
form (Stirling) are applicable. For the single unit cell the number of arrangements for octahedral atoms is (18)!/(18@)!(18 - lBO)!. The ratio of the natural logarithms of these is, accordingly
+
In Wrn - (3)(6.02 X loz3)[e In e (1 - e) In (1 - e) ] ~In wc In [18!/(188)!(18 - lse)!] =
1.16
x
1023
where Wrn is the arrangement per gram-atom and wo fs the arrangement per unit cell. The ratio for tetrahedral occupancy is arrived a t in a similar fashion and equals 1.69 X loz3. (The reason these ratios are not equal to (3)(6.02 X lOZ3)/l8and (6)(6.02 X 1OZ3)/24, respectively, is, of course, partly due to the failure of Stirling’s approximation for small numbers. Since 8, = Bo the variable terms would cancel save for this failure .) The above reasoning may be shown empirically to apply with sufficient accuracy to the calculation of In w,/ln wo when the number of arrangements is restricted by geometric (etc.) factors and simple combinatorial formulas cannot be used. The proof of this is simply whether In wo may be represented to the desired accuracy by a complex combinatorial formula containing only terms in e so that In w,/ln wc is approximately constant for a given value of n over a range of compositions. It is readily found that curve B, for example, of Fig. 2 may be substantially duplicated in this way. Specifically one finds from Fig. 1 that the maximum interstitial content under the restrictions cited is n = 1.5, corresponding to the filling of half of the eighteen octahedral sites. If one calculates the number of arrangements based on nine sites, 9!/(9e)!(9 - ge)!, and multiplies each value of In w, so found, by 3/2, one obtains a curve which is practically superposable on’curve B of Fig. 2. I n the same fashion curve F of Fig. 2 may be approximately reproduced by multiplying the natural log of the combinatorial formula based on the limiting total occupancy, vix., 12!/(120)!(12 - 120)! by the empirical factor 4/3. These empirical factors are obtained by noting that the average number of sites available is greater than that based on maximum occupancy, Le., on a combinatorial formula based on nine or twelve sites, except as this maximum occupancy is approached. Therefore the number of arrangements will be greater than that given by the combinatorial formula. For the octahedral case, when n = nrnax/2= 0.75, there may be seven sites available, or there may be as few as ______~
(10) These enumerations for curves B and F, Fig. 2, were performed by Hewes, Holz, and Willard Co., Cambridge, Mass.
CRYSTALLOGliAPHIC
REQUIREMENTS 13 B O D Y - C E N T E R E D
four sites. The average number is approximately that explected from maximum occupancy. For the tetrahedral case, one would expect only six sites to be available when n = nmax/2 == 1.5, whereas there may be as many as eight, hence the factor 4/3. Even though such considerations lack mathematical rigor, the resulting equations agree surprisingly well with the enumerated data and they are far less cumbersome than the rigorous alternatives. These simple relationships also aplpear to be greatly superior in accuracy to those now in use (e.g., ref. 12), partly because usage often ignores the considerable effect of geometrical restrictions on the number of arrangements, or assumes that occupation of a site excludes a constant number of other sites. If we consider more carefully the effect of geometrical restrictions on occupancy of sites, it is clear that on adding the first atom to an octahedral site in a single structural unit (Fig. l), either two or four other sites in that unit will be prevented from subsequent occupancy (excluded) by the cited restrictions, depending on whether 1,he occupied site is in an edge or a fa,ce centered poaitjon. The weighted mean number of sites so excluded is %2/3. It is obviously true that because of these restrictions the number of available sites is reduced rapidly as sites are occupied. Let D, be the total number of available Bites and y the nuinber of occupied sites. It may be shown from the enumeration used to obtain curve B of Fig. 2 that D, varies with y in the sequence shown in Table I as y increases frorn 0 to 11. This table shows that for y = 1 the added atom renders 32/3 sites unavailable, which is just the weighted mean number found on inspection. The second atom to enter renders 3.40 sites unavailable, etc. The values of D,are defined baythe relation
CUBIC
HYDRIDES
1099
Table I : Number of Available Sites D as a Function of Occupancy y with Occupancy Restricted --Octahedral
D,,
21
sites---Tetrahedral Sites eliminated per added atom D,
24
18.QO
0
4.00
3.67 1
14.33
2
11.16
3
8.78
4
6.77
5
5.72
6
5.11
7
4.47
8
3. ‘75
9
2.92
10
2.00
11
1.00
12
0.00
sites-Sites eliminated per added atom
20 3.46
3.40 16.54
2.96
2.15 13.58
2.53
2.01 11.05
2.14
1.05
8.91 1.82
0.61 7.09
1.44
0.64 5.55
1.33
0.72 4.22
1.16
0.83 3.06
1.06
0.92 2.00
1.00
1.00 1.00
1.00
1.00 0.00
The point of the foregoing is that when site occupation is restricted, the number of arrangements is not simply combinatorial. The two widely used methods for handling such computations are not particularly accurate. The dashed curve of Fig. 3 indicates what happens if one uses a combinatorial formula of the type suggested, for example, by ref. 12. Here the number of arrangements is based on the formula
or
w = -(18)(18 -- k)(18 - 2k). . . (18 - ( 1 ~- 1)k) Y!
Y=Y
w
=
Il Duly! Y-0
which is equivalent to a combinatorial formula where the denominator is D - 1, D - 2, etc.) The second and fourth columins of Table I give the values of D,. Tlhe significance of the third and fifth columns requires a further word of explanation. The fractional values are averages over all arrangemerits for a given y, some of which do not allow an additional atom to enter. The isuccessive values of D, may be closely approximated by a third-order polynomial based on the easily enumerated number of arrangements fior y = 0, y = I, y = 11, and y = 12.”
where k , the number of sites excluded per site occupied, is taken here as 32/3, and the terms in the numerator are supposed to represent the successive number of sites available. The approximation suggested above and shown by the solid curve of Fig. 3 is closer but still not entirely accurate, uiz., a combinatorial formula based on the maximum number of sites which can be filled. Note the comparison of curve B of Fig. 2 and curve B’ of (11) R.Willard, private communication. (12) R. Speiser and J. W. Bpretnak, Trans. A I M E , 47, 493 (1955); Cj,
p. 497.
Volume 68, Number 6 M a y , io64
1100
THOMAS R. P. GIBB,JR.
Y 0
3 I
6
9
12
I
814,
Figure 3. Enumerated number of arrangements (small circles) of atoms in octahedral sites (Fig. 1) as a function of the number y of sites occupied and as a function of the occupied fraction 0 of the initially available eighteen sites. Compare with curve B of Fig. 2. The dashed curve is obtained with k = 32/s = constant number of sites made unavailable by occupation of a single site. The solid curve is (12/11)[12!/g!(12 - y)!].
Fig. 3. Both curves use the same enumeration, but owing to the fact that an occupied face-centered site increases n in hlH, by l / 4 whereas an occupied edgecentered site increases n by l / ~ , the curves appear different depending on whether n or y is used as the abscissa. This distinction must be made when a
The Journal of Physical Chemistry
single unit cell is considered, but disappears when a mole is involved. While it is possible to obtain an accurate expression for coniputing the number of arrangements, this expression is cumbersome. The purpose of this article fs primarily to clarify the relation between a sufficiently small structural unit and the assemblage of these units to make a mole of solid. Two semiempirical formulas have been suggested which provide a means of calculating the configurational entropy due to interstitial atoms in a b.c.c. lattice when site occupancy is limited in two particular ways. Site-preference energies, as estimated from ligand field or similar calculations probably determine whether an entering atom takes up an octahedral or a tetrahedral site ; however configurational entropy is by no means unimportant in the choice of energetically equivalent structures. In the case of so-called interstitial hydrides the configurational entropy may also control the stoichiometry. It is of interest that curves B and F of Fig. 2 show maximum values of configurational entropy to occur a t hlHo.,s and MH,.,B for octahedral and tetrahedral sites, respectively, and further that the configurational entropy for hydrogen contents below 0.6 is higher for octahedral than for tetrahedral occupation of the interstices of a b.c.c. metal.
Acknowledgment. The author is indebted to G. and R. Willard for help and guidance in connection with the mathematics, and to T. B. Douglas for helpful criticism. A portion of this work was supported by the U. S. Atomic Energy Commission.
