1238
JERRY [CONTRIBUTION FROM THE
DEPARTMEXT O F CHEMISTRY,
DONOHUE
Vol. 85
UNIVERSITY O F SOUTHERN CALIFORNIA, LOS ASGELES
7 , CALIF.]
The Relationship between Atomic Radii in Close-packed and Body-centered Cubic Structures BY JERRY DONOHUE RECEIVED NOVEMBER 2, 1962 Data on transitions between the body-centered cubic structure and either the cubic or hexagonal close-packed structure for sixteen elements have been examined, and it is found that the equation R( 1) R ( n ) = 0.300 log n does not satisfactorily account for the observed interatomic distances. On the other hand, the ratios R( 12)/R(8) between the atomic radii in the two structures all lie quite close to the value of 1.029 expected if the volume change on transition is zero
-
Introduction In a now classic paper Paulingl proposed the equation
~ m . ~ / m o l eVIiquid , = 19.718 ~ m . ~ / m o l e From . these we obtain a0 = 3.978 A. for the bcc form and (assuming colao = 1.634, the same value ob2erved5 a t a higher R ( 1 ) - R ( n ) = 0.300 log TZ (1) pressure) ac = 3.535, co = 5.776 A. for the hcp form. to relate the univalent radius, R(1), of an atom to the These data refer to the triple point bcc-hcp-liquid. radius when the valence is n. This relation was not The other extreme of the transition line between bcc used, however, to calculate the atomic radius apand hcp occurs a t 102 kg. cm.-2 and O'K., a t which propriate t o coordination number 12, ;.e., for cubic or point it was found that A V (solidb,, + liquid) = A V hexagonal close-packed structures, from those observed (SOlidhcp + liquid) = 0.840 ~ m . ~ / m O l eVliquid , = in body-centered cubic structures, where the coordina20.900 ~ r n . ~ / m o l e .These values give, for the bcc form tion number is 8, because the differences between bodya. = 4.054 A., and for the hcp form a. = 3.612, co centered cubic and close-packed radii for Fe, Ti, Zr, = 5.898A. and T1, which crystallize with both structures, did not Helium-4 also exhibits complex phase behaviorJ5 fit the equation. Instead, an empirical curve based on and different from that of helium-3. The p V T measdata obtained from these four metals was given, from urements of Grilly and Mills7 give, a t 1.73"K. and which the values of AR = R(12) - R(8) as a function 29.01 atm., molar volumes of 20.928 ~ m for. the ~ bcc of R(8)could be read. The assumptions made in calcuform and 20.737 ~ mfor. the ~ hcp form. The resulting lating AR with (1) are that the valence is the same in lattice constants are a. = 4.111 A. for bcc, and (assumboth structures, and that while only the 12 nearest ing colao= 1.629, the same value obseryeds a t a slightly neighbors need be considered in the close-packed struchigher pressure) a0 = 3.655, co = 5.954 A. for hcp. tures, the bond forming of both the 8 nearest and 6 next Lithium is bcc a t room temperature and below, but nearest neighbors in the body-centered cubic structures a t low temperatures both the ccp and hcp forms also must be taken into account. Somewhat later Thewlis2 have been observed. The transition bcc hcp, on pointed out that the points used to obtain the empirical cooling, is spontaneous and partial a t about 78'K. curve were based on data a t different temperatures in The partial transition bcc + ccp is induced by cold the case of Ti, Zr and T1 and on an unreliable extrapoworking a t 78'K. Observe$ lattice constants a t 78'K. lated value3 in the case of Fe, and stated that when are: for b ~ ca0, ~= 3.4831 A.; for C C P , ~a. = 4.379 8.; allowance for thermal expansion was made, little or no for hcp,loao= 3.104, co = 5.081 A. modification of eq. 1 was needed. However, the Sodium undergoes a spontaneous, partial transformaextrapolations used by Thewlis are also unreliable, intion from bcc to hcp on cooling below036"K. At 5°K. asmuch as the thermal expansion coefficients not only the lattice constantslo are a. = 4.225 A. for bcc and a0 were apparently assumed constant, but also the ex= 3.767, co = 6.154 A. for hcp. trapolations were very large, ;.e., from %O0, goo', Calcium of sufficiently high purity undergoes one 867' and 262' to room temperature. Moreover, there transition between room temperature and the melting are now available data on twelve more elements which point, transforming a t about 464' from ccp to bcg.. crystallize with both structures. Accordingly, it is The lattice constant of the bcc form11a t 467' is 4.480 A. possible to make a more extensive test of eq. 1 than has The observed lattice constant of the ccp form" a t 26' been done heretofore. of 5.5884 A. together with the average thermal expanLattice Constants sion coefficdent11m12 of 22.3 X between 0 ' and 371' give 5.644 A. for the lattice constant a t 467'. Some of the numerical data in this section may Strontium, which is ccp a t room temperature, changes be found in P e a r ~ o n ,but ~ the original literature cito the hcp structure a t about 215'13 or 235',14 then,at tations are also given. Lattice constants are in Angabout 535'14 or to a bcc form which persists up strom units throughout, the factor 1.00202 having to the melting point. The lattice constant at' 25' is been used to convert from kX where necessary. The 6.0849 A. The lattice constants of the two high temabbreviations bcc, ccp and hcp are used to denote perature forms have been measured only approximately, the body-centered cubic, cubic close-packed and hexand a t only one temperature each, 366' apart. Howagonal close-packed structures, respectively. Greek letter designations have not been used, as their use is ever, the dilatometric datal4 may be used to calculate the appropriate crystal structure data : between 0 ' sometimes confusing. Helium-3 exhibits complicated phase b e h a ~ i o r . ~ and 535' the linear expansion is 11.6 X lo+, and a t the The equations of Grilly and Mills6 give the following (7) E. R. Grilly and R. L. Mills, A n n . Phys. (to be published); quoted in quantities relative to the bcc + hcp transition a t ref. 8. (8) A . F. S c h u c h and R. L. Mills, Phys. Rev. L e t l e u s , 8 , 469 (1962). 3.1-18'K.: p = 140.45 kg. cm.-2, Ab' (solidb,, + liquid) (9) E. A. Owen and G. I. Williams, Pvoc. Phys. SOL.,8 6 7 , 895 (195.1). = 0.767 ~ m . ~ / m o l eAv , (SOlidhcp --t liquid) = 0.89'7 (IO) C. S. Barrett, Acta Cryst., 9, 671 (1956). T h e values ao = 3.111
-
( I ) L. Pauling, J . Ana. Chem. Soc.. 69, 542 (1947). ( 2 ) J . Thewlis, i b i d . , 7 5 , 2279 (1953). ( 3 ) Stuuklurbericht, 4 , 228 (1936). (1) W. B. Pearson, "A Handbook of Lattice Spacings and Structures of hletals a n d Alloys," Pergamon Press, London, 1958. A . F. Schuch a n d R . L. Mills, Advan. Cryogenic Eng., 7 , 311 (1900). ( 0 ) E . R. Grilly a n d R . I,. 3Iills, A n i f . P h y s . , 8 , 1 (1956).
and cg = 5.093 A. given by Barrett have been adjusted t o ta ke account of the fact t h a t he used t h e value 3.491 A. for the lattice constznt of bcc Li as an internal standard in his study of the hcp form. (11) B. T. Bernstein and J , F. Smith, Acla Ci'ysl., 12, 419 (1959). (12) P. G . Cath and 0 . L. v. Steenis, 2. techn. P h y s . , 17, 239 (1936). (13) E. A. Sheldon and A. J. K i n g , Acta Cryst., 6, 100 (1953). (14) E. Rinck, Compt. ueiad. acod. sci. Pavis, 234, 845 (1952).
hlay 5 , 1963
ATOMICRADII IN ELEMENTS
hcp + bcc transition there is a linear contraction of 0.9 X These data, combined with the ccp lattice constant quoted above, give, for the hcp form (assuming cola0 = 1.635, the value obseryed13a t 248') a o = 4 . 3 4 7 , c o =7 . 1 1 2 A . , a n d a o = 4 . 8 7 9 A . f o r t h e b c c form, both a t 535'. (These values are in fair agreement with thq directly measured values13 of a0 = 4.32, co = 7.06 A. for hcp a t 248' and a. = 4.85 8. for bcc a t
614'.) Lanthanurnl5is ccp a t room temperature, and an hcp form (with a doubled co) may coexist with this up to 292'. The atomic volume of the hcp form is less than 0.5% larger than that of the ccp form. At 868' there is a transition ccp + bcc. The lattice constant for the bcc form is 4.25 A. a t 881'. Data for the ccp form obtained in the range of021' to 598' extrapolate to a lattice constant of 5.341 A. a t 881'. The extrapolation function includes terms in t and t 2 . Cerium's is ccp a t room temperature and bcc above 730". At 742" the bcc lattice constant is 4.11 8. Data for the ccp form from the range 13' to 619' extrapolate to a lattice constant of 5.187 A. a t 742'. The extrapolation function includes terms in t , t 2 and t3.
P r a s e o d y m i ~ r nis~ double-hcp ~ a t room temperature and transforms to the bcc form a t 798" with lattice constant 4.13 A. a t 814'. Data for the hexagonal form from the range 22' t o 449' extrapolate to the lattice constants a0 = 3.684, co = 11.936 A. a t 814'. The extrapolation functions are linear in t. NeodymiumI6 is also double-hcp a t room temperature and transforms to the bcc form a t 868". The lattice constant is 4.13 A. a t 882'. Data for the hexagonal form from the range 20' to 696' extrapolate to a. = 3.688, co = 11.956 k . a t 882'. The extrapolation functions contain terms in t , t 2 and t3. Ytterbiumlj is ccp a t room temperature. Atmospheric impurities stabilize an hcp structure above 260" which transforms t o a bcc form a t 7;O0, a t which temperature the lattice constant is 4.44 A. Data for the ccp form from 23' to 598' extrapolate to ao = 5.596 A.a t 720'. The extrapolation function includes terms in t , t 2 and t 3 . The atomic volume of the hcp form is larger than that of the ccp form by O.SOj, a t about 260°, the difference gradually decreasing to 0.3% a t about 4 10'. Thallium transforms from hcp to bcc a t about 23OO0. At 262' the lattice constant of the bcc form16is 3.882 A. Thermal data on the hcp form17 a t 18', 102' and 192' extrapolate to a0 = 3.489, co = 5.541 A. a t 262'. Titanium is hcp a t 25", with a. = 2.9504 A., c(, = 4.6833 At about 882", there is a transition hcp + bcc, but the thermal expansion data on the hcp form are ~ n r e l i a b l e 'because ~ of contamination above GOO' by Si, 0 and N, the lattice constants above room temperature showing considerable scatter. Extrapolation to 100% of Ti for the bcc phases of Ti-CrIZ0TiFe,? I , 2 2 Ti-Mn,23 T ~ - - M o and ~ ~ Ti-ZrIZ5 all a t room temperature, gives the respective values a. = 3.284, 3.296, 3.281, 3,286 and 3.287 A.; average a. = 3.287 8., a.d. = 0.004A. The respective observed composition (1.7) J. J. Hanak, Dissertation, Iowa State Univrrsity, 19.59. (16) H. Lipson and A. K. Stokes, S a t a r e , 148, 437 (1941).
A. Schneider and G. Heymer, Z . anoyg. C h e o f . , 286, 118 (195G). H. T.Clark, J . M e t o i s , 1, 588 (1949). R . I,. P. Berry and G. V. Raynor, Research, 6 , 218 (1953). P. Duwez and J. L. Taylor, Tyans. A m . S O L .Me!., 4 4 , 495 (1952). (21) B. W. Levinger, J. .Metals, 5 , 195 (1953). (22) H. W. U'orner, J . Insl. M e t . , 7 9 , 173 (1951); ref. 4, p. 662. (23) n. J. Maykuth, R . H. Ogden and R . I. Jaff6, J . Metals, 5 , 225 (1953); ref. 4, p. 750. (24) M. Hansen, E. L. Kamen, H. D. Kessler and 11. J. McPherson, J . .2le!nls, 5 , 881 (1951); ref. 4, p. 703. ( 2 5 ) P. D u w e z , J . Z i i s l . M e t . , 80, 525 (1B52). (17) (18) (19) (20)
1239
TABLE I EXTRAPOLATION O F LATTICE CONSTANTS --Ti-Cr-Atom
--Ti-Fe-Atom
70
% of Fe, p
-ao, A,-p Obsd. Calcd. 6 3.260 3 . 2 6 1 8 3.254 3 . 2 5 3 10 3.245 3.245 12 3.236 3 . 2 3 7 14 3.230 3.230 16 3.222 3.222 a0 = 3 284 3.885 x l o - ' $ a.d. = 0.001 A.
of Cr,
--Ti-MoAtom
-
Ti-Zr
T1
A,-Calcd.
4.40 3.255 3.257 5.00 3.249 3.253 5.2 3.260 3.221 6.68 3 . 2 3 7 3.239 12.0 3 . 2 1 0 3.202 12.07 3.195 3.201 13.86 3 . 1 8 6 3.191 14.5 3.195 3.187 15.39 3 . 1 7 5 3.183 16.5 3 . 1 8 0 3.179 20.2 3.160 3.161 ao = 3.296 9.5 X 10-89 1.4 X 10 - 4 p 2 a.d. = 0.005 A.
+
Ti-Mn Atom
p
of MO,
p
+
Atom
% of M n ,
%
-ao, A,-Obsd. Calcd. 11.5 3.250 3.251 1 8 . 0 3.237 3 . 2 3 4 33.5 3 . 1 9 8 3.199 42.5 3.184 3.181 53.5 3.165 3.165 66.5 3 . 1 4 9 3.150 82.0 3.142 3 . 1 4 1 100.0 3.140 3.140 a0 = 3.286 - 3.2 X 10-1 p 1.74 x 10-1 p 2 a . d . = 0.001 A.
-ao,
Ohsd.
-ao,
Obsd.
A,---
70
of Zr, -ao, p Obsd. Calcd.
5.5 3.242 3.246 30 6.2 3.244 3.242 40 8.8 3.233 3.227 50 60 12.0 3.213 3.213 13.4 3.207 3.207 70 a0 = 13.4 3.203 3.207 18.1 3.199 3.190 18.1 3.183 3.190 ao = 3.281 - 7 X 10-1 p 1.1 x 10-4 p * a.d. = 0.004 A.
+
A,-Calcd.
3.377 3.377 3.407 3.407 3.437 3.437 3.467 3.467 3.497 3.497 3.287 f 0 . 0 0 3 p
-ao, Obsd.
1,
'C.
18 102 192 262
A.Calcd.
3.456 3.469 3.481
3.456 3.469 3.481 . .. 3.489 ao = 3 . 4 5 3 i17 X 10-bt 12 X 10-812
-
--CO,A.---
t , "C.
Obsd. 5.530 5.535 5,539
Calcd. 18 5.530 102 5.535 192 5,539 262 ... 5.541 CD = 5.529 7 x 10-61 - 87 X
+
10-912
ranges, in atom % Ti, are 84-94%, 79.8-95.6%, 81.994.5%, 0-88.5% and 30 to 70%. The first and last extrapolations are linear, the other three quadratic. Zirconium undergoes a transition hcp + bcc a t about 865'. At 862' the observedz6lattice constant of the bcc phase is 3.6090 A.,while those calculatedz6 for the hcp form by the use of quadratic extrapolation of data collected between 10' and 579' are a0 = 3.2491, co = 5.20108. Macroscopic m e a s ~ r e r n e n t sgive ~ ~ a volume decrease of O.66yOa t the transition; this result, when combined with the foregoing hcp lattice constants, gives 3.615 A. for a0 of the bcc form. At 25' the lattice constants of the hcp phasez8are a0 = 3.2312, co = 5.1477 A. The bcc form is not stable a t room temperature] but extrapolation of data from bcc Zr-TiZ5 (30 to 70 atom % Zr) and bcc Zr-U29 (40 to 90 atom % Zr) to 100% of Zr both give 3.587 A. for the value of a. for bcc Zr a t room temperature. Thorium transforms a t about 1450" from ccp to bcc. The respective 1at;ice constants3oa t that temperature are 5.180and 4.11A4. Manganese transforms from ccp to bcc a t about 1130'. The respective lattice constants are313.568 and 3.080 A. The dilatometrically observed32 volume change a t the transition is +0.90%; this figure, together with the above ccp lattice constant, gives the value 3.079 8. foraoof the bcc phase. (2(i) R. B. Russell, J . Metals, 6, 1045 (19.54). (27) G. R . Skinner a n d H. L. Johnston, J . Chem. P h v s . , 21, 1383 (1953). (28) R. B. Russell, J . AppE. P h y s . , 2 4 , 232 (1953). (29) D. Summers-Smith, J . I n s f . M e l . , 85, 277 (1955). (30) P. Chiotti, J . Electrochem. Soc., 101, 567 (1954). (31) Z. Basinski and J. W. Christian, Proc. R o y . Soc. (London), 8 2 2 9 , 554 (1954). This paper presents t h e lattice constants in graphs from which t h e quoted values were read. (32) H. Yoshisaki, Sci. R e p . Tohzrku I m p . Univ., 126, 182 (1937).
1240
DONOHUE
JERRY
I
I
Vol. 85
TABLE I1 ATOMICRADII IN CLOSE-PACKED (R12) ASD BODY-CENTERED CUBIC( R 8 ) STRUCTURES R(12)" -
R(8)
R(12)
Sr La Ce Pr Nd Yb T1 Ti
1.723 1,755 1.780 1.508 1,508 1,830 1.940 2.113 1.840 1,780 1.788 1.788 1.923 1.681 1 423
1.768 0.045 1.806 051 1.826 ,046 1.548 040 1,553 ,045 1.884 054 1.995 055 2.176 062 1.888 ,048 1.834 054 1.837 049 1.839 ,051 1.979 066 1.729 048 1.461 038
Zr Zr
1.563 1.614 1 , 5 5 3 1.602
Zr Th Mn Mn Fe Fe
1.565 1.780 1.334 1.333 1.258 1.269
He-3 He-3 He-4 Li Li Na
Ca
0
directly observed
W v i 0 observed volume chonge at tronsition 0 via thermal expansion of low temperature form
0
O'
I20
via alloys
1.40
,.bo
1.60
2.;0
2;o
2.40
R (8 1,
Fig. 1.-The
relation between AR
=
R(12)- R ( 8 ) and R ( 8 ) .
Iron undergoes a transition bcc + ccp a t 916". At that temperature, the respective lattice constants33 are 2.9044 and 3.6468 A. At 1388' there is a second transition ccp + bcc, the lattice constants33 being 3.6869 and 2.9315 A. Additional numerical data involving some of the extrapolated data are presented in Table I. Discussion The lattice constants in the foregoing section may be used to calculate the atomic radii in the various allotropes. The results are presented in Table 11. Values of A R vs. R(8) are shown in Fig. 1, in which it may be seen that while the revised data for Fe and Ti are in satisfactory agreement with eq. 1, those for T1 and Zr are not. Moreover, the data for the twelve additional elements show rather poor agreement with the predictions of eq. 1, especially a t the larger values of R(8). It would thus appear that eq. 1 is not appropriate for calculating the values of R(12) from the directly observed values of R(8) for those elements which crystallize only with the bcc structure. This result also raises doubt as to whether eq. 1 should therefore be used, as has been widely done, in the interpretation of interatomic distances in intermetallic compounds. It is interesting that the relation R(12)= k R ( 8 )
(2)
with k = 61/221/3/3= 1.029 reproduces the observed data satisfactorily. This function results from the assumption that transition from the close-packed struc(:13) 2. S. Basinski, W. Hume-Rothery and A. L. Sutton, P i w R o y . Soc. (London), A229, 459 (1955).
1.614 1.831 1.368 1.368 1 289 1 303
ARa
R(8)
Remarks
1 026 1 029 1 026 1 027 1 030 1 030 1 029 1 029 1 026 1 030 1 027 1 029
At 3.148'K., from p V T d a t a At OOK., from p V T d a t a At 1.73'K., from p V T d a t a At 78'K., from ccp bcc bcc At 78'K., from hcp At 5'K. At 467OC., R(12) from thermal d a t a At 635'C., via thermal d a t a At 8 8 I 0 C . , R(12) from thermal d a t a At 74ZoC., R(12) from thermal d a t a At 814OC., R(12) from thermal d a t a At 88ZoC., R(12) from thermal d a t a At 72OoC., R(12) from thermal d a t a At 26ZoC., R(12) from thermal d a t a At room temperature, R ( 8 ) from alloys At 862OC., R(12)from thermal d a t a At room temperature, R(8) from alloys At 862"C., from obsd. A V , hcp bcc At 1450'C. At 113OoC. At ca. 113OoC., from obsd. A V At 916°C. .4t 1388'C.
1 029 1 028 1 027
,061 049
1 033 1 032
049 052 034 034 032 034
1 031 1 029 1 025 1 026 1 025 1 027
--
-
These quantities were calculated from the lattice constants given in the text in order t o avoid rounding off errors.
tures to the bcc structure occurs with zero change in volume. For AV of plus and minus 1%)the values of k are 1.032 and 1.025, respectively; all except one of the points of Fig. 1 lie within these limits. This near constancy of the concentration of electrons implies that the total bond forming power per atom, i.e., the valence, is essentially unchanged during the transition. It is noteworthy that the A V = 0 approximation and eq. 2 are valid for elements with widely differing properties, ;.e., from the alkali metals to the transition metals and even a group 0 element, helium, and under widely different conditions, including the temperatures a t which the transitions take place, a t room temperature (by extrapolation) and in certain alloys (also by extrapolation). In the foregoing discussion no assumptions were made concerning the valences of atoms and their dependence on interatomic distances. The gradual divergence from the predictions of eq. 1 of the observed values of AR as R(8) increases suggests that although the equation may be useful in the consideration of distances between the smaller atoms, some modification will be required in the case of the larger atoms, but the development of such a relation is outside the scope of the present paper. Acknowledgments.-This work was supported by a grant from the National Science Foundation. The author wishes to thank Prof. Linus Pauling for several suggestions and for helpful discussion.
