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The debye temperature and radius ratio of laves phases
Scripta
METALLURGICA
Vol. 2, pp. 631-634,
1968
Pergamon
Press,
Inc.
Printed in the United S t a t e s
THE DEBYE
TEMPERATURE
AND RADIUS RATIO OF LAVES PHASES*
R . R. J o s e p h + a n d K. A. G s c h n e i d n e r ,
Institute
for Atomic
Research
Jr.
and Department
Iowa State University,
Ames,
Iowa
of M e t a l l u r g y 50010
(Received September 9. 1968) I n a s t u d y of t h e l o w t e m p e r a t u r e phase compounds
it was observed
r a t i o of t h e c o m p o n e n t Debye temperature
atoms.
B-B metal
B.
that the Debye temperature
When the radius
of t h e L a v e s
p u r e A t h a n t h a t of p u r e
s p e c i f i c h e a t ( 1 . 4 ° to 8 ° K ) of s o m e l a n t h a n i d e
phase was found to be closer
The converse
The B atoms
the
to t h e D e b y e t e m p e r a t u r e
lie on the corners
of t e t r a h e d r a .
where
of
ratio indicated
the A atom is larger
In the cubic Laves phases,
a r e j o i n e d p o i n t to p o i n t w i t h t h e a r r a n g e m e n t
holes in the lattice for the A atoms. above arrangement
occurs
In this ideal case there
The closest
p a c k i n g of h a r d s p h e r e s
i f t h e r a t i o of t h e r a d i u s
are no AB contacts
of t h e t e t r a h e d r a
Compounds,
ratio closer
C15,
leaving
of t w o s i z e s i n t h e
and only A-A and B-B contacts.
When the ratio
and when the ratio is
are only B-B and A-B contacts. however,
from the ideal value. t i o n of t h e c o m p o u n d ,
than the
of t h e A a t o m to t h e B a t o m i s 1. 225.
than the ideal value there are only A-A and A-B contacts,
l e s s t h a n 1. 225 t h e r e
a radius
contacts,
contacts.
these tetrahedra
is larger
with the radius
A-A metal
was also noted when the radius
Laves .phases always have the AB 2 stoichiometry B atom.
correlated
ratio indicated
Laves
are known to form from
But it is thought that these atoms t o 1. 2 2 5 .
If the radius
atoms
with radius
undergo
ratios
far different
size adjustment
ratio rA/r B is greater
to a c h i e v e
t h a n 1. 225 u p o n f o r m a -
the A atom contracts
a n d t h e B a t o m e x p a n d s . 1, 2 I t i s u s u a l l y f o u n d rA that the larger atom undergoes the larger size change.3 W e a s s u m e d f o r t h e c a s e of ~ > r i d e a l t h a t t h e l a r g e r a t o m u n d e r g o e s a l l of t h e s i z e c h a n g e s o t h a t t h e r a d i u s of t h e A a t o m {rA c a l ) i n t h e c o m p o u n d
can be calculated
Using this value for the A metal r a t i o r A c a l / r B to d e t e r m i n e
*
radius
whether
Work was performed in the Ames C o n t r i b u t i o n No. 2 3 9 5 .
+Present address: Components Poughkeepsie, New York.
from the lattice parameter,
and the pure B metal
radius
there are A-A or B-B contacts
Laboratory,
Division,
U. S. A t o m i c
International
631
Energy
Business
a,
rAcal
= a~r3/8.
we calculated
the radius
in the Laves phases.
Commission,
Machine
Corporation,
632
DEBYE TEMPERATURE
A N D RADIUS RATIO OF LAVES PHASES
A n a n a l y s i s of o u r r e s u l t s Laves phases
of the c o m p o n e n t m e t a l s mental results
of t h e s e c o m p o u n d s .
and the r a d i u s r a t i o s
I n T a b l e I we h a v e s u m m a r i z e d
In a d d i t i o n to the L a v e s p h a s e D e b y e t e m p e r a t u r e s ,
r a t i o s and the p u r e c o m p o n e n t m e t a l D e b y e t e m p e r a t u r e s
A s c a n be s e e n f r o m the t a b l e , Debye temperature ideal value.
s p e c i f i c h e a t s of s o m e l a n t h a n i d e
b e t w e e n the D e b y e t e m p e r a t u r e s
our experi-
a l o n g w i t h a l l of t h e e x i s t i n g d a t a k n o w n to u s on t h e D e b y e t e m p e r a t u r e s
Laves phase compounds. and r A c a l / r B
on t h e l o w t e m p e r a t u r e
suggested a correlation
Vol. 2, No, 11
the D e b y e t e m p e r a t u r e
of
the r A / r B
a r e l i s t e d i n T a b l e I.
of t h e L a v e s p h a s e i s c l o s e r to the
of t h e A m e t a l c o m p o n e n t if r A c a l / r B is e q u a l to o r g r e a t e r
t h a n the
Th u s it is implied that the lattice d y n a m i c s near absolute zero are controlled
by A - A contacts, which is in a g r e e m e n t with the ratio being larger than ideal.
When
rAcal/r B is less than ideal the B a t o m s are in contact and control the lattice dynamics, w hich is in accord with the data in Table I. O n e would not expect the D e b y e temperature of the c o m p o u n d to be in perfect a g r e e m e n t with e B if there are B - B contacts or with e A if there are A - A
contacts, since there are always s o m e A - B contacts in the Laves phases ex-
cept at the ideal radius ratio. O n e m a y w o n d e r if the D e b y e temperature of the c o m p o u n d is s o m e average value of the two c o m p o n e n t metals, e.g., I/3(e A + 2 eB). served D e b y e temperatures,
e LP'
But it is found that only one of the ob-
is in better a g r e e m e n t with the average value than with
the D e b y e temperature of the c o m p o n e n t metal which is specified by the radius ratio.
Thus
w e find the correlation as outlined above holds well although w e are a w a r e that these c o m pounds represent a very small fraction of the total n u m b e r Furthermore, I. 225.
of k n o w n L a v e s phases (~ 300).
it should be noted that of the c o m p o u n d s listed in Table I, all have rA/r B >
Although a majority of L a v e s phase c o m p o u n d s have r A / r B > I. 225, there are m a n y
c o m p o u n d s w h i c h have radius ratios less than the ideal value.
W e a s s u m e that these will
also fit the above m e n t i o n e d correlation, but this m u s t await experimental verification. This correlation, if correct, implies several important factors concerning L a v e s phase c o m p o u n d s .
At low t e m p e r a t u r e s near absolute zero the dominant contribution to the
lattice vibrations are due to the B - B or A - A contacts.
contacts with a m i n o r contribution due to A - B
U p o n heating it would be expected, however, that as higher frequencies are ex-
cited, A - B and A - A
(or B-B} contributions b e c o m e
important and B-B (or A - A ) contacts no
longer dominate. If the pure metal a t o m s w h i c h c o m b i n e to f o r m a Laves phase adjust their mutual sizes so that in the c o m p o u n d
their radius ratio is that of the ideal value, then w e would not
expect to find the above correlation.
Furthermore,
it is noted if the above correlation is
correct for c o m p o u n d s w h i c h have r A / r B > 1.22 and B - B contacts {i.e. r A c a l / r B < 1.22), the A a t o m s m u s t either over c o m p r e s s to give a radius ratio less than ideal, or the B a t o m s undergo a slight expansion, while the A a t o m s c o m p r e s s s o m e w h a t to give rAcal/r B < I. 22. Clearly s o m e of the ideas concerning L a v e s phases m u s t be revised in vlew of the above correlation. It is hoped that this w o r k will stimulate other scientists w h o are working on, or are interested in L a v e s phase c o m p o u n d s to e x a m i n e their results or to start n e w w o r k in light of this correlation.
Vol. 2, No. Ii
DEBYE
TEMPERATURE
A N D RADIUS RATIO O F L A V E S P H A S E S
633
Acknowledgement T h e a u t h o r s w i s h to t h a n k t h e i r c o - w o r k e r s their interesting
and stimulating
the D e b y e t e m p e r a t u r e s published results
discussions
J. F . S m i t h a n d O. D. M c M a s t e r s
a n d G. W. S h a n n e t t e f o r c a l c u l a t i n g
from elastic constant data and for permission
for
s o m e of
to u s e s o m e of h i s u n -
o n H f C o 2 a n d Z r C o 2.
References
I.
R. L. Berry and G. V. Raynor, Acta Cryst. 6, 178 (1953).
2.
W. H u m e - R o t h e r y and G. V. Raynor, of Metals, London, England, 1962).
3.
G. E. R. Schulze, Z. Krist. III, Z49 (1959).
4.
K. A. Gschneidner,
5.
A. S u m e r and J. F. Smith, J. AppI. Phys. 38, 2283 (1962).
6.
P. I. Slick, C. W. M a s s e n a and R. S. Craig, J. C h e m .
7.
C. H. Cheng, J. Phys. C h e m . Solids 28, 413 (1967).
8.
R. R. Joseph and K. A. Gschneidner, Jr., to be published.
9.
M. Dixon, M. Aoyagi, R. S. Craig and W. E. Wallace, Paper presented at 6th Rare Earth R e s e a r c h Conference, Gatlinburg, Tenn. (May, 1967) p. 546 in Conference preprints.
The Structure of Metals and Alloys, (Institute
Jr., Solid State Physics 16, 276 (1964).
Phys. 43, 2788 (1965).
634
DEBYE TEMPERATURE
A N D RADIUS RATIO O F LAVES PHASES
Vol. 2, No. 11
TABLE I C o m p a r i s o n of D e b y e T e m p e r a t u r e s
of the L a v e s P h a s e ( e L p ) and T h e i r
C o m p o n e n t M e t a l s (e A and eB) Compound ABz
Debye Temperatures Type
rA/rB
rAcal/rB
0Lp
(°K)
OAa
O;
c
Z34
396
e6 ]
396
342
Ref
C o m p o u n d s w i t h r A c a l / r B l e s s t h a n i d e a l ( I . Z2) CaMg 2
C_14
1.23
1.19
426 b
MgCu Z
C_15
I. 25
I. 19
ft 332 336 bd
HfCo 2
C_I 5
1.26
1. Z0
368 b
f
256
452
ZrCo 2
C15
1.28
1.20
420 b
f
289
452
LaPt 2
CI 5
I. 35
I. Zl
236 d
8
14Z
234
C o m p o u n d s w i t h rAcal/r B greater than ideal (I. 22) CeRu z
C__I5
I. Z5
1.22
171 d
8
Z39 g
600
LaRu z
C_I 5
I. 40
I. 24
160 d
8
14Z
600
CeNi 2
C15
1.48
1.25
227 d
9
146
427
a b c d
A s g i v e n b y Ref. 4. Debye temperature
determined from elastic constant data.
C a l c u l a t e d f r o m e l a s t i c c o n s t a n t s at 100°K b y G. W. S h a n n e t t e , A m e s L a b o r a t o r y , I o w a f r o m d a t a r e p o r t e d b y R e f . 5. Debye temperature
Ames,
determined from specific heat data.
e C a l c u l a t e d f r o m e l a s t i c c o n s t a n t s at 80°K b y G. W. S h a n n e t t e , A m e s L a b o r a t o r y , A m e s , I o w a f r o m d a t a r e p o r t e d b y R e f . 7. f G. W. S h a n n e t t e , P r i v a t e c o m m u n i c a t i o n , A m e s L a b o r a t o r y , A m e s , I o w a (1968). g E s t i m a t e d f r o m L i n d e m a n n e q u a t i o n 4 a s s u m i n g m e l t i n g p o i n t of t e t r a v a l e n t Ce is l a r g e r t h a n t h a t of L a b y a s m u c h a s Hi i s l a r g e r than t h a t of Lu.
METALLURGICA
Vol. 2, pp. 631-634,
1968
Pergamon
Press,
Inc.
Printed in the United S t a t e s
THE DEBYE
TEMPERATURE
AND RADIUS RATIO OF LAVES PHASES*
R . R. J o s e p h + a n d K. A. G s c h n e i d n e r ,
Institute
for Atomic
Research
Jr.
and Department
Iowa State University,
Ames,
Iowa
of M e t a l l u r g y 50010
(Received September 9. 1968) I n a s t u d y of t h e l o w t e m p e r a t u r e phase compounds
it was observed
r a t i o of t h e c o m p o n e n t Debye temperature
atoms.
B-B metal
B.
that the Debye temperature
When the radius
of t h e L a v e s
p u r e A t h a n t h a t of p u r e
s p e c i f i c h e a t ( 1 . 4 ° to 8 ° K ) of s o m e l a n t h a n i d e
phase was found to be closer
The converse
The B atoms
the
to t h e D e b y e t e m p e r a t u r e
lie on the corners
of t e t r a h e d r a .
where
of
ratio indicated
the A atom is larger
In the cubic Laves phases,
a r e j o i n e d p o i n t to p o i n t w i t h t h e a r r a n g e m e n t
holes in the lattice for the A atoms. above arrangement
occurs
In this ideal case there
The closest
p a c k i n g of h a r d s p h e r e s
i f t h e r a t i o of t h e r a d i u s
are no AB contacts
of t h e t e t r a h e d r a
Compounds,
ratio closer
C15,
leaving
of t w o s i z e s i n t h e
and only A-A and B-B contacts.
When the ratio
and when the ratio is
are only B-B and A-B contacts. however,
from the ideal value. t i o n of t h e c o m p o u n d ,
than the
of t h e A a t o m to t h e B a t o m i s 1. 225.
than the ideal value there are only A-A and A-B contacts,
l e s s t h a n 1. 225 t h e r e
a radius
contacts,
contacts.
these tetrahedra
is larger
with the radius
A-A metal
was also noted when the radius
Laves .phases always have the AB 2 stoichiometry B atom.
correlated
ratio indicated
Laves
are known to form from
But it is thought that these atoms t o 1. 2 2 5 .
If the radius
atoms
with radius
undergo
ratios
far different
size adjustment
ratio rA/r B is greater
to a c h i e v e
t h a n 1. 225 u p o n f o r m a -
the A atom contracts
a n d t h e B a t o m e x p a n d s . 1, 2 I t i s u s u a l l y f o u n d rA that the larger atom undergoes the larger size change.3 W e a s s u m e d f o r t h e c a s e of ~ > r i d e a l t h a t t h e l a r g e r a t o m u n d e r g o e s a l l of t h e s i z e c h a n g e s o t h a t t h e r a d i u s of t h e A a t o m {rA c a l ) i n t h e c o m p o u n d
can be calculated
Using this value for the A metal r a t i o r A c a l / r B to d e t e r m i n e
*
radius
whether
Work was performed in the Ames C o n t r i b u t i o n No. 2 3 9 5 .
+Present address: Components Poughkeepsie, New York.
from the lattice parameter,
and the pure B metal
radius
there are A-A or B-B contacts
Laboratory,
Division,
U. S. A t o m i c
International
631
Energy
Business
a,
rAcal
= a~r3/8.
we calculated
the radius
in the Laves phases.
Commission,
Machine
Corporation,
632
DEBYE TEMPERATURE
A N D RADIUS RATIO OF LAVES PHASES
A n a n a l y s i s of o u r r e s u l t s Laves phases
of the c o m p o n e n t m e t a l s mental results
of t h e s e c o m p o u n d s .
and the r a d i u s r a t i o s
I n T a b l e I we h a v e s u m m a r i z e d
In a d d i t i o n to the L a v e s p h a s e D e b y e t e m p e r a t u r e s ,
r a t i o s and the p u r e c o m p o n e n t m e t a l D e b y e t e m p e r a t u r e s
A s c a n be s e e n f r o m the t a b l e , Debye temperature ideal value.
s p e c i f i c h e a t s of s o m e l a n t h a n i d e
b e t w e e n the D e b y e t e m p e r a t u r e s
our experi-
a l o n g w i t h a l l of t h e e x i s t i n g d a t a k n o w n to u s on t h e D e b y e t e m p e r a t u r e s
Laves phase compounds. and r A c a l / r B
on t h e l o w t e m p e r a t u r e
suggested a correlation
Vol. 2, No, 11
the D e b y e t e m p e r a t u r e
of
the r A / r B
a r e l i s t e d i n T a b l e I.
of t h e L a v e s p h a s e i s c l o s e r to the
of t h e A m e t a l c o m p o n e n t if r A c a l / r B is e q u a l to o r g r e a t e r
t h a n the
Th u s it is implied that the lattice d y n a m i c s near absolute zero are controlled
by A - A contacts, which is in a g r e e m e n t with the ratio being larger than ideal.
When
rAcal/r B is less than ideal the B a t o m s are in contact and control the lattice dynamics, w hich is in accord with the data in Table I. O n e would not expect the D e b y e temperature of the c o m p o u n d to be in perfect a g r e e m e n t with e B if there are B - B contacts or with e A if there are A - A
contacts, since there are always s o m e A - B contacts in the Laves phases ex-
cept at the ideal radius ratio. O n e m a y w o n d e r if the D e b y e temperature of the c o m p o u n d is s o m e average value of the two c o m p o n e n t metals, e.g., I/3(e A + 2 eB). served D e b y e temperatures,
e LP'
But it is found that only one of the ob-
is in better a g r e e m e n t with the average value than with
the D e b y e temperature of the c o m p o n e n t metal which is specified by the radius ratio.
Thus
w e find the correlation as outlined above holds well although w e are a w a r e that these c o m pounds represent a very small fraction of the total n u m b e r Furthermore, I. 225.
of k n o w n L a v e s phases (~ 300).
it should be noted that of the c o m p o u n d s listed in Table I, all have rA/r B >
Although a majority of L a v e s phase c o m p o u n d s have r A / r B > I. 225, there are m a n y
c o m p o u n d s w h i c h have radius ratios less than the ideal value.
W e a s s u m e that these will
also fit the above m e n t i o n e d correlation, but this m u s t await experimental verification. This correlation, if correct, implies several important factors concerning L a v e s phase c o m p o u n d s .
At low t e m p e r a t u r e s near absolute zero the dominant contribution to the
lattice vibrations are due to the B - B or A - A contacts.
contacts with a m i n o r contribution due to A - B
U p o n heating it would be expected, however, that as higher frequencies are ex-
cited, A - B and A - A
(or B-B} contributions b e c o m e
important and B-B (or A - A ) contacts no
longer dominate. If the pure metal a t o m s w h i c h c o m b i n e to f o r m a Laves phase adjust their mutual sizes so that in the c o m p o u n d
their radius ratio is that of the ideal value, then w e would not
expect to find the above correlation.
Furthermore,
it is noted if the above correlation is
correct for c o m p o u n d s w h i c h have r A / r B > 1.22 and B - B contacts {i.e. r A c a l / r B < 1.22), the A a t o m s m u s t either over c o m p r e s s to give a radius ratio less than ideal, or the B a t o m s undergo a slight expansion, while the A a t o m s c o m p r e s s s o m e w h a t to give rAcal/r B < I. 22. Clearly s o m e of the ideas concerning L a v e s phases m u s t be revised in vlew of the above correlation. It is hoped that this w o r k will stimulate other scientists w h o are working on, or are interested in L a v e s phase c o m p o u n d s to e x a m i n e their results or to start n e w w o r k in light of this correlation.
Vol. 2, No. Ii
DEBYE
TEMPERATURE
A N D RADIUS RATIO O F L A V E S P H A S E S
633
Acknowledgement T h e a u t h o r s w i s h to t h a n k t h e i r c o - w o r k e r s their interesting
and stimulating
the D e b y e t e m p e r a t u r e s published results
discussions
J. F . S m i t h a n d O. D. M c M a s t e r s
a n d G. W. S h a n n e t t e f o r c a l c u l a t i n g
from elastic constant data and for permission
for
s o m e of
to u s e s o m e of h i s u n -
o n H f C o 2 a n d Z r C o 2.
References
I.
R. L. Berry and G. V. Raynor, Acta Cryst. 6, 178 (1953).
2.
W. H u m e - R o t h e r y and G. V. Raynor, of Metals, London, England, 1962).
3.
G. E. R. Schulze, Z. Krist. III, Z49 (1959).
4.
K. A. Gschneidner,
5.
A. S u m e r and J. F. Smith, J. AppI. Phys. 38, 2283 (1962).
6.
P. I. Slick, C. W. M a s s e n a and R. S. Craig, J. C h e m .
7.
C. H. Cheng, J. Phys. C h e m . Solids 28, 413 (1967).
8.
R. R. Joseph and K. A. Gschneidner, Jr., to be published.
9.
M. Dixon, M. Aoyagi, R. S. Craig and W. E. Wallace, Paper presented at 6th Rare Earth R e s e a r c h Conference, Gatlinburg, Tenn. (May, 1967) p. 546 in Conference preprints.
The Structure of Metals and Alloys, (Institute
Jr., Solid State Physics 16, 276 (1964).
Phys. 43, 2788 (1965).
634
DEBYE TEMPERATURE
A N D RADIUS RATIO O F LAVES PHASES
Vol. 2, No. 11
TABLE I C o m p a r i s o n of D e b y e T e m p e r a t u r e s
of the L a v e s P h a s e ( e L p ) and T h e i r
C o m p o n e n t M e t a l s (e A and eB) Compound ABz
Debye Temperatures Type
rA/rB
rAcal/rB
0Lp
(°K)
OAa
O;
c
Z34
396
e6 ]
396
342
Ref
C o m p o u n d s w i t h r A c a l / r B l e s s t h a n i d e a l ( I . Z2) CaMg 2
C_14
1.23
1.19
426 b
MgCu Z
C_15
I. 25
I. 19
ft 332 336 bd
HfCo 2
C_I 5
1.26
1. Z0
368 b
f
256
452
ZrCo 2
C15
1.28
1.20
420 b
f
289
452
LaPt 2
CI 5
I. 35
I. Zl
236 d
8
14Z
234
C o m p o u n d s w i t h rAcal/r B greater than ideal (I. 22) CeRu z
C__I5
I. Z5
1.22
171 d
8
Z39 g
600
LaRu z
C_I 5
I. 40
I. 24
160 d
8
14Z
600
CeNi 2
C15
1.48
1.25
227 d
9
146
427
a b c d
A s g i v e n b y Ref. 4. Debye temperature
determined from elastic constant data.
C a l c u l a t e d f r o m e l a s t i c c o n s t a n t s at 100°K b y G. W. S h a n n e t t e , A m e s L a b o r a t o r y , I o w a f r o m d a t a r e p o r t e d b y R e f . 5. Debye temperature
Ames,
determined from specific heat data.
e C a l c u l a t e d f r o m e l a s t i c c o n s t a n t s at 80°K b y G. W. S h a n n e t t e , A m e s L a b o r a t o r y , A m e s , I o w a f r o m d a t a r e p o r t e d b y R e f . 7. f G. W. S h a n n e t t e , P r i v a t e c o m m u n i c a t i o n , A m e s L a b o r a t o r y , A m e s , I o w a (1968). g E s t i m a t e d f r o m L i n d e m a n n e q u a t i o n 4 a s s u m i n g m e l t i n g p o i n t of t e t r a v a l e n t Ce is l a r g e r t h a n t h a t of L a b y a s m u c h a s Hi i s l a r g e r than t h a t of Lu.