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Some observations on the crystal structures of the rare-earth metals and alloys
Journal of the Less-Common Metals Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
336
Some observations on the crystal structures of the rare-earth
metals
and alloys The rare-earth metals comprise a useful group of elements for studying the factors which influence the crystal structure of metals and alloys. An interesting sequence of crystal structures occurs in the rare-earth metal period and in the intra rare-earth binary systems between the light (atomic number Z< 63) and heavy (2 >63) rare-earth metals. At room temperature (or just below in the case of /L cerium) the elements from lanthanum to neodymium have the double hexagonal crystal structure (ABAC stacking sequence of close-packed planes), samarium has a rhombohedral structure which may be regarded as a hexagonal structure with an ABABCBCAC stacking sequence of close-packed planes: the elements from gadolinium to lutetium (with the exception of ytterbium) have the close-packed hexagonal structure (ABAB stacking sequence of close-packed planes). A similar sequence of crystal structures occurs in the intra rare-earth alloy systems, Ce-Nd, Ce-Smr, Ce-Gd2, Ce-Tb 1, Ce-Ho 3, and Ce-Er 4 which are considered in this note. HODGES5 has discussed the structures of the rare-earth metals and alloys in terms of the pseudopotential and perturbation theory and has shown that there is an approximately linear variation between the proportion of hexagonal stacking sequence* and the deviation of the axial ratio from ideality in the systems, Y-Ce, Y-Pr, Gd-Ce and Cd-Pr 2. This treatment did not enable prediction of the ranges of composition at which the various hexagonal modifications occur in a particular rareearth alloy system. GSCHNEIDNER AND VALLETTA~ have discussed the crystal structures of the rare-earth metals in terms of the participation of the 4f-electrons in the bonding of the crystal. The ratio of the metallic radius to the 4f-radius was taken as a measure of the 4_f participation in the bonding process and it was shown that the various hexagonal modifications encountered in the rare-earth metals and alloys occur in particular ranges of radius ratios. It is possible, by this method, to predict the range of compositions of the various stacking sequences in a particular rare-earth binary system. GSCHNEIDNER AND VALLETTA6 were also able to explain qualitatively, the behaviour of rare-earth metals under hydrostatic pressure. The present note reports a correlation between the axial ratios of the hexagonal rare-earth metals and alloys and the atomic number. In the case of an intra rare-earth binary system, (components A and B) the axial ratios have been plotted as a function of the mean atomic number (&) defined as:
zm= &
[&?A+ (100~-)ZB]
where a is the atomic percent of the A-component and ZA and Zn are the atomic numbers of the A and B components, respectively. The lattice spacings of the pure rare-earth metals are summarised in Table I * The proportion of hexagonal 3 and Q, respectively.
sequence in the double
J. Less-Common Metals, 17 (1969) 336-339
hexagonal
and
%-type
structures
is
337
SHORT COMMIJNICATIONS
and the axial ratios* are plotted as a function of the atomic number in Fig. I together with the axial ratios of some intra rare-earth metal alloys in the systems, Ce-Nd, Ce-Sm 1, Ce-Gd 2, Ce-Tb 1, Ce-Ho 3 and C,e-Er 4 which are plotted as a function of the average atomic number (2,) defined above. It can be seen that, in general, the axial ratios of the pure metals and the cerium alloys lie on a common curve with a pronounced minimum at an atomic number of between 68 and 69. TXBLE
I
THE Z+.TTICE PARAMETERS
OF THE
PURE
RARE-EARTH
METALS
AND
OF A h-7.7%
C‘2 ALLOY
AT
ROOM TEMPERATURE* _
___I__
.Ildal __.._____
.-.._______..-
a-Spacing (kXl _____..- - _.____
..
lkXl
Axial ratio (da)
.4 tolnic nuvnbzv
c-Spaciq
Reference
a-La
3.762
I*.‘34
I.612
57
9
!:Z
3.666 3.6640 3.6490 3.6208 4.572
iX~778 11.807 11.77’ .26.178
1.611 I.607
58 59 60
10 2
ir-?id
n-Pm x-Sm q-Eu** XX-GCl
!X-Tb S-D? ~-HO x-Er %:-Tm n-Yb** <-I-u cx-Lu-7.794
3.6265
Ce
3.5986 3.5858 3.5724 3.5538 3.5355 5.4736 3.5028 3.5r9o -_-
I.613 1.607
5.7697 5.6858 5.6423 5.61 j8 5.5816 5.5596
I.5910 I.5800 I.5735
61 62 63 64 65 66
1.j720
67
5.5554 5.5590
I .586o T.5797
1.5706 I.5725
-
I
1 Present work 2 I Present work 3
68 69 70 7’ 70
Ptesent Present Present Present
work work work work
* The axial ratios of the d.hex. and Sm-type structures have been adjusted to correspond twice the distance between close-packed planes by dividing by L and 4.5, respectively. ** Eu and Yb have b.c.c. and f.c.c. structure, respectively.
with
The c.p.hex. structure in the above alloy systems has a limiting mean atomic number of about 63.5, i.e., close to the atomic number of gadolinium (64). The limiting axial ratio is about 1.587 and this is less than the axial ratio of a-gadolinium (1.591). Unlike the rare-earths with atomic numbers in excess of 64, however, gadolinium is in a ferromagnetic state at, or just below room temperature, and this could account for the fact that the axial ratio of gadolinium exceeds the critical value of 1.587, exhibited by the intra rare-earth metal alloys. These observations are consistent with the instability of the c.p.hex. structure of Lx-gadolinium with respect to pressure? and cold works. The variation of the axial ratio with atomic number (Fig. I) for the c.p.hex. alloys indicates that a Lu-Ce alloy with a value of 2, of 70, should have an axial ratio between that of Tm (1.572) and Lu (1.586). The lattice spacings of a Lu-7.7:/o Ce alloy (Z&=70) are reported in Table I and it can be seen that the axial ratio of this alloy has an intermediate value of 1.580. The alloys with the Sm-type structure exist over a range of atomic numbers * For purposes of comparison, the axial ratios of the d.hex. and Sm-type structures have been adjusted to correspond with twice the distance between close-packed planes by dividing by 2 and 4.5, respectively. J. Less-C0~~0~
Metak,
17
(1969)
336-339
SHORTCOMMUNICATIONS
338
from about 62 to 63. This indicates that a hypothetical, trivalent form of europium would have a Sm-type structure and it might be possible to form such a phase at ultra-high pressures. The alloys with the d.hex. structure exist over the range of 2, from about 59.0 to 61.5 and this indicates that the probable structure of metallic promethium (61) is d.hex. which is in agreement with the prediction of GSCHNEIDNER AND VALLETTA~. The atomic number of promethium is close to the limit of the d.hex. range and this indicates that the d.hex. structure of this metal could be heavily faulted. The axial ratio of p-cerium lies below the general variation of c/a with 2,.
x590-
1.580*
1.570b Ce Pr Nd R Sm Eu Gd Tb Dy Ho Er lm Yb Lu 575659606162,63&46566676869x)71
Atomic number (2)
Fig. I. The variation of axial ratio with atomic number of the pure rare-earth metals and of some cerium-rare-earth atloys. @ Pure rare earths x Ce-Cd alloys A Lu-7.7 at.% Ce alloy v Ce-Tb alloys Q Ce-Nd alloys A Ce-Ho alloys m Ce-Sm alloys + Ce-Er alloys
It should be remembered, however, that the effective valency of p-cerium is slightly in excess of the trivalent rare earths and this could account for the deviation of the axial ratio from the general behaviour. The axial ratio of a hypothetical 3-valent form of cerium, obtained by extrapolating the a- and c-spacings of a number of d.hex. Ce rare-earth alloys to 100% Ce is 1.611 +O.OOI. The composition at which the Sm-type alloys occur in the Y-Ce alloy system2 suggests that yttrium behaves like a heavy rare-earth metal with an atomic number of 69, i.e. simiIar to thulium. J. Less-Common
Metals, 17 (1969) 336-339
SHORT
339
COMMUNICATIONS
A similar plot of axial ratio against 2, heavy
rare-earth
compositions
alloy systems
at which
can be obtained
and it is possible,
the various
stacking
for the praseodymium/
by this method,
sequences
to predict
occur in a particular
the intra
rare-earth metal alloy system, simply from a consideration of the atomic numbers of the constituent metals. The correlation reported in this note provides an empirical method for predicting the structures of alloy phases in intra rare-earth metal binary systems. If the variation of the axial ratios of the hexagonal structures in these systems reflect the interaction of the Fermi surface of these alloys with the Brillouin zone faces then the correlation of c/a with Zm might indicate is applicable to the intra rare-earth alloys. Rare Earth Research Group, Department of Physical Metallurgy Science
that
some form of rigid band model
I. R. G. V.
ad
HARRIS RAYNOR
of Materials,
University of Birmingham, Birmingham I 2 3 4 5 6 7 8 Q 10
(Gt. Britain)
J. D. SPEIGHT, 1. R. HARRIS AND G. V. RAYNOR, J. Less-Common Metals, 15 (1958) 317. I. R. HARRIS, C. C. KOCH AND G. V. RAY~OR, J. Less-Common Metals, II (1966) 436. J. D. SPEIGHT AND 1. R. HARRIS, to be published. M. NORMAN, I. R. HARRIS AND G. V. RAYNOR, J. Less-Common Metals, 13 (1967) 24. C. H. HODGES, Acta Met., r5 (1~67) 1787. K. A. GSCHNEIDNER AND R. M. VALLETTA, Acta Met., 16 (1~68) 477. A. JAYARAMAN AND R. C. SHERWOOD, Phys. Rev. Letters, II (1964) 22. I. R. HARRIS, C. C. KOCH AND G. V. RAYNOR, J. Less-Common Metals, 12 (1967) 239. F. H. SPEDDING, A. H. DAANE AND K. W. HERRMANN, Acta Cryst., 9 (1956) 599, K. A. GSCHNEIDNER, R. 0. ELLIOTT AND Ii. R. MCDONALD, J. Phys. Chem. Solids, 23 (1962) 555.
Received
November
7th, 1968 J. Less-Common
Metals,
17 (1969)
336-339
336
Some observations on the crystal structures of the rare-earth
metals
and alloys The rare-earth metals comprise a useful group of elements for studying the factors which influence the crystal structure of metals and alloys. An interesting sequence of crystal structures occurs in the rare-earth metal period and in the intra rare-earth binary systems between the light (atomic number Z< 63) and heavy (2 >63) rare-earth metals. At room temperature (or just below in the case of /L cerium) the elements from lanthanum to neodymium have the double hexagonal crystal structure (ABAC stacking sequence of close-packed planes), samarium has a rhombohedral structure which may be regarded as a hexagonal structure with an ABABCBCAC stacking sequence of close-packed planes: the elements from gadolinium to lutetium (with the exception of ytterbium) have the close-packed hexagonal structure (ABAB stacking sequence of close-packed planes). A similar sequence of crystal structures occurs in the intra rare-earth alloy systems, Ce-Nd, Ce-Smr, Ce-Gd2, Ce-Tb 1, Ce-Ho 3, and Ce-Er 4 which are considered in this note. HODGES5 has discussed the structures of the rare-earth metals and alloys in terms of the pseudopotential and perturbation theory and has shown that there is an approximately linear variation between the proportion of hexagonal stacking sequence* and the deviation of the axial ratio from ideality in the systems, Y-Ce, Y-Pr, Gd-Ce and Cd-Pr 2. This treatment did not enable prediction of the ranges of composition at which the various hexagonal modifications occur in a particular rareearth alloy system. GSCHNEIDNER AND VALLETTA~ have discussed the crystal structures of the rare-earth metals in terms of the participation of the 4f-electrons in the bonding of the crystal. The ratio of the metallic radius to the 4f-radius was taken as a measure of the 4_f participation in the bonding process and it was shown that the various hexagonal modifications encountered in the rare-earth metals and alloys occur in particular ranges of radius ratios. It is possible, by this method, to predict the range of compositions of the various stacking sequences in a particular rare-earth binary system. GSCHNEIDNER AND VALLETTA6 were also able to explain qualitatively, the behaviour of rare-earth metals under hydrostatic pressure. The present note reports a correlation between the axial ratios of the hexagonal rare-earth metals and alloys and the atomic number. In the case of an intra rare-earth binary system, (components A and B) the axial ratios have been plotted as a function of the mean atomic number (&) defined as:
zm= &
[&?A+ (100~-)ZB]
where a is the atomic percent of the A-component and ZA and Zn are the atomic numbers of the A and B components, respectively. The lattice spacings of the pure rare-earth metals are summarised in Table I * The proportion of hexagonal 3 and Q, respectively.
sequence in the double
J. Less-Common Metals, 17 (1969) 336-339
hexagonal
and
%-type
structures
is
337
SHORT COMMIJNICATIONS
and the axial ratios* are plotted as a function of the atomic number in Fig. I together with the axial ratios of some intra rare-earth metal alloys in the systems, Ce-Nd, Ce-Sm 1, Ce-Gd 2, Ce-Tb 1, Ce-Ho 3 and C,e-Er 4 which are plotted as a function of the average atomic number (2,) defined above. It can be seen that, in general, the axial ratios of the pure metals and the cerium alloys lie on a common curve with a pronounced minimum at an atomic number of between 68 and 69. TXBLE
I
THE Z+.TTICE PARAMETERS
OF THE
PURE
RARE-EARTH
METALS
AND
OF A h-7.7%
C‘2 ALLOY
AT
ROOM TEMPERATURE* _
___I__
.Ildal __.._____
.-.._______..-
a-Spacing (kXl _____..- - _.____
..
lkXl
Axial ratio (da)
.4 tolnic nuvnbzv
c-Spaciq
Reference
a-La
3.762
I*.‘34
I.612
57
9
!:Z
3.666 3.6640 3.6490 3.6208 4.572
iX~778 11.807 11.77’ .26.178
1.611 I.607
58 59 60
10 2
ir-?id
n-Pm x-Sm q-Eu** XX-GCl
!X-Tb S-D? ~-HO x-Er %:-Tm n-Yb** <-I-u cx-Lu-7.794
3.6265
Ce
3.5986 3.5858 3.5724 3.5538 3.5355 5.4736 3.5028 3.5r9o -_-
I.613 1.607
5.7697 5.6858 5.6423 5.61 j8 5.5816 5.5596
I.5910 I.5800 I.5735
61 62 63 64 65 66
1.j720
67
5.5554 5.5590
I .586o T.5797
1.5706 I.5725
-
I
1 Present work 2 I Present work 3
68 69 70 7’ 70
Ptesent Present Present Present
work work work work
* The axial ratios of the d.hex. and Sm-type structures have been adjusted to correspond twice the distance between close-packed planes by dividing by L and 4.5, respectively. ** Eu and Yb have b.c.c. and f.c.c. structure, respectively.
with
The c.p.hex. structure in the above alloy systems has a limiting mean atomic number of about 63.5, i.e., close to the atomic number of gadolinium (64). The limiting axial ratio is about 1.587 and this is less than the axial ratio of a-gadolinium (1.591). Unlike the rare-earths with atomic numbers in excess of 64, however, gadolinium is in a ferromagnetic state at, or just below room temperature, and this could account for the fact that the axial ratio of gadolinium exceeds the critical value of 1.587, exhibited by the intra rare-earth metal alloys. These observations are consistent with the instability of the c.p.hex. structure of Lx-gadolinium with respect to pressure? and cold works. The variation of the axial ratio with atomic number (Fig. I) for the c.p.hex. alloys indicates that a Lu-Ce alloy with a value of 2, of 70, should have an axial ratio between that of Tm (1.572) and Lu (1.586). The lattice spacings of a Lu-7.7:/o Ce alloy (Z&=70) are reported in Table I and it can be seen that the axial ratio of this alloy has an intermediate value of 1.580. The alloys with the Sm-type structure exist over a range of atomic numbers * For purposes of comparison, the axial ratios of the d.hex. and Sm-type structures have been adjusted to correspond with twice the distance between close-packed planes by dividing by 2 and 4.5, respectively. J. Less-C0~~0~
Metak,
17
(1969)
336-339
SHORTCOMMUNICATIONS
338
from about 62 to 63. This indicates that a hypothetical, trivalent form of europium would have a Sm-type structure and it might be possible to form such a phase at ultra-high pressures. The alloys with the d.hex. structure exist over the range of 2, from about 59.0 to 61.5 and this indicates that the probable structure of metallic promethium (61) is d.hex. which is in agreement with the prediction of GSCHNEIDNER AND VALLETTA~. The atomic number of promethium is close to the limit of the d.hex. range and this indicates that the d.hex. structure of this metal could be heavily faulted. The axial ratio of p-cerium lies below the general variation of c/a with 2,.
x590-
1.580*
1.570b Ce Pr Nd R Sm Eu Gd Tb Dy Ho Er lm Yb Lu 575659606162,63&46566676869x)71
Atomic number (2)
Fig. I. The variation of axial ratio with atomic number of the pure rare-earth metals and of some cerium-rare-earth atloys. @ Pure rare earths x Ce-Cd alloys A Lu-7.7 at.% Ce alloy v Ce-Tb alloys Q Ce-Nd alloys A Ce-Ho alloys m Ce-Sm alloys + Ce-Er alloys
It should be remembered, however, that the effective valency of p-cerium is slightly in excess of the trivalent rare earths and this could account for the deviation of the axial ratio from the general behaviour. The axial ratio of a hypothetical 3-valent form of cerium, obtained by extrapolating the a- and c-spacings of a number of d.hex. Ce rare-earth alloys to 100% Ce is 1.611 +O.OOI. The composition at which the Sm-type alloys occur in the Y-Ce alloy system2 suggests that yttrium behaves like a heavy rare-earth metal with an atomic number of 69, i.e. simiIar to thulium. J. Less-Common
Metals, 17 (1969) 336-339
SHORT
339
COMMUNICATIONS
A similar plot of axial ratio against 2, heavy
rare-earth
compositions
alloy systems
at which
can be obtained
and it is possible,
the various
stacking
for the praseodymium/
by this method,
sequences
to predict
occur in a particular
the intra
rare-earth metal alloy system, simply from a consideration of the atomic numbers of the constituent metals. The correlation reported in this note provides an empirical method for predicting the structures of alloy phases in intra rare-earth metal binary systems. If the variation of the axial ratios of the hexagonal structures in these systems reflect the interaction of the Fermi surface of these alloys with the Brillouin zone faces then the correlation of c/a with Zm might indicate is applicable to the intra rare-earth alloys. Rare Earth Research Group, Department of Physical Metallurgy Science
that
some form of rigid band model
I. R. G. V.
ad
HARRIS RAYNOR
of Materials,
University of Birmingham, Birmingham I 2 3 4 5 6 7 8 Q 10
(Gt. Britain)
J. D. SPEIGHT, 1. R. HARRIS AND G. V. RAYNOR, J. Less-Common Metals, 15 (1958) 317. I. R. HARRIS, C. C. KOCH AND G. V. RAY~OR, J. Less-Common Metals, II (1966) 436. J. D. SPEIGHT AND 1. R. HARRIS, to be published. M. NORMAN, I. R. HARRIS AND G. V. RAYNOR, J. Less-Common Metals, 13 (1967) 24. C. H. HODGES, Acta Met., r5 (1~67) 1787. K. A. GSCHNEIDNER AND R. M. VALLETTA, Acta Met., 16 (1~68) 477. A. JAYARAMAN AND R. C. SHERWOOD, Phys. Rev. Letters, II (1964) 22. I. R. HARRIS, C. C. KOCH AND G. V. RAYNOR, J. Less-Common Metals, 12 (1967) 239. F. H. SPEDDING, A. H. DAANE AND K. W. HERRMANN, Acta Cryst., 9 (1956) 599, K. A. GSCHNEIDNER, R. 0. ELLIOTT AND Ii. R. MCDONALD, J. Phys. Chem. Solids, 23 (1962) 555.
Received
November
7th, 1968 J. Less-Common
Metals,
17 (1969)
336-339