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An initial comparison of the elastic properties of Dy and Tb-50%Ho
Solid State Communications, Vol. 9, pp. 857—859, 1971.
Pergamon Press.
Printed in Great Britain
AN INITIAL COMPARISON OF THE ELASTIC PROPERTIES OF Dy AND Tb—50%Ho R.G. Jordan Centre for Materials Science, University of Birmingham, Birmingham 15, England
(Received 9 March 1971 by C.W. McCombie)
The temperature variation of the c axis lattice parameter of Tb—50%Ho is compared to that of Dy. The results confirm the similarity of the elastic properties suggested by previous magnetic measurements.
RECENT magnetization measurements by Spedding, 1 and neutron diffraction Jordan Williams Ito and Jordan2 on single studies and by Spedding, crystal Tb—Ho alloys (hcp) indicated some rather remarkable similarities between the magnetic parameters of Tb—50%Ho and the pure element Dy. Spedding et al. suggested that the closeness of their de Gennes factor might account for the similarities of their magnetic parameters near the paramagnetic to antiferromagnetic transition. 2 The similarities between the temperature variation of the modulation vectors in the antiferromagnetic region were explained by the fact that the temperature variation of the superzone splitting of the :onduction bands3 were similar for Dy and Tb—50%Ho.2 Spedding et al. further suggested that, as the magneto-elastic effect is thought to be the driving force at the order—order transition as well as a contributing factor to the temperature dependence of the modulation vector,4 it followed that the elastic properties were probably similar also, in view of the similarity between the Curie temperatures of Dy and Tb—50%Ho.2 The purpose of this present communication is to present initial results of a comparison of the elastic properties of Dy and Tb—50%Ho.
Cu Ka~and K~ Data wereand taken from the (002), (004) and (006) 2. reflections a cosOcot6 extrapolation used to obtain accurate lattice parameters. Absolute accuracies are estimated to be about ±0.0006 A.
C
C
-
—
‘.~
5t5
—
5~2 —
----~.
—
CT - -
-
—
-~
—
_________
0
-
20c
FIG. 1. c axis lattice parameter versus temperature for Dy and Tb—50%Ho. The broken lines ‘normal’ contraction, calculated from a Tb—50%Ho. o Dy, from reference 5.
The temperature variation of the c axis lattice parameters of Tb—50%Ho and Dy, from Darnell and Moore,5 are shown in Fig. 1. From room temperature down to the neighbourhood of the Ned points, ‘normal’ contraction occurs for both metals, as indicated by the broken lines. These lines are of the form:
The c axis lattice parameter of a single crystal Tb-.50%Ho sample was measured from about 60°K to 320°K on a Picker X-ray diffractometer using an MRC model X—86GC low temperature attachment. The radiation employed was 857
858
THE ELASTIC PROPERTIES OF Dy AND Tb—50%Ho
=
11
c
.4.C~(T).T] =
0
c 0(l
a~.T)
+
161]
(1)
calculated from the Gruneisen relation, where A is a constant and C~(T)is the specific heat in the Debye approximation. Values of c0 and A for Dy and Tb—50%Ho, determined from a least squares fit of the data above about 200°Kto equation (1), are shown in Table 1. The expansion 4 below Néel pointvariation is due toofmagnetic effects, and thethe temperature the strains in this region:
!
=
Vol. 9, No. 12
~a.l
—
where
~al
and
=
-~
A1(1
__J__
-E~_A
4~/3
c’~,
~a.2
=
_
~
equation
A1
=
.±~_
ii
(1)
and
_____________________________________
--
-,
A
Cc(A)
Tb—50%Ho D
(1
—
~,)
cos
—
ç~i)
A
3cos b)
i-
—
5.6385 5.6301
2.068 2 439
A2
io~ x
106
2
~
~2
A (1
—
costti)
and 2ca
Metal
cos
+
a
CT)/CT
where c is the measured lattice parameter and CT that calculated from equation (1) for Dy and Tb—50°~Ho are compared in Fig. 2. Ft is apparent Table 1. Constants for
(2)
3
2(1
(c
2
+
=
—~--
\~1O~, J~(T) .J.. -
~.—
L
I~ L~(T)]
2
~1D~J~(T)
2c~2 ~ I 2’3 22 L~(T)] As the effective spins of Dy and Tb—50%Ho are the same and the reduced temperature variation 2 it of the reduced also the same, follows that themoment reducedare temperature variation
-
of the spin correlation functions6 Jt(T,’T\) I —
~ — —
—
— —
—
—
—
2.C—
—
—
-
0~
C6
0~
FIG. 2. c axis strain versus temperature for Dy and Tb—50%Ho. The solid line is the variation of m2, from reference 2. a Tb—50%Ho, T\ =- l83K. o Dy, from reference 5, T~ = 179°K that in the antiferromagnetic region the strains are identical, although below the Curies temperatures there appears to be a discrepancy of about S per cent. Using the notation of Evenson and Liu,4 the strain in the antiferromagnetic region is given by:
and Lv(T ‘Tv) are also the same for Dy and Tb—50%Ho. Therefore, in view of the similarity of the strains in the antiferromagnetic region, it follows from (2) that the combination of the magneto-elastic coupling coefficients. D] , and the elastic constants, C~, should also be the same for Dy and Tb—50%Ho in this temperature region. This result is rather interesting in view of the different nature of the two metals. If the combination of the coupling terms and elastic constants for Dy and Tb—50%Ho remain the same below their Curie points, one would expect the strains to be the same also.4 The differences in the data below the Curie points might be accounted for by a difference in the coupling terms or elastic constants between Dy and Tb—50%Ho in the ferromagnetic tegion. Such differences would not be surprising in view of the fact that the easy magnetization direction in Dy is a whereas in Tb—50%Ho it is b’2. The main temperature variation in (2) arises from the correlation functions which vary
Vol. 9, No. 12
THE ELASTIC PROPERTIES OF Dy AND Tb—50%Ho
approximately as the square of the magnetization.6 The solid line through the data in Fig. 2 is the experimentally determined variation of m2, from reference 2, normalized to the data at T/TN = 0.5.
progress, as well as determinations of the elastic constants in the magnetically ordered regions of Tb—50%Ho. Acknowledgements The author wishes to thank Professor A.J.C. Wilson for the use of the X-ray diffractometer, and Dr. I. Langford and Mr. H. Edwards their useful advice and assistance in thefortaking of the data. —
In order to extend the conclusions of this work, further measurements of the corresponding strains along the hexagonal a and b axes are in -
.
859
REFERENCES 1. 2.
SPEDDING F.H., JORDAN R.G. and WILLIAMS R.W., J. Chem. Phys. 51, 509 (1969). SPEDDING F.H., ITO Y. and JORDAN R.G., J. Chem. Phys. 53, 1455 (1970).
3. 4.
ELLIOTT R.J. and WEDGEWOOD F.A., Proc. Phys. Soc. (London) 84, 63 (1964). EVENSON W.E. and LIU S.H., Phys. Rev. 178, 783 (1969).
5.
DARNELL F.J. and MOORE E.P., J. app!. Phys. 34, 1337 (1963).
6.
CALLEN E. and CALLEN H.B., Phys. Rev. 139, 455 (1965).
Le variation de la temperature du paramétre crystallin c de Tb—50%Ho est compare de celui de Dy. Les données confirment les similarités entre des propriétés élastiques des deux matériaux qui ont été indiquées par les mesurages magnétiques antérieurs.
Pergamon Press.
Printed in Great Britain
AN INITIAL COMPARISON OF THE ELASTIC PROPERTIES OF Dy AND Tb—50%Ho R.G. Jordan Centre for Materials Science, University of Birmingham, Birmingham 15, England
(Received 9 March 1971 by C.W. McCombie)
The temperature variation of the c axis lattice parameter of Tb—50%Ho is compared to that of Dy. The results confirm the similarity of the elastic properties suggested by previous magnetic measurements.
RECENT magnetization measurements by Spedding, 1 and neutron diffraction Jordan Williams Ito and Jordan2 on single studies and by Spedding, crystal Tb—Ho alloys (hcp) indicated some rather remarkable similarities between the magnetic parameters of Tb—50%Ho and the pure element Dy. Spedding et al. suggested that the closeness of their de Gennes factor might account for the similarities of their magnetic parameters near the paramagnetic to antiferromagnetic transition. 2 The similarities between the temperature variation of the modulation vectors in the antiferromagnetic region were explained by the fact that the temperature variation of the superzone splitting of the :onduction bands3 were similar for Dy and Tb—50%Ho.2 Spedding et al. further suggested that, as the magneto-elastic effect is thought to be the driving force at the order—order transition as well as a contributing factor to the temperature dependence of the modulation vector,4 it followed that the elastic properties were probably similar also, in view of the similarity between the Curie temperatures of Dy and Tb—50%Ho.2 The purpose of this present communication is to present initial results of a comparison of the elastic properties of Dy and Tb—50%Ho.
Cu Ka~and K~ Data wereand taken from the (002), (004) and (006) 2. reflections a cosOcot6 extrapolation used to obtain accurate lattice parameters. Absolute accuracies are estimated to be about ±0.0006 A.
C
C
-
—
‘.~
5t5
—
5~2 —
----~.
—
CT - -
-
—
-~
—
_________
0
-
20c
FIG. 1. c axis lattice parameter versus temperature for Dy and Tb—50%Ho. The broken lines ‘normal’ contraction, calculated from a Tb—50%Ho. o Dy, from reference 5.
The temperature variation of the c axis lattice parameters of Tb—50%Ho and Dy, from Darnell and Moore,5 are shown in Fig. 1. From room temperature down to the neighbourhood of the Ned points, ‘normal’ contraction occurs for both metals, as indicated by the broken lines. These lines are of the form:
The c axis lattice parameter of a single crystal Tb-.50%Ho sample was measured from about 60°K to 320°K on a Picker X-ray diffractometer using an MRC model X—86GC low temperature attachment. The radiation employed was 857
858
THE ELASTIC PROPERTIES OF Dy AND Tb—50%Ho
=
11
c
.4.C~(T).T] =
0
c 0(l
a~.T)
+
161]
(1)
calculated from the Gruneisen relation, where A is a constant and C~(T)is the specific heat in the Debye approximation. Values of c0 and A for Dy and Tb—50%Ho, determined from a least squares fit of the data above about 200°Kto equation (1), are shown in Table 1. The expansion 4 below Néel pointvariation is due toofmagnetic effects, and thethe temperature the strains in this region:
!
=
Vol. 9, No. 12
~a.l
—
where
~al
and
=
-~
A1(1
__J__
-E~_A
4~/3
c’~,
~a.2
=
_
~
equation
A1
=
.±~_
ii
(1)
and
_____________________________________
--
-,
A
Cc(A)
Tb—50%Ho D
(1
—
~,)
cos
—
ç~i)
A
3cos b)
i-
—
5.6385 5.6301
2.068 2 439
A2
io~ x
106
2
~
~2
A (1
—
costti)
and 2ca
Metal
cos
+
a
CT)/CT
where c is the measured lattice parameter and CT that calculated from equation (1) for Dy and Tb—50°~Ho are compared in Fig. 2. Ft is apparent Table 1. Constants for
(2)
3
2(1
(c
2
+
=
—~--
\~1O~, J~(T) .J.. -
~.—
L
I~ L~(T)]
2
~1D~J~(T)
2c~2 ~ I 2’3 22 L~(T)] As the effective spins of Dy and Tb—50%Ho are the same and the reduced temperature variation 2 it of the reduced also the same, follows that themoment reducedare temperature variation
-
of the spin correlation functions6 Jt(T,’T\) I —
~ — —
—
— —
—
—
—
2.C—
—
—
-
0~
C6
0~
FIG. 2. c axis strain versus temperature for Dy and Tb—50%Ho. The solid line is the variation of m2, from reference 2. a Tb—50%Ho, T\ =- l83K. o Dy, from reference 5, T~ = 179°K that in the antiferromagnetic region the strains are identical, although below the Curies temperatures there appears to be a discrepancy of about S per cent. Using the notation of Evenson and Liu,4 the strain in the antiferromagnetic region is given by:
and Lv(T ‘Tv) are also the same for Dy and Tb—50%Ho. Therefore, in view of the similarity of the strains in the antiferromagnetic region, it follows from (2) that the combination of the magneto-elastic coupling coefficients. D] , and the elastic constants, C~, should also be the same for Dy and Tb—50%Ho in this temperature region. This result is rather interesting in view of the different nature of the two metals. If the combination of the coupling terms and elastic constants for Dy and Tb—50%Ho remain the same below their Curie points, one would expect the strains to be the same also.4 The differences in the data below the Curie points might be accounted for by a difference in the coupling terms or elastic constants between Dy and Tb—50%Ho in the ferromagnetic tegion. Such differences would not be surprising in view of the fact that the easy magnetization direction in Dy is a whereas in Tb—50%Ho it is b’2. The main temperature variation in (2) arises from the correlation functions which vary
Vol. 9, No. 12
THE ELASTIC PROPERTIES OF Dy AND Tb—50%Ho
approximately as the square of the magnetization.6 The solid line through the data in Fig. 2 is the experimentally determined variation of m2, from reference 2, normalized to the data at T/TN = 0.5.
progress, as well as determinations of the elastic constants in the magnetically ordered regions of Tb—50%Ho. Acknowledgements The author wishes to thank Professor A.J.C. Wilson for the use of the X-ray diffractometer, and Dr. I. Langford and Mr. H. Edwards their useful advice and assistance in thefortaking of the data. —
In order to extend the conclusions of this work, further measurements of the corresponding strains along the hexagonal a and b axes are in -
.
859
REFERENCES 1. 2.
SPEDDING F.H., JORDAN R.G. and WILLIAMS R.W., J. Chem. Phys. 51, 509 (1969). SPEDDING F.H., ITO Y. and JORDAN R.G., J. Chem. Phys. 53, 1455 (1970).
3. 4.
ELLIOTT R.J. and WEDGEWOOD F.A., Proc. Phys. Soc. (London) 84, 63 (1964). EVENSON W.E. and LIU S.H., Phys. Rev. 178, 783 (1969).
5.
DARNELL F.J. and MOORE E.P., J. app!. Phys. 34, 1337 (1963).
6.
CALLEN E. and CALLEN H.B., Phys. Rev. 139, 455 (1965).
Le variation de la temperature du paramétre crystallin c de Tb—50%Ho est compare de celui de Dy. Les données confirment les similarités entre des propriétés élastiques des deux matériaux qui ont été indiquées par les mesurages magnétiques antérieurs.