- Email: [email protected]
Quadrupolar induced structural transitions in TmCd and TmZn
Journal of Magnetism and Magnetic Materials 13 (1979) 198-200 0 North-Holland Publishing Company
SECTION 11: RARE EARTHS QUADRUPOLAR
INDUCED STRUCTURAL
TRANSITIONS
IN TmCd AND TmZn
B. LUTHI, R. SOMMER Physikalisches Institut der Universitcit, Robert-Mayer-Str.
2-4,
D-6000 Frankfurt am Main, Fed. Rep. Germany
and P. MORIN CNRS Laboratoire Louis Niel, 166 X 38042
Grenoble Cedex, France
Received 14 March 1979
TmCd and TmZn exhibit pure quadrupolar structural phase transitions at 3.16 K and 8.55 K respectively. The magnetoelastic and quadrupolar coupling constants for these two compounds have been determined by different methods. The structural phase transition temperature 7’a is strongly field dependent. We show the measured magnetic phase diagram T#$!B) together with the calculated one. The strong first order phase transitions at T, are due to the F5 groundstate of the Tm ion in these materials.
The CsCl-structure materials TmCd and TmZn have been investigated intensively in recent years because of the occurrence of a structural transition at 3.16 K and 8.55 K respectively. The first investigations [1,2] showed in both cases a strong softening of the cr 1 cl2 elastic mode. In fig. 1 we give the temperature dependence of these modes together with a theoretical fit using the model Hamiltonian.
0 25
(1) Here e3 = (2e,, - e,, - e )/G is the tetragonal strain, 08 = 2Jz - Jz - J,,fYthe quadrupole operator of the Tm3+ ion. The first term describes the magnetoelastic interaction and the second one a quadrupolar interaction which is probably mediated by conduction electrons in these metallic systems. Within molecular field theory, eq. (1) leads to a temperature dependence of the elastic constants of the form [3,4] : (Cl1 -
xl
100
150
I 7 200
o(
_ ---_co(T) --_
------_
2
0
50
100
150
TX)0
(W
dNxs c12)/2
= cc11
-
c12)OD
-
I 1 -gxs’
(2)
Fig. 1. Elastic constants (~11 - cr 2)/2 for TmCd and TmZn as a function of temperature. Full lines are calculated curves based on eq. (2) with coupling constants taken from table 1. Dashed lines are background elastic constants [(cl r - c,f2)/2]O.
where xg is the single-ion strain susceptibility and g’ = ZbGij. Eq. (2) gives a very good fit to the experimen198
B. Ltithi et al. / Quadrupolar induced structural transifions
199
Table 1 Ko (mK) T,
TC
W
_ 8.12 K
0.88 K 1.2 K
X
Elastic
Ta
T,(B)
Parastriction
&/co
g’
9.2 21.9
11.3 25.5
24.3 mK
161
12 25 [71
0.7 mK 3.6 mK
[51
11.3 25.1 this work
TmCd TmZn References
3.16 K 8.55 K
-0.33 -0.3 1
tal results for both substances as shown in fig. 1. Coupling constants and other parameters are listed in table 1. The large values of g’ compared with gqN/ce make these structural transitions typical quadrupolar induced ones, compared with strain induced cooperative Jahn-Teller transitions. We took a I’s groundstate for both samples [S]. It has been observed [l] that the structural transition Ta is strongly dependent on applied magnetic fields. We have made a quantitative study of this effect. In fig. 2 we show T,(B) for both substances. The magnetic field was applied along a (001).direction. Different experimental methods were used to get this mag-
8.5 mK
netic phase diagram (magnetization, electrical resistivity and elastic constant measurements). For the calculation of the phase diagram T,(B) we use the following molecular field Hamiltonian per Tm3+-ion:
(3) K. = giN/co +g’ obtained by different methods is listed in table 1 (elastic constant measurements [S], transition temperature T, [6] and parastriction mea-
-4
T,,.6.6 0
30 -
20 -
-3
TmZn
\
l_--/ B=OT
To(K)
20
0 0
TmZn
10
I
---__ M
-2 -1
I
_
_
2
T&PEtATUR:
c
K
( K1
IN
QI T, = 6.6K
.
‘f-“-I
0
I
I
I
1
2
3
40 =---------
.
30 -
--.y
TmZn B =
-4
‘\
\
\\
0.1T
-3 \
20 -
-2
10 -
-1
L
L
0 (T)
Fig. 2. Magnetic phase diagram for TmCd and TmZn. T,(B) determined experimentally by magnetization (o), elastic constants (A) and electrical resistivity (0). FUR lines are theoretical curves as described in text.
I
I
I
2
4
6
8
TEMPERATURE
___-__ 10 (K)
_ 12
Fig. 3. Calculated order parameters M = (J,) (dotted lines) and Q = (0:) (full lines) as a function of temperature for B = 0 and B = 0.1 T for TmZn.
B. L&hi et al. / Quadrupolar induced structural transitions
200
surements [7]). h is the exchange constant which we take as -0.69 K for TmZn and = 0 for TmCd. We then calculate Q = ( 0;) as the statistical average over the crystal field states and likewise M = V >.Since the Tm3+-CEF energy levels are likewise functions of Q and M we solve these coupled equations numerically by iteration. A typical result for the order parameters Q and M in the case of TmZn is shown for B = 0 and 0.1 T in fig. 3. The Ke and h values are chosen such that they reproduce the magnetic and structural transition temperatures T,, T, rather closely. One notices from table 1 that K, is almost identical to the value obtained from other measurements, which shows the consistency of our description of the phase transition. The results in fig. 3 indicate that the magnetic and structural transition temperatures already coincide for small magnetic fields. They remain together up to higher fields as shown in fig. 4. The strong first order nature for B = 0 is a consequence of the Fs groundstate [8,9]. For higher fields the discontinuity at the phase transition decreases as shown in fig. 4 for the structural order parameter Q. At high enough fields the discontinuity disappear completely and one can no longer speak of a phase transition. This behaviour is analogous to a ferromagnet in a magnetic field. The full lines in fig. 2 give the results of the calculation for T,(B). The agreement with experiment is better in TmCd than it is for TmZn although in both Q
To(K)
=
8.8
11.9
13.815D 16.1
50
40
2
4
6 8 10 TEMPERATURE
12 (K)
I&
16
18
Fig. 4. Calculated structural order parameter Q = (0: ) as a function of temperature and magnetic field for TmZn.
cases the essential features are well reproduced. Both substances TmCd and TmZn are by now fairly welt characterized. By taking for both TmZn and TmCd a Fs groundstate, one can explain elastic constants, magnetization, first order transition and T,(B). The coupling parameters g2, g’, K. are now rather well determined by different experiments as listed in table 1. This internal consistency enables one now to go a step further and to ask for a microscopic description of the quadrupolar coupling etc. A step in this direction has been made by studying dynamical effects and transport processes. Ultrasonic attenuation studies near the structural phase transition and electrical resistivity measurements over a large temperature region for TmCd, TmZn enable one to extract the microscopic aspherical Coulomb charge scattering constant and to study relaxation processes in these systems [lo].
Acknowledgements Part of this research was supported by the Sonderforschungsbereich 65, Frankfurt-Darmstadt. The calculations were performed at the Hochschulrechenzentrum Frankfurt.
References [l] B. Liithi, M.E. Mullen, K. Andres, E. Bucher and J.P. Maita, Phys. Rev. B8 (1973) 2639. [2] P. Morin, A. Waintal and B. Liithi, Phys. Rev. B14 (1976) 2972. 131 P. Levy, J. Phys. C6 (1973) 3545. [4] B. Liithi, AIP Conf. Proc. 34 (1976) 7. [S] B. Ltithi, R. Sommer and P. Morin, 3. de Phys. Phys. C5 (1979) 139. [6] P. Morin, J. Rouchy and D. Schmitt, Phys. Rev. B17 (1978) 3684. [7] P. Morin, D. Schmitt and E. du Tremolet de Lacheisserie, Phys. Lett. 69A (1978) 217. [8] M. Kataoka and J. Kanamori, J. Phys. Sot. Japan 32 (1972) 113. [9] Y. Kino, B. Liithi and M.E. Mullen, J. Phys. Sot. Japan 33 (1972) 687. [lo] K.M. Leung, D.L. Huber and B. Liithi, J. Appl. Phys. 50 (1979) 1831.
SECTION 11: RARE EARTHS QUADRUPOLAR
INDUCED STRUCTURAL
TRANSITIONS
IN TmCd AND TmZn
B. LUTHI, R. SOMMER Physikalisches Institut der Universitcit, Robert-Mayer-Str.
2-4,
D-6000 Frankfurt am Main, Fed. Rep. Germany
and P. MORIN CNRS Laboratoire Louis Niel, 166 X 38042
Grenoble Cedex, France
Received 14 March 1979
TmCd and TmZn exhibit pure quadrupolar structural phase transitions at 3.16 K and 8.55 K respectively. The magnetoelastic and quadrupolar coupling constants for these two compounds have been determined by different methods. The structural phase transition temperature 7’a is strongly field dependent. We show the measured magnetic phase diagram T#$!B) together with the calculated one. The strong first order phase transitions at T, are due to the F5 groundstate of the Tm ion in these materials.
The CsCl-structure materials TmCd and TmZn have been investigated intensively in recent years because of the occurrence of a structural transition at 3.16 K and 8.55 K respectively. The first investigations [1,2] showed in both cases a strong softening of the cr 1 cl2 elastic mode. In fig. 1 we give the temperature dependence of these modes together with a theoretical fit using the model Hamiltonian.
0 25
(1) Here e3 = (2e,, - e,, - e )/G is the tetragonal strain, 08 = 2Jz - Jz - J,,fYthe quadrupole operator of the Tm3+ ion. The first term describes the magnetoelastic interaction and the second one a quadrupolar interaction which is probably mediated by conduction electrons in these metallic systems. Within molecular field theory, eq. (1) leads to a temperature dependence of the elastic constants of the form [3,4] : (Cl1 -
xl
100
150
I 7 200
o(
_ ---_co(T) --_
------_
2
0
50
100
150
TX)0
(W
dNxs c12)/2
= cc11
-
c12)OD
-
I 1 -gxs’
(2)
Fig. 1. Elastic constants (~11 - cr 2)/2 for TmCd and TmZn as a function of temperature. Full lines are calculated curves based on eq. (2) with coupling constants taken from table 1. Dashed lines are background elastic constants [(cl r - c,f2)/2]O.
where xg is the single-ion strain susceptibility and g’ = ZbGij. Eq. (2) gives a very good fit to the experimen198
B. Ltithi et al. / Quadrupolar induced structural transifions
199
Table 1 Ko (mK) T,
TC
W
_ 8.12 K
0.88 K 1.2 K
X
Elastic
Ta
T,(B)
Parastriction
&/co
g’
9.2 21.9
11.3 25.5
24.3 mK
161
12 25 [71
0.7 mK 3.6 mK
[51
11.3 25.1 this work
TmCd TmZn References
3.16 K 8.55 K
-0.33 -0.3 1
tal results for both substances as shown in fig. 1. Coupling constants and other parameters are listed in table 1. The large values of g’ compared with gqN/ce make these structural transitions typical quadrupolar induced ones, compared with strain induced cooperative Jahn-Teller transitions. We took a I’s groundstate for both samples [S]. It has been observed [l] that the structural transition Ta is strongly dependent on applied magnetic fields. We have made a quantitative study of this effect. In fig. 2 we show T,(B) for both substances. The magnetic field was applied along a (001).direction. Different experimental methods were used to get this mag-
8.5 mK
netic phase diagram (magnetization, electrical resistivity and elastic constant measurements). For the calculation of the phase diagram T,(B) we use the following molecular field Hamiltonian per Tm3+-ion:
(3) K. = giN/co +g’ obtained by different methods is listed in table 1 (elastic constant measurements [S], transition temperature T, [6] and parastriction mea-
-4
T,,.6.6 0
30 -
20 -
-3
TmZn
\
l_--/ B=OT
To(K)
20
0 0
TmZn
10
I
---__ M
-2 -1
I
_
_
2
T&PEtATUR:
c
K
( K1
IN
QI T, = 6.6K
.
‘f-“-I
0
I
I
I
1
2
3
40 =---------
.
30 -
--.y
TmZn B =
-4
‘\
\
\\
0.1T
-3 \
20 -
-2
10 -
-1
L
L
0 (T)
Fig. 2. Magnetic phase diagram for TmCd and TmZn. T,(B) determined experimentally by magnetization (o), elastic constants (A) and electrical resistivity (0). FUR lines are theoretical curves as described in text.
I
I
I
2
4
6
8
TEMPERATURE
___-__ 10 (K)
_ 12
Fig. 3. Calculated order parameters M = (J,) (dotted lines) and Q = (0:) (full lines) as a function of temperature for B = 0 and B = 0.1 T for TmZn.
B. L&hi et al. / Quadrupolar induced structural transitions
200
surements [7]). h is the exchange constant which we take as -0.69 K for TmZn and = 0 for TmCd. We then calculate Q = ( 0;) as the statistical average over the crystal field states and likewise M = V >.Since the Tm3+-CEF energy levels are likewise functions of Q and M we solve these coupled equations numerically by iteration. A typical result for the order parameters Q and M in the case of TmZn is shown for B = 0 and 0.1 T in fig. 3. The Ke and h values are chosen such that they reproduce the magnetic and structural transition temperatures T,, T, rather closely. One notices from table 1 that K, is almost identical to the value obtained from other measurements, which shows the consistency of our description of the phase transition. The results in fig. 3 indicate that the magnetic and structural transition temperatures already coincide for small magnetic fields. They remain together up to higher fields as shown in fig. 4. The strong first order nature for B = 0 is a consequence of the Fs groundstate [8,9]. For higher fields the discontinuity at the phase transition decreases as shown in fig. 4 for the structural order parameter Q. At high enough fields the discontinuity disappear completely and one can no longer speak of a phase transition. This behaviour is analogous to a ferromagnet in a magnetic field. The full lines in fig. 2 give the results of the calculation for T,(B). The agreement with experiment is better in TmCd than it is for TmZn although in both Q
To(K)
=
8.8
11.9
13.815D 16.1
50
40
2
4
6 8 10 TEMPERATURE
12 (K)
I&
16
18
Fig. 4. Calculated structural order parameter Q = (0: ) as a function of temperature and magnetic field for TmZn.
cases the essential features are well reproduced. Both substances TmCd and TmZn are by now fairly welt characterized. By taking for both TmZn and TmCd a Fs groundstate, one can explain elastic constants, magnetization, first order transition and T,(B). The coupling parameters g2, g’, K. are now rather well determined by different experiments as listed in table 1. This internal consistency enables one now to go a step further and to ask for a microscopic description of the quadrupolar coupling etc. A step in this direction has been made by studying dynamical effects and transport processes. Ultrasonic attenuation studies near the structural phase transition and electrical resistivity measurements over a large temperature region for TmCd, TmZn enable one to extract the microscopic aspherical Coulomb charge scattering constant and to study relaxation processes in these systems [lo].
Acknowledgements Part of this research was supported by the Sonderforschungsbereich 65, Frankfurt-Darmstadt. The calculations were performed at the Hochschulrechenzentrum Frankfurt.
References [l] B. Liithi, M.E. Mullen, K. Andres, E. Bucher and J.P. Maita, Phys. Rev. B8 (1973) 2639. [2] P. Morin, A. Waintal and B. Liithi, Phys. Rev. B14 (1976) 2972. 131 P. Levy, J. Phys. C6 (1973) 3545. [4] B. Liithi, AIP Conf. Proc. 34 (1976) 7. [S] B. Ltithi, R. Sommer and P. Morin, 3. de Phys. Phys. C5 (1979) 139. [6] P. Morin, J. Rouchy and D. Schmitt, Phys. Rev. B17 (1978) 3684. [7] P. Morin, D. Schmitt and E. du Tremolet de Lacheisserie, Phys. Lett. 69A (1978) 217. [8] M. Kataoka and J. Kanamori, J. Phys. Sot. Japan 32 (1972) 113. [9] Y. Kino, B. Liithi and M.E. Mullen, J. Phys. Sot. Japan 33 (1972) 687. [lo] K.M. Leung, D.L. Huber and B. Liithi, J. Appl. Phys. 50 (1979) 1831.