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Magnetic excitations in TmCu
287
Journal of Magnetism and Magnetic Materials 40 (1984) 287-292 North-Holland, Amsterdam
MAGNETIC EXCITATIONS IN TmCu P. MORIN, D. SCHMITT Laboratoire Louis N$el, CNRS 166X, 38042 Grenoble-c$dex, France
and C. VETTIER Institut Laue- Langevin, 156X, 38042 Grenoble-c~dex, France
Received 13 June 1983; in revised form 30 August 1983
The magnetic excitations have been studied in the rare-earth intermetallic compound TmCu by means of inelastic neutron scattering. Dispersion curves have been followed through the cubic Brillouln zone, in particular an intense and dispersive transverse branch: this latter presents a lifting of the degeneracy along particular directions. Within a generalized susceptibility formalism, this is related to the coupling of the excitations at q and q + Q, where Q = ( n / a , 'rr/a, 0) is the antiferromagnetic propagation vector in TmCu. The inter-ionic isotropic bilinear exchange parameters are deduced and compared with those of isomorphous compounds.
1. Introduction
Among the cubic rare-earth intermetallic compounds, those with the thulium ion have received a particular attention over the last few years. This is motivated by the competition in such compounds between rather weak Heisenberg interactions and possibly large quadrupolar interactions, which explained the occurrence of the quadrupole phase transition above the magnetic ordering temperature, observed in TmZn [1] (To = 8.6 K, T~= 8.1 K) and in TmCd [2] (To = 3.2 K, no magnetic ordering). This favourable situation arises because of the combination of a small de Gennes factor and strong orbital effects for the thulium ion [3]. In addition, the large two-ion quadrupolar interactions observed in the above cubic (CsCl-type structure) compounds seem to be mediated by the conduction electrons [4] similar to the usual isotropic bilinear R u d e r m a n - K i t t e l - K a s u y a Yoshida-type (RKKY) interactions. In TmCu the quadrupole interactions, although weaker than in TmZn and TmCd, are still noticeable, as evidenced by various experiments performed in the paramagnetic phase [5]. On the other hand, due to the different number of con-
duction electrons [6], the ordering of the magnetic moments is not ferromagnetic as in TmZn, but antiferromagnetic at low temperature, with a propagation vector Q = 2~r/a(½, ½, 0) and a moment pointing along the [001] axis [7]. In addition, this structure is replaced between Tt = 6.7 K and the Ntel temperature Tr~ = 7.7 K by an amplitudemodulated structure with an incommensurate propagation vector Qm = 2~r/a(½ + 0.064, ½, 0). The two transitions are first order, the quadrupole interactions being responsible for the first-order character of the transition at Ts [8]. Below Tt, in agreement with the magnetic ordering, the 4f quadrupoles are fully ordered, driving a coherent tetragonal distortion ( c / a - 1 = - 7 × 10 -3) via the magnetoelastic coupling. The various experiments performed in the paramagnetic phase of TmCu have been consistently described within the molecular-field approximation by means of an Hamiltonian including the Crystalline Electric Field (CEF), the Zeeman coupling, the Heisenberg bilineax interactions and the quadrupolar (magnetoelastic and two-ion) interactions. The knowledge of all the involved parameters allows us to study the magnetic properties in the ordered phase, in particular the mag-
0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
288
1>. Morin et al. / Magnetic excitatwns in TmCu
netic excitations. Such experiments have already been performed in some isomorphous compounds, namely in the ferromagnets H o Z n [9] and TbZn [10], the antiferromagnet ErCu [11] and in the quadrupole system TmcLu~_cZn [12]. They have shown the long-range character of the isotropic bilinear interactions, as expected for a RKKY-type coupling. In section 2, the results of inelastic neutron scattering experiments on TmCu are presented. An interpretation is attempted in section 3.
6
T
F
s
="
~4 2
0 X
~
r
t
M
T
R
s
X
z
Mr
^
R
Fig. 1. Dispersion curves of the magnetic excitations in T m C u at 1.7 K; continuous lines are calculated from the best fit of the transverse (T) mode.
2. Experimental results 2.1. Experiment The TmCu single crystal was obtained by the Bridgman method, both components being sealed in stoichiometric proportions in a tantalum crucible. A 10 × 10 × 2 mm 3 platelet, cut perpendicularly to the [110] direction was used in order to investigate successively the (001) and (110) scattering planes. The inelastic neutron scattering experiment was carried out on the IN2 spectrometer of the ILL high flux reactor in Grenoble. The (002) reflection of pyrolytic graphite was used as monochromator and analyser. The experiment was operated in the constant Q-mode with a fixed incident energy of 14.7 meV and a resolution of 0.7 meV. A first set of measurements was performed with a (001) scattering plane in order to investigate the zone around the incommensurate propagation vector Qm at various temperatures. A second experiment was carded out with a (110) scattering plane for completing the dispersion curves at low temperature along the main symmetry directions of the cubic Brillouin zone.
excitations. Typical scans are shown in fig. 2. Particular attention was paid to the excitations at the vectors Q and Qm" However, the lowest energy mode being around 1 meV could not be resolved from the intense elastic line (magnetic Bragg peak) within our experimental conditions, and this prevented precise measurement of this low energy mode. A second weaker and much less dispersive branch exists at T = 1.7 K around an energy transfer of 3.8 meV; it does not seem to be split, the highest energy transfer occurring at F and X points (4.0 meV) and the lowest one at M point (3.6 meV). Some q-points have been also investigated above the N~el temperature: a single CEF transition is observed around an energy transfer of 1.6 meV a
TmCu T=1.7K r~
(o5 o o)
too ~
'X"
/
#1 (1,o~o~)1
.
(o.s.o5.ORS)
1
2
v
2.2. Results The dispersion curves obtained at T = 1.7 K are shown in fig. 1. An intense and dispersive branch was followed in the energy range 0.9-2.2 meV (10-25 K); it is split in the whole cubic Brillouin zone except along the [111] direction FR. A broad not-resolved branch is observed in the center of XM, suggesting the existence of more than two
/
A
~,o.2s.o~
'°°I l\ 1
2
3
4
1 2 3 t~ ENERGY (meV)
3
4
Fig. 2. Typical scans in T m C u at 1.7 K for several q-points; lines are least-squares fits.
289
P. Morin et al. / Magnetic excitations in TmCu
value a little lower than-ihe previous determination of 1.8 meV [5]. The magnetic Bragg reflections at Q and Qm have been investigated as a function of the temperature. The two first-order transitions at Tt = 6.7 K and TN = 7.7 K [7] have been confirmed. Between Tt and T, only the modulated phase exists on the single crystal, in contrast to the previous neutron diffraction experiment, in which both phases coexisted. This was because the polycrystal was not annealed (the same situation was found on TmAg [13]). On the other hand, using a single crystal allowed us to determine with accuracy the value of the incommensurate propagation vector Qm = 2'rr/a(½ + z, ½, 0) with z = 0.0640 + 0.0003. Third-order satellites have also been observed at q = 2,rr/a(½ + 3~', ½, 0) with an intensity ratio of 4 × 10 -3 with respect to the first-order satellite (no fifth-order satellites were detected within the experimental uncertainties): the presence of higher harmonics indicates that the modulation of the magnetic moments amplitude is not purely sinusoidal, which may be due to the non-linear dependence of the magnetic moment magnitude on the exchange field. Finally, a longitudinal elastic q-scan at T = 1.7 K around the nuclear Bragg peak 2~r/a(0, 2, 0) provided an estimation of the tetragonal distortion, c / a - 1 = - 8 × 10 -3, in good agreement with neutron diffraction on a polycrystal [7].
3. Interpretation
~ ( Bi1) _- -
The formalism describing the magnetic excitations originates from the generalized susceptibility theory, which is particularly well-adapted to rareearth compounds where CEF effects are strong [14]. Here, only isotropic bilinear interactions have been taken into account:
(1)
i j~i
the factor ½ being included because each site is counted twice in the summation. In the case o f TmCu, the magnetic moments are aligned along a
--
E J(iJ)Jz(i)CJz(J))
= -J(Q)Jz(i)CJz(i)).
(2)
An antiferromagnetic bihnear exchange coefficient 0~ may be related to the Fourier transform of the coupling coefficients in a way analogous to the paramagnetic exchange coefficient 0* [14] (which is related to J(0)): J(J+
1) =J(Q)= ~" J(/j) e'Q"q j~i
(3)
In addition to the above bilinear interaction, the total one-ion Hamiltonian includes the CEF Hamiltonian ~CEF and the quadrupolar Hamiltonian &C'Q.According to the cubic symmetry.,~cev is expressed in the fourfold axes system as [15]: EF
~--"
WvX( OO+ 5o: )
i
+ W(l"lxl----~)(O° 21064)r
(4)
The quadrupolar Hamiltonian HQ contains a magnetoelastic and a two-ion part, which are isomorphous in the molecular field approximation [8]: ~i)=
3.1. Theory
= - ½ E E J(o)J(O .J(y),
fourfold axis below Tt, e.g. the z-axis, and two sublattices A and B have to be considered, with (J~(B)) =-(J~(A)). Therefore, applying the molecular field approximation leads to the following one-ion bilinear Hamiltonian ~g's,:
_ G1COO(i))oO(i).
(5)
It is worth noting that, for the quadrupoles, no distinction between the two sublattices A and B is needed for TmCu, since (O°(A)) = COO(B)). The self-consistent diagonalization of the total one-ion Hamiltonian yields the eigenvalues o~,, and the CEF eigenfunctions In(i)) which depend on the sublattice. However, their expansion coefficients in the usual fJ, Ms) basis may be easily related from symmetry considerations:
CJ, MjIn(B)) = CJ , - MjIn(A)),
(6)
from which we obtain the following relations between the matrix elements of the operators J,
290
P. Morin et al. / Magnetic excitations in TmCu
(a = z, +, - ) for both sublattices:
with
J~m,(A) = ( m ( A ) I J A n ( A ) ) = - J , , , ( B ) ,
:±
mn ( A ) =
(7)
( m (A)IJ±ln (A)) ----J:r m. (B).
D(q,o~) = [1 + ¼(J(q) + J(q + Q ) ) g A - ] X
Considering the inter-ion Hamiltonian leads, within the random phase approximation, to the following expression for the Fourier transform of the generalized susceptibilities G~a(q,~0) [16]:
) = g B(
)_
½S(q ) ggz( c : +
¼(J(q) +J(q + Q ) ) g ~ - ]
-~(J(q)-J(q+Q))2gl-g~-.
3.2. Generalized susceptibifity formalism
G;e(q,
[1 +
)
(13)
It is worth noting that only q appears in eq. (11), while the excitations at q and q + Q are coupled for the transverse modes [11,17] with the same energies given by the condition D(q,~0)= D(q + Q,~0) = 0.
3.3. Analysis of the data
½J(q + Q )g~( Gff~- G~/J) ¼S(q)[g~,+(GT, 1~+
GB" )
-¼J(q + Q )[ g~+ ( G~ ~ - G~ B) +g~-(G;/~-G~")]
(8)
and the same relation with A ~ B. This expression includes single-ion susceptibilities:
g~'(o~) = ~ J~m"(A)Ja"m(A)(fm - f " ) mn
tO --
(9)
O ) n Jr" 0 2 m
which are the solutions of the equations of motion without interactions (fro are the Boltzman factors of the CEF level ]m)). Because of the relations (6) and of the selection rules on the matrix elements (eq. (7)), the only non-zero single-ion susceptibilities are for TmCu: g~[(~o) = gf3Z(~o), g~ ~:(~o) = g~ ±(~o).
The coefficients involved in the one-ion Hamiltonian have been previously obtained [5] and constitute the starting point of the present analysis: W = 1.4 K, x = -0.42, G 1 = 11 mK. In addition, an antiferromagnetic exchange parameter @~ = 13 K has been used to account for the average energy of the observed branches; it leads to a calculated N~el temperature of 9 K, slightly higher than the experimental one. The first dispersive branch is then identified as a transverse excitation involving the first excited CEF level and the second branch around 3.8 meV as a longitudinal one involving the third excited level. The ratio of the square of the corresponding matrix elements is about 3. As seen in section 3.2 the transverse excitations for q and q + Q are coupled and have the same energy. Thus, in the cubic Brillouin zone (fig. 3) the points M = (0.5, 0.5, 0) and M' = (0.5, 0, 0.5) (in 2~r/a units) are not longer equivalent since F is
(10)
This allows one to decouple the magnetic excitation modes into one longitudinal mode M'
Gff(q,~) = G~Z(q,o~)=
g~'Z(°~) 1 + g~:(~)J(q)
Z°
and two transverse modes: G ~ ~ (q,o~ ) = G ~ ±(q,~o )
+ ½g¢
J(q + O)]
D(q,a})
"
(11)
(12)
I
;y
Z M Fig. 3. Cubic Brillouin zone with notation associated with the tetragonal magnetic symmetry.
P. Morin et al. / Magnetic excitations in TmCu
291
Table 1 Isotropic bilinear exchange coefficients J(ij) with neighbours defined by rq = (h, k, l)a in TmCu (positive values mean ferromagnetic coupling): also given are the values for ErCu [11], normalized to Tm by the ratio of ( g j - 1 ) 2 for both rare-earths hk l
J(ij) (mK)
TmCu ErCu
100
110
111
200
210
211
220
- 199.0 - 28.8
- 20.5 - 8.6
18.4 30.4
33.2 20.1
- 7.5 - 1.4
9.2 - 8.5
- 7.9 - 0.4
coupled with M but not with M'. Keeping the cubic Brillouin zone for the dispersion curves (fig. 1), this allows one to identify the points M and M' among the two observed excitations: having the same energy transfer as F the highest mode corresponds to M. In the same way the point X' is identified as having the same energy transfer as R. In addition some branches are verified to be identical (FX' and MR), or symmetrical with respect to their centre (I'M or RX'). The inter-ion coupling coefficients J(/j) have been obtained by a least-squares procedure. At least 7 neighbours (see table 1) are necessary for describing the splitting at X- and M-points. Taking up to 10 neighbours does not improve significantly the fit; in particular, the calculated energy at M' is always a little higher than the experimental one. On the other hand, calculating the longitudinal excitation energies with the above coefficients leads to a good agreement with experiment except at R. Compared with the results on the isomorphous compound ErCu [11], the relative magnitude of successive J(ij)'s is very different (see table 1), this cannot be accounted for by a simple R K K Y theory, and suggests that the band structures for both compounds differ, perhaps because of the larger distortion in TmCu ( - 8 x 1 0 -3) than in ErCu (less than 10-3). Some disagreements appear when relating J(q) with the static properties. Indeed, if O~ is well-described starting from J(M) (O~°,,o= 10 K), this is not the case of the paramagnetic exchange coefficient O*~lc.= - 1 6 K, whereas the experimental value is - 3K [5]. In addition, the value of J(R) is larger than that of J(M) leading to an antiferromagnetic ordering with Q = R a t T N ~ 18 K. This situation remains for all the fits we have been made. Considering the existence of anisotropic bilin-
ear interactions, as in TbP [18], does not seem to be able to improve the fits. Indeed, these anisotropic interactions would lead to an additional lifting of the degeneracy along the directions FX and FM' [16], which was not observed experimentally. In summary, the inelastic neutron scattering which has been performed on TmCu has clearly shown the lifting of the degeneracy of a transverse magnetic excitation along particular directions of the cubic Brillouin zone. This has been well accounted for, at least qualitatively, by a generalized susceptibility formalism including only isotropic bilinear exchange interactions for an antiferromagnet. The degeneracy is raised by the coupling between the magnetic excitations at q and q + Q [17]. Quantitatively, some problems remain unexplained at this time, in particular why J(q) is maximum at q = R and not q = M, as expected for a ( ~ 0 ) - t y p e magnetic structure. Nevertheless, the deduced inter-ionic exchange parameters have a relative variation as a function of the neighbours, which is very different from those of the isomorphous compound ErCu. This could indicate that the bilinear exchange coupling J(q) is a strong specific property of each compound and that it appears to be useless to search for any scaling law for such pair interactions.
References [1] P. Morin, J. Rouchy and D. Schmitt, Phys. Rev. B17 (1978) 3684. [2] R. A16onard and P. Morin, Phys. Rev. B19 (1979) 3868. [3] D. Schmitt and P.M. Levy, J. Magn. Magn. Mat. 31-34 (1983) 249. [4] P.M. Levy, P. Morin and D. Schmitt, Phys. Rev. Lett. 42 (1979) 1417. [5] C. Jaussaud, P. Morin and D. Schmitt, J. Magn. Magn. Mat. 22 (1980) 98.
292
P. Morin et al. / Magnetic excitations in TmCu
[6] J. Sakurai, Y. Kubo, T. Kondo, J. Pierre and E.F. Bertaut, J. Phys. Chem. Solids 34 (1973) 1305. [7] P. Morin and D. Schmitt, J. Magn. Magn. Mat. 21 (1980) 243. [8] P. Morin and D. Schmitt, Phys. Rev. B27 (1983) 4412. [9] J. Pierre, P. Morin, D. Schmitt and B. Hennion, J. Phys. F 7 (1977) 1965. [10] Y. Hamaguchi, H. Betsuyaku and S. Funahashi, J. Magn. Magn. Mat. 15-18 (1980) 377. [11] J. Pierre, P. Morin, D. Schmitt and B. Hennion, J. de Phys. 39 (1978) 793. [12] P. Morin, D. Sehmitt and C. Vettier, J. Phys. F 11 (1981) 1487.
[13] P. Morin and D. Schmitt, J. Magn. Magn. Mat. 28 (1982) 188. [14] P. Morin, D. Schmitt, C. Vettier and J. Rossat-Mignod, J. Phys. F 10 (1980) 1575. [15] K.R. Lea, J.M. Leask and W.P. Wolf, J. Phys. Chem. Solids 23 (1962) 1381. [16] P. Morin and D. Schmitt, to be published. [17] J. Rossat-Mignod, D. Delac6te, J.M. Effantin, C. Vettier and O. Vogt, Physica 120B (1983) 163. [18] K. Knorr, A. Loidl and J.K. Kjems, in: Crystalline Electric Field and Structural Effects in f-Electron Systems, eds. J.E. Crow, R.P. Guertin and T.W. Mihalisin (Plenum Press, New York, 1980) p. 141.
Journal of Magnetism and Magnetic Materials 40 (1984) 287-292 North-Holland, Amsterdam
MAGNETIC EXCITATIONS IN TmCu P. MORIN, D. SCHMITT Laboratoire Louis N$el, CNRS 166X, 38042 Grenoble-c$dex, France
and C. VETTIER Institut Laue- Langevin, 156X, 38042 Grenoble-c~dex, France
Received 13 June 1983; in revised form 30 August 1983
The magnetic excitations have been studied in the rare-earth intermetallic compound TmCu by means of inelastic neutron scattering. Dispersion curves have been followed through the cubic Brillouln zone, in particular an intense and dispersive transverse branch: this latter presents a lifting of the degeneracy along particular directions. Within a generalized susceptibility formalism, this is related to the coupling of the excitations at q and q + Q, where Q = ( n / a , 'rr/a, 0) is the antiferromagnetic propagation vector in TmCu. The inter-ionic isotropic bilinear exchange parameters are deduced and compared with those of isomorphous compounds.
1. Introduction
Among the cubic rare-earth intermetallic compounds, those with the thulium ion have received a particular attention over the last few years. This is motivated by the competition in such compounds between rather weak Heisenberg interactions and possibly large quadrupolar interactions, which explained the occurrence of the quadrupole phase transition above the magnetic ordering temperature, observed in TmZn [1] (To = 8.6 K, T~= 8.1 K) and in TmCd [2] (To = 3.2 K, no magnetic ordering). This favourable situation arises because of the combination of a small de Gennes factor and strong orbital effects for the thulium ion [3]. In addition, the large two-ion quadrupolar interactions observed in the above cubic (CsCl-type structure) compounds seem to be mediated by the conduction electrons [4] similar to the usual isotropic bilinear R u d e r m a n - K i t t e l - K a s u y a Yoshida-type (RKKY) interactions. In TmCu the quadrupole interactions, although weaker than in TmZn and TmCd, are still noticeable, as evidenced by various experiments performed in the paramagnetic phase [5]. On the other hand, due to the different number of con-
duction electrons [6], the ordering of the magnetic moments is not ferromagnetic as in TmZn, but antiferromagnetic at low temperature, with a propagation vector Q = 2~r/a(½, ½, 0) and a moment pointing along the [001] axis [7]. In addition, this structure is replaced between Tt = 6.7 K and the Ntel temperature Tr~ = 7.7 K by an amplitudemodulated structure with an incommensurate propagation vector Qm = 2~r/a(½ + 0.064, ½, 0). The two transitions are first order, the quadrupole interactions being responsible for the first-order character of the transition at Ts [8]. Below Tt, in agreement with the magnetic ordering, the 4f quadrupoles are fully ordered, driving a coherent tetragonal distortion ( c / a - 1 = - 7 × 10 -3) via the magnetoelastic coupling. The various experiments performed in the paramagnetic phase of TmCu have been consistently described within the molecular-field approximation by means of an Hamiltonian including the Crystalline Electric Field (CEF), the Zeeman coupling, the Heisenberg bilineax interactions and the quadrupolar (magnetoelastic and two-ion) interactions. The knowledge of all the involved parameters allows us to study the magnetic properties in the ordered phase, in particular the mag-
0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
288
1>. Morin et al. / Magnetic excitatwns in TmCu
netic excitations. Such experiments have already been performed in some isomorphous compounds, namely in the ferromagnets H o Z n [9] and TbZn [10], the antiferromagnet ErCu [11] and in the quadrupole system TmcLu~_cZn [12]. They have shown the long-range character of the isotropic bilinear interactions, as expected for a RKKY-type coupling. In section 2, the results of inelastic neutron scattering experiments on TmCu are presented. An interpretation is attempted in section 3.
6
T
F
s
="
~4 2
0 X
~
r
t
M
T
R
s
X
z
Mr
^
R
Fig. 1. Dispersion curves of the magnetic excitations in T m C u at 1.7 K; continuous lines are calculated from the best fit of the transverse (T) mode.
2. Experimental results 2.1. Experiment The TmCu single crystal was obtained by the Bridgman method, both components being sealed in stoichiometric proportions in a tantalum crucible. A 10 × 10 × 2 mm 3 platelet, cut perpendicularly to the [110] direction was used in order to investigate successively the (001) and (110) scattering planes. The inelastic neutron scattering experiment was carried out on the IN2 spectrometer of the ILL high flux reactor in Grenoble. The (002) reflection of pyrolytic graphite was used as monochromator and analyser. The experiment was operated in the constant Q-mode with a fixed incident energy of 14.7 meV and a resolution of 0.7 meV. A first set of measurements was performed with a (001) scattering plane in order to investigate the zone around the incommensurate propagation vector Qm at various temperatures. A second experiment was carded out with a (110) scattering plane for completing the dispersion curves at low temperature along the main symmetry directions of the cubic Brillouin zone.
excitations. Typical scans are shown in fig. 2. Particular attention was paid to the excitations at the vectors Q and Qm" However, the lowest energy mode being around 1 meV could not be resolved from the intense elastic line (magnetic Bragg peak) within our experimental conditions, and this prevented precise measurement of this low energy mode. A second weaker and much less dispersive branch exists at T = 1.7 K around an energy transfer of 3.8 meV; it does not seem to be split, the highest energy transfer occurring at F and X points (4.0 meV) and the lowest one at M point (3.6 meV). Some q-points have been also investigated above the N~el temperature: a single CEF transition is observed around an energy transfer of 1.6 meV a
TmCu T=1.7K r~
(o5 o o)
too ~
'X"
/
#1 (1,o~o~)1
.
(o.s.o5.ORS)
1
2
v
2.2. Results The dispersion curves obtained at T = 1.7 K are shown in fig. 1. An intense and dispersive branch was followed in the energy range 0.9-2.2 meV (10-25 K); it is split in the whole cubic Brillouin zone except along the [111] direction FR. A broad not-resolved branch is observed in the center of XM, suggesting the existence of more than two
/
A
~,o.2s.o~
'°°I l\ 1
2
3
4
1 2 3 t~ ENERGY (meV)
3
4
Fig. 2. Typical scans in T m C u at 1.7 K for several q-points; lines are least-squares fits.
289
P. Morin et al. / Magnetic excitations in TmCu
value a little lower than-ihe previous determination of 1.8 meV [5]. The magnetic Bragg reflections at Q and Qm have been investigated as a function of the temperature. The two first-order transitions at Tt = 6.7 K and TN = 7.7 K [7] have been confirmed. Between Tt and T, only the modulated phase exists on the single crystal, in contrast to the previous neutron diffraction experiment, in which both phases coexisted. This was because the polycrystal was not annealed (the same situation was found on TmAg [13]). On the other hand, using a single crystal allowed us to determine with accuracy the value of the incommensurate propagation vector Qm = 2'rr/a(½ + z, ½, 0) with z = 0.0640 + 0.0003. Third-order satellites have also been observed at q = 2,rr/a(½ + 3~', ½, 0) with an intensity ratio of 4 × 10 -3 with respect to the first-order satellite (no fifth-order satellites were detected within the experimental uncertainties): the presence of higher harmonics indicates that the modulation of the magnetic moments amplitude is not purely sinusoidal, which may be due to the non-linear dependence of the magnetic moment magnitude on the exchange field. Finally, a longitudinal elastic q-scan at T = 1.7 K around the nuclear Bragg peak 2~r/a(0, 2, 0) provided an estimation of the tetragonal distortion, c / a - 1 = - 8 × 10 -3, in good agreement with neutron diffraction on a polycrystal [7].
3. Interpretation
~ ( Bi1) _- -
The formalism describing the magnetic excitations originates from the generalized susceptibility theory, which is particularly well-adapted to rareearth compounds where CEF effects are strong [14]. Here, only isotropic bilinear interactions have been taken into account:
(1)
i j~i
the factor ½ being included because each site is counted twice in the summation. In the case o f TmCu, the magnetic moments are aligned along a
--
E J(iJ)Jz(i)CJz(J))
= -J(Q)Jz(i)CJz(i)).
(2)
An antiferromagnetic bihnear exchange coefficient 0~ may be related to the Fourier transform of the coupling coefficients in a way analogous to the paramagnetic exchange coefficient 0* [14] (which is related to J(0)): J(J+
1) =J(Q)= ~" J(/j) e'Q"q j~i
(3)
In addition to the above bilinear interaction, the total one-ion Hamiltonian includes the CEF Hamiltonian ~CEF and the quadrupolar Hamiltonian &C'Q.According to the cubic symmetry.,~cev is expressed in the fourfold axes system as [15]: EF
~--"
WvX( OO+ 5o: )
i
+ W(l"lxl----~)(O° 21064)r
(4)
The quadrupolar Hamiltonian HQ contains a magnetoelastic and a two-ion part, which are isomorphous in the molecular field approximation [8]: ~i)=
3.1. Theory
= - ½ E E J(o)J(O .J(y),
fourfold axis below Tt, e.g. the z-axis, and two sublattices A and B have to be considered, with (J~(B)) =-(J~(A)). Therefore, applying the molecular field approximation leads to the following one-ion bilinear Hamiltonian ~g's,:
_ G1COO(i))oO(i).
(5)
It is worth noting that, for the quadrupoles, no distinction between the two sublattices A and B is needed for TmCu, since (O°(A)) = COO(B)). The self-consistent diagonalization of the total one-ion Hamiltonian yields the eigenvalues o~,, and the CEF eigenfunctions In(i)) which depend on the sublattice. However, their expansion coefficients in the usual fJ, Ms) basis may be easily related from symmetry considerations:
CJ, MjIn(B)) = CJ , - MjIn(A)),
(6)
from which we obtain the following relations between the matrix elements of the operators J,
290
P. Morin et al. / Magnetic excitations in TmCu
(a = z, +, - ) for both sublattices:
with
J~m,(A) = ( m ( A ) I J A n ( A ) ) = - J , , , ( B ) ,
:±
mn ( A ) =
(7)
( m (A)IJ±ln (A)) ----J:r m. (B).
D(q,o~) = [1 + ¼(J(q) + J(q + Q ) ) g A - ] X
Considering the inter-ion Hamiltonian leads, within the random phase approximation, to the following expression for the Fourier transform of the generalized susceptibilities G~a(q,~0) [16]:
) = g B(
)_
½S(q ) ggz( c : +
¼(J(q) +J(q + Q ) ) g ~ - ]
-~(J(q)-J(q+Q))2gl-g~-.
3.2. Generalized susceptibifity formalism
G;e(q,
[1 +
)
(13)
It is worth noting that only q appears in eq. (11), while the excitations at q and q + Q are coupled for the transverse modes [11,17] with the same energies given by the condition D(q,~0)= D(q + Q,~0) = 0.
3.3. Analysis of the data
½J(q + Q )g~( Gff~- G~/J) ¼S(q)[g~,+(GT, 1~+
GB" )
-¼J(q + Q )[ g~+ ( G~ ~ - G~ B) +g~-(G;/~-G~")]
(8)
and the same relation with A ~ B. This expression includes single-ion susceptibilities:
g~'(o~) = ~ J~m"(A)Ja"m(A)(fm - f " ) mn
tO --
(9)
O ) n Jr" 0 2 m
which are the solutions of the equations of motion without interactions (fro are the Boltzman factors of the CEF level ]m)). Because of the relations (6) and of the selection rules on the matrix elements (eq. (7)), the only non-zero single-ion susceptibilities are for TmCu: g~[(~o) = gf3Z(~o), g~ ~:(~o) = g~ ±(~o).
The coefficients involved in the one-ion Hamiltonian have been previously obtained [5] and constitute the starting point of the present analysis: W = 1.4 K, x = -0.42, G 1 = 11 mK. In addition, an antiferromagnetic exchange parameter @~ = 13 K has been used to account for the average energy of the observed branches; it leads to a calculated N~el temperature of 9 K, slightly higher than the experimental one. The first dispersive branch is then identified as a transverse excitation involving the first excited CEF level and the second branch around 3.8 meV as a longitudinal one involving the third excited level. The ratio of the square of the corresponding matrix elements is about 3. As seen in section 3.2 the transverse excitations for q and q + Q are coupled and have the same energy. Thus, in the cubic Brillouin zone (fig. 3) the points M = (0.5, 0.5, 0) and M' = (0.5, 0, 0.5) (in 2~r/a units) are not longer equivalent since F is
(10)
This allows one to decouple the magnetic excitation modes into one longitudinal mode M'
Gff(q,~) = G~Z(q,o~)=
g~'Z(°~) 1 + g~:(~)J(q)
Z°
and two transverse modes: G ~ ~ (q,o~ ) = G ~ ±(q,~o )
+ ½g¢
J(q + O)]
D(q,a})
"
(11)
(12)
I
;y
Z M Fig. 3. Cubic Brillouin zone with notation associated with the tetragonal magnetic symmetry.
P. Morin et al. / Magnetic excitations in TmCu
291
Table 1 Isotropic bilinear exchange coefficients J(ij) with neighbours defined by rq = (h, k, l)a in TmCu (positive values mean ferromagnetic coupling): also given are the values for ErCu [11], normalized to Tm by the ratio of ( g j - 1 ) 2 for both rare-earths hk l
J(ij) (mK)
TmCu ErCu
100
110
111
200
210
211
220
- 199.0 - 28.8
- 20.5 - 8.6
18.4 30.4
33.2 20.1
- 7.5 - 1.4
9.2 - 8.5
- 7.9 - 0.4
coupled with M but not with M'. Keeping the cubic Brillouin zone for the dispersion curves (fig. 1), this allows one to identify the points M and M' among the two observed excitations: having the same energy transfer as F the highest mode corresponds to M. In the same way the point X' is identified as having the same energy transfer as R. In addition some branches are verified to be identical (FX' and MR), or symmetrical with respect to their centre (I'M or RX'). The inter-ion coupling coefficients J(/j) have been obtained by a least-squares procedure. At least 7 neighbours (see table 1) are necessary for describing the splitting at X- and M-points. Taking up to 10 neighbours does not improve significantly the fit; in particular, the calculated energy at M' is always a little higher than the experimental one. On the other hand, calculating the longitudinal excitation energies with the above coefficients leads to a good agreement with experiment except at R. Compared with the results on the isomorphous compound ErCu [11], the relative magnitude of successive J(ij)'s is very different (see table 1), this cannot be accounted for by a simple R K K Y theory, and suggests that the band structures for both compounds differ, perhaps because of the larger distortion in TmCu ( - 8 x 1 0 -3) than in ErCu (less than 10-3). Some disagreements appear when relating J(q) with the static properties. Indeed, if O~ is well-described starting from J(M) (O~°,,o= 10 K), this is not the case of the paramagnetic exchange coefficient O*~lc.= - 1 6 K, whereas the experimental value is - 3K [5]. In addition, the value of J(R) is larger than that of J(M) leading to an antiferromagnetic ordering with Q = R a t T N ~ 18 K. This situation remains for all the fits we have been made. Considering the existence of anisotropic bilin-
ear interactions, as in TbP [18], does not seem to be able to improve the fits. Indeed, these anisotropic interactions would lead to an additional lifting of the degeneracy along the directions FX and FM' [16], which was not observed experimentally. In summary, the inelastic neutron scattering which has been performed on TmCu has clearly shown the lifting of the degeneracy of a transverse magnetic excitation along particular directions of the cubic Brillouin zone. This has been well accounted for, at least qualitatively, by a generalized susceptibility formalism including only isotropic bilinear exchange interactions for an antiferromagnet. The degeneracy is raised by the coupling between the magnetic excitations at q and q + Q [17]. Quantitatively, some problems remain unexplained at this time, in particular why J(q) is maximum at q = R and not q = M, as expected for a ( ~ 0 ) - t y p e magnetic structure. Nevertheless, the deduced inter-ionic exchange parameters have a relative variation as a function of the neighbours, which is very different from those of the isomorphous compound ErCu. This could indicate that the bilinear exchange coupling J(q) is a strong specific property of each compound and that it appears to be useless to search for any scaling law for such pair interactions.
References [1] P. Morin, J. Rouchy and D. Schmitt, Phys. Rev. B17 (1978) 3684. [2] R. A16onard and P. Morin, Phys. Rev. B19 (1979) 3868. [3] D. Schmitt and P.M. Levy, J. Magn. Magn. Mat. 31-34 (1983) 249. [4] P.M. Levy, P. Morin and D. Schmitt, Phys. Rev. Lett. 42 (1979) 1417. [5] C. Jaussaud, P. Morin and D. Schmitt, J. Magn. Magn. Mat. 22 (1980) 98.
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P. Morin et al. / Magnetic excitations in TmCu
[6] J. Sakurai, Y. Kubo, T. Kondo, J. Pierre and E.F. Bertaut, J. Phys. Chem. Solids 34 (1973) 1305. [7] P. Morin and D. Schmitt, J. Magn. Magn. Mat. 21 (1980) 243. [8] P. Morin and D. Schmitt, Phys. Rev. B27 (1983) 4412. [9] J. Pierre, P. Morin, D. Schmitt and B. Hennion, J. Phys. F 7 (1977) 1965. [10] Y. Hamaguchi, H. Betsuyaku and S. Funahashi, J. Magn. Magn. Mat. 15-18 (1980) 377. [11] J. Pierre, P. Morin, D. Schmitt and B. Hennion, J. de Phys. 39 (1978) 793. [12] P. Morin, D. Sehmitt and C. Vettier, J. Phys. F 11 (1981) 1487.
[13] P. Morin and D. Schmitt, J. Magn. Magn. Mat. 28 (1982) 188. [14] P. Morin, D. Schmitt, C. Vettier and J. Rossat-Mignod, J. Phys. F 10 (1980) 1575. [15] K.R. Lea, J.M. Leask and W.P. Wolf, J. Phys. Chem. Solids 23 (1962) 1381. [16] P. Morin and D. Schmitt, to be published. [17] J. Rossat-Mignod, D. Delac6te, J.M. Effantin, C. Vettier and O. Vogt, Physica 120B (1983) 163. [18] K. Knorr, A. Loidl and J.K. Kjems, in: Crystalline Electric Field and Structural Effects in f-Electron Systems, eds. J.E. Crow, R.P. Guertin and T.W. Mihalisin (Plenum Press, New York, 1980) p. 141.