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Inelastic neutron scattering of the mixed dimer YbDyBr3-9
Physica B 180 & 181 (1992) North-Holland
Inelastic M.A.
PHYSICA li:
206-208
neutron
Aebersold”,
H.U.
scattering Giidel”,
of the mixed dimer YbDyBrz
A. Furrer”
and H. Blank’
“Institut
fiir Anorganischr Chemw, Univrrsitiit Bern. CH-.3000 Bern 9, Switzerland hLuhoratorium fiir Neutronenstreuung. ETH Ziirich. CH-.$232 Villigen PSI. Switzerland ‘Institut Laue- Langwin. 38042 Grenoble Cedex. France
Magnetic excitations of the mixed crystal Cs,Yb, ,Dy,, ,Br,, were measured by high-resolution inelastic neutron scattering on the instrument INS at the ILL. Grenoble. The YbDyBr: dimers show magnetic excitations at 0.058 and 0.101 meV with clear-cut Q dependencies. This can be understood in terms of a simple theoretical model in which two Kramer’s doublets are coupled-by anisotropic exchange
1. Introduction
A lot of work has been done in order to get a better understanding of exchange interactions in dimers of transition metal (TM) ions, and only lately dimers of rare earth (RE) metal ions have been investigated [ 11. Whereas the exchange effects in TM dimers can be large, thus leading to considerable splittings of the energy levels in the ground state, the interactions between the well-shielded f-electrons of RE dimers are very small. We found values for the ground state splitting of 0.36 and 0.08 meV in the Yb,Br:and Dy,Br:~- dimers, respectively [2]. Here we report results obtained on the mixed dimer YbDyBrz by inelastic neutron scattering (INS). These dimers are conveniently prepared by doping ten percent Dy into Cs,Yb,Br,. The Yb,Br: dimers of the host lattice have only one magnetic excitation at 0.36 meV, which is well known and well characterised [l]. This leaves a very large spectral range for the possible observation of magnetic YbDyBrz excitations. Cs,Yb,Br,, crystallises in the space group R% [3]. The point symmetry of the RE ions is C,,, and the crystal-field ground state is a Kramer’s doublet for both YbZ’ and Dy’+. This ground state is separated from the next higher crystal-field level by more than 24meV for Yb” and by approximately 1.8 meV for Dyzi. In YbDyBrzthe trigonally distorted YbB$ and DyBrzoctahedra are sharing a common face. 2. Inelastic
neutron
scattering
The INS experiments were performed on the timeof-flight spectrometer INS at the Institut LaueLangevin, Grenoble. The energy of the incoming neutrons was fixed at E = 1.28 meV, giving rise to a resolution of 27 IJ-eV. The polycrystalline sample was filled under He atmosphere into a plate-like Al-container of dimension 4 x 40 x 45 mm’. Standard procedures were used for correction of the data. Figure 1 shows 1.8 K spectra for two selected scatOY21-4526/92/$05.00
0
1992 - Elsevier
Science
Publishers
0 = 0.27A-1
-0.2
-0.1
0.0 E[meVJ
0. I
0.2
Fig. 1. Energy spectra of neutrons scattered from Cs,Yb, ,Dy,, lBru for Q = 0.27 8, ’ (above) and Q = 0.94 ?X ’ (below) at 1.8 K. The lines are the result of a Gaussian least-squares fit.
tering vectors Q. There are no Yb,Brz excitations in this energy range, and the observed peaks can be attributed to the YbDyBri dimers. The transitions at 0.101 and 0.058 meV, on the energy loss side, show distinct Q dependencies. While the intensity of the high energy peak is decreasing with increasing Q, the low energy peak shows the opposite behaviour. Their intensity ratio is plotted as a function of Q in fig. 2. 3. Analysis The dimer interaction Hamiltonian of two ion5 with an effective spin S, = $ can be expressed in general form as [4]:
B.V. All rights
reserved
M. A. Aebersold
0.5c y i 0.0
I
0.0
0.2
et al. I Inelastic neutron scattering of YbDyBrl-
Y I 06
I 0.4
I 0.8
0 [A-l]
I
I
I
I
1.0
1.2
1.4
Fig. 2. Experimental (A, with estimated uncertainties) and calculated intensity ratio of the 0.058 and 0.101 meV peaks. For the calculations formula (13) of ref. [S] was used with the parameters J = IO.121 meV and r = -0.18.
if,, = -2J[rS,&
+ (1 - Irj)StS:]
appears that two transitions coincide in the 0.058 meV peak. This leaves three possible r values in the diagram of fig. 3, two of which can be eliminated by the following intensity considerations. The differential neutron cross-section for dimer transitions of this kind has been formulated in ref. [5]. Very distinct Q dependencies are expected for the three possible transitions between the dimer level, I,, TZ and IX in fig. 3. Using formulae (13)-( 16) of ref. [5] we get [6]: for I,
for
-r?:
rzc-fri
(1)
this notation we have the Heisenberg model for r = *l, the Ising model for r = 0 and the xy model for r = -0.5. Diagonalization of eq. (1) gives rise to the energy level diagram shown in fig. 3. In both the Heisenberg and the Ising limits we expect to observe only one INS transition. Our experimental spectra clearly show two peaks, and we can immediately eliminate the Heisenberg and Ising limits. The lower energy peak occurs at roughly half the energy of the higher energy peak. It thus
sin( QR)
In
I
207
I
-e3R” for
1
(3)
r, - ri
sin( QR) +F? 1 where F(Q) is the magnetic form factor and the factors in square brackets are so-called interference terms. The expressions eqs. (2)-(4) are plotted in fig. 4. The observed decrease of the 0.101 meV peak, with identifies it as the increasing Q, unambiguously Ir +-+I7 transition. This places the Ii level between the two others, and the broken line in fig. 3 indicates the only possible solution in the energy diagram. It corresponds to: J = 10.121 meV and r = -0.18. A complete data analysis gives a somewhat better fit for a positive sign of J, corresponding to a I3 ground level. A test of the above result is obtained from a quantitative comparison of the experimental and calculated
1
7
I Il,il> \
05
Fig. 3. Energy level diagram of an exchange-coupled of spin-! ions according to eq. (1) for .I = 1.
0
dimer
Fig. 4. Calculated eqs. (Z)-(4).
Q dependence
of intensities,
according
to
208
M. A. Aebencsold et al. I Inelastic neutron scattering
Q dependence of the intensity ratio of the two peaks. The cross-section for the various transitions was calculated with the above parameter values, using eq. (13) in ref. [5]. The comparison with the experiment is shown in fig. 2. Considering the fact that there is no adjustable parameter, the agreement is quite good. Another test is a comparison of the experimental and calculated ratio of the energy-loss to energy-gain intensities at 1.8 and 6 K. At Q = 0.7 A- ’ and T = 1.8 K the experimental ratio is 1.8 (calculated: 1.9) and at 6 K it is 1.1 (calculated: 1.1). These comparisons confirm that our model and the parameters are essentially correct. 4. Conclusion The ground state of the mixed dimer YbDyB$ can be described with the most simple theoretical model. Both ions have Kramer’s doublet ground states, and treating them as effective spin- 4 states with the general anisotropic Hamiltonian eq. (1) yields a good description of the energies and wave functions of the dimer levels. The exchange is strongly anisotropic. lying close to but clearly not at the Ising limit. It is interesting to compare our result with those obtained for homonuclear dimers Yb,Br:and
of YbDyBri
Dy,B$ [l, 21. Their ground state splittings of 0.36 and 0.08 meV, respectively, could be interpreted by a Heisenberg model. No evidence was found in either compound for more than one transition, which would indicate anisotropy. It makes intuitive sense that the J-value found for the mixed dimer lies between the values found for the pure dimers. Acknowledgement This work was financially National Science Foundation.
supported
by the Swiss
References
111 H.U. Giidel,
A. Furrer and H. Blank, Inorg. Chem. 29 (1990) 4081. [21A. Diinni, A. Furrer. H. Blank, A. Heidemann and H.U. Giidel, J. de Phys. 49 (1988) C8-1.513. Mater. Res. Bull. IS (31 G. Meyer and A. Schonemund. (1980) 89. E.R. Krausz and H. Blank. 141 A. Furrer. H.U. Gtidel. Phys. Rev. Lett. 64 (1990) 68. H.U. Giidel and J. Darriet, Less-Common 151 A. Furrer, Metals 111 ( 1985) 223. 161 There are the following printing errors in eqs. (13), (IS) and (16) of ref. [S]: a,, and a,,Z should be replaced by 0: and a:,, , respectively.
Inelastic M.A.
PHYSICA li:
206-208
neutron
Aebersold”,
H.U.
scattering Giidel”,
of the mixed dimer YbDyBrz
A. Furrer”
and H. Blank’
“Institut
fiir Anorganischr Chemw, Univrrsitiit Bern. CH-.3000 Bern 9, Switzerland hLuhoratorium fiir Neutronenstreuung. ETH Ziirich. CH-.$232 Villigen PSI. Switzerland ‘Institut Laue- Langwin. 38042 Grenoble Cedex. France
Magnetic excitations of the mixed crystal Cs,Yb, ,Dy,, ,Br,, were measured by high-resolution inelastic neutron scattering on the instrument INS at the ILL. Grenoble. The YbDyBr: dimers show magnetic excitations at 0.058 and 0.101 meV with clear-cut Q dependencies. This can be understood in terms of a simple theoretical model in which two Kramer’s doublets are coupled-by anisotropic exchange
1. Introduction
A lot of work has been done in order to get a better understanding of exchange interactions in dimers of transition metal (TM) ions, and only lately dimers of rare earth (RE) metal ions have been investigated [ 11. Whereas the exchange effects in TM dimers can be large, thus leading to considerable splittings of the energy levels in the ground state, the interactions between the well-shielded f-electrons of RE dimers are very small. We found values for the ground state splitting of 0.36 and 0.08 meV in the Yb,Br:and Dy,Br:~- dimers, respectively [2]. Here we report results obtained on the mixed dimer YbDyBrz by inelastic neutron scattering (INS). These dimers are conveniently prepared by doping ten percent Dy into Cs,Yb,Br,. The Yb,Br: dimers of the host lattice have only one magnetic excitation at 0.36 meV, which is well known and well characterised [l]. This leaves a very large spectral range for the possible observation of magnetic YbDyBrz excitations. Cs,Yb,Br,, crystallises in the space group R% [3]. The point symmetry of the RE ions is C,,, and the crystal-field ground state is a Kramer’s doublet for both YbZ’ and Dy’+. This ground state is separated from the next higher crystal-field level by more than 24meV for Yb” and by approximately 1.8 meV for Dyzi. In YbDyBrzthe trigonally distorted YbB$ and DyBrzoctahedra are sharing a common face. 2. Inelastic
neutron
scattering
The INS experiments were performed on the timeof-flight spectrometer INS at the Institut LaueLangevin, Grenoble. The energy of the incoming neutrons was fixed at E = 1.28 meV, giving rise to a resolution of 27 IJ-eV. The polycrystalline sample was filled under He atmosphere into a plate-like Al-container of dimension 4 x 40 x 45 mm’. Standard procedures were used for correction of the data. Figure 1 shows 1.8 K spectra for two selected scatOY21-4526/92/$05.00
0
1992 - Elsevier
Science
Publishers
0 = 0.27A-1
-0.2
-0.1
0.0 E[meVJ
0. I
0.2
Fig. 1. Energy spectra of neutrons scattered from Cs,Yb, ,Dy,, lBru for Q = 0.27 8, ’ (above) and Q = 0.94 ?X ’ (below) at 1.8 K. The lines are the result of a Gaussian least-squares fit.
tering vectors Q. There are no Yb,Brz excitations in this energy range, and the observed peaks can be attributed to the YbDyBri dimers. The transitions at 0.101 and 0.058 meV, on the energy loss side, show distinct Q dependencies. While the intensity of the high energy peak is decreasing with increasing Q, the low energy peak shows the opposite behaviour. Their intensity ratio is plotted as a function of Q in fig. 2. 3. Analysis The dimer interaction Hamiltonian of two ion5 with an effective spin S, = $ can be expressed in general form as [4]:
B.V. All rights
reserved
M. A. Aebersold
0.5c y i 0.0
I
0.0
0.2
et al. I Inelastic neutron scattering of YbDyBrl-
Y I 06
I 0.4
I 0.8
0 [A-l]
I
I
I
I
1.0
1.2
1.4
Fig. 2. Experimental (A, with estimated uncertainties) and calculated intensity ratio of the 0.058 and 0.101 meV peaks. For the calculations formula (13) of ref. [S] was used with the parameters J = IO.121 meV and r = -0.18.
if,, = -2J[rS,&
+ (1 - Irj)StS:]
appears that two transitions coincide in the 0.058 meV peak. This leaves three possible r values in the diagram of fig. 3, two of which can be eliminated by the following intensity considerations. The differential neutron cross-section for dimer transitions of this kind has been formulated in ref. [5]. Very distinct Q dependencies are expected for the three possible transitions between the dimer level, I,, TZ and IX in fig. 3. Using formulae (13)-( 16) of ref. [5] we get [6]: for I,
for
-r?:
rzc-fri
(1)
this notation we have the Heisenberg model for r = *l, the Ising model for r = 0 and the xy model for r = -0.5. Diagonalization of eq. (1) gives rise to the energy level diagram shown in fig. 3. In both the Heisenberg and the Ising limits we expect to observe only one INS transition. Our experimental spectra clearly show two peaks, and we can immediately eliminate the Heisenberg and Ising limits. The lower energy peak occurs at roughly half the energy of the higher energy peak. It thus
sin( QR)
In
I
207
I
-e3R” for
1
(3)
r, - ri
sin( QR) +F? 1 where F(Q) is the magnetic form factor and the factors in square brackets are so-called interference terms. The expressions eqs. (2)-(4) are plotted in fig. 4. The observed decrease of the 0.101 meV peak, with identifies it as the increasing Q, unambiguously Ir +-+I7 transition. This places the Ii level between the two others, and the broken line in fig. 3 indicates the only possible solution in the energy diagram. It corresponds to: J = 10.121 meV and r = -0.18. A complete data analysis gives a somewhat better fit for a positive sign of J, corresponding to a I3 ground level. A test of the above result is obtained from a quantitative comparison of the experimental and calculated
1
7
I Il,il> \
05
Fig. 3. Energy level diagram of an exchange-coupled of spin-! ions according to eq. (1) for .I = 1.
0
dimer
Fig. 4. Calculated eqs. (Z)-(4).
Q dependence
of intensities,
according
to
208
M. A. Aebencsold et al. I Inelastic neutron scattering
Q dependence of the intensity ratio of the two peaks. The cross-section for the various transitions was calculated with the above parameter values, using eq. (13) in ref. [5]. The comparison with the experiment is shown in fig. 2. Considering the fact that there is no adjustable parameter, the agreement is quite good. Another test is a comparison of the experimental and calculated ratio of the energy-loss to energy-gain intensities at 1.8 and 6 K. At Q = 0.7 A- ’ and T = 1.8 K the experimental ratio is 1.8 (calculated: 1.9) and at 6 K it is 1.1 (calculated: 1.1). These comparisons confirm that our model and the parameters are essentially correct. 4. Conclusion The ground state of the mixed dimer YbDyB$ can be described with the most simple theoretical model. Both ions have Kramer’s doublet ground states, and treating them as effective spin- 4 states with the general anisotropic Hamiltonian eq. (1) yields a good description of the energies and wave functions of the dimer levels. The exchange is strongly anisotropic. lying close to but clearly not at the Ising limit. It is interesting to compare our result with those obtained for homonuclear dimers Yb,Br:and
of YbDyBri
Dy,B$ [l, 21. Their ground state splittings of 0.36 and 0.08 meV, respectively, could be interpreted by a Heisenberg model. No evidence was found in either compound for more than one transition, which would indicate anisotropy. It makes intuitive sense that the J-value found for the mixed dimer lies between the values found for the pure dimers. Acknowledgement This work was financially National Science Foundation.
supported
by the Swiss
References
111 H.U. Giidel,
A. Furrer and H. Blank, Inorg. Chem. 29 (1990) 4081. [21A. Diinni, A. Furrer. H. Blank, A. Heidemann and H.U. Giidel, J. de Phys. 49 (1988) C8-1.513. Mater. Res. Bull. IS (31 G. Meyer and A. Schonemund. (1980) 89. E.R. Krausz and H. Blank. 141 A. Furrer. H.U. Gtidel. Phys. Rev. Lett. 64 (1990) 68. H.U. Giidel and J. Darriet, Less-Common 151 A. Furrer, Metals 111 ( 1985) 223. 161 There are the following printing errors in eqs. (13), (IS) and (16) of ref. [S]: a,, and a,,Z should be replaced by 0: and a:,, , respectively.