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Coupled electron-nuclear magnetism and neutron diffraction
Physica 136B (1986) 379-384 North-Holland, Amsterdam
Chapter 13 Magneticordering COUPLED ELECTRON-NUCLEAR MAGNETISM AND NEUTRON DIFFRACTION S. KAWARAZAKI Department of Physics, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan
and J. A R T H U R Solid State Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Invited paper It often occurs that, in a rare-earth compound having a singlet electronic ground state, a long-range magnetically ordered state is formed by coupling of the nuclear spins with the electrons through the hyperfine interaction, at millikelvin temperatures. Neutron scattering has been used to measure the details of the magnetization processes of both the electrons and the nuclear spins in the coupled state in PrSn3, HoVO4, and PrCu 2. The polarized nuclear spins could be observed through spin dependent nuclear scattering by 14~pr an 165H0. The experimental results for PrSn 3 and HoVO 4 can be explained very well based on mean field considerations. For PrCuz, a recent experiment indicates that a sinusoidal screw structure is realized simultaneously both in the electronic and nuclear spin systems of this material. Such a magnetic structure requires a more refined theoretical explanation.
I. Introduction
the magnetic scattering length due to the atomic magnetic moment Pe is
In addition to being a powerful tool for studying electronic magnetism, neutron diffraction can be used to study the magnetism of nuclear spin systems. The spin-dependent nuclear scattering length is expressed in a quantum operator form as
[1] b+-b#n
--
-
(I. or) = 0.27(/**. o'),
2 I- + 1
(1)
where 1
0.27
b+-b _
_
21+1
-
1
(2)
and b + and b - are the nuclear scattering lengths of the neutron-nucleus coupled states with total spin 1 + ½ and I - ½, respectively, or is the Pauli operator of neutron spin. The quantity /~* is called the pseudo-nuclear moment and is a convenient measure of the magnitude of/~,. Eq. (2) gives /~n* in Bohr magnetons if b + and b - are measured in units of 10 -12 cm. For comparison,
=
•
(3)
where Pez is the component of pe perpendicular to the scattering vector K, and f0¢) is the form factor of the magnetic electrons. Unlike p,, Pe depends on K through the factors f(•) and/x~_L, and this makes it possible to measure ~*n a n d / ~ separately even when both of them coexist. This capability of neutron scattering is especially useful for the study of hyperfine-coupled electronnuclear magnetism in singlet electronic ground state compounds. In this paper, we describe the results of our neutron experiments in this field. Since descriptions of the first sequence of experiments with PrSn 3 [2], PrCu 2 [3], and HoVO 4 [4] have already been presented, we give only a brief summary of the results of those experiments in this section, and spend the following sections describing the recent new experiments with PrCu 2. In singlet electronic ground state compounds,
0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
380
S. Kawarazaki and J. Arthur / Coupled electron-nuclear magnetism
the electronic system will undergo a spontaneous ordering only if the magnitude of the magnetic interaction of the electrons (K) exceeds a certain threshold value (D). However, even when K < D, the existence of the nuclear spin and the hyperfine interaction can give rise to a phase transition which involves simultaneous development of polarization in both the electronic and the nuclear systems. The descriptions of the phase transitions in the extreme cases of K>> D or K ~ D are rather simple [5]. For K>> D, the electronic system undergoes a self-induced ordering at relatively high temperature and the nuclear spins polarize gradually at lower temperatures in the hyperfine field, maintaining a thermal distribution. In fig. 1, the peak intensity of Bragg scattering due to the polarized nuclear spins of 141pr in PrSn 3 is shown as a function of temperature. Cubic PrSn 3 undergoes an antiferromagnetic ordering at 8.6 K. The magnetic propagation vector is (1 0 0) and the moment is parallel to this vector. In this magnetic structure, at (1 + h 0 0) positions in reciprocal space Pe vanishes because #e± in eq. (3) is zero. Thus, at these positions,
only scattering due to the polarized nuclear spins is observed. This is an ideal case in which one can separately observe the nuclear polarization in a state where it is coupled with electronic polarization. This experiment determined the value of /z*n for 14]pr to be/~n* = -0.17-+ 0.009 (/ZB). For K ~ D, magnetic ordering is well described as a spontaneous polarization of the sublevels which are characteristic of the nuclear spin states. The mean field theory predicts that the magnetization of the electron system and that of the nuclear system occur simultaneously with similar temperature dependences, and behave as an ordinary second order phase transition [5]. The antiferromagnetic ordering that occurs in HoVO 4 at 4.5 inK, first observed in a susceptibility experiment [6], is believed to be an example of the case K ~ D. Bleaney has considered this phase transition in terms of the dipole-dipole interaction of the electrons and predicted a possible magnetic structure. In a neutron diffraction experiment carried out at temperatures down to 2.7 mK [4], several diffraction peaks were observed at positions in reciprocal space which are consistent ORNL-DWG 8 5 - 1 3 2 9 9
20
~3TT
1
I 1 1 rill
T
I
I 1 [llT
l
-
I -1
17 I I I U
/
e~--,~... ! I
o
-"
X
PrSn 3
(3/20 O)
E tD 03
k z O O
z I0 (:3 EE
l--
142K+
I.aJ Z
I i lll
I
I01
I
I kLIII
_~_
I
I I II11
I0 ?
I__J
lO 3
I I III1
10 4
TEMPERATURE (mK) Fig. 1. Temperature dependence of the purely nuclear magnetic ( ~ 0 0) peak intensity of PrSn 3. The solid curve represents the square of the magnitude of the nuclear spin polarization predicted by the Boltzmann population function.
S. Kawarazaki and J. Arthur / Coupled electron-nuclear magnetism
with this magnetic structure. Fig. 2 shows that the external magnetic field dependences of the Bragg intensities of the (100) and (001) peak across the field-induced spin flop transition are also consistent with the predictions. Evidence for the existence of nuclear polarization in this material was found by measuring the ratio of the integrated intensities of the (001) and (009) peaks. The electronic contribution to the (009) reflection is much less than to the (001) due to the reduction in the form factor, while the nuclear contribution is the same in both reflections. The measured ratio was in good agreement with calculations based on the known/~ n*of the Ho nucleus and the magnitude of the electronic polarization estimated by Bleaney. For K ~ D, the molecular field theory predicts that the magnetization processes of the nuclear and electronic systems are not similar and depend very sensitively on small changes in the value of K. Near the transition temperature, the electronic polarization changes much less rapidly as a function of temperature than in the cases where K>>D or K ~ D. Two materials, dhcp Pr and PrCu2, are believed to belong to this category. Many neutron experiments have been made with dhcp Pr [8], and it has been shown that the Bragg intensity due to the long range magnetic order in
I
o ,\
--Z,
,',___ / o- ""-o
/
Z D 0
( '.
.
z 0
~
°" o
HS. T
~3 LIJ Z
x, • bg ", . . . . . . . . . . . . . . . . .
_
~
H
o,
I ~ _ _
o "'"o.
I 100
d
200 EXTERNAL
this material has a peculiar temperature dependence. Although this ordering is believed to be due to coupled nuclei and electrons, there has been as yet no definitive experimental evidence of a development of nuclear polarization in this material. PrCu 2 has an orthorhombic CeCuz-type structure with space group Imma. A susceptibility experiment has indicated an antiferromagnetic ordering below 54 mK [9]. In our previous neutron experiment [3], several antiferromagnetic Bragg reflections were observed below 58 mK in the a*-c* reciprocal plane, the magnetic propagation vector being ~"= 0.24 a* + 0.68c*. The existence of nuclear polarization in this material was, however, not proven. It is of interest to separate the nuclear and electronic polarization processes in materials like dhcp Pr and PrCu 2 and consider the results on the basis of the molecular field theory. In the following sections, we describe the results of a neutron diffraction experiment on PrCu 2 which was designed to elucidate the magnetic structure of the ordered phase. The details of the experimental data and of its analysis will appear elsewhere, together with the results of an attempt to separate the polarization processes of the nuclear and electronic systems.
2. Experiment
I"
o
x
381
i
300 FIELD (Oe}
o
o _-o o-o-o-o-o-o~-o-o-o-o~ _
_
_o
i
i
ZOO
500
_
Fig. 2. Peak intensities of the (100) and (001) magnetic Bragg scattering in HoVO 4 measured at 3.5 mK as a function of the external magnetic field, across the field-induced spin-flop transition. The insert is the behavior expected from the model magnetic structure given by Bleaney.
A new single crystal of P r C u 2 was made by the method described in ref. 3; however, a Z r O 2 crucible was used instead of an A I 2 0 3 crucible. Although the Z r O 2 crucible was more reactive with the melt of the alloy than the A I 2 0 3 crucible, the quality of the single crystal obtained was better than the quality of the previous one. Possible contamination by Zr, Ca and Si from the crucible was checked by the method of arc emission spectroscopy, and it was confirmed that the concentrations of these elements were all less than 100ppm. The crystal was mounted in a helium dilution refrigerator on the HB1 neutron spectrometer at the HFIR at Oak Ridge, and aligned with the a*-c* reciprocal plane as the horizontal scattering plane.
382
S. Kawarazaki and J. Arthur / Coupled electron-nuclear magnetism
3. Results and discussion
3.1. Transition temperature
The temperature dependence of the intensity of the magnetic Bragg peak (0.76, 0, 1.68) is shown in fig. 3. The transition temperature, T N, deduced from these data is 38.5 ± 0.5 mK. This is much lower than TN measured in the previous experiment using a different crystal, which was 58 mK. However, the magnetic structure- that is, the locations of the magnetic Bragg peaks in reciprocal space - has not changed at all. As mentioned in section 2, we have checked for impurity contamination of the crystal, but since we have not checked for all possible contaminations, there remains a slight possibility that the change in Tn is due to an impurity effect. However, as is well known from the studies of dhcp Pr, the magnetism of singlet electronic ground state compounds with K ~ D is very sensitive to the quality of the crystal and to stress in the crystal. There is a good possibility that the two crystals have differing amounts of lattice defects and/or residual stress caused by friction with the crucible wall while they were growing, and that this causes the difference in T N .
deduce the direction of polarization. For this purpose, the most useful technique is polarization analysis of the scattered neutrons [1]. Under the assumption that ~ ] ] I , one can determine the direction/.t~± and I relative to the direction of the neutron polarization by measuring the flipping ratio of the scattered neutrons. In fig. 4, a rocking curve of the magnetic peak (1.24, 0, 0.32) is shown, taken with the spectrometer's spin flipper on (yielding the spin-flip scattering by the sample) and off (giving the nonspin-flip scattering). The polarization of the neutrons was vertical. Correcting for the imperfect polarization of the incident neutrons, we obtain a value for the ratio of the spin-flip scattering to the total scattering of 1.0 ± 0.03. We therefore conclude that the magnetic polarization lies in the a*-c* reciprocal plane. 3.3. Intensity analysis
Since the polarization is confined to the a*-c* plane, the modulation of the polarization can be generally written as P = P~e, cos(~" • r i + 4 ' i ) + P~et3 sin(7 • r i +
~bi), (4)
3.2. Polarization analysis z,O0-
Once the fundamental magnetic propagation vector has been determined, the next task is to
18 mK °~o /~f~
FLIPPER -o- ON
d~ ~1 .24 0.32
c 3000 0
PrCu2
80 \. R60 0
c
( Q76,0,168 )
o
200-
/
\
•
i]1 C U
"OO •
o, O
•
o
\
~2o
15' \
o Z I
0
O
z 100-
\
~0
o
0
10
30 40 Temperature ( mK )
50
Fig. 3. Temperature dependence of the (0.76, 0, 1.68) peak intensity of PrCu 2. The broken line is a guide to the eye.
16' Scattering
1'6 1~7 Angle(deg
19' )
Fig. 4. Spin-flip scattering (flipper on) and nonspin-flip scattering (off) measured at the (1.24, 0, 0.32) magnetic Bragg position of PrCu 2. The conclusion that all scattering is spin-flip scattering indicates the magnetic moments are aligned in the a*-c* plane.
383
S. Kawarazald and J. Arthur / Coupled electron-nuclear magnetism
where e, and e~ are the unit vectors of a Cartesian coordinate system in the a*-c* plane. If one takes the orthorhombic crystal structure of PrCu 2 into account, one can reasonably expect the magnetism of this material to be highly uniaxially anisotropic. Thus, we first attempted a leastsquare fit of the integrated intensities of 19 peaks assuming a uniaxial polarization (sinusoidal structure). For the direction of G , we considered the cases e~lla*, e, llc*, eoll~" and e~±z. After correction for the Lorentz factor, the integrated intensity of the nth magnetic peak is written
--~2rr2 2 2 2 In = ~ r c ~ f ~ q ~ x - 2f~q.x + 1} ,
T II \,
/
\
(a)
\
/
/" ~.\ /
\/ //
'\
~
~\ \
~
. . . . . . . . .
(s)
where f is the form factor and F G is the structure factor of the reciprocal lattice point to which the magnetic peak belongs, q is the ratio/Zex//Ze, and is equal to sin 0, cos 0, sin ~, or cos 5, respectively, for eolla*, e.llc*, e.ll¢, or e, ±~', where 0 and are the angles between K and a* and between K and ~-, respectively. The normalization factor A and the relative magnitude of the electronic polarization to the nuclear polarization, x-ItG(To)/tZ*n(To) ], at the temperature To where the data were taken are the fitting parameters for each q. We found that the fit obtained with q = sin 0 was significantly better than with the other choices for q. (We intend to publish the details of our analysis elsewhere.) This result indicates that the sinusoidal structure with polarization parallel to the a* axis is the most likely structure. We then tested the effect of a superposition of polarization parallel to the c* axis on this structure and found no improvement in the fit. The polarization parallel to the a* axis indicated by the present experiment is quite consistent with the result of an inelastic neutron experiment using PrCu 2 at approximately 5 K by Kjems et al. [10]. They observed an intense crystal field transition along the b* and c* axes, and the absence of this transition along the a* axis uniquely defined it as a J~ transition. The best fit value of the parameter x is 3.22. If one assumes that this value of x does not change down to 0 K, then the saturated electronic moment is calculated to be/ze(0) = 3.22/z*(0) = 0.55
(b) Fig. 5. Possible modulation structures of nuclear spins in PrCuv a) Nuclear spins are modulated sinusoidally with paramagnetic motion which persists down to 0 K. b) Nuclear spins align in an antiphase domain in the hyperfine field. /xB. This value needs to be explained theoretically in connection with the transition temperature. These results show that in PrCu 2 the electronic and nuclear polarizations develop simultaneously below TN. However, more work is necessary to finally determine the magnetic structure. In a sinusoidal structure the polarization has different magnitudes on different atoms. This is possible for the electronic system because it is an induced moment system. On the other hand, the nuclear spins have to be accompanied with a paramagnetic motion of the Iy and I z components to be modulated sinusoidally (see fig. 5). This motion could be a quantum-zero-point motion if it persists down to 0 K. However, it seems more likely that all the nuclear spins align completely in the hyperfine field at low enough temperature. One should be able to determine the actual nuclear spin structure by measuring the (very weak) higher harmonics of the magnetic Bragg scattering.
Acknowledgements We wish to thank Dr. Pradeep Kumar and Dr.
384
S. Kawarazaki and J. Arthur / Coupled electron-nuclear magnetism
W.G. Stirling for helpful discussions of nuclear magnetism in P r C u 2. This research was carried out at the Oak Ridge National Laboratory under the U.S.-Japan Cooperative Program on Neutron Scattering and sponsored in part by the Division of Materials Sciences, U.S. Department of Energy under contract DE-AC05-840R214000 with Martin Marietta Energy Systems, Inc. References [1] W. Marshall and S.W. Lovesey, Thermal Neutron Scattering (Oxford Univ. Press, London, 1971). [2] S. Kawarazaki, N. Kunitomi, Y. Morii, H. Suzuki, R.M. Moon and R.M. Nicklow, Solid State Commun. 49 (1984) 1147.
[3] S. Kawarazaki, N. Kunitomi, R.M. Moon, R.M. Nicklow and H. Suzuki, Solid State Commun. 49 (1984) 1153. [4] H. Suzuki, T. Otsuka, S. Kawarazaki, N. Kunitomi, R.M. Moon and R.M. Nicklow, Solid State Commun. 49 (1984) 1157. [5] T. Murao, J. Phys. C16 (1983) 335. [6] H. Suzuki, N. Nambudripad, B. Bleaney, A.L. Allsop, G.J. Bowden, I.A. Campbell and N.J. Stone, J. Phys. (Paris) 39 (1978) C6-800. [7] B. Bleaney, Proc. Roy. Soc. London A370 (1980) 313. [8] K.A. McEwen, W.G. Stirling, and C. Vettier, in: Crystalline Electric Field Effects in f-Electron Magnetism, R.P. Guertin, W. Suski and Z. Zolnierek, eds. (Plenum, New York, 1982). [9] K. Andres, E. Bucher, J.P. Maita and A.S. Cooper, Phys. Rev. Lett. 28 (1972) 1652. [10] J.K. Kjems, H.R. Ott, S.M. Shapiro and K. Andres, J. de Physique C6 (1978) 1010.
Chapter 13 Magneticordering COUPLED ELECTRON-NUCLEAR MAGNETISM AND NEUTRON DIFFRACTION S. KAWARAZAKI Department of Physics, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan
and J. A R T H U R Solid State Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Invited paper It often occurs that, in a rare-earth compound having a singlet electronic ground state, a long-range magnetically ordered state is formed by coupling of the nuclear spins with the electrons through the hyperfine interaction, at millikelvin temperatures. Neutron scattering has been used to measure the details of the magnetization processes of both the electrons and the nuclear spins in the coupled state in PrSn3, HoVO4, and PrCu 2. The polarized nuclear spins could be observed through spin dependent nuclear scattering by 14~pr an 165H0. The experimental results for PrSn 3 and HoVO 4 can be explained very well based on mean field considerations. For PrCuz, a recent experiment indicates that a sinusoidal screw structure is realized simultaneously both in the electronic and nuclear spin systems of this material. Such a magnetic structure requires a more refined theoretical explanation.
I. Introduction
the magnetic scattering length due to the atomic magnetic moment Pe is
In addition to being a powerful tool for studying electronic magnetism, neutron diffraction can be used to study the magnetism of nuclear spin systems. The spin-dependent nuclear scattering length is expressed in a quantum operator form as
[1] b+-b#n
--
-
(I. or) = 0.27(/**. o'),
2 I- + 1
(1)
where 1
0.27
b+-b _
_
21+1
-
1
(2)
and b + and b - are the nuclear scattering lengths of the neutron-nucleus coupled states with total spin 1 + ½ and I - ½, respectively, or is the Pauli operator of neutron spin. The quantity /~* is called the pseudo-nuclear moment and is a convenient measure of the magnitude of/~,. Eq. (2) gives /~n* in Bohr magnetons if b + and b - are measured in units of 10 -12 cm. For comparison,
=
•
(3)
where Pez is the component of pe perpendicular to the scattering vector K, and f0¢) is the form factor of the magnetic electrons. Unlike p,, Pe depends on K through the factors f(•) and/x~_L, and this makes it possible to measure ~*n a n d / ~ separately even when both of them coexist. This capability of neutron scattering is especially useful for the study of hyperfine-coupled electronnuclear magnetism in singlet electronic ground state compounds. In this paper, we describe the results of our neutron experiments in this field. Since descriptions of the first sequence of experiments with PrSn 3 [2], PrCu 2 [3], and HoVO 4 [4] have already been presented, we give only a brief summary of the results of those experiments in this section, and spend the following sections describing the recent new experiments with PrCu 2. In singlet electronic ground state compounds,
0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
380
S. Kawarazaki and J. Arthur / Coupled electron-nuclear magnetism
the electronic system will undergo a spontaneous ordering only if the magnitude of the magnetic interaction of the electrons (K) exceeds a certain threshold value (D). However, even when K < D, the existence of the nuclear spin and the hyperfine interaction can give rise to a phase transition which involves simultaneous development of polarization in both the electronic and the nuclear systems. The descriptions of the phase transitions in the extreme cases of K>> D or K ~ D are rather simple [5]. For K>> D, the electronic system undergoes a self-induced ordering at relatively high temperature and the nuclear spins polarize gradually at lower temperatures in the hyperfine field, maintaining a thermal distribution. In fig. 1, the peak intensity of Bragg scattering due to the polarized nuclear spins of 141pr in PrSn 3 is shown as a function of temperature. Cubic PrSn 3 undergoes an antiferromagnetic ordering at 8.6 K. The magnetic propagation vector is (1 0 0) and the moment is parallel to this vector. In this magnetic structure, at (1 + h 0 0) positions in reciprocal space Pe vanishes because #e± in eq. (3) is zero. Thus, at these positions,
only scattering due to the polarized nuclear spins is observed. This is an ideal case in which one can separately observe the nuclear polarization in a state where it is coupled with electronic polarization. This experiment determined the value of /z*n for 14]pr to be/~n* = -0.17-+ 0.009 (/ZB). For K ~ D, magnetic ordering is well described as a spontaneous polarization of the sublevels which are characteristic of the nuclear spin states. The mean field theory predicts that the magnetization of the electron system and that of the nuclear system occur simultaneously with similar temperature dependences, and behave as an ordinary second order phase transition [5]. The antiferromagnetic ordering that occurs in HoVO 4 at 4.5 inK, first observed in a susceptibility experiment [6], is believed to be an example of the case K ~ D. Bleaney has considered this phase transition in terms of the dipole-dipole interaction of the electrons and predicted a possible magnetic structure. In a neutron diffraction experiment carried out at temperatures down to 2.7 mK [4], several diffraction peaks were observed at positions in reciprocal space which are consistent ORNL-DWG 8 5 - 1 3 2 9 9
20
~3TT
1
I 1 1 rill
T
I
I 1 [llT
l
-
I -1
17 I I I U
/
e~--,~... ! I
o
-"
X
PrSn 3
(3/20 O)
E tD 03
k z O O
z I0 (:3 EE
l--
142K+
I.aJ Z
I i lll
I
I01
I
I kLIII
_~_
I
I I II11
I0 ?
I__J
lO 3
I I III1
10 4
TEMPERATURE (mK) Fig. 1. Temperature dependence of the purely nuclear magnetic ( ~ 0 0) peak intensity of PrSn 3. The solid curve represents the square of the magnitude of the nuclear spin polarization predicted by the Boltzmann population function.
S. Kawarazaki and J. Arthur / Coupled electron-nuclear magnetism
with this magnetic structure. Fig. 2 shows that the external magnetic field dependences of the Bragg intensities of the (100) and (001) peak across the field-induced spin flop transition are also consistent with the predictions. Evidence for the existence of nuclear polarization in this material was found by measuring the ratio of the integrated intensities of the (001) and (009) peaks. The electronic contribution to the (009) reflection is much less than to the (001) due to the reduction in the form factor, while the nuclear contribution is the same in both reflections. The measured ratio was in good agreement with calculations based on the known/~ n*of the Ho nucleus and the magnitude of the electronic polarization estimated by Bleaney. For K ~ D, the molecular field theory predicts that the magnetization processes of the nuclear and electronic systems are not similar and depend very sensitively on small changes in the value of K. Near the transition temperature, the electronic polarization changes much less rapidly as a function of temperature than in the cases where K>>D or K ~ D. Two materials, dhcp Pr and PrCu2, are believed to belong to this category. Many neutron experiments have been made with dhcp Pr [8], and it has been shown that the Bragg intensity due to the long range magnetic order in
I
o ,\
--Z,
,',___ / o- ""-o
/
Z D 0
( '.
.
z 0
~
°" o
HS. T
~3 LIJ Z
x, • bg ", . . . . . . . . . . . . . . . . .
_
~
H
o,
I ~ _ _
o "'"o.
I 100
d
200 EXTERNAL
this material has a peculiar temperature dependence. Although this ordering is believed to be due to coupled nuclei and electrons, there has been as yet no definitive experimental evidence of a development of nuclear polarization in this material. PrCu 2 has an orthorhombic CeCuz-type structure with space group Imma. A susceptibility experiment has indicated an antiferromagnetic ordering below 54 mK [9]. In our previous neutron experiment [3], several antiferromagnetic Bragg reflections were observed below 58 mK in the a*-c* reciprocal plane, the magnetic propagation vector being ~"= 0.24 a* + 0.68c*. The existence of nuclear polarization in this material was, however, not proven. It is of interest to separate the nuclear and electronic polarization processes in materials like dhcp Pr and PrCu 2 and consider the results on the basis of the molecular field theory. In the following sections, we describe the results of a neutron diffraction experiment on PrCu 2 which was designed to elucidate the magnetic structure of the ordered phase. The details of the experimental data and of its analysis will appear elsewhere, together with the results of an attempt to separate the polarization processes of the nuclear and electronic systems.
2. Experiment
I"
o
x
381
i
300 FIELD (Oe}
o
o _-o o-o-o-o-o-o~-o-o-o-o~ _
_
_o
i
i
ZOO
500
_
Fig. 2. Peak intensities of the (100) and (001) magnetic Bragg scattering in HoVO 4 measured at 3.5 mK as a function of the external magnetic field, across the field-induced spin-flop transition. The insert is the behavior expected from the model magnetic structure given by Bleaney.
A new single crystal of P r C u 2 was made by the method described in ref. 3; however, a Z r O 2 crucible was used instead of an A I 2 0 3 crucible. Although the Z r O 2 crucible was more reactive with the melt of the alloy than the A I 2 0 3 crucible, the quality of the single crystal obtained was better than the quality of the previous one. Possible contamination by Zr, Ca and Si from the crucible was checked by the method of arc emission spectroscopy, and it was confirmed that the concentrations of these elements were all less than 100ppm. The crystal was mounted in a helium dilution refrigerator on the HB1 neutron spectrometer at the HFIR at Oak Ridge, and aligned with the a*-c* reciprocal plane as the horizontal scattering plane.
382
S. Kawarazaki and J. Arthur / Coupled electron-nuclear magnetism
3. Results and discussion
3.1. Transition temperature
The temperature dependence of the intensity of the magnetic Bragg peak (0.76, 0, 1.68) is shown in fig. 3. The transition temperature, T N, deduced from these data is 38.5 ± 0.5 mK. This is much lower than TN measured in the previous experiment using a different crystal, which was 58 mK. However, the magnetic structure- that is, the locations of the magnetic Bragg peaks in reciprocal space - has not changed at all. As mentioned in section 2, we have checked for impurity contamination of the crystal, but since we have not checked for all possible contaminations, there remains a slight possibility that the change in Tn is due to an impurity effect. However, as is well known from the studies of dhcp Pr, the magnetism of singlet electronic ground state compounds with K ~ D is very sensitive to the quality of the crystal and to stress in the crystal. There is a good possibility that the two crystals have differing amounts of lattice defects and/or residual stress caused by friction with the crucible wall while they were growing, and that this causes the difference in T N .
deduce the direction of polarization. For this purpose, the most useful technique is polarization analysis of the scattered neutrons [1]. Under the assumption that ~ ] ] I , one can determine the direction/.t~± and I relative to the direction of the neutron polarization by measuring the flipping ratio of the scattered neutrons. In fig. 4, a rocking curve of the magnetic peak (1.24, 0, 0.32) is shown, taken with the spectrometer's spin flipper on (yielding the spin-flip scattering by the sample) and off (giving the nonspin-flip scattering). The polarization of the neutrons was vertical. Correcting for the imperfect polarization of the incident neutrons, we obtain a value for the ratio of the spin-flip scattering to the total scattering of 1.0 ± 0.03. We therefore conclude that the magnetic polarization lies in the a*-c* reciprocal plane. 3.3. Intensity analysis
Since the polarization is confined to the a*-c* plane, the modulation of the polarization can be generally written as P = P~e, cos(~" • r i + 4 ' i ) + P~et3 sin(7 • r i +
~bi), (4)
3.2. Polarization analysis z,O0-
Once the fundamental magnetic propagation vector has been determined, the next task is to
18 mK °~o /~f~
FLIPPER -o- ON
d~ ~1 .24 0.32
c 3000 0
PrCu2
80 \. R60 0
c
( Q76,0,168 )
o
200-
/
\
•
i]1 C U
"OO •
o, O
•
o
\
~2o
15' \
o Z I
0
O
z 100-
\
~0
o
0
10
30 40 Temperature ( mK )
50
Fig. 3. Temperature dependence of the (0.76, 0, 1.68) peak intensity of PrCu 2. The broken line is a guide to the eye.
16' Scattering
1'6 1~7 Angle(deg
19' )
Fig. 4. Spin-flip scattering (flipper on) and nonspin-flip scattering (off) measured at the (1.24, 0, 0.32) magnetic Bragg position of PrCu 2. The conclusion that all scattering is spin-flip scattering indicates the magnetic moments are aligned in the a*-c* plane.
383
S. Kawarazald and J. Arthur / Coupled electron-nuclear magnetism
where e, and e~ are the unit vectors of a Cartesian coordinate system in the a*-c* plane. If one takes the orthorhombic crystal structure of PrCu 2 into account, one can reasonably expect the magnetism of this material to be highly uniaxially anisotropic. Thus, we first attempted a leastsquare fit of the integrated intensities of 19 peaks assuming a uniaxial polarization (sinusoidal structure). For the direction of G , we considered the cases e~lla*, e, llc*, eoll~" and e~±z. After correction for the Lorentz factor, the integrated intensity of the nth magnetic peak is written
--~2rr2 2 2 2 In = ~ r c ~ f ~ q ~ x - 2f~q.x + 1} ,
T II \,
/
\
(a)
\
/
/" ~.\ /
\/ //
'\
~
~\ \
~
. . . . . . . . .
(s)
where f is the form factor and F G is the structure factor of the reciprocal lattice point to which the magnetic peak belongs, q is the ratio/Zex//Ze, and is equal to sin 0, cos 0, sin ~, or cos 5, respectively, for eolla*, e.llc*, e.ll¢, or e, ±~', where 0 and are the angles between K and a* and between K and ~-, respectively. The normalization factor A and the relative magnitude of the electronic polarization to the nuclear polarization, x-ItG(To)/tZ*n(To) ], at the temperature To where the data were taken are the fitting parameters for each q. We found that the fit obtained with q = sin 0 was significantly better than with the other choices for q. (We intend to publish the details of our analysis elsewhere.) This result indicates that the sinusoidal structure with polarization parallel to the a* axis is the most likely structure. We then tested the effect of a superposition of polarization parallel to the c* axis on this structure and found no improvement in the fit. The polarization parallel to the a* axis indicated by the present experiment is quite consistent with the result of an inelastic neutron experiment using PrCu 2 at approximately 5 K by Kjems et al. [10]. They observed an intense crystal field transition along the b* and c* axes, and the absence of this transition along the a* axis uniquely defined it as a J~ transition. The best fit value of the parameter x is 3.22. If one assumes that this value of x does not change down to 0 K, then the saturated electronic moment is calculated to be/ze(0) = 3.22/z*(0) = 0.55
(b) Fig. 5. Possible modulation structures of nuclear spins in PrCuv a) Nuclear spins are modulated sinusoidally with paramagnetic motion which persists down to 0 K. b) Nuclear spins align in an antiphase domain in the hyperfine field. /xB. This value needs to be explained theoretically in connection with the transition temperature. These results show that in PrCu 2 the electronic and nuclear polarizations develop simultaneously below TN. However, more work is necessary to finally determine the magnetic structure. In a sinusoidal structure the polarization has different magnitudes on different atoms. This is possible for the electronic system because it is an induced moment system. On the other hand, the nuclear spins have to be accompanied with a paramagnetic motion of the Iy and I z components to be modulated sinusoidally (see fig. 5). This motion could be a quantum-zero-point motion if it persists down to 0 K. However, it seems more likely that all the nuclear spins align completely in the hyperfine field at low enough temperature. One should be able to determine the actual nuclear spin structure by measuring the (very weak) higher harmonics of the magnetic Bragg scattering.
Acknowledgements We wish to thank Dr. Pradeep Kumar and Dr.
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W.G. Stirling for helpful discussions of nuclear magnetism in P r C u 2. This research was carried out at the Oak Ridge National Laboratory under the U.S.-Japan Cooperative Program on Neutron Scattering and sponsored in part by the Division of Materials Sciences, U.S. Department of Energy under contract DE-AC05-840R214000 with Martin Marietta Energy Systems, Inc. References [1] W. Marshall and S.W. Lovesey, Thermal Neutron Scattering (Oxford Univ. Press, London, 1971). [2] S. Kawarazaki, N. Kunitomi, Y. Morii, H. Suzuki, R.M. Moon and R.M. Nicklow, Solid State Commun. 49 (1984) 1147.
[3] S. Kawarazaki, N. Kunitomi, R.M. Moon, R.M. Nicklow and H. Suzuki, Solid State Commun. 49 (1984) 1153. [4] H. Suzuki, T. Otsuka, S. Kawarazaki, N. Kunitomi, R.M. Moon and R.M. Nicklow, Solid State Commun. 49 (1984) 1157. [5] T. Murao, J. Phys. C16 (1983) 335. [6] H. Suzuki, N. Nambudripad, B. Bleaney, A.L. Allsop, G.J. Bowden, I.A. Campbell and N.J. Stone, J. Phys. (Paris) 39 (1978) C6-800. [7] B. Bleaney, Proc. Roy. Soc. London A370 (1980) 313. [8] K.A. McEwen, W.G. Stirling, and C. Vettier, in: Crystalline Electric Field Effects in f-Electron Magnetism, R.P. Guertin, W. Suski and Z. Zolnierek, eds. (Plenum, New York, 1982). [9] K. Andres, E. Bucher, J.P. Maita and A.S. Cooper, Phys. Rev. Lett. 28 (1972) 1652. [10] J.K. Kjems, H.R. Ott, S.M. Shapiro and K. Andres, J. de Physique C6 (1978) 1010.