Department of Pure and Applied Physics, University of Salford, Salford M5 4WT, UK. fInstitut Laue-Langevin, BP 156, 38042 Grenoble Cedex Francc. Received ...
J. Phys. C : Solid State Phys., 14 (1981) 157-165. Printed in Great Britain
Magnetic ordering in praseodymium at millikelvin temperatures K A McEwent and W G Stirling$ ?Department of Pure and Applied Physics, University of Salford, Salford M5 4WT, U K fInstitut Laue-Langevin, BP 156, 38042 Grenoble Cedex Francc
Received 11 July 1980
Abstract. Using thermal neutron scattering techniques, the development of magnetic ordering in single-crystal DHCP praseodymium has been studied over the temperature range 0.03-4.2 K. The intensity of the broad elastic peak around the wavevector 0.1 1 rloo(which has been observed in previous studies of Pr) increased steadily as the temperature was reduced. In addition. new satellite reflections originating from a sinusoidally modulated magnetic structure with wavevector 0.13 T~~~ were observed at temperatures well below 1 K. The magnetic transition is believed to be driven by an enhancement of the exchange interaction via the hyperfine interaction. No temperature dependence of the magnetic excitation energies between 4.2 K and 0.03 K was detected.
1. Introduction It is well established that the ions in DHCP praseodymium metal experience a crystalline electric field which splits the J = 4 ground multiplet to produce singlet ground states at both the locally hexagonal and cubic sites. For the hexagonal site ions, the ground state lJ, = 0) is separated by an energy A = 3.2 meV (37 K) from the lJ, = k 1) doublet state (see Jensen (1979) or McEwen (1978) for recent reviews). For singlet ground state systems to exhibit magnetic ordering as T -+ 0, the ratio of the exchange coupling J ( q )to A must exceed a critical value, specifically q = 4u2J(q)/A 2 1where (xisthe matrix element linking ground and excited states (Trammel1 1963). The detailed analysis by Houmann et al (1979) of the magnetic excitations in Pr at T 5 K reveals that the coupling between hexagonal ions is largest for q 0.25 A-' along TM (-0,125 tIo0) but does not exceedO92-0.95. Clearly the ratio q may be increased by suitable modifications to either A or J(q). By applying an appropriate uniaxial stress to Pr we have removed the degeneracy of the excited doublet states (i.e. effectively reduced A) and observed magnetic ordering (TN dependent on applied stress) with the wavevector Q = 0,127 z l o o (McEwen et al 1978, 1979). In a series of papers, Murao (1971, 1972, 1975, 1979) and others (Andres 1973, Triplett and White 1973) have demonstrated the importance of the hyperfine interaction in singlet ground state systems at very low temperatures, particularly when y is near the threshold value of unity. We consider the Hamiltonian for a system of electronic spins (Ji)and nuclear spins (Ii):
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K A McEwen and W G Stirling
where the terms describe, in turn, the crystal-field potential, the exchange interaction and the hyperfine interaction. In the molecular-field approximation the Hamiltonian for the ith ion becomes: Hi =
Vy - (2 1Jij(Jj)+ A I , ) . J i
(2)
J
and it can be seen that the effective coupling between the electronic moments is slightly increased as the nuclei become polarised (i.e. when (I) # 0). It is then possible for ordering of the combined electronic-moment-nuclear-moment system to occur at values of y just below unity. Values of TN calculated from (2) are extremely sensitive totheinputparametersA,AandJ,but modelcalculationsfor 14'Pr(I = 5 / 2 , A 50 mK) indicate that (a) ordering will occur at a temperature between 10 m K and 100 mK, and (b)the saturation moment will be of the order of 0.1 pg per atom. In a study ofthe specific heat of Pr, Lindelof et a1 (1975) observed that as the temperature of their single-crystal sample was reduced from 200 m K t o 30 mK, its heat capacity increased more rapidly than the expected T-' dependence arising from the nuclear heat capacity. They ascribed their results to the high-temperature side of a lambda anomaly arising from a cooperative ordering at about 30 mK. Since neutron scattering studies provide direct and definitive evidence of magnetic ordering, we have examined the elastic and inelastic scattering of neutrons from Pr at temperatures in the millikelvin range, guided by our observations of magnetic ordering under uniaxial stress. A 'satellite' reflection has been observed which increases rapidly in intensity as the temperature decreases. 'The magnetic excitations d o not exhibit any obvious temperature dependence below 4.2 K.
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2. Experimental details The single crystal used was spark-cut from the same ingot as was used in studies of the magnetic ordering under uniaxial stress. Its dimensions were 6 x 6 x 4.5 111111.' The crystal had not been subjected to any known uniaxial stress. It was mounted with its a direction [I101 vertical in an Oxford Instruments dilution refrigerator:i. The sample was mounted in an aluminium container with approximately 0.5" of liquid 4He around the crystal. The temperature was measured by a carbon resistor close to the sample; this resistor was calibrated at the lowest temperature by measuring the anisotropy of y-rays emitted by a small 6oCosource inside the sample container. The rapid change in observed satellite intensity (to be described below) indicated good thermal contact between the sample and the copper block of the cryostat, via the liquid 4He around the sample. The dilution refrigerator was mounted on the triple-axis crystal spectrometer IN2 at the high-flux reactor of the Institut Laue-Langevin, Grenoble. Measurements were made with pyrolytic graphite monochromator and analyser crystals, using a fixed incident wavelength of 2.36%I . ;a pyrolytic graphite filter in the incident beam suppressed higher-order contamination. Normally, the collimator angles were 60-40-6G60 (min) but some measurements were also made with 60-4&3&30.
t We are indebted to the Neutron Division, Rutherford and Appleton Laboratories for this refrigerator.
159
Magnetic ordering in praseodymium at millikelvin temperatures 3. Results and discussion
A series of elastic scattering scans through the (001)DHCP reciprocal lattice point, parallel to the [loo] direction (8) was made at various temperatures (see Figure 1). At 4.2K, a weak broad peak at Q , = 0.11 zloowas observed. This peak has been observed previously when it has been described as a ‘central peak‘. As the temperature was reduced the intensity of this peak increased steadily. However, at temperatures below T 1 K, a second peak appeared at Qz = 0.128 zloo and, as can be seen from figure 1, the intensity of this ‘satellite peak’ increased more rapidly.
-
11000 2 7
I
l
l
0
0
1
800-
(1001
-
600 LT,
m
LOO
200
W
-
200 -
+
f 2ool ,e1
0-
mK
s o
LOO
z
2001 0
0.05
0 10
LOO 200 0
015
Wavevector
0 05
0 10
015
lrlu)
Figure I . Elastic scans through the reciprocal lattice position (0.1.0, 1) as a function of temperature. The ‘central peak’ or ’broad peak’ IS seen at a wavevector of about 0.11, while a ‘satellite peak’ appears at a wavevector of 0,128. The full lines are the sum of two gaussians
fitted to the data, as explained in the text.
The data were fitted by a least-squares analysis to the sum of two gaussian functions plus a background: the heights, widths and positions deduced for the two peaks are tabulated in table 1. The peak intensities of the two peaks are plotted as a function of temperature in figure 2, which shows that the satellite peak develops rapidiy below a characteristic temperature of about 400 mK. The width of the Bragg peaks are in accord with the calculated instrumental resolution (approx 0.02 A- I). Thus the measured width of the Qzpeak (see table 1)indicates long-range order. However, our analysis shows that the Q, peak is consistently some four times broader than the instrumental resolution and thus has an inherent width of 0.08 A-,.
160
K A McEwen and W G Stirling Table 1. Parameters of the two gaussians fitted by least-squares analysis to the experimental data. The widths and centres are quoted in reciprocal lattice units (474aJ3 = 1.974 & I ) . The typical accuracy of the fits is: for Ql, height F12, width F0.003, centre i 0 . 0 0 2 ; for Q,, height & 30, width .tO,OOO5, centre +00003.
Gaussian l(Ql) Temperature Height Width Centre (mK) (Counts/l69s) (rlu) (rlu) ____ 0.118 361 0.045 30 0.116 262 0.043 200 0,113 270 230 0.037 0,112 117 0,038 480 0.1 11 0,040 950 164 1250 150 0.031 0,106 0.110 4200 138 0.040
Gaussian 2 (Q,) Height Width Centre (Counts/l69s) (rlu) (rW 845 554 407 216 185 143
0009 0.008 0.010 0.009 0,009 0.012
0.131 0.130 0.131 0,129 0,129 0,128
It may be noted that both peaks exhibit a slight shift in wavevector with decreasing temperature: the Qz peak moves further from the commensurable wavevector 0.125 rlo0. Additional scans parallel to [loo] through the reciprocal lattice points (003), (005), (007),(loo), (101) and (102) also revealed peaks at Q, and Qz from these positions (see
800
7 1
Satellite
i
IQ,)
-
- 600-
. 2 2001 O-0
C c
2
Central peak IQ,)
LOO
200
I 0
A A I
05
I
1 Temperature [ K )
-w.-.v+’
4
-I
Figure2. Temperature dependence of the peak intensities of the ‘central peak’ ( Q 1 )and ‘satel-
lite’ (Q,).
figure 3). Measurements were continued out to the Brillouin zone boundary and further scans were made parallel to [Ool] through (001) to (002) and from (100) to (101):no magnetic peaks were detected in these scans.
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Magnetic ordering in praseodymium at lnillikelvin temperatures
0 0
I9021 30 mK
300 -
O
(1021
200-
i q03 1
0 0
100 -
200mK
OL
m I
I
0.85
_I
095
0.90
C
e
c
3
o
-016
u
,
-012
-0.08 r ~ o 8
I
,
I
10.12
,
01
I
Wavevector
I
I
0.8 5
+016'
0.95
0.90
(rlui
Figure 3. Elastic scans near several reciprocal lattice points. The full lines are the sum of the two fitted gaussians except ior the (q00)scan which was fitted to only one gaussian function.
3.1. T h e satellite peaks with wavevector Q ,
The peaks at Q, may be understood to arise from an incommensurable magnetic structure comprising moments (on the hexagonal sites) parallel to [ 1001which are sinusoidally modulated along this direction, i.e. m h = ml16sin(Qz.R
+ $).
(3)
The scan (q00)shown in figure 3 could be fitted by a single weak peak at (1 - Q,, 0,O). The presence of this peak is particularly significant since the neutron scattering orientation factor (m.m* - ]I;..mI2) is zero for i?along [loo]. Therefore we deduce that the magnetic moments have a component perpendicular to 8.r and write mh
= mil6 sin(Q,.
R
+ 4,,) + m,ri
. +4
sin(Q, R
~ .
,
(4)
Assuming the magnetic structure can be described by (4) we have estimated m and m, by a comparison of the satellite peak intensities with that of the (100)nuclear Bragg peak, using the ratio : Ip;f/I;y(( = P34bi,) (m: mi cos2 c() If(K)ZI IF(Z,IZ (5) where p o is the magnetic scattering length (= 0,269 x m). mlI and m, are in Bohr magnetons per atom, a is the angle between the c direction and scattering vector K,
+
t The strong uniaxial magnetic anisotropy may be assumed to confine the moments to the basal plane.
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K A McEwen and W G Stirling
Values for the m) for and b,, is the nuclear scattering length (= 0-44 x magnetic form factorf(rc) were deduced from the measurements of Lebech et a1 (1979). F ( z ) is the structure factor for the reciprocal lattice vector 7, given by
F(z) =
exp(i z. rn) n
where the summation is over only the magnetic atoms within the unit cell, i.e. those at the hexagonal sites alone. Since no correction was made for possible extinction of the (100) Bragg reflection, the calculations produce upper limits for the moments. - q5b between the coniponents in Unfortunately, the phase difference Aq5 = (4) cannot be measured with unpolarised_neutrons. Aq5 = 0 produces a simple canting ofthe modulated moments away from the b directions :Aq5 = 4 2 , which may be favoured by entropy considerations, leads to an elliptical modulation of the moments. A least-squares fit to ( 5 ) using data from the measured satellite reflections (see table 2) yields ml, = (0947 & 0.003) pB,
mL = (0.014 & 0.002)pB.
(7)
Table 2. Ratio of the measured integrated intensities of various magnetic satellites with the (100) Bragg peak intensity. In connection with equation (5), the final column represents m: + m: cos' a. Magnetic satellite reflection
Imag
(Q,, 0, 1) (Qz, 0, 3 ) (Q2,0,7) (1 - Qz,0,O) (1 - Q,,O, 1) (1 - Qz, 0, 2)
(7.61 024) x (7.39 & 0.30) x (5.60 i 0.60) x (065 5 0.09) x (0.25 k 0.05) x (1.96 i. 021) x
*
0,808 0.974 0,995 0 0.087 0,277
0.978 0,892 0.614 0.886 0.877 0.850
4 4 4 3 1 3
(208 k 0.7) x (222 0.9) x 1 0 - ~ (24.4 k 2.6) x (2.6 k 0.4) x (3.1 0.6) x (8.2 0.9) x 10-4
The satisfactory quality of the fit supports our assumption that the moments have a localised electronic (i.e. 4f) nature. Since the crystal structure has three equivalent &directions,we assume that two additional sets of magnetic satellite reflections remain to be found in positions out of the Qz, 0, l), horizontal scattering plane. For example, in addition to the peak observed (4 we anticipate peaks would also be found at (0, & Qz, 1) and (& QZ,T QZ,1). Direct evidence for these satellite reflections will be sought in further experiments, using a crystal with the ab plane horizontal. If the satellite intensities we have measured arise from domains each occupying only one-third of the crystal's volume, then the moment at the hexagonal sites is given by p = J3m with components: p , ,= (0.082 & 0*005)pB
pL
=
(0.025 F 0*004)pB.
(8)
The magnetic structure of Pr at low temperature is thus remarkably similar to that of Nd which, in the temperature range between 18.6 K and 7.5 K, exhibits moments parallel and perpendicular to equivalent b directions (Lebech et a1 1979, Moon et a1 1979). Moreover the ratio (pL/pl,) = 0-30 for Pr is comparable with the value (0.24) obtained for Nd at 10 K. However, in Nd the ordering wavevector is close to Q = 0.15 zloo
Magnetic ordering in praseodymium at rnillikelvin temperatures
163
at TNand reduces with decreasing temperature until it locks in to Q = 0.125 zloo around 10 K. 3.2. T h e broad peaks with wavevector Q ,
At 1 K and above the wavevector of these peaks is Q1 = 0108 zloo (=n/4a). There is a distinct shift in Q , as the intensity of the satellite peak increases. The q-widths of the peaks are several times greater than the instrumental resolution. The corresponding correlation length is some 80 8, or 22 lattice spacings. Our data (see table 1) suggest that this correlation length decreases somewhat as the temperature is reduced. We observe that these broad peaks coexist with the satellite peaks in the ordered phase, Our studies of Pr under stress also demonstrate that the broad peak remains in the presence of much larger ordered moments (> lpB). Therefore, we are reluctant to describe the Q, peak as a clear precursor of the long-range magnetic ordering (i.e. a ‘central peak’). Jensen (1979) has suggested that the Q, peak may be due to an ordering of the ions close to the surface of the crystal. However, the integrated intensity (at 30 mK) of the (Q, 0 1) peak is 2.1 times that of the (Q, 0 1) peak. Whilst this intensity is equivalent to a modulated moment amplitude of 0.12 pB distributed over all the hexagonal sites in the crystal, a moment of some 300 times as much within an 80 8, thick surface layer would be required to produce the same intensity. Finally, we note that the existence of the broad peak at a wavevector close to that of the long-range ordering wavevector has obscured the recognition, in previous experiments, of the onset of ordering in Pr. Careful re-examination of the elastic scattering studies of Pr at 410 mK reported by Lebech et a1 (1974, 1975) reveals that these data are consistent with our present studies on a completely different crystal. However, the statistical uncertainties in the data of Lebecn et al were too great to allow an unambiguous distinction between the ordering peak and the broad peak.
3.3. Magnetic excitations Inelastic scattering scans were made at 30 mK at several wavevectors around the minimum energy (at q N 012) of the longitudinal optic exciton dispersion curve along TM. The excitation energies were indistinguishable from those measured at 4.2 K. For example, the energies of the LO and TO modes are respectively 1.08 meV and 1.70 meV at (0.1,0,1)at both 30 mKand4.2 K. This result may be compared with our observations of the excitations under uniaxial stress. The relative elastic scattering intensities at 30 mK of the (Q,Ol) peak, the (Q,Ol) peak and the (100) Bragg peak are reproduced by an applied uniaxial stress of approximately 40 MPa at 4.5 K. However, in this case, the energy of the lower (LO) exciton mode is reduced to 0.83 meV at q = 0.1, whilst that of the upper (TO) mode is raised to 1.82meV. Thus while the elastic satellite appears at the same wavevector either (i) by applying uniaxial stress or (ii) by reducing the temperature sufficiently, the magnetic excitations behave differently in the two situations. The dispersion of the exciton optic branches along TM is given by
*
*
E: = A 2 - 2 A m ? ) K(q)l - [J’M K’(q)I) (9) where J(q), J’(q) and K(q),K’(q) denote isotropic and anisotropic exchange parameters, respectively (Jensen 1979). In (i) the shift of the crystal-field levels with stress and the reduction in A (see 5 1)appear directly as a change in the exciton energies. At zero stress, changes in these energies as T + 30mK are minimal: the induced moment is small
164
K A McEwen and W G Stirling
and implies correspondingly small changes in the matrix element (a) and exchange parameters which appear in (9). 4. Conclusion
At temperatures well below 1 K, Pr develops a long-range magnetically ordered structure. Our initial study of this structure reveals a sinusoidal modulation, with wave, moments parallel and perpendicular to Q 2 . Further studies vector Q 2 = 0.13 t I o 0of are planned to establish the multidomain or possible ‘triple-q’ nature of this structure. Whilst the magnitude of the ordered moment is consistent with that expected from Murao’s theory of hyperfine enhanced magnetic ordering in a singlet ground state system, the transition temperature is substantially higher than predicted. Our crystal contains approximatelyO.Ol% Nd, which we do not consider significant. This conclusion, however, awaits confirmation when Pr with lower Nd content becomes available. The ordering ofthe combined electron-nuclear spin system will give rise to a modulation of the nuclear scattering length of Pr which will, in principle, modify both the relevant Bragg and satellite peak intensities. Such effects, believed to be small, may be investigated using polarised neutron techniques. The origin of the broad peaks with wavevector Q1 = 0.11 z l o o remains puzzling. However, it is now clear that they do not transform continuously, with decreasing temperature, into the magnetic satellite peaks: the Q1peaks persist in the ordered phase. No changes in the energies of the magnetic excitations observed in the paramagnetic phase (at 4.2 K) were detected at 30 mK, in contrast to their behaviour when Pr is subjected to uniaxial stress. This illustrates the different mechanisms driving the phase transition in the two situations. Acknowledgments
We wish to thank D Brochier, J L Ragazzoni and P Suttling for their enthusiastic assistance in the operation of the dilution refrigerator, and J P Benoit for the calibration of the 6oCothermometer. We are also grateful for the help and advice of S Burke, K Gobrecht, P A Hilton, R Pynn, A Turberfield and C Vettier. Financial assistance was provided by the Science Research Council. KAM wishes to thank the ILL for the use of its facilities.
References Andres K 1973 Phys. Rev. B 14295-300 Houmann J G, Rainford B D, Jensen J and Mackintosh A R 1979 Phys. Rev. B 20 1105-18 Jensen J 1979 J . Physique 40 C5 1-7 Lebech B, Ais-Nielsen J and McEwen K A 1979 Phys. Rev. Lett. 43 65-7 Lebech B, Houmann J G and Chapellier M 1974 Ris0 Rep. No. 320 pp 18-9 Lebech B, McEwen K A and Lindgard P A 1975 J . Phys. C: Solid SI. Phys. 8 1 6 8 4 9 6 Lebech B, Rainford B D, Brown P J and Wedgwood F A 1979 J . Mugn. Mugn. Mater. 14 298-300 Lindelof P E, Miller I E and Pickett G R 1975 Phvs. Rev. Lett. 35 1297-9 McEwen K A 1978 Handbook on the Physits cinti Chemistry o f R a r e Earths ed K A Uschneider Jr (Amsterdam: North-Holland) pp 41 1-88 McEwen K A, Stirling W G and Vettier C 1978 Phys. Rev. Lett. 41 343-6 McEwen K A, Vettier C and Stirling W G 1979 J . Physique 40 C5 26-7
Magnetic ordering in praseodymium at millikelvin teinperatures Moon R M, Koehler W C, Sinha S K, Stassis C and Kline G R 1979 Phys. Rev. Lett. 43 6 2 4 MuraoT 1971 J . Phys. Soc. Japan 31 683-90 - 1972 J . Phys. Soc. Japan 33 33-8 __ 1975 J . Phys. Soc. Japan 39 50-7, 1629-30 - 1979 J . Phys. Soc. Japan 46 40-4 Trammel1 G T 1963 Phys. Rev. 131 932-48 Triplett B B and White R M 1973 Phys. Rev. B7 4938-41
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