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Polarized neutron study of the RA12 compound
308
Journal of Magnetism and Magnetic Materials 24 (1981) 308- 324 North-Holland Publishing Company
POLARIZED NEUTRON STUDY OF THE I. Magnetization density in NdAI2
RAI2COMPOUND
J.X. BOUCHERLE DRF.CENG, BP 85X, 38041 Grenoble Cedex, France
and J. SCHWEIZER DRF.CENG and ILL, BP 156X, 38042 Grenoble Cedex, France
Received 19 May 1980 The magnetization density of the ferromagnet NdA12 has been studied by diffraction of polarized neutrons on single crystals. The data have been treated as well in direct space using Fourier transformation techniques as in reciprocal space with the tensor operator method. The two treatments give the same result: besides the localized magnetization due to the 4f electrons of Nd, and corresponding to 2.61 t~B, there exists another magnetization, opposite to the first one, which fluctuates throughout the cell with a pronounced extremum around the Nd atoms. Its integrated value is -0.14 ~tB/Nd. The 4f electron wave function corresponding to the localized magnetization has been determined.
1. Introduction The RA12 intermetallic compounds, where R is a rare earth element, are particularly well suited for the study of rare earth magnetism: 1) They form a complete series from LaA12 to LuA12, all the lanthanides combining with aluminium to produce this compound with the same structure. 2) The crystal structure is the simple cubic Laves phase with prototype MgCu2, the rare earth occupying one atomic site only. 3) With the exception of cerium the compounds including trivalent magnetic rare earth are all ferromagnetic. These features explain why the RA12 series has been extensively studied by all kinds of methods. Among these methods neutron diffraction is the most suitable for characterizing the magnetism of the rare earth atoms. In addition to their capacity for revealing magnetic structures, the neutrons are able to describe the magnetization distribution of each atom in the crystal. For this kind of investigation the most sensitive method is diffraction of polarized neutrons by ferromagnetic crystals. In this method one compares directly the amplitude of magnetic scattering to that of nuclear scattering and simple 0304-8853/81/0000-0000/$02.75 © 1981 North-Holland
crystal structures, such as RA12, are desirable. We have studied RA12 compounds with several lanthanides. We report here the investigations on the first of them, NdA12, chosen because important crystal field effects are expected, and we explain the experimental method and the data treatments which are common to all the studies on other RA12 intermetallics. For the investigations on other compounds, publication of which will follow, we shall concentrate on the physical problem particular to each of these compounds and on the results of the study. We devote section 2 to the experimental part, then in section 3 we explain the way to reduce the data, which leads to magnetic structure factors. Section 4 shows how it is possible to determine the ground state of the rare earth ion and, finally, section 5 discusses the magnetization density in NdA12.
2. Experimental NdA12 is a ferromagnet [ 1 - 4 ] with a Curie temperature 79 K and [100] as easy axis. A previous neutron diffraction experiment with powder sample [5] has confirmed the ferromagnetic structure and
309
J.X. Boucherle, J. Schweizer /RAI2 compound and polarized neutrons. I
found a magnetic moment (2.5 +_0.1)/a B for Nd. The present study has been carried out on single crystals using both unpolarized and polarized neutrons. 2.1. Single crystal preparation
All the compounds RA12 have a congruent melting in the vicinity of 1500°C. However, the reaction between rare earth and aluminum is strongly exothermic. To prevent crucible contamination, we have separated the 2 processes: alloy preparation and crystal growth. Alloys were prepared by rf levitation technique melting together 99.9% purity rare earth ingots wrapped in 99.999% purity aluminum foils. The melting occurred in a helium atmosphere, without any crucible, to avoid contamination. The melt was then quenched in a copper mould. The alloys were carefully analyzed by X-rays. Different compositions have been prepared to detect a possible composition range around the stoichiometry RAI~. But alloys on both side of that composition, apart from the presence of impurities, did not show any lattice constant difference, indicating no composition range. Crystals were grown by Bridgmann technique in cone-shaped tungsten crucibles: tungsten appeared better adapted than other materials, causing minimum contamination in the final product (120 ppm W is a typical analysis result). The heating was provided by a rf coil and a graphite susceptor. Displacement velocities were around 2 cm/h. One important problem concerns mechanical strains occurring when the crystals cooled down to room temperature. They were so large that the crystals were completely broken. To avoid them we wrapped the inside part of the crucible with a 0.025 mm tungsten foil. This was enough to accommodate the dilatations. Two platelets were cut from the same NdAI~ crystal: sample A and sample B, both parallel to plane (110) with dimensions 0.6 X 1.2 X 2.0 mm 3 for sample A and 0.9 × 2.2 × 3.5 mm 3 for sample B. Direction [001] was parallel to the long edge in both cases. The surfaces of the platelets have been carefully polished to reduce neutron depolarization.
2.2. Neutron diffraction: unpolarized and polarized neutrons
For a polarized neutron beam, the diffraction cross section of centrosymmetrical crystals depends on the value and sign of the polarization P [ P = (nt - n$)/ (nf + n$)]. )
oY(H) = FN(H) 2 + 2PFN(tDFMz(I-I) + [FM (H)[ 2 ,
(1)
)
where FN (H) and FM (H) are the nuclear and magnetic structure factors corresponding to the scattering vector H, and FMz(H) the FM(H) component parallel to the polarization axis z. For unpolarized neutrons (P = 0), the cross term disappears and o°(H) = FN(/-/) u + IFM(/t)I 2 •
(2)
For neutrons which are perfectly polarized in the + direction (P = + 1), then in the opposite direction (P = - 1 ) one can define for each Bragg reflection a flipping ratio R - °+(H) - F~ + 2FNFMz + IF--MM12 o-(It)
F~ - 2FNFMz+ I~MMIa '
(3)
In general (see appendix A) it is possible to define 7(//) = FMz (H)/sin20 _ IFM (//)l/sin O FN(H) FN (//) '
(4)
where 0 is the angle between H and the polarization axis. The flipping ratio R can then be written: R
7(/02 + 27(//) + 1/sin:O 7(t02 - 27(//) + 1/sin2® "
(5)
From eq. (5) we obtain 7(H) =R + 1 V ( R + I ~ 2 R - 1 -+ L \ R ~ - I /
1 71/2 sin~OJ
(6)
the selection between the two possible solutions being generally straightforward. In the case of imperfect polarization P, imperfect flipping efficiency and in the presence of a X/2 contamination of the beam, formula (6) is slightly modified. Its expression has been given in ref. [6]. 2.2.1. Unpolarized neutron diffraction In order to know accurately the nuclear structure factors of NdA12 at 4.2 K, the intensities of (hkO) Bragg reflections for unpolarized neutrons have been
310
J.X. Boucherle, J. Schweizer / RAI 2 compound and polarized neutrons. I
recorded up to sin 0/X = 1.0 A -1. This experiment was performed at helium temperature with a 16.5 kOe vertical magnetic field applied on the sample. The wavelength 0.736 A was obtained from a germanium monochromator (115) reflection, which allowed a good resolution at high angles. As the data in the literature concerning neutron absorption by Nd are incoherent, the linear absorption coefficient was measured on the NdA12 sample itself at the same wavelength. It was found to be p = 0.446 cm- 1. 2.2.2. Polarized neutron diffraction Polarized neutron experiments have been carried out on 3 different diffractometers: DN2 at the reactor Melusine of the Grenoble Nuclear Centre and D3 and D5 at the High Flux Reactor of the ILL. The last of these, located on the hot source, allows the use o f short wavelengths. On these 3 diffractometers neutrons were polarized by a CoFe monochromator and the polarization was reversed by a rf coil. All the measurements have been performed at helium temperature with the easy axis of the crystal [001] vertical, parallel to the applied field. The flipping ratios R for most of the Bragg reflections were measured with the counter in the horizontal plane, up to sin 0 / k = 1.58 A -t . For the other reflections (up to hk3) the counter was tilted in a vertical plane, which is possible on the 3 diffractometers. The wavelength, polarization o f the beam, flipping efficiency, contamination k/2 and applied field corresponding to the NdA12 experiments are recorded in table 1. It should be noted that contamination k/2 was reduced by use of filters: Pu for ?~= 1.05 A and Er for ?, = 0.84 and 0.74 A.
Depolarization of neutrons through the sample was evaluated with a Cu2MnA1 analyzing crystal located behind the sample. It was found to be negligible. Each flipping ratio measurement was performed with a rotation of the crystal around the scattering vector to detect any possible multiple scattering effect. Any suspicious result was rejected. Finally some measurements have also been carried out at 1.5 K. As no difference was detected between the flipping ratios at 1.5 and 4.2 K, it was deduced that the effects due to a possible polarization of the Nd nuclei were negligible. This corresponds to the low concentration in natural neodymium of isotopes with a nuclear moment.
3. Data reduction The aim o f the neutron diffraction experiments on NdA12 is the accurate knowledge of the magnetic structure factors. As they are obtained from both 7(H) and FN (H), besides the intrinsic experimental uncertainties, two other sources affect the final results. - uncertainties related to the calculated values FN (H) due to insufficient knowledge of the crystal structure parameters; - uncertainties due to extinction which appears in all the scattering processes, and which affects the experimental 7(//) values. We shall show in the first part o f this section how the crystal structure parameters were determined at 4.2 K using b o t h polarized and unpolarized neutron experiments. In the second part it will be explained that the measurement of flipping ratios R ( H ) at dif-
Table 1 Experimental conditions for NdAI2 on the polarized neutron diffractometers: wavelength (h), polarization (P), flipping efficiency (E), contribution hi2 (C) and applied field (/4). J(h) represents the neutron flux in the incident beam at wavelength h Diffractometer
h (A)
P
E
1 J(h/2) C= ~ J(X~
H (kOe)
12
DN2
1.10
0.985
0.990
0.0010
D3
0.80
0.970 (5)
0.9960 (4)
0.0009 (4)
16.5
D5
0.50 0.74 0.84 1.05
0.9665 0.9845 0.9895 0.9836
0.9975 0.9927 0.9911 0.9915
0.00046 0.00092 0.00033 0.00234
16.5
(12) (12) (5) (10)
(33) (20) (13) (20)
(5) (3) (2) (5)
J.X. Boucherle, J. Schweizer / RAl2 compound and polarized neutrons. I
ferent wavelengths allows a good determination of the parameters which control extinction, allowing a precise knowledge of 7(//) and then of the magnetic structure factor. 3.1. Crystal structure and nuclear structure factors
The RA12 compounds crystallize in the fcc Laves phase with 8 formulae in the cubic cell. In the centrosymmetrical description of space group Fd3m, atoms are located as follows: R in -+ (1/8, 1/8, 1/8), A1 in (1/2, 1/2, 1/2); (1/2, 1/4, 1/4); (1/4, 1/2, 1/4); (1/4, 1/4, 1/2). The structure, illustrated in fig. 1, represents a diamond type stacking of the rare earth in which is imbricated an aluminum tetrahedral network. For such a structure the different types of reflections are reported in table 2. One may note that for type b the A1 contribution cancels, for type d the rare earth contribution cancels and that for type g, like (442) which is allowed by the space group, both contributions dis-
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appear due to the special position of the atoms. It must be noted that these cancellations are complete only if the atoms themselves are symmetrical enough, for instance when they are completely spherical. At low temperature, below the Curie point, most of the RA12 exhibit a small distortion. NdA12 becomes slightly quadratic [7] but the formulae of table 2 are still correct. In spite of the fact that atoms are in special positions, accurate expression for the nuclear structure factors requires a good knowledge of the thermal vibration parameters at 4.2 K for both atoms and also a correct value for their scattering length (see table 2). This last point is achieved for aluminum (bAl = 0.3449(9) × 10 -15 cm) but not for Nd. All these unknown quantities have been accurately determined from the neutron experiments at 4.2 K. As they include both magnetic and nuclear contributions, the separation of the nuclear structure factors implies combining both unpolarized and polarized neutron results. The intensities of the Bragg reflections measured with unpolarized neutrons at 4.2 K are equal to I(H) = K [FN(H) 2 + IFM(/~)12]y(//).
(7)
Introducing the extinction coefficient y and the ~, values deduced from the polarized neutron experiment (as will be seen in the next paragraph), the FM's disappear
(~ Nd o At
1(11) = K E N (//)2 [1 + 7(H) 2 sin20]y(/-/).
(8)
We determined the scale factor K, the Nd Fermi
length bNd and the 2 thermal vibration parameters BAI and BNd. We found bNd = 0.769(5) × 10 -15 cm, BNd = 0.12(4) A 2 , BAI = 0.40(7) A s .
o ~
s~
o~.
Agreement between observed and calculated intensities is reported in table 3. It should be noted that the value found for bNd is in very good agreement with the recent determinations as discussed in ref. [8]. 3. 2. Extinction treatment and magnetic structure factors
Fig. 1. Crystal structure of the Laves phase NdAI2 in the centrosymmetrical representation.
Among the corrections of the polarized neutron data, imperfect polarization, flipping efficiency and
J.X. Boucherle, J. Schweizer / RAI 2 compound and polarized neutrons. I
312
Table 2 T h e different type o f reflections for the RA12 Laves phase structure Conditions
h, k, l
Type
"4n+2 4n+ 2 4n
h+k+l=8n+4
-8f R b
h + k + I = 8n
+8fR
4n+ 2 4n+ 2 4n+2 h = 2n k = 2n l = 2n
an 4n 4n
F~
d
+16fAl
h + k + I = 8n + 4
e
- 8 ( f R - 2fAl)
h+k+l=8n
f
+ 8 ( f R + 2fAl)
n~0
g
0
4n
4n 4n an 4n 4n+2
8[IR -IAI)
= 8n +-3 (h + k ) and (h + l) = 4n + 2
h + k+l
h=2n+l k=2n+l 1=2n+1
+8C&_.rAl)
h+k+l=8n+_ l (h + k) and (h + l) = 4n h+k+l=8n+_3 (h + k) and (h + I) = 4n h+k+l=8n+_l • (h + k) and (h + l) = 4n + 2
,
/
sin20~
(
f R = bR e x p ~ - B R - - ~ - - / , f A l = bAl exp k - B A I
sin20~ h2 ,].
Table 3 Agreement b e t w e e n the observed and calculated intensities after the nuclear refinement on NdAI 2 h
k
l
2 2 0 4 0 0 4 4 0 12 4 0 14 2 0 10 10 0 14 6 0 16 0 0
sin 0 / h
3'
Absorption
lobs
0.177 0.250 0.354 0.792 0.886 0.886 0.954 1.003
0.841 (6) 6.54 (29) 0.374 (2) 0.159 (2) 0.237 (4) 0.209 (7) 0.186 (5) 0.108 (3)
1.041 1.052 1.041 1.053 1.063 1.048 1.057 1.069
22 242 8 331 49657 37 789 11530 11 838 10515 29 798
(573) (741) (1490) (756) (280) (393) (316) (894)
Icalc
Y
21 974 8 298 48795 35 899 11 923 11 801 11 356 29 866
0.937 0.988 0.915 0.958 0.985 0.986 0.987 0.966
ZX. Boucherle, Z Schweizer / RAl2 compound and polarized neutrons. I X/2 contamination are straightforward. This is not the case for extinction, which corresponds to a weakening o f the beam by the diffraction process: b y amplitude coupling in the case o f primary extinction and by intensity coupling in t h e case o f secondary extinction. Following the theoretical treatment o f Zachariasen [9] modified by Becker Coppens [10] 2 parameters can be defined which characterize the mosaic crystal: dimension o f perfect blocks t and parameter g which corresponds to the angular disorientation A o f the perfect blocks W(A) = x/2g exp(-2zrg 2 A2).
(9)
Expression (3) for the flipping ratio R becomes, in the first order R(H) = °+(H)[1 - x ; o-(/~[1
- x s l = RoYR
(10)
- x ~ - x23
with + 2 IXa(F~-+ 2FNFMz+iFMI2)I t 2 sin 20 x~ =~V 2 sin 20 X
(11)
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Table 4 Agreement between the observed and calculated R values after the extinction parameters refinement on crystal A h k 1
h
Rob s
Rcal
YR
4 4 4 4
4 4 4 4
0 0 0 0
0.500 0.740 0.840 1.052
4.448 (42) 4.506(35) 4.500(30) 4.264(31)
4.451 4.496 4.485 4.288
0.984 0.964 0.953 0.925
8 8 8 8
0 0 0 0
0 0 0 0
0.500 0.740 0.840 1.052
3.178(53) 3.219(47) 3.346(47) 3.169(40)
3.249 3.269 3.261 3.153
0.988 0.973 0.964 0.942
extinction is o f type I according to Zachariasen's classification [9]. For crystal A we had only measured R values at different wavelengths for 2 reflections. It was imposTable 5 Agreement between the observed and calculated R values after the extinction parameters refinement on crystal B
and Xs=~
•
V 2sin20
X ~
+~g~] _]
"
,
(12)
where T is the mean path in the crystal and V the volume o f the crystal cell. F r o m these expressions it is clear that the experimental values R ( H ) depend on the wavelength, extinction being minimum for small X. It is therefore possible, by comparing the values R(It, X), measured for the same reflections at different wavelengths, to determine the parameters t and g which rule extinction, considering as unknown the magnetic structure factors FM 's. This has been done for b o t h crystals NdA12. For crystal B, 25 experimental R values corresponding to 8 different (hkl) Bragg reflections and 4 wavelengths have made it possible to determine t = (14.7 + 0.4)/a, g = 369 + 54
(mosaic r~ = 1/2x/~g = 2.6').
This corresponds to a case where extinction is b o t h o f primary and secondary type and where secondary
h k I
h
Rob s
Rcal
YR
2 2 2 2
2 2 2 2
0 0 0 0
0.500 0.740 0.840 1.052
33.62 (2.69) 47.10 (2.85) 55.08 (2.14) 42.10(2.99)
37.295 48.006 53.867 40.975
0.991 0.973 0.962 0.946
4 4 4 4
4 4 4 4
0 0 0 0
0.500 0.740 0.840 1.052
4.299 4.237 4.152 3.814
(37) (31) (27) (34)
4.293 4.229 4.157 3.822
0.967 0.924 0.900 0.840
8 8 8 8
0 0 0 0
0 0 0 0
0.500 0.740 0.840 1.052
3.185 3.146 3.082 2.903
(26) (23) (21) (17)
3.182 3.140 3.096 2.897
0.975 0.942 0.922 0.872
6 6 0 6 6 0 6 6 0
0.740 0.840 1.052
7.698 (111) 7.944 (116) 7.630(192)
7.805 7.887 7.459
0.976 0.968 0.949
8 4 0 8 4 0
0.740 0.840
3.160(110) 3.333 (57)
3.274 3.303
0.994 0.993
10 2 0 10 2 0
0.740 0.840
5.422 (34) 5.348 (59)
5.397 5.423
0.980 0.972
0 0 0 0
0.500 0.740 0.840 1.052
2.020(18) 2.040 (24) 2.011 (17) 1.936 (18)
2.031 2.021 2.008 1.940
0.987 0.971 0.961 0.934
12 4 0 12 4 0
0.500 0.740
1.828 (18) 1.852 (19)
1.843 1.835
0.990 0.976
8 8 8 8
8 8 8 8
314
J.X. Boucherle, J. Schweizer / RAl2 compound and polarized neutrons. 1
sible to d e t e r m i n e b o t h t and g i.e. impossible to separate primary f r o m secondary e x t i n c t i o n . We fixed 't at 10/a and d e t e r m i n e d g = 293 -+ 100 ( r / = 3.3'). The agreement b e t w e e n observed and calculated R values r e p o r t e d in tables 4 and 5 for crystals A and B is remarkable and gives full c o n f i d e n c e in the m e t h o d . Then, using the values d e t e r m i n e d for the extinction parameters, 7 values have been calculated, Cor-
rected f r o m e x t i n c t i o n , and the magnetic structure factors have been d e d u c e d f r o m eq. (6). The validity o f this t r e a t m e n t which assumes relation (4) is fulfilled, is d e m o n s t r a t e d in appendix A. Considering that almost all the magnetic scattering came f r o m the Nd atoms, F M ' s values have been multiplied by exp(+BNd sin 2 0/)~ 2 ) to r e m o v e thermal smearing. Values o f F N ' s , 7's and FM'S are reported in table 6.
Table 6 Nuclear structure factors, ~ ratios and magnetic structure factors of NdA12 h k l 2 4 4 6 8 6 10 8 10 12 14 10 14 16 12 14 16 18 18 14 20 18 16 20 24 24 1 3 3 5 5 2 4 6 6 1 3 3
2 0 0 0 4 0 2 0 0 0 6 0 2 0 8 0 6 0 4 0 2 0 10 0 6 0 0 0 12 0 10 0 8 0 2 0 6 0 14 0 4 0 10 0 16 0 12 0 0 0 8 0 1 1 1 1 3 1 1 1 3 1 2 2 2 2 2 2 6 2 1 3 1 3 3 3
sin O/h 0.177 0.251 0.354 0.396 0.501 0.532 0.639 0.709 0.731 0.793 0.886 0.886 0.955 1.003 1.064 1.078 1.121 1.135 1.189 1.241 1.278 1.290 1.418 1.462 1.504 1.585 0.109 0.208 0.273 0.326 0.371 0.217 0.307 0.416 0.546 0.208 0.273 0.326
T b e f b f b b f b f b b b f b b f b b b f b f f f f a c a c a d b d d c a c
F N (10 -12 cm) -6.128 (40) -0.723 (42) 11.303 (41) 6.032 (39) 10.951 (39) -5.938 (37) -5.846 (36) 10.288 (37) 5.755 (35) 9.976 (35) 5.577 (33) -5.577 (33) -5.490 (32) 9.113 (32) 8.847 (31) 5.312 (30) 8.591 (30) -5.237 (30) 5.155 (28) -5.075 (27) 7.881 (27) 4.996 (26) 7.248 (24) 7.053 (24) 6.864 (23) 6.507 (21) -1.597 (29) -7.038 (29) 1.632 (29) 6.937 (28) 1.665 (28) 5.415 (14) 6.080 (39) 5.149 (12) 4.896 (11) -7.038 (29) 1.632 (29) 6.937 (28)
.y 0.841 (6) 6.54 (29) 0.374 (2) 0.653 (4) 0.300 (1) 0.494 (3) 0.414 (3) 0.182 (2) 0.309 (3) 0.159 (2) 0.237 (4) 0.209 (7) 0.186 (5) 0.108 (3) 0 075 (8) 0.129 (10) 0.064 (7) 0.140 (11) 0.103 (12) 0.058 (13) 0.049 (13) 0.050 (15) 0.030 (9) 0.033 (14) 0.051 (10) 0.034 (12) 2.285 (13) 0.513 (3) 2.075 (16) 0.454 (4) 1.787 (22) -0.0097 (9) 0.736 (4) -0.0041 (10) 0.0012 (9) 0.496 (13) 1.958 (61) 0.433 (5)
F M × exp(W) (10 -12 cm) -5.176 (42) -4.725 (53) 4.305 (60) 4.020 (34) 3.385 (16) -3.041 (22) -2.544 (22) 1.999 (16) 1.901 (21) 1.710 (23) 1.455 (25) -1.284 (41) -1.144 (27) 1.117 (30) 0.766 (70) 0.793 (53) 0.646 (62) -0.863 (55) 0.634 (62) -0.358 (64) 0.469 (101) 0.308 (75) 0.274 (63) 0.303 (99) 0.467 (70) 0.301 (78) -3.650 (69) -3.610 (23) 3.387 (63) 3.148 (28) 2.975 (64) -0.0523 (50) • 4.473 (38) -0.0210 (50) 0.0060 (45) -3.492 (93) 3.196 (115) 3.006 (38)
J.X. Boucherle, J. Schweizer / R A I 2 compound and polarized neutrons. I
315
4. Analysis in the reciprocal space: the description of the 4f magnetization density
4-fold axis completely remove the degeneracy. The wave function of the ground state is
To examine the experimental results, there are two possible approaches. One method consists of building the magnetization density in the direct space by Fourier transformation. It is presented in the next section. A second method is the use of a model which gives magnetic structure factors comparable to the experimental ones. In the case of RA12 compounds such a method is very suitable: the magnetization density originates a priori from the electrons of the 4f shell, and can be very well described by an atomic model.
1~) = a91219/2, 9/2)
+ all= 19/2, 1/2) + a_7/219/2, - 7 / 2 ) .
(14)
Magnetization measurements performed on a single crystal [3-12] give the values of the crystal field and exchange parameters and make it possible to calculate the wave function of the ground state (table 7(a)). The splitting between this level and the first excited one is of the order of 70 K. The atomic model is now well defined, and it is possible to calculate its contribution to the magnetic structure factors.
4.1. The state o f the Nd 3+ ion in NdAI2 The neodymium ion has three electrons on the 4f shell. The ground state is described by the quantum numbers: S = 3/2,
L = 6,
J = L - S = 9/2.
The higher multiplets are high enough to be neglected. The wave function can be written:
J~) = ~ aMI9/2, M). M
(13)
The crystalline electric field corresponding to the Td symmetry of the environment decomposes the tenfold degenerated multiplet 9/2 into a doublet F 6 and two quartets Fs. Different experiments [3,11,12] show that the ground state is the Kramers doublet F 6 . The exchange and applied fields parallel to the
4.2. Calculation o f the magnetic structure factors: the tensor operator method In the case of a rare earth atom, due to the large orbital contribution, the usual approximations cannot be used for the calculation of the magnetic form factor. A complete calculation must be done. It can be performed by the tensor operator method developed by Johnston Lovesey Rimmer [13,14,15]. They define the magnetic scattering amplitude of one atom as a vector E(H) defined by: E(H) = 7e2 (•lDtl•),
(15)
mc 2
where D ± is the operator representing the magnetic interaction neutron electron including the spin part
Table 7 Ground state wave function at 4.2 K and with an applied field o f 16.5 kOe: I~) = a9/219/2, 9/2> + al/219/2, 1/2> + a_7/219/2, -7/2). (a) Calculated with parameters fitted from magnetization measurements on a single crystal [3], (b) same calculation but with a polarisation o f - 0 . 1 4 ~B due to the conduction electrons [ 19], (c) calculated with parameters coming from magnetic excitation spectrum [ 11 ], (d) fitted from form factor measurements (this work) Conditions
ag/2
all 2
a_7/2
# 0zB)
Magnetization measurements (a)
0.853
Magnetization measurements and polarization o f conduction electrons (b)
0.520
-0.049
2.47(2)
0.880
0.473
0.045
2.61(2)
0.040
2.61
Magnetic excitation spectrum (c)
0.880
0.480
Fitted wave function (d)
0.885(18)
0.451 (77)
-0.112(180)
2.61(2)
J.X. Boucherle, J. Schweizer / RAI 2 compound and polarized neutrons. I
316
and the orbital part:
4. 3. Ground state wave function
D±= -
At 4.2 K, the temperature of the experiment, only the ground state is populated and the magnetic scattering amplitudes involve only this level. With the radial integrals <]K>known from atomic calculations and with the ground state coefficients aM, the magnetic scattering amplitude can be expanded as:
~
v electrons
X
exp(2irrHrv)
A svA IHI
hlH[ Ini Ap
.
(16)
It should be noted that with this definition the magnetic structure factor becomes:
FM(n) =
El(H) exp(2irrHr]) exp(-W/). (17) 2
atom ]
Using the electronic expansion of the wave function (13) and the spherical components of the magnetic scattering amplitude E(H) E 0 =E z ,
-1 E, = ~ - ( E x + iEy),
E_,
1
(18)
(Ex- We),
one gets for these spherical components Eq (/t) =
-23'e2
"
mc 2 K~,,Q,, X/r4~ "~ ''([t) X
,
, aMaM' (A (K"K') + B(K"K')}
XI ( K "Q" K Q ' ' [lq>.
(19)
The spherical harmonics Y~:: (/I) depend on the two angles ® and ¢P which characterize the orientation of the scattering vector H. They represent the anisotropic aspect of the form factor. The A(K", K') and B(K", K') terms correspond to the orbital and spin parts of the magnetic interaction with the neutron. They can be separated out in one part depending on the f electron configuration of the free ion by introducing coefficientsA' and C' (see appendix B) and in another part depending on the radial distribution of the f electrons involving the one electron radial integrals (]K>:
(JK) = ? r2 lf(r)12]K(Hr) dr. 0
(20)
Eo (1t) = M~M,a~taM' HMM' (I0,
(21)
where HMM'(H)is independent of the value of the coefficients aM [16]. Several experiments on rare earth metals have shown that relativistic calculations are well adapted to the description of the radial densities of 4f electrons [ 17]. The radial integrals have been taken from a Dirac-Fock calculation [ 18]. In a first step the calculation of the structure factors was done with the wave function obtained from magnetization measurements (see table 7(a)). The results are compared to the experiment in table 8(b). The moment agrees well with the value measured by magnetization measurements (/a = 2.47/~B) but the calculated structure factors are far too small, leading to important discrepancies. It is clear that this wave function cannot describe correctly the ground state of the 4f electrons and that it is necessary to adjust the coefficients aM . For this purpose a least square minimization .method was used and this permitted a refinement of the parameters aM of the wave function by minimizing the quantity G Wi l e o (H)iob s - E o ( ~ i c a l ] 2. i
In this refinement the reflections (hkO) of the horizontal plane were taken into account. In this case the form factor ~f(H) is proportional to Eo(//):/af(H) = -~Eo(H). The agreement between observed and calculated pf(/-/) is very good except for the first 4 reflections including (000). Finally these reflections were withdrawn from the refinement process (table 8(c) and fig. 2). The coefficients aM obtained in this way are reported in table 7(d). They correspond to a wave function which describes correctly the magnetic contribution of the 4f electrons. However, the important differences observed at low sin 0/X can be attrib-
J.X. Boucherle, J. Sch weizer / RAl2 compound and polarized neutrons. I
Table 8 Magnetic structure factors in (VB/atom), comparison between (a) experimental values and (b) calculated values for the wave function deduced from magnetization measurements and (c) for the fitted wave function (reflections with * were not used in the refinement process) h
k
l
sin0/X
i
,uf (,u:)
o ,~.0
pf(~B/atom) observation (a)
317
I
calculation (b)
calculation (c)
2.0
I
o
NdAi 2
0
[001]
t~D
0 2 4 4 6 8 6 10 8 10 12 10 14 14 16 12 14 16 18 18 14 20 18 16 20 24 24
0 0 2 0 0 0 4 0 2 0 0 0 6 0 2 0 8 0 6 0 4 0 10 0 2 0 6 0 0 0 12 0 10 0 8 0 2 0 6 0 14 0 4 0 10 0 16 0 12 0 0 0 8 0
0.000 0.177 0.251 0.354 0.396 0.501 0.532 0.639 0.709 0.731 0.793 0.886 0.886 0.955 1.003 1.064 1.078 1.121 1.135 1.189 1.241 1.278 1.290 1.418 1.462 1.504 1.585
2.470(20) 2.401(19) 2.192(25) 1.997(28) 1.864(16) 1.570(7) 1.410(10) 1.180(10) 0.927(7) 0.882(10) 0.793(11) 0.596(19) 0.675(12) 0.531(13) 0.518(14) 0.355(32) 0.368(25) 0.300(29) 0.400(26) 0.294(29) 0.166(30) 0.217(47) 0.143(35) 0.127(32) 0.141(45) 0.216(29) 0.140(36)
Reliability factor (excluding the first four reflections)
2.471 2.319 2.171 1.881 1.769 1.474 1.341 1.102 0.877 0.846 0.758 0.540 0.628 0.490 0.483 0.318 0.314 0.301 0.349 0.276 0.179 0.239 0.178 0.096 0.100 0.153 0.099
2.608* 2.449* 2.294* 1.991" 1.872 1.559 1.474 1.166 0.934 0.900 0.803 0.580 0.662 0.521 0.508 0.344 0.339 0.322 0.366 0.292 0.197 0.251 0.192 0.107 0.110 0.158 0.105
5.84%
1.48%
uted to another magnetization contribution superimposed on the 4f one and more delocalized than the 4f density. This extra density can represent a polarization of the conduction band by the 4f spin as has been observed in numerous metallic compounds. It is now very fruitful to compare the refined 4f wave function with that obtained from other experiments. The first case is a new interpretation of magnetization measurements [19] taking into account an extra polarization Pd = --0.14 #a. The wave function of the
0 • •
0 ~0 i~0 ~0 0~.
~ exp. 0
o cal.
o~o 1.0
o
oo ~© o ~C~I O 0 P--~CO~I'o 0-0.0 ¢N 0 cDO~r I I i
'I 6
I
0 i
~o
"®
O.O
0.5
110- ~
(%'3 115
sin
0//k
Fig. 2. Observed and calculated magnetic structure factors (in PB/atom).
ground state (table 7(b)) is now in excellent agreement with our results. Another determination comes from inelastic neutron scattering [11]. From dispersion curves of magnetic excitations at low temperature, the wave function of the ground was deduced; it is reported in table 7(c). This determination is also an agreement with our results. Let us note, however, that in contrast to the other methods, the determination of the wave function based on the neutron diffraction data determines directly the coefficients aM, without assuming anything on the magnetic interactions which enter the Hamiltonian. Therefore it is more reliable. In the case of NdA12, the agreement between the different 4f ground state determinations corroborate the inter-
318
J.X. goucherle, J. Schweizer / RAI 2 compound and polarized neutrons. 1
pretation of an extra magnetization density added to the 4f density. However, to confirm the differences in localization of the two densities and to specify the "diffuse" extra density another analysis of the neutron data has been undertaken in direct space.
5. Analysis in the direct space: magnetization density maps The magnetic structure factors we have measured are the Fourier components of the magnetization density. When the observations are collected in a plane, it is possible to build the projection in direct space: -boo
-boo
Table 9 Magnetic structure factors F M and errors used for the calculation of the (001) density map (in 10-l 2 cm); x F M measured by magnetization experiments, o F M measured by integrated intensities, + F M calculated from refined wave function
x o
+
+
1 ~ =~- ~ k=~- ~o FM(hkO) m (x, y ) = -ff X exp[-2irr(hx + k y ) ] .
5.1. Magnetization density projected on the plane (001) To build up the observed density map projected on the (001) plane all the structure factors (hkO) are necessary. In neutron diffraction experiments, two types of limitations appear. The first one concerns the diffraction technique which prevents the measurement of reflections too far in reciprocal space. To reduce this problem, short wavelengths have been used. ~/second limitation comes from the polarized neutron technique and concerns reflections with very small nuclear structure factors. The number of such reflections is small. Those at low values of sin 0/~ have been taken from integrated intensity measurements, the other ones just calculated from the previous 4f model. With all the structure factors of the horizontal plane up to sin 0/~ = 1.42 A -1 (table 9), the projected density around the rare earth atoms was established very precisely. However, for the regions between the magnetic atoms, where the density is close to zero, the truncation of the Fourier series produces small but non-significant oscillations. In order to eliminate this effect, an averaging of the density was performed by integration on a small surface
+
+
+ +
+ +
h
k
l
sin0/h
0 2 4 4 6 8 6 8 10 8 10 12 12 14 10 12 14 16 16 12 14 16 18 18 14 16 20 20 18 20 22 16 20 24 24
0 2 0 4 2 0 6 4 2 8 6 0 4 2 10 8 6 0 4 12 10 8 2 6 14 12 0 4 10 8 2 16 12 0 8
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.000 0.177 0.251 0.354 0.396 0.501 0.532 0.561 0.639 0.709 0.731 0.752 0.793 0.886 0.886 0.904 0.955 1.003 1.034 1.064 1.078 1.121 1.135 1.189 1.241 1.253 1.253 1.278 1.290 1.350 1.384 1.418 1.462 1.504 1.585
F M × exp(W) (10-12 cm) 5.325(43) -5.176(42) -4.725(53) 4.305(60) 4.020(34) 3.385(16) -3.041(22) -2.919 -2.544(22) 1.999(16) 1.901(21) -1.971 1.710(23) 1.455(25) -1.284(41) -1.217 -1.144(27) 1.117(30) -0.974 0.766(70) 0.793(53) 0.646(62) -0.863(55) 0.634(62) -0.358(64) -0.422 -0.604 0.469(101) -0.308(75) -0.394 0.439 0.274(68) 0.303(99) 0.467(70) 0.301(78)
around each point. The map in fig. 3 presents the projected density. It corresponds to two treatments: one zone around the magnetic atoms above the zero contour where the density is not averaged and one zone between the atoms where the density is averaged over a square with an edge 28 such as 26/a = 0.08. The equidensity contours have been drawn from 17.5/~B/ A 2 to - 0 . 0 2 / z B / A 2 . The uncertainties are less than
J.X. Boucherle, .1..Schweizer / RAI 2 compound and polarized neutrons. I a
319
the localized density corresponds to the 4 f electrons, but we shall not assume in the following treatment any knowledge of the spatial extension of the 4f electrons. We shall just consider that the 4f density falls rapidly and is zero between the Nd atoms. Considering the magnetic density as a sum of 2 parts, one localized and one more diffuse:
m(r) = mL (r) + mD (r), the magnetic structure factors can be written as the sum of 2 contributions:
FM(I[)= FML (H) + FMD(II).
! 2 Fig. 3. Magnetisation density projected along the [001] direction (in/zB/A2 ). •
(1OO)
0.01/IB/)k 2 except in very symmetric positions where they reach 0.02 ~B/A 2 . The two main features of the map are: i) almost all the density is concentrated on the position of the neodymium atoms and presents a slight cubic deviation from a spherical distribution. This is expected for the 4f electrons magnetization density. ii) Between the magnetic atoms, the density is not zero and fluctuates throughout the cell. One observes smooth variations with a small positive peak on the aluminum position and negative values between the atoms. Such an extra density, which is attributed to a polarization of conduction electrons, is clearly evidenced between the Nd atoms, but is masked by the high value of the 4f magnetization near these atoms. To reveal it completely we were led to remove from the total density map the part localized on the magnetic atoms.
5.2. Separation between "localized" and "diffuse" part Our aim is to withdraw from the total magnetization density the part which is localized around the Nd atoms. This result, briefly presented elsewhere [20], will be discussed here with more details. We know that
The separation between the 2 densities can be effected, it being noted that the slowly varying diffuse density contributes only to low sin 0/~ Bragg reflections: only the first FMD(~ ) are different from zero. For the other reflections, FMD(/"0 = 0 and the experimental values F u (H) correspond to the localized part FML(H). For the first reflections with FMD (H) sa 0 one determines by a least square refinement the localized parts FML (H). These are the values which, together with the other structure factors (where FM = FML) provide a "localized" Fourier map where the density between the neodymium atoms is the flattest and the closest to zero. The sum over interatomic points ro was minimized: ~i [m(ri) ] 2. Two descriptions of the interatomic zone have been tried, which are represented in fig. 4. In the first one (fig. 4a) the sum was performed over a ring centered on the neodymium atom. In the second one the sum corresponds to the hatched surface of fig. 4b. It appears that it is not necessary to refine FML (H) beyond reflection (440) for which sin 0/~, = 0.254 A-1. The results are compared in table 10 with the experimental values and with the values obtained in the previous section using the 4f wave function fitting. It is very satisfactory to see that the two integration methods in direct space give very close results for the localized part of the structure factors. But the most interesting result is that this localized structure factors agree very well with the 4f structure factors calculated in the last section for the ground state I¢) = 0.88519/2, 9/2) + 0.45119/2, 1/2) - 0.11219/2, - 7 / 2 ) .
320
J.X. Boucherle, J. Schweizer / RA I2 compound and polarized neutrons. I
"
•
(a)
)
(b)
Fig. 4. Surface of integration used in the "localized", "delocalized" scheme; (a) rings around one Nd atom, (b) surface between Nd atoms.
The validity o f the separation between "localized" and "diffuse" part is well supported since the two methods based on completely different and independent grounds give the same answer. 5. 3. "Diffuse" magnetization density
The diffuse structure factors FMD (H) were obtained b y subtracting the 4f contribution calculated for the ground state from the experimental structure
factors (table 11). Most of the contribution to FMD(//) appear for reflections with sin 0/~ < 0.30 A -1. With all the significantly non-zero Fourier coefficients we can build the magnetization density maps. For a comparison with the total projected map (fig. 4) we present as a first step the "diffuse" density projected on the (001) plane (fig. 5). Between the neodymium atoms, this density is very close to that observed in the total map. But, in addition, a minimum appears clearly at the position of the n e o d y m i u m
Table 10 Total and "localized" magnetic structure factors for the first four reflections (in/~B/atom)
Total magnetic structure factor
"Localized" structure factors
Conditions
(000)
(220)
(400)
(440)
Experiment
2.470(20)
2.401(19)
2.192(25)
1.997(28)
Ring around Nd atom
2.629(35)
2.445(24)
2.282(15)
1.973(6)
Surface between Nd atoms
2.615(26)
2.434(28)
2.281(10)
1.974(25)
2.608
2.449
2.294
1.991
Localized-delocalized scheme
4f contribution I¢) =0.88519/2> + 0.45111/2) - 0.1121-7/2)
J.X. Boucherle, J. Schweizer
I
RAI2 compound and polarized neutrons. I
321
Table 11 Magnetic structure factors o f the first reflections and "diffuse" contribution h k I
sin 0IX
T
0 2 4 4 1 3 3 5 5 2 4 6 6 1 3 3
0.000 0.177 0.251 0.354 0.109 0.208 0.273 0.326 0.371 0.217 0.307 0.416 0.546 0.208 0.273 0.326
b e f a c a c a d b d d c a c
0 2 0 4 1 1 3 1 3 2 2 2 6 1 1 3
0 0 0 0 1 1 1 1 1 2 2 2 2 3 3 3
F o b s (10 -12 cm)
Fca I (10 -12 cm)
5.325(43) -5.176(42) -4.725(53) 4.305(60) -3.650(69) -3.610(23) 3.387(63) 3.148(28) 2.975(64) -0.0523(50) 4.473(38) -0.0210(50) 0.0060(45) -3.492(93) 3.196(115) 3.006(38)
FMD (10 -12 cm)
5.627 -5.281 -4.946 4.296 -3.852 -3.602 -3.371 -3.161 2.946 4.478 -3.398 3.190 2.994
atoms. It reaches - 0 . 0 9 / a a / A 2 and is too broad to be attributed to 4f electrons. To make sure that the extrema which appear in the projection map on the atomic positions (negative on Nd and positive on A1) correspond to these atoms, also
-0.302 0.105 0.221 0.202 -0.0523 -0.0210 -
magnetization density was built up corresponding to a (110) plane. The diffuse density section is represented in fig. 6. It is very clear that negative peaks are on the neodymium atoms and positive ones on the aluminum atoms.
in 3-dimensional space, a section map of the diffuse a
/.," / / / ~--
...,///
" \ ~ ",'~.-J ;
,
\
/ ///,
. "x
',
, .,o:....,,,..;.
I
[\".
:.:
/
'
i i ,
: ONd
,
/
. \
\
-
\
/
/
\
,
,.._~
- - 1
,' / - ~
~
/ /
N ~
'
".
."
.;
',
-,;'...,'
[
."x
L
t
~'l
~
~'
.... J
"0ol]
,/ .""
\ ".
J I .:/
i AI,.
,' , ;
.'
,, | ,
\ ' . . ~ ,,
",.
..... / /
"'" /
i ', x~l
//
~
) I":",,L/:"
\
, I
J
\ "", • "-
' "" '
.".",,X Y"-.YJ/,."
V..'//
I //
/
,/ ,,'/
/
"
...
/ "
/
J
'.
t
/
;
\\\\ \ ~ ' \ \ *-"~ J / ////
\
~
_.~,X~','..... ; ", ,
~k~\ / /
I
I I
4o ,r\
I
'--./
..'" ii 11_15 \\\\
/././-~"',
.o
\
,
1
, ' , , / / / { " NO',\/
I
l.t(x w
"-'" \'~ \ \ ~
,/,,,I j/,| /I.'l
/
i
0oo1
Fig. 5. Diffuse part of the magnetisation density projected along the [001] directions (in p B / A 2, errors ~0.01 pB/A2).
'
[,,o] Fig. 6. Section of the diffuse part of the magnetisation density by a (110) plane (in 10 -3 pB/A 3, errors ~-5 X 10 -3 #B/A3).
322
J.X. Boucherle, J. Schweizer/ RAI2 compound and polarized neutrons. I
5.4. Polarization of the conduction electrons The observed "diffuse" magnetization which is present in the whole cell can be understood as a polarization of the conduction electrons. However, the main part of this density is on the magnetic atoms with a localization close to that of atomic shells. The peak on the rare earth position is a direct observation of the coupling between 4f electrons and conduction electron spins which is at the origin of magnetic ordering in such metallic compounds. Usually it is admitted that the coupling takes place between spins and is represented by the Hamiltonian = -2FsS with F positive. In the case of rare earth atoms, J is a good quantum number and the coupling must be written:
~c = - 2 r ( g j - 1)sJ. As gj is less than one for neodymium, the negative peak observed in the Fourier maps corresponds to a parallel coupling between spins. Such a result is in very good agreement with previous results obtained for rare earth metals (for example in gadolinium [17]).
6. Conclusion In this paper, which is the first of a series dealing with the study of the RA12 compounds by polarized neutron diffraction, we have emphasized the methods used for all the series: measurement, data reduction and result analysis. In addition to the methods, the results observed for the magnetization density of Nd in NdA12 are of importance and will be also met for all the other compounds of the series. The experimental magnetization can be accounted for by 2 different contributions: one, corresponding to the 4f electrons, is very localized around the atomic position while the second, more diffuse, fluctuates throughout the cell. As regards the 4f magnetization, its analysis allows the determination of the ground state of the Nd a+ ion. This determination is based on the shape of the electronic magnetic cloud: it refines directly the coefficients of the wave function and does not assume any model for the magnetic interactions as
the other methods generally do. The resulting ground state yields a 4f magnetic moment which is larger than that determined by macroscopic measurements, the difference corresponding to the diffuse magnetization. This diffuse magnetization corresponds to "non-4f" electrons polarized by the 4f shell. It expands all over the cell but is mainly concentrated around the Nd atoms, with a sign opposite to that of the 4f electrons. Less localized than the 4f shell, the origin of the corresponding electrons has still to be clarified.
Appendix A
Validity of the 7 ratio treatment The diffraction cross section (1) differs from the usual Halpern and Johnson [21] form: oF(H) = FN(H) 2 + 2PFN (H)FM(H) + FM(H) 2 ,
(A.1)
where the magnetic structure factor was supposed to be scalar. Therefore the usual definition 7(H) = FM(II)/FN (H)
(A.2)
cannot be applied to solve eq. (3)~ This corresponds to the vectorial character of FM (H). One experiment which measures R(H) only cannot determine all the components of the vector. However, if the following condition is fulfilled )
F ~ ( H ) = sin OIFM(H)[,
(A.3)
definition (4) may be applied to 7(H), which solves eq. (3). In terms of the spherical components Eq(H) of the atomic scattering amplitude condition (A.3) becomes Eo(H): = sin2®[Eo(H) z - 2EI(H)E-1 (H)] •
(A.4)
In the present experiment on NdA12, this condition is either exactly fulfilled for symmetry reasons or verified with enough accuracy to solve eq. (3). For (hkO) reflections O = 1r/2 and eq. (A.4) reduces to e , ( H ) = 0.
(A.5)
The axis 4 vertical implies AM = M' - M = 4 (even) which imposes in eq. (19) Q' even for non-zero Clebsch-Gordan coefficients (K'Q'JM'IJM). Because of the second coefficient (K"Q"K'Q'I lq), for q = 1,
J.X. Boucherle,J. Schweizer/ RAI2 compoundand polarizedneutrons.I Table 12 Validity of the 3' ratio treatment for reflection with l ~ 0, calculation ofD = E~(H)/tg2 ® - 2 IE1 (H)I2 h k 1
sin 0/h
®
E0 (~B/at)
LEll D ~uB/at) (X 106)
1 3 1 3 3 4 3 5 5 5 5
0.109 0.208 0.208 0.273 0.273 0.307 0.326 0.326 0.371 0.371 0.411
54.74 72.45 25.23 76.74 46.51 65.90 54.74 78.90 80.27 59.53 62.77
1.6848 2.1478 0.4049 2.0952 1.1012 1.7309 1.3096 1.9968 1.8774 1.3642 1.3569
0.8423 0.4803 0.6069 0.3491 0.7387 0.5475 0.6547 0.2770 0.2276 0.5675 0.4938
1 1 1 3 1 2 3 l 3 1 3
1 1 3 1 3 2 3 1 1 3 3
-8 -7 -5 -4 -16 -41 -4 -12 -26 -66 -158
323
It is possible to write:
A(K", K') =A'(K", K')[(]K,+I) + (/K'-I)], where the coefficients A'(K", K'), independent of the (]K> integrals, represent only the symmetry properties of the ion. There exists a relation between A ( K ' - 1, K') and A (K'+ 1, K') which results in the following equation in terms of A':
A'(K'- 1,K')_{K' + I'~I/2 A'(K'+I,K') \ K' ] " The spin component
B(K", K') is non-zero for
K'
~< 2l + 1,
K" even
~< 2l + 2,
K ' - 1 ~< K" ~< K' + 1. Q" should be odd. As K" is even ( see appendix B) the only spherical harmonics Y~',',(O,~) which appear to include cos O and therefore are zero for O = zr/2. This verifies exactly condition (A.5). For the (hkl) reflections, measured in this work, with l :~ 0 we have calculated the quantities D = E0(/-/)2/tg20 - 21E1 (//)12 corresponding to the wave function I¢) = 0.88519/2) + 0.45111/2) 0.1121-7/2 >as determined from the neutron data. They are reported in table 12. It is clear that these quantities, which should be zero to obey condition (A.3), are small enough to be neglected.
In general K' has no parity constraint. But within one term with L ' = L, S' = S, J ' = J, K' is odd and K" = K ' -+ 1. One may write
B(K'- 1 , K ' ) = C ' ( K ' - 1, K')(/K- 1) iK ' - I ( K ' + 1) X [3(2K'+ 1)] ,/2
× (]K,+,)iK'+IF K'(K'+I)I"2 L3(-5 + DJ
Orbit and spin contribution to Eq (It) The coefficients A (K", K') and B(K", K') which represent the orbital and spin part of the magnetic interaction of the neutron, have been reexpressed by Lander et al. [22]. The orbital component A (K", K') is non-zero only for K'odd
~<2l+ 1,
K" even
~< 2l,
K'-
'
with
B(K'-I,K')
{K'+ 1) '/2
B(K'+
Appendix B
+C'(K'+ 1 , K ' )
"
The structure of terms A (K", K') and B(K", K') has been expressed in table 13 for a rare earth ion in terms o f coefficient A ' and C'. These coefficients have been tabulated for the ground states of all the rare earth ions in ref. [22]. For ion Nd 3+, they are the following: A'(0, 1 ) = - 2 . 1 1 0 6 ,
C'(0, 1)= 1.3568,
A'(2, 3) = 0.3476,
C'(2, 1) = 0.1357,
A'(4, 5) = - 0 . 1 5 6 8 ,
C'(2, 3) = - 0 . 0 9 9 6 , C'(4, 3) = 0.2369,
I~
with l = 3 for the rare earth atoms.
C'(4, 5 ) = - 0 . 1 0 0 1 , C'(6, 5) = 0.8585, C'(6, 7) = - 0 . 1 9 4 8 .
324
J.X. Boucherle, J. Schweizer / RAI 2 compound and polarized neutrons. I
Table 13 Expression ofA(K", K') and
B(K", K')
K'
K"
A (K", K')
B(K", K')
0
A'(0, 1) [(/o> +I
c'(o, 1) 2 - c'(2, 1) ~
2
x/~-~A'(0, 1)[(/o> + (/2)]
X~[C'(0,1)~(/0>-C'(2,1)~<]2) 1
2
A'(2, 3)[(12)+ (/4)]
-C'(2, 3) ~
4
"v/~A'( 2, 3)[(J2)+ (/4)]
4
A'(4, 5)[(14) + (/6)1
X/~-~[-C'(2, 3) V~--21- c'(6, 5) v'~ (/6>
6
X//~A'( 4, 5)[(/4> + (]6>]
I
4
(/2> + C'(4,
3
5
6
X/~[C'(4, S) A < j 4 ) - C ' ( 6 , s)x l/---~II(]6>1 7 - C ' ( 6 7) "~~ ( / 6 > ,
7 8
X/~-[-C'(6, 7) x/~ (16>1
References [1] H.J. Williams, J.H. Wernick, E.A. Nesbitt and R.C. Sherwood, J. Phys. Soc. Japan 17 suppl. B1 (1962). [2] W.M. Swift and W.E. Wallace, J. Phys. Chem. Solids 29 (1968) 2053. [3] B. Barbara, J.X. Boucherle and M. Rossignol, Phys. Stat. Sol. (a) 25 (1974) 165. [4] G.J. Cock, L.W. Roeland, H.G. Purwins, E. Walker and A. Furrer, Solid State Commun. 15 (1974) 845. [5] N. Nereson, C. Olsen and G. Arnold, J. Appl. Phys. 37 (1968) 4575. [6] J. Schweizer and F. Tasset, J. Phys. F10 (1980) 2799. [7] B. Barbara, M.F. Rossignol and M. Uehara, Physica 86-88B (1977) 183. [8] J.X. Boucherle and J. Schweizer, Acta Cryst. B31 (1975) 2745. [9] W.H. Zachariasen, Acta Cryst. 23 (1967) 558. [10] P. Becker and P. Coppens, Acta Cryst. A30 (1974) 129. [11] J.G. Houmann, P. Bak, H.G. Purwins and E. Walker, J. Phys. C7 (1974) 2691.
[12] M.F. Rossignol, Thesis, Univ. Grenoble (1980). [13] D.F. Johnston, Proc. Phys. Soc. 88 (1966) 37. [14] S.W. Lovesey and D.E. Rimmer, Rep. Prog. Phys. 32 (1969) 333. [15] W. Marshall and S.W. Lovesey, Theory of Thermal Neutron Scattering (Oxford, Clarendon, 1971). [16] B. Barbara, J.X. Boucherle, J.P. Desclaux, M.F. Rossignol and J. Schweizer, Crystal Field Effects in Metals and Alloys, ed. A. Furrer (Plenum, New York, 1971) p. 168. [ 17] R.M. Moon, W.C. Koehler, J.W. Cable and H.R. Child, Phys. Rev. B5 (1972) 997. [18] A.J. Freeman and J.P. Desclaux, J. Magn. Magn. Mat. 12 (1979) 11. [ 19] B. Barbara and M.F. Rossignol, private communication. [20] J.X. Boucherle and J. Schweizer, Physica 86-88B (1977) 174. [21] O. Halpetn and M.H. Johnson, Phys. Rev. 55 (1939)
898. [22] G.H. LanderandT.O. Brun,J. Chem.Phys.53 (1970) 1387.
Journal of Magnetism and Magnetic Materials 24 (1981) 308- 324 North-Holland Publishing Company
POLARIZED NEUTRON STUDY OF THE I. Magnetization density in NdAI2
RAI2COMPOUND
J.X. BOUCHERLE DRF.CENG, BP 85X, 38041 Grenoble Cedex, France
and J. SCHWEIZER DRF.CENG and ILL, BP 156X, 38042 Grenoble Cedex, France
Received 19 May 1980 The magnetization density of the ferromagnet NdA12 has been studied by diffraction of polarized neutrons on single crystals. The data have been treated as well in direct space using Fourier transformation techniques as in reciprocal space with the tensor operator method. The two treatments give the same result: besides the localized magnetization due to the 4f electrons of Nd, and corresponding to 2.61 t~B, there exists another magnetization, opposite to the first one, which fluctuates throughout the cell with a pronounced extremum around the Nd atoms. Its integrated value is -0.14 ~tB/Nd. The 4f electron wave function corresponding to the localized magnetization has been determined.
1. Introduction The RA12 intermetallic compounds, where R is a rare earth element, are particularly well suited for the study of rare earth magnetism: 1) They form a complete series from LaA12 to LuA12, all the lanthanides combining with aluminium to produce this compound with the same structure. 2) The crystal structure is the simple cubic Laves phase with prototype MgCu2, the rare earth occupying one atomic site only. 3) With the exception of cerium the compounds including trivalent magnetic rare earth are all ferromagnetic. These features explain why the RA12 series has been extensively studied by all kinds of methods. Among these methods neutron diffraction is the most suitable for characterizing the magnetism of the rare earth atoms. In addition to their capacity for revealing magnetic structures, the neutrons are able to describe the magnetization distribution of each atom in the crystal. For this kind of investigation the most sensitive method is diffraction of polarized neutrons by ferromagnetic crystals. In this method one compares directly the amplitude of magnetic scattering to that of nuclear scattering and simple 0304-8853/81/0000-0000/$02.75 © 1981 North-Holland
crystal structures, such as RA12, are desirable. We have studied RA12 compounds with several lanthanides. We report here the investigations on the first of them, NdA12, chosen because important crystal field effects are expected, and we explain the experimental method and the data treatments which are common to all the studies on other RA12 intermetallics. For the investigations on other compounds, publication of which will follow, we shall concentrate on the physical problem particular to each of these compounds and on the results of the study. We devote section 2 to the experimental part, then in section 3 we explain the way to reduce the data, which leads to magnetic structure factors. Section 4 shows how it is possible to determine the ground state of the rare earth ion and, finally, section 5 discusses the magnetization density in NdA12.
2. Experimental NdA12 is a ferromagnet [ 1 - 4 ] with a Curie temperature 79 K and [100] as easy axis. A previous neutron diffraction experiment with powder sample [5] has confirmed the ferromagnetic structure and
309
J.X. Boucherle, J. Schweizer /RAI2 compound and polarized neutrons. I
found a magnetic moment (2.5 +_0.1)/a B for Nd. The present study has been carried out on single crystals using both unpolarized and polarized neutrons. 2.1. Single crystal preparation
All the compounds RA12 have a congruent melting in the vicinity of 1500°C. However, the reaction between rare earth and aluminum is strongly exothermic. To prevent crucible contamination, we have separated the 2 processes: alloy preparation and crystal growth. Alloys were prepared by rf levitation technique melting together 99.9% purity rare earth ingots wrapped in 99.999% purity aluminum foils. The melting occurred in a helium atmosphere, without any crucible, to avoid contamination. The melt was then quenched in a copper mould. The alloys were carefully analyzed by X-rays. Different compositions have been prepared to detect a possible composition range around the stoichiometry RAI~. But alloys on both side of that composition, apart from the presence of impurities, did not show any lattice constant difference, indicating no composition range. Crystals were grown by Bridgmann technique in cone-shaped tungsten crucibles: tungsten appeared better adapted than other materials, causing minimum contamination in the final product (120 ppm W is a typical analysis result). The heating was provided by a rf coil and a graphite susceptor. Displacement velocities were around 2 cm/h. One important problem concerns mechanical strains occurring when the crystals cooled down to room temperature. They were so large that the crystals were completely broken. To avoid them we wrapped the inside part of the crucible with a 0.025 mm tungsten foil. This was enough to accommodate the dilatations. Two platelets were cut from the same NdAI~ crystal: sample A and sample B, both parallel to plane (110) with dimensions 0.6 X 1.2 X 2.0 mm 3 for sample A and 0.9 × 2.2 × 3.5 mm 3 for sample B. Direction [001] was parallel to the long edge in both cases. The surfaces of the platelets have been carefully polished to reduce neutron depolarization.
2.2. Neutron diffraction: unpolarized and polarized neutrons
For a polarized neutron beam, the diffraction cross section of centrosymmetrical crystals depends on the value and sign of the polarization P [ P = (nt - n$)/ (nf + n$)]. )
oY(H) = FN(H) 2 + 2PFN(tDFMz(I-I) + [FM (H)[ 2 ,
(1)
)
where FN (H) and FM (H) are the nuclear and magnetic structure factors corresponding to the scattering vector H, and FMz(H) the FM(H) component parallel to the polarization axis z. For unpolarized neutrons (P = 0), the cross term disappears and o°(H) = FN(/-/) u + IFM(/t)I 2 •
(2)
For neutrons which are perfectly polarized in the + direction (P = + 1), then in the opposite direction (P = - 1 ) one can define for each Bragg reflection a flipping ratio R - °+(H) - F~ + 2FNFMz + IF--MM12 o-(It)
F~ - 2FNFMz+ I~MMIa '
(3)
In general (see appendix A) it is possible to define 7(//) = FMz (H)/sin20 _ IFM (//)l/sin O FN(H) FN (//) '
(4)
where 0 is the angle between H and the polarization axis. The flipping ratio R can then be written: R
7(/02 + 27(//) + 1/sin:O 7(t02 - 27(//) + 1/sin2® "
(5)
From eq. (5) we obtain 7(H) =R + 1 V ( R + I ~ 2 R - 1 -+ L \ R ~ - I /
1 71/2 sin~OJ
(6)
the selection between the two possible solutions being generally straightforward. In the case of imperfect polarization P, imperfect flipping efficiency and in the presence of a X/2 contamination of the beam, formula (6) is slightly modified. Its expression has been given in ref. [6]. 2.2.1. Unpolarized neutron diffraction In order to know accurately the nuclear structure factors of NdA12 at 4.2 K, the intensities of (hkO) Bragg reflections for unpolarized neutrons have been
310
J.X. Boucherle, J. Schweizer / RAI 2 compound and polarized neutrons. I
recorded up to sin 0/X = 1.0 A -1. This experiment was performed at helium temperature with a 16.5 kOe vertical magnetic field applied on the sample. The wavelength 0.736 A was obtained from a germanium monochromator (115) reflection, which allowed a good resolution at high angles. As the data in the literature concerning neutron absorption by Nd are incoherent, the linear absorption coefficient was measured on the NdA12 sample itself at the same wavelength. It was found to be p = 0.446 cm- 1. 2.2.2. Polarized neutron diffraction Polarized neutron experiments have been carried out on 3 different diffractometers: DN2 at the reactor Melusine of the Grenoble Nuclear Centre and D3 and D5 at the High Flux Reactor of the ILL. The last of these, located on the hot source, allows the use o f short wavelengths. On these 3 diffractometers neutrons were polarized by a CoFe monochromator and the polarization was reversed by a rf coil. All the measurements have been performed at helium temperature with the easy axis of the crystal [001] vertical, parallel to the applied field. The flipping ratios R for most of the Bragg reflections were measured with the counter in the horizontal plane, up to sin 0 / k = 1.58 A -t . For the other reflections (up to hk3) the counter was tilted in a vertical plane, which is possible on the 3 diffractometers. The wavelength, polarization o f the beam, flipping efficiency, contamination k/2 and applied field corresponding to the NdA12 experiments are recorded in table 1. It should be noted that contamination k/2 was reduced by use of filters: Pu for ?~= 1.05 A and Er for ?, = 0.84 and 0.74 A.
Depolarization of neutrons through the sample was evaluated with a Cu2MnA1 analyzing crystal located behind the sample. It was found to be negligible. Each flipping ratio measurement was performed with a rotation of the crystal around the scattering vector to detect any possible multiple scattering effect. Any suspicious result was rejected. Finally some measurements have also been carried out at 1.5 K. As no difference was detected between the flipping ratios at 1.5 and 4.2 K, it was deduced that the effects due to a possible polarization of the Nd nuclei were negligible. This corresponds to the low concentration in natural neodymium of isotopes with a nuclear moment.
3. Data reduction The aim o f the neutron diffraction experiments on NdA12 is the accurate knowledge of the magnetic structure factors. As they are obtained from both 7(H) and FN (H), besides the intrinsic experimental uncertainties, two other sources affect the final results. - uncertainties related to the calculated values FN (H) due to insufficient knowledge of the crystal structure parameters; - uncertainties due to extinction which appears in all the scattering processes, and which affects the experimental 7(//) values. We shall show in the first part o f this section how the crystal structure parameters were determined at 4.2 K using b o t h polarized and unpolarized neutron experiments. In the second part it will be explained that the measurement of flipping ratios R ( H ) at dif-
Table 1 Experimental conditions for NdAI2 on the polarized neutron diffractometers: wavelength (h), polarization (P), flipping efficiency (E), contribution hi2 (C) and applied field (/4). J(h) represents the neutron flux in the incident beam at wavelength h Diffractometer
h (A)
P
E
1 J(h/2) C= ~ J(X~
H (kOe)
12
DN2
1.10
0.985
0.990
0.0010
D3
0.80
0.970 (5)
0.9960 (4)
0.0009 (4)
16.5
D5
0.50 0.74 0.84 1.05
0.9665 0.9845 0.9895 0.9836
0.9975 0.9927 0.9911 0.9915
0.00046 0.00092 0.00033 0.00234
16.5
(12) (12) (5) (10)
(33) (20) (13) (20)
(5) (3) (2) (5)
J.X. Boucherle, J. Schweizer / RAl2 compound and polarized neutrons. I
ferent wavelengths allows a good determination of the parameters which control extinction, allowing a precise knowledge of 7(//) and then of the magnetic structure factor. 3.1. Crystal structure and nuclear structure factors
The RA12 compounds crystallize in the fcc Laves phase with 8 formulae in the cubic cell. In the centrosymmetrical description of space group Fd3m, atoms are located as follows: R in -+ (1/8, 1/8, 1/8), A1 in (1/2, 1/2, 1/2); (1/2, 1/4, 1/4); (1/4, 1/2, 1/4); (1/4, 1/4, 1/2). The structure, illustrated in fig. 1, represents a diamond type stacking of the rare earth in which is imbricated an aluminum tetrahedral network. For such a structure the different types of reflections are reported in table 2. One may note that for type b the A1 contribution cancels, for type d the rare earth contribution cancels and that for type g, like (442) which is allowed by the space group, both contributions dis-
311
appear due to the special position of the atoms. It must be noted that these cancellations are complete only if the atoms themselves are symmetrical enough, for instance when they are completely spherical. At low temperature, below the Curie point, most of the RA12 exhibit a small distortion. NdA12 becomes slightly quadratic [7] but the formulae of table 2 are still correct. In spite of the fact that atoms are in special positions, accurate expression for the nuclear structure factors requires a good knowledge of the thermal vibration parameters at 4.2 K for both atoms and also a correct value for their scattering length (see table 2). This last point is achieved for aluminum (bAl = 0.3449(9) × 10 -15 cm) but not for Nd. All these unknown quantities have been accurately determined from the neutron experiments at 4.2 K. As they include both magnetic and nuclear contributions, the separation of the nuclear structure factors implies combining both unpolarized and polarized neutron results. The intensities of the Bragg reflections measured with unpolarized neutrons at 4.2 K are equal to I(H) = K [FN(H) 2 + IFM(/~)12]y(//).
(7)
Introducing the extinction coefficient y and the ~, values deduced from the polarized neutron experiment (as will be seen in the next paragraph), the FM's disappear
(~ Nd o At
1(11) = K E N (//)2 [1 + 7(H) 2 sin20]y(/-/).
(8)
We determined the scale factor K, the Nd Fermi
length bNd and the 2 thermal vibration parameters BAI and BNd. We found bNd = 0.769(5) × 10 -15 cm, BNd = 0.12(4) A 2 , BAI = 0.40(7) A s .
o ~
s~
o~.
Agreement between observed and calculated intensities is reported in table 3. It should be noted that the value found for bNd is in very good agreement with the recent determinations as discussed in ref. [8]. 3. 2. Extinction treatment and magnetic structure factors
Fig. 1. Crystal structure of the Laves phase NdAI2 in the centrosymmetrical representation.
Among the corrections of the polarized neutron data, imperfect polarization, flipping efficiency and
J.X. Boucherle, J. Schweizer / RAI 2 compound and polarized neutrons. I
312
Table 2 T h e different type o f reflections for the RA12 Laves phase structure Conditions
h, k, l
Type
"4n+2 4n+ 2 4n
h+k+l=8n+4
-8f R b
h + k + I = 8n
+8fR
4n+ 2 4n+ 2 4n+2 h = 2n k = 2n l = 2n
an 4n 4n
F~
d
+16fAl
h + k + I = 8n + 4
e
- 8 ( f R - 2fAl)
h+k+l=8n
f
+ 8 ( f R + 2fAl)
n~0
g
0
4n
4n 4n an 4n 4n+2
8[IR -IAI)
= 8n +-3 (h + k ) and (h + l) = 4n + 2
h + k+l
h=2n+l k=2n+l 1=2n+1
+8C&_.rAl)
h+k+l=8n+_ l (h + k) and (h + l) = 4n h+k+l=8n+_3 (h + k) and (h + I) = 4n h+k+l=8n+_l • (h + k) and (h + l) = 4n + 2
,
/
sin20~
(
f R = bR e x p ~ - B R - - ~ - - / , f A l = bAl exp k - B A I
sin20~ h2 ,].
Table 3 Agreement b e t w e e n the observed and calculated intensities after the nuclear refinement on NdAI 2 h
k
l
2 2 0 4 0 0 4 4 0 12 4 0 14 2 0 10 10 0 14 6 0 16 0 0
sin 0 / h
3'
Absorption
lobs
0.177 0.250 0.354 0.792 0.886 0.886 0.954 1.003
0.841 (6) 6.54 (29) 0.374 (2) 0.159 (2) 0.237 (4) 0.209 (7) 0.186 (5) 0.108 (3)
1.041 1.052 1.041 1.053 1.063 1.048 1.057 1.069
22 242 8 331 49657 37 789 11530 11 838 10515 29 798
(573) (741) (1490) (756) (280) (393) (316) (894)
Icalc
Y
21 974 8 298 48795 35 899 11 923 11 801 11 356 29 866
0.937 0.988 0.915 0.958 0.985 0.986 0.987 0.966
ZX. Boucherle, Z Schweizer / RAl2 compound and polarized neutrons. I X/2 contamination are straightforward. This is not the case for extinction, which corresponds to a weakening o f the beam by the diffraction process: b y amplitude coupling in the case o f primary extinction and by intensity coupling in t h e case o f secondary extinction. Following the theoretical treatment o f Zachariasen [9] modified by Becker Coppens [10] 2 parameters can be defined which characterize the mosaic crystal: dimension o f perfect blocks t and parameter g which corresponds to the angular disorientation A o f the perfect blocks W(A) = x/2g exp(-2zrg 2 A2).
(9)
Expression (3) for the flipping ratio R becomes, in the first order R(H) = °+(H)[1 - x ; o-(/~[1
- x s l = RoYR
(10)
- x ~ - x23
with + 2 IXa(F~-+ 2FNFMz+iFMI2)I t 2 sin 20 x~ =~V 2 sin 20 X
(11)
313
Table 4 Agreement between the observed and calculated R values after the extinction parameters refinement on crystal A h k 1
h
Rob s
Rcal
YR
4 4 4 4
4 4 4 4
0 0 0 0
0.500 0.740 0.840 1.052
4.448 (42) 4.506(35) 4.500(30) 4.264(31)
4.451 4.496 4.485 4.288
0.984 0.964 0.953 0.925
8 8 8 8
0 0 0 0
0 0 0 0
0.500 0.740 0.840 1.052
3.178(53) 3.219(47) 3.346(47) 3.169(40)
3.249 3.269 3.261 3.153
0.988 0.973 0.964 0.942
extinction is o f type I according to Zachariasen's classification [9]. For crystal A we had only measured R values at different wavelengths for 2 reflections. It was imposTable 5 Agreement between the observed and calculated R values after the extinction parameters refinement on crystal B
and Xs=~
•
V 2sin20
X ~
+~g~] _]
"
,
(12)
where T is the mean path in the crystal and V the volume o f the crystal cell. F r o m these expressions it is clear that the experimental values R ( H ) depend on the wavelength, extinction being minimum for small X. It is therefore possible, by comparing the values R(It, X), measured for the same reflections at different wavelengths, to determine the parameters t and g which rule extinction, considering as unknown the magnetic structure factors FM 's. This has been done for b o t h crystals NdA12. For crystal B, 25 experimental R values corresponding to 8 different (hkl) Bragg reflections and 4 wavelengths have made it possible to determine t = (14.7 + 0.4)/a, g = 369 + 54
(mosaic r~ = 1/2x/~g = 2.6').
This corresponds to a case where extinction is b o t h o f primary and secondary type and where secondary
h k I
h
Rob s
Rcal
YR
2 2 2 2
2 2 2 2
0 0 0 0
0.500 0.740 0.840 1.052
33.62 (2.69) 47.10 (2.85) 55.08 (2.14) 42.10(2.99)
37.295 48.006 53.867 40.975
0.991 0.973 0.962 0.946
4 4 4 4
4 4 4 4
0 0 0 0
0.500 0.740 0.840 1.052
4.299 4.237 4.152 3.814
(37) (31) (27) (34)
4.293 4.229 4.157 3.822
0.967 0.924 0.900 0.840
8 8 8 8
0 0 0 0
0 0 0 0
0.500 0.740 0.840 1.052
3.185 3.146 3.082 2.903
(26) (23) (21) (17)
3.182 3.140 3.096 2.897
0.975 0.942 0.922 0.872
6 6 0 6 6 0 6 6 0
0.740 0.840 1.052
7.698 (111) 7.944 (116) 7.630(192)
7.805 7.887 7.459
0.976 0.968 0.949
8 4 0 8 4 0
0.740 0.840
3.160(110) 3.333 (57)
3.274 3.303
0.994 0.993
10 2 0 10 2 0
0.740 0.840
5.422 (34) 5.348 (59)
5.397 5.423
0.980 0.972
0 0 0 0
0.500 0.740 0.840 1.052
2.020(18) 2.040 (24) 2.011 (17) 1.936 (18)
2.031 2.021 2.008 1.940
0.987 0.971 0.961 0.934
12 4 0 12 4 0
0.500 0.740
1.828 (18) 1.852 (19)
1.843 1.835
0.990 0.976
8 8 8 8
8 8 8 8
314
J.X. Boucherle, J. Schweizer / RAl2 compound and polarized neutrons. 1
sible to d e t e r m i n e b o t h t and g i.e. impossible to separate primary f r o m secondary e x t i n c t i o n . We fixed 't at 10/a and d e t e r m i n e d g = 293 -+ 100 ( r / = 3.3'). The agreement b e t w e e n observed and calculated R values r e p o r t e d in tables 4 and 5 for crystals A and B is remarkable and gives full c o n f i d e n c e in the m e t h o d . Then, using the values d e t e r m i n e d for the extinction parameters, 7 values have been calculated, Cor-
rected f r o m e x t i n c t i o n , and the magnetic structure factors have been d e d u c e d f r o m eq. (6). The validity o f this t r e a t m e n t which assumes relation (4) is fulfilled, is d e m o n s t r a t e d in appendix A. Considering that almost all the magnetic scattering came f r o m the Nd atoms, F M ' s values have been multiplied by exp(+BNd sin 2 0/)~ 2 ) to r e m o v e thermal smearing. Values o f F N ' s , 7's and FM'S are reported in table 6.
Table 6 Nuclear structure factors, ~ ratios and magnetic structure factors of NdA12 h k l 2 4 4 6 8 6 10 8 10 12 14 10 14 16 12 14 16 18 18 14 20 18 16 20 24 24 1 3 3 5 5 2 4 6 6 1 3 3
2 0 0 0 4 0 2 0 0 0 6 0 2 0 8 0 6 0 4 0 2 0 10 0 6 0 0 0 12 0 10 0 8 0 2 0 6 0 14 0 4 0 10 0 16 0 12 0 0 0 8 0 1 1 1 1 3 1 1 1 3 1 2 2 2 2 2 2 6 2 1 3 1 3 3 3
sin O/h 0.177 0.251 0.354 0.396 0.501 0.532 0.639 0.709 0.731 0.793 0.886 0.886 0.955 1.003 1.064 1.078 1.121 1.135 1.189 1.241 1.278 1.290 1.418 1.462 1.504 1.585 0.109 0.208 0.273 0.326 0.371 0.217 0.307 0.416 0.546 0.208 0.273 0.326
T b e f b f b b f b f b b b f b b f b b b f b f f f f a c a c a d b d d c a c
F N (10 -12 cm) -6.128 (40) -0.723 (42) 11.303 (41) 6.032 (39) 10.951 (39) -5.938 (37) -5.846 (36) 10.288 (37) 5.755 (35) 9.976 (35) 5.577 (33) -5.577 (33) -5.490 (32) 9.113 (32) 8.847 (31) 5.312 (30) 8.591 (30) -5.237 (30) 5.155 (28) -5.075 (27) 7.881 (27) 4.996 (26) 7.248 (24) 7.053 (24) 6.864 (23) 6.507 (21) -1.597 (29) -7.038 (29) 1.632 (29) 6.937 (28) 1.665 (28) 5.415 (14) 6.080 (39) 5.149 (12) 4.896 (11) -7.038 (29) 1.632 (29) 6.937 (28)
.y 0.841 (6) 6.54 (29) 0.374 (2) 0.653 (4) 0.300 (1) 0.494 (3) 0.414 (3) 0.182 (2) 0.309 (3) 0.159 (2) 0.237 (4) 0.209 (7) 0.186 (5) 0.108 (3) 0 075 (8) 0.129 (10) 0.064 (7) 0.140 (11) 0.103 (12) 0.058 (13) 0.049 (13) 0.050 (15) 0.030 (9) 0.033 (14) 0.051 (10) 0.034 (12) 2.285 (13) 0.513 (3) 2.075 (16) 0.454 (4) 1.787 (22) -0.0097 (9) 0.736 (4) -0.0041 (10) 0.0012 (9) 0.496 (13) 1.958 (61) 0.433 (5)
F M × exp(W) (10 -12 cm) -5.176 (42) -4.725 (53) 4.305 (60) 4.020 (34) 3.385 (16) -3.041 (22) -2.544 (22) 1.999 (16) 1.901 (21) 1.710 (23) 1.455 (25) -1.284 (41) -1.144 (27) 1.117 (30) 0.766 (70) 0.793 (53) 0.646 (62) -0.863 (55) 0.634 (62) -0.358 (64) 0.469 (101) 0.308 (75) 0.274 (63) 0.303 (99) 0.467 (70) 0.301 (78) -3.650 (69) -3.610 (23) 3.387 (63) 3.148 (28) 2.975 (64) -0.0523 (50) • 4.473 (38) -0.0210 (50) 0.0060 (45) -3.492 (93) 3.196 (115) 3.006 (38)
J.X. Boucherle, J. Schweizer / R A I 2 compound and polarized neutrons. I
315
4. Analysis in the reciprocal space: the description of the 4f magnetization density
4-fold axis completely remove the degeneracy. The wave function of the ground state is
To examine the experimental results, there are two possible approaches. One method consists of building the magnetization density in the direct space by Fourier transformation. It is presented in the next section. A second method is the use of a model which gives magnetic structure factors comparable to the experimental ones. In the case of RA12 compounds such a method is very suitable: the magnetization density originates a priori from the electrons of the 4f shell, and can be very well described by an atomic model.
1~) = a91219/2, 9/2)
+ all= 19/2, 1/2) + a_7/219/2, - 7 / 2 ) .
(14)
Magnetization measurements performed on a single crystal [3-12] give the values of the crystal field and exchange parameters and make it possible to calculate the wave function of the ground state (table 7(a)). The splitting between this level and the first excited one is of the order of 70 K. The atomic model is now well defined, and it is possible to calculate its contribution to the magnetic structure factors.
4.1. The state o f the Nd 3+ ion in NdAI2 The neodymium ion has three electrons on the 4f shell. The ground state is described by the quantum numbers: S = 3/2,
L = 6,
J = L - S = 9/2.
The higher multiplets are high enough to be neglected. The wave function can be written:
J~) = ~ aMI9/2, M). M
(13)
The crystalline electric field corresponding to the Td symmetry of the environment decomposes the tenfold degenerated multiplet 9/2 into a doublet F 6 and two quartets Fs. Different experiments [3,11,12] show that the ground state is the Kramers doublet F 6 . The exchange and applied fields parallel to the
4.2. Calculation o f the magnetic structure factors: the tensor operator method In the case of a rare earth atom, due to the large orbital contribution, the usual approximations cannot be used for the calculation of the magnetic form factor. A complete calculation must be done. It can be performed by the tensor operator method developed by Johnston Lovesey Rimmer [13,14,15]. They define the magnetic scattering amplitude of one atom as a vector E(H) defined by: E(H) = 7e2 (•lDtl•),
(15)
mc 2
where D ± is the operator representing the magnetic interaction neutron electron including the spin part
Table 7 Ground state wave function at 4.2 K and with an applied field o f 16.5 kOe: I~) = a9/219/2, 9/2> + al/219/2, 1/2> + a_7/219/2, -7/2). (a) Calculated with parameters fitted from magnetization measurements on a single crystal [3], (b) same calculation but with a polarisation o f - 0 . 1 4 ~B due to the conduction electrons [ 19], (c) calculated with parameters coming from magnetic excitation spectrum [ 11 ], (d) fitted from form factor measurements (this work) Conditions
ag/2
all 2
a_7/2
# 0zB)
Magnetization measurements (a)
0.853
Magnetization measurements and polarization o f conduction electrons (b)
0.520
-0.049
2.47(2)
0.880
0.473
0.045
2.61(2)
0.040
2.61
Magnetic excitation spectrum (c)
0.880
0.480
Fitted wave function (d)
0.885(18)
0.451 (77)
-0.112(180)
2.61(2)
J.X. Boucherle, J. Schweizer / RAI 2 compound and polarized neutrons. I
316
and the orbital part:
4. 3. Ground state wave function
D±= -
At 4.2 K, the temperature of the experiment, only the ground state is populated and the magnetic scattering amplitudes involve only this level. With the radial integrals <]K>known from atomic calculations and with the ground state coefficients aM, the magnetic scattering amplitude can be expanded as:
~
v electrons
X
exp(2irrHrv)
A svA IHI
hlH[ Ini Ap
.
(16)
It should be noted that with this definition the magnetic structure factor becomes:
FM(n) =
El(H) exp(2irrHr]) exp(-W/). (17) 2
atom ]
Using the electronic expansion of the wave function (13) and the spherical components of the magnetic scattering amplitude E(H) E 0 =E z ,
-1 E, = ~ - ( E x + iEy),
E_,
1
(18)
(Ex- We),
one gets for these spherical components Eq (/t) =
-23'e2
"
mc 2 K~,,Q,, X/r4~ "~ ''([t) X
,
, aMaM' (A (K"K') + B(K"K')}
X
(19)
The spherical harmonics Y~:: (/I) depend on the two angles ® and ¢P which characterize the orientation of the scattering vector H. They represent the anisotropic aspect of the form factor. The A(K", K') and B(K", K') terms correspond to the orbital and spin parts of the magnetic interaction with the neutron. They can be separated out in one part depending on the f electron configuration of the free ion by introducing coefficientsA' and C' (see appendix B) and in another part depending on the radial distribution of the f electrons involving the one electron radial integrals (]K>:
(JK) = ? r2 lf(r)12]K(Hr) dr. 0
(20)
Eo (1t) = M~M,a~taM' HMM' (I0,
(21)
where HMM'(H)is independent of the value of the coefficients aM [16]. Several experiments on rare earth metals have shown that relativistic calculations are well adapted to the description of the radial densities of 4f electrons [ 17]. The radial integrals have been taken from a Dirac-Fock calculation [ 18]. In a first step the calculation of the structure factors was done with the wave function obtained from magnetization measurements (see table 7(a)). The results are compared to the experiment in table 8(b). The moment agrees well with the value measured by magnetization measurements (/a = 2.47/~B) but the calculated structure factors are far too small, leading to important discrepancies. It is clear that this wave function cannot describe correctly the ground state of the 4f electrons and that it is necessary to adjust the coefficients aM . For this purpose a least square minimization .method was used and this permitted a refinement of the parameters aM of the wave function by minimizing the quantity G Wi l e o (H)iob s - E o ( ~ i c a l ] 2. i
In this refinement the reflections (hkO) of the horizontal plane were taken into account. In this case the form factor ~f(H) is proportional to Eo(//):/af(H) = -~Eo(H). The agreement between observed and calculated pf(/-/) is very good except for the first 4 reflections including (000). Finally these reflections were withdrawn from the refinement process (table 8(c) and fig. 2). The coefficients aM obtained in this way are reported in table 7(d). They correspond to a wave function which describes correctly the magnetic contribution of the 4f electrons. However, the important differences observed at low sin 0/X can be attrib-
J.X. Boucherle, J. Sch weizer / RAl2 compound and polarized neutrons. I
Table 8 Magnetic structure factors in (VB/atom), comparison between (a) experimental values and (b) calculated values for the wave function deduced from magnetization measurements and (c) for the fitted wave function (reflections with * were not used in the refinement process) h
k
l
sin0/X
i
,uf (,u:)
o ,~.0
pf(~B/atom) observation (a)
317
I
calculation (b)
calculation (c)
2.0
I
o
NdAi 2
0
[001]
t~D
0 2 4 4 6 8 6 10 8 10 12 10 14 14 16 12 14 16 18 18 14 20 18 16 20 24 24
0 0 2 0 0 0 4 0 2 0 0 0 6 0 2 0 8 0 6 0 4 0 10 0 2 0 6 0 0 0 12 0 10 0 8 0 2 0 6 0 14 0 4 0 10 0 16 0 12 0 0 0 8 0
0.000 0.177 0.251 0.354 0.396 0.501 0.532 0.639 0.709 0.731 0.793 0.886 0.886 0.955 1.003 1.064 1.078 1.121 1.135 1.189 1.241 1.278 1.290 1.418 1.462 1.504 1.585
2.470(20) 2.401(19) 2.192(25) 1.997(28) 1.864(16) 1.570(7) 1.410(10) 1.180(10) 0.927(7) 0.882(10) 0.793(11) 0.596(19) 0.675(12) 0.531(13) 0.518(14) 0.355(32) 0.368(25) 0.300(29) 0.400(26) 0.294(29) 0.166(30) 0.217(47) 0.143(35) 0.127(32) 0.141(45) 0.216(29) 0.140(36)
Reliability factor (excluding the first four reflections)
2.471 2.319 2.171 1.881 1.769 1.474 1.341 1.102 0.877 0.846 0.758 0.540 0.628 0.490 0.483 0.318 0.314 0.301 0.349 0.276 0.179 0.239 0.178 0.096 0.100 0.153 0.099
2.608* 2.449* 2.294* 1.991" 1.872 1.559 1.474 1.166 0.934 0.900 0.803 0.580 0.662 0.521 0.508 0.344 0.339 0.322 0.366 0.292 0.197 0.251 0.192 0.107 0.110 0.158 0.105
5.84%
1.48%
uted to another magnetization contribution superimposed on the 4f one and more delocalized than the 4f density. This extra density can represent a polarization of the conduction band by the 4f spin as has been observed in numerous metallic compounds. It is now very fruitful to compare the refined 4f wave function with that obtained from other experiments. The first case is a new interpretation of magnetization measurements [19] taking into account an extra polarization Pd = --0.14 #a. The wave function of the
0 • •
0 ~0 i~0 ~0 0~.
~ exp. 0
o cal.
o~o 1.0
o
oo ~© o ~C~I O 0 P--~CO~I'o 0-0.0 ¢N 0 cDO~r I I i
'I 6
I
0 i
~o
"®
O.O
0.5
110- ~
(%'3 115
sin
0//k
Fig. 2. Observed and calculated magnetic structure factors (in PB/atom).
ground state (table 7(b)) is now in excellent agreement with our results. Another determination comes from inelastic neutron scattering [11]. From dispersion curves of magnetic excitations at low temperature, the wave function of the ground was deduced; it is reported in table 7(c). This determination is also an agreement with our results. Let us note, however, that in contrast to the other methods, the determination of the wave function based on the neutron diffraction data determines directly the coefficients aM, without assuming anything on the magnetic interactions which enter the Hamiltonian. Therefore it is more reliable. In the case of NdA12, the agreement between the different 4f ground state determinations corroborate the inter-
318
J.X. goucherle, J. Schweizer / RAI 2 compound and polarized neutrons. 1
pretation of an extra magnetization density added to the 4f density. However, to confirm the differences in localization of the two densities and to specify the "diffuse" extra density another analysis of the neutron data has been undertaken in direct space.
5. Analysis in the direct space: magnetization density maps The magnetic structure factors we have measured are the Fourier components of the magnetization density. When the observations are collected in a plane, it is possible to build the projection in direct space: -boo
-boo
Table 9 Magnetic structure factors F M and errors used for the calculation of the (001) density map (in 10-l 2 cm); x F M measured by magnetization experiments, o F M measured by integrated intensities, + F M calculated from refined wave function
x o
+
+
1 ~ =~- ~ k=~- ~o FM(hkO) m (x, y ) = -ff X exp[-2irr(hx + k y ) ] .
5.1. Magnetization density projected on the plane (001) To build up the observed density map projected on the (001) plane all the structure factors (hkO) are necessary. In neutron diffraction experiments, two types of limitations appear. The first one concerns the diffraction technique which prevents the measurement of reflections too far in reciprocal space. To reduce this problem, short wavelengths have been used. ~/second limitation comes from the polarized neutron technique and concerns reflections with very small nuclear structure factors. The number of such reflections is small. Those at low values of sin 0/~ have been taken from integrated intensity measurements, the other ones just calculated from the previous 4f model. With all the structure factors of the horizontal plane up to sin 0/~ = 1.42 A -1 (table 9), the projected density around the rare earth atoms was established very precisely. However, for the regions between the magnetic atoms, where the density is close to zero, the truncation of the Fourier series produces small but non-significant oscillations. In order to eliminate this effect, an averaging of the density was performed by integration on a small surface
+
+
+ +
+ +
h
k
l
sin0/h
0 2 4 4 6 8 6 8 10 8 10 12 12 14 10 12 14 16 16 12 14 16 18 18 14 16 20 20 18 20 22 16 20 24 24
0 2 0 4 2 0 6 4 2 8 6 0 4 2 10 8 6 0 4 12 10 8 2 6 14 12 0 4 10 8 2 16 12 0 8
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.000 0.177 0.251 0.354 0.396 0.501 0.532 0.561 0.639 0.709 0.731 0.752 0.793 0.886 0.886 0.904 0.955 1.003 1.034 1.064 1.078 1.121 1.135 1.189 1.241 1.253 1.253 1.278 1.290 1.350 1.384 1.418 1.462 1.504 1.585
F M × exp(W) (10-12 cm) 5.325(43) -5.176(42) -4.725(53) 4.305(60) 4.020(34) 3.385(16) -3.041(22) -2.919 -2.544(22) 1.999(16) 1.901(21) -1.971 1.710(23) 1.455(25) -1.284(41) -1.217 -1.144(27) 1.117(30) -0.974 0.766(70) 0.793(53) 0.646(62) -0.863(55) 0.634(62) -0.358(64) -0.422 -0.604 0.469(101) -0.308(75) -0.394 0.439 0.274(68) 0.303(99) 0.467(70) 0.301(78)
around each point. The map in fig. 3 presents the projected density. It corresponds to two treatments: one zone around the magnetic atoms above the zero contour where the density is not averaged and one zone between the atoms where the density is averaged over a square with an edge 28 such as 26/a = 0.08. The equidensity contours have been drawn from 17.5/~B/ A 2 to - 0 . 0 2 / z B / A 2 . The uncertainties are less than
J.X. Boucherle, .1..Schweizer / RAI 2 compound and polarized neutrons. I a
319
the localized density corresponds to the 4 f electrons, but we shall not assume in the following treatment any knowledge of the spatial extension of the 4f electrons. We shall just consider that the 4f density falls rapidly and is zero between the Nd atoms. Considering the magnetic density as a sum of 2 parts, one localized and one more diffuse:
m(r) = mL (r) + mD (r), the magnetic structure factors can be written as the sum of 2 contributions:
FM(I[)= FML (H) + FMD(II).
! 2 Fig. 3. Magnetisation density projected along the [001] direction (in/zB/A2 ). •
(1OO)
0.01/IB/)k 2 except in very symmetric positions where they reach 0.02 ~B/A 2 . The two main features of the map are: i) almost all the density is concentrated on the position of the neodymium atoms and presents a slight cubic deviation from a spherical distribution. This is expected for the 4f electrons magnetization density. ii) Between the magnetic atoms, the density is not zero and fluctuates throughout the cell. One observes smooth variations with a small positive peak on the aluminum position and negative values between the atoms. Such an extra density, which is attributed to a polarization of conduction electrons, is clearly evidenced between the Nd atoms, but is masked by the high value of the 4f magnetization near these atoms. To reveal it completely we were led to remove from the total density map the part localized on the magnetic atoms.
5.2. Separation between "localized" and "diffuse" part Our aim is to withdraw from the total magnetization density the part which is localized around the Nd atoms. This result, briefly presented elsewhere [20], will be discussed here with more details. We know that
The separation between the 2 densities can be effected, it being noted that the slowly varying diffuse density contributes only to low sin 0/~ Bragg reflections: only the first FMD(~ ) are different from zero. For the other reflections, FMD(/"0 = 0 and the experimental values F u (H) correspond to the localized part FML(H). For the first reflections with FMD (H) sa 0 one determines by a least square refinement the localized parts FML (H). These are the values which, together with the other structure factors (where FM = FML) provide a "localized" Fourier map where the density between the neodymium atoms is the flattest and the closest to zero. The sum over interatomic points ro was minimized: ~i [m(ri) ] 2. Two descriptions of the interatomic zone have been tried, which are represented in fig. 4. In the first one (fig. 4a) the sum was performed over a ring centered on the neodymium atom. In the second one the sum corresponds to the hatched surface of fig. 4b. It appears that it is not necessary to refine FML (H) beyond reflection (440) for which sin 0/~, = 0.254 A-1. The results are compared in table 10 with the experimental values and with the values obtained in the previous section using the 4f wave function fitting. It is very satisfactory to see that the two integration methods in direct space give very close results for the localized part of the structure factors. But the most interesting result is that this localized structure factors agree very well with the 4f structure factors calculated in the last section for the ground state I¢) = 0.88519/2, 9/2) + 0.45119/2, 1/2) - 0.11219/2, - 7 / 2 ) .
320
J.X. Boucherle, J. Schweizer / RA I2 compound and polarized neutrons. I
"
•
(a)
)
(b)
Fig. 4. Surface of integration used in the "localized", "delocalized" scheme; (a) rings around one Nd atom, (b) surface between Nd atoms.
The validity o f the separation between "localized" and "diffuse" part is well supported since the two methods based on completely different and independent grounds give the same answer. 5. 3. "Diffuse" magnetization density
The diffuse structure factors FMD (H) were obtained b y subtracting the 4f contribution calculated for the ground state from the experimental structure
factors (table 11). Most of the contribution to FMD(//) appear for reflections with sin 0/~ < 0.30 A -1. With all the significantly non-zero Fourier coefficients we can build the magnetization density maps. For a comparison with the total projected map (fig. 4) we present as a first step the "diffuse" density projected on the (001) plane (fig. 5). Between the neodymium atoms, this density is very close to that observed in the total map. But, in addition, a minimum appears clearly at the position of the n e o d y m i u m
Table 10 Total and "localized" magnetic structure factors for the first four reflections (in/~B/atom)
Total magnetic structure factor
"Localized" structure factors
Conditions
(000)
(220)
(400)
(440)
Experiment
2.470(20)
2.401(19)
2.192(25)
1.997(28)
Ring around Nd atom
2.629(35)
2.445(24)
2.282(15)
1.973(6)
Surface between Nd atoms
2.615(26)
2.434(28)
2.281(10)
1.974(25)
2.608
2.449
2.294
1.991
Localized-delocalized scheme
4f contribution I¢) =0.88519/2> + 0.45111/2) - 0.1121-7/2)
J.X. Boucherle, J. Schweizer
I
RAI2 compound and polarized neutrons. I
321
Table 11 Magnetic structure factors o f the first reflections and "diffuse" contribution h k I
sin 0IX
T
0 2 4 4 1 3 3 5 5 2 4 6 6 1 3 3
0.000 0.177 0.251 0.354 0.109 0.208 0.273 0.326 0.371 0.217 0.307 0.416 0.546 0.208 0.273 0.326
b e f a c a c a d b d d c a c
0 2 0 4 1 1 3 1 3 2 2 2 6 1 1 3
0 0 0 0 1 1 1 1 1 2 2 2 2 3 3 3
F o b s (10 -12 cm)
Fca I (10 -12 cm)
5.325(43) -5.176(42) -4.725(53) 4.305(60) -3.650(69) -3.610(23) 3.387(63) 3.148(28) 2.975(64) -0.0523(50) 4.473(38) -0.0210(50) 0.0060(45) -3.492(93) 3.196(115) 3.006(38)
FMD (10 -12 cm)
5.627 -5.281 -4.946 4.296 -3.852 -3.602 -3.371 -3.161 2.946 4.478 -3.398 3.190 2.994
atoms. It reaches - 0 . 0 9 / a a / A 2 and is too broad to be attributed to 4f electrons. To make sure that the extrema which appear in the projection map on the atomic positions (negative on Nd and positive on A1) correspond to these atoms, also
-0.302 0.105 0.221 0.202 -0.0523 -0.0210 -
magnetization density was built up corresponding to a (110) plane. The diffuse density section is represented in fig. 6. It is very clear that negative peaks are on the neodymium atoms and positive ones on the aluminum atoms.
in 3-dimensional space, a section map of the diffuse a
/.," / / / ~--
...,///
" \ ~ ",'~.-J ;
,
\
/ ///,
. "x
',
, .,o:....,,,..;.
I
[\".
:.:
/
'
i i ,
: ONd
,
/
. \
\
-
\
/
/
\
,
,.._~
- - 1
,' / - ~
~
/ /
N ~
'
".
."
.;
',
-,;'...,'
[
."x
L
t
~'l
~
~'
.... J
"0ol]
,/ .""
\ ".
J I .:/
i AI,.
,' , ;
.'
,, | ,
\ ' . . ~ ,,
",.
..... / /
"'" /
i ', x~l
//
~
) I":",,L/:"
\
, I
J
\ "", • "-
' "" '
.".",,X Y"-.YJ/,."
V..'//
I //
/
,/ ,,'/
/
"
...
/ "
/
J
'.
t
/
;
\\\\ \ ~ ' \ \ *-"~ J / ////
\
~
_.~,X~','..... ; ", ,
~k~\ / /
I
I I
4o ,r\
I
'--./
..'" ii 11_15 \\\\
/././-~"',
.o
\
,
1
, ' , , / / / { " NO',\/
I
l.t(x w
"-'" \'~ \ \ ~
,/,,,I j/,| /I.'l
/
i
0oo1
Fig. 5. Diffuse part of the magnetisation density projected along the [001] directions (in p B / A 2, errors ~0.01 pB/A2).
'
[,,o] Fig. 6. Section of the diffuse part of the magnetisation density by a (110) plane (in 10 -3 pB/A 3, errors ~-5 X 10 -3 #B/A3).
322
J.X. Boucherle, J. Schweizer/ RAI2 compound and polarized neutrons. I
5.4. Polarization of the conduction electrons The observed "diffuse" magnetization which is present in the whole cell can be understood as a polarization of the conduction electrons. However, the main part of this density is on the magnetic atoms with a localization close to that of atomic shells. The peak on the rare earth position is a direct observation of the coupling between 4f electrons and conduction electron spins which is at the origin of magnetic ordering in such metallic compounds. Usually it is admitted that the coupling takes place between spins and is represented by the Hamiltonian = -2FsS with F positive. In the case of rare earth atoms, J is a good quantum number and the coupling must be written:
~c = - 2 r ( g j - 1)sJ. As gj is less than one for neodymium, the negative peak observed in the Fourier maps corresponds to a parallel coupling between spins. Such a result is in very good agreement with previous results obtained for rare earth metals (for example in gadolinium [17]).
6. Conclusion In this paper, which is the first of a series dealing with the study of the RA12 compounds by polarized neutron diffraction, we have emphasized the methods used for all the series: measurement, data reduction and result analysis. In addition to the methods, the results observed for the magnetization density of Nd in NdA12 are of importance and will be also met for all the other compounds of the series. The experimental magnetization can be accounted for by 2 different contributions: one, corresponding to the 4f electrons, is very localized around the atomic position while the second, more diffuse, fluctuates throughout the cell. As regards the 4f magnetization, its analysis allows the determination of the ground state of the Nd a+ ion. This determination is based on the shape of the electronic magnetic cloud: it refines directly the coefficients of the wave function and does not assume any model for the magnetic interactions as
the other methods generally do. The resulting ground state yields a 4f magnetic moment which is larger than that determined by macroscopic measurements, the difference corresponding to the diffuse magnetization. This diffuse magnetization corresponds to "non-4f" electrons polarized by the 4f shell. It expands all over the cell but is mainly concentrated around the Nd atoms, with a sign opposite to that of the 4f electrons. Less localized than the 4f shell, the origin of the corresponding electrons has still to be clarified.
Appendix A
Validity of the 7 ratio treatment The diffraction cross section (1) differs from the usual Halpern and Johnson [21] form: oF(H) = FN(H) 2 + 2PFN (H)FM(H) + FM(H) 2 ,
(A.1)
where the magnetic structure factor was supposed to be scalar. Therefore the usual definition 7(H) = FM(II)/FN (H)
(A.2)
cannot be applied to solve eq. (3)~ This corresponds to the vectorial character of FM (H). One experiment which measures R(H) only cannot determine all the components of the vector. However, if the following condition is fulfilled )
F ~ ( H ) = sin OIFM(H)[,
(A.3)
definition (4) may be applied to 7(H), which solves eq. (3). In terms of the spherical components Eq(H) of the atomic scattering amplitude condition (A.3) becomes Eo(H): = sin2®[Eo(H) z - 2EI(H)E-1 (H)] •
(A.4)
In the present experiment on NdA12, this condition is either exactly fulfilled for symmetry reasons or verified with enough accuracy to solve eq. (3). For (hkO) reflections O = 1r/2 and eq. (A.4) reduces to e , ( H ) = 0.
(A.5)
The axis 4 vertical implies AM = M' - M = 4 (even) which imposes in eq. (19) Q' even for non-zero Clebsch-Gordan coefficients (K'Q'JM'IJM). Because of the second coefficient (K"Q"K'Q'I lq), for q = 1,
J.X. Boucherle,J. Schweizer/ RAI2 compoundand polarizedneutrons.I Table 12 Validity of the 3' ratio treatment for reflection with l ~ 0, calculation ofD = E~(H)/tg2 ® - 2 IE1 (H)I2 h k 1
sin 0/h
®
E0 (~B/at)
LEll D ~uB/at) (X 106)
1 3 1 3 3 4 3 5 5 5 5
0.109 0.208 0.208 0.273 0.273 0.307 0.326 0.326 0.371 0.371 0.411
54.74 72.45 25.23 76.74 46.51 65.90 54.74 78.90 80.27 59.53 62.77
1.6848 2.1478 0.4049 2.0952 1.1012 1.7309 1.3096 1.9968 1.8774 1.3642 1.3569
0.8423 0.4803 0.6069 0.3491 0.7387 0.5475 0.6547 0.2770 0.2276 0.5675 0.4938
1 1 1 3 1 2 3 l 3 1 3
1 1 3 1 3 2 3 1 1 3 3
-8 -7 -5 -4 -16 -41 -4 -12 -26 -66 -158
323
It is possible to write:
A(K", K') =A'(K", K')[(]K,+I) + (/K'-I)], where the coefficients A'(K", K'), independent of the (]K> integrals, represent only the symmetry properties of the ion. There exists a relation between A ( K ' - 1, K') and A (K'+ 1, K') which results in the following equation in terms of A':
A'(K'- 1,K')_{K' + I'~I/2 A'(K'+I,K') \ K' ] " The spin component
B(K", K') is non-zero for
K'
~< 2l + 1,
K" even
~< 2l + 2,
K ' - 1 ~< K" ~< K' + 1. Q" should be odd. As K" is even ( see appendix B) the only spherical harmonics Y~',',(O,~) which appear to include cos O and therefore are zero for O = zr/2. This verifies exactly condition (A.5). For the (hkl) reflections, measured in this work, with l :~ 0 we have calculated the quantities D = E0(/-/)2/tg20 - 21E1 (//)12 corresponding to the wave function I¢) = 0.88519/2) + 0.45111/2) 0.1121-7/2 >as determined from the neutron data. They are reported in table 12. It is clear that these quantities, which should be zero to obey condition (A.3), are small enough to be neglected.
In general K' has no parity constraint. But within one term with L ' = L, S' = S, J ' = J, K' is odd and K" = K ' -+ 1. One may write
B(K'- 1 , K ' ) = C ' ( K ' - 1, K')(/K- 1) iK ' - I ( K ' + 1) X [3(2K'+ 1)] ,/2
× (]K,+,)iK'+IF K'(K'+I)I"2 L3(-5 + DJ
Orbit and spin contribution to Eq (It) The coefficients A (K", K') and B(K", K') which represent the orbital and spin part of the magnetic interaction of the neutron, have been reexpressed by Lander et al. [22]. The orbital component A (K", K') is non-zero only for K'odd
~<2l+ 1,
K" even
~< 2l,
K'-
'
with
B(K'-I,K')
{K'+ 1) '/2
B(K'+
Appendix B
+C'(K'+ 1 , K ' )
"
The structure of terms A (K", K') and B(K", K') has been expressed in table 13 for a rare earth ion in terms o f coefficient A ' and C'. These coefficients have been tabulated for the ground states of all the rare earth ions in ref. [22]. For ion Nd 3+, they are the following: A'(0, 1 ) = - 2 . 1 1 0 6 ,
C'(0, 1)= 1.3568,
A'(2, 3) = 0.3476,
C'(2, 1) = 0.1357,
A'(4, 5) = - 0 . 1 5 6 8 ,
C'(2, 3) = - 0 . 0 9 9 6 , C'(4, 3) = 0.2369,
I~
with l = 3 for the rare earth atoms.
C'(4, 5 ) = - 0 . 1 0 0 1 , C'(6, 5) = 0.8585, C'(6, 7) = - 0 . 1 9 4 8 .
324
J.X. Boucherle, J. Schweizer / RAI 2 compound and polarized neutrons. I
Table 13 Expression ofA(K", K') and
B(K", K')
K'
K"
A (K", K')
B(K", K')
0
A'(0, 1) [(/o> +
c'(o, 1) 2
2
x/~-~A'(0, 1)[(/o> + (/2)]
X~[C'(0,1)~(/0>-C'(2,1)~<]2) 1
2
A'(2, 3)[(12)+ (/4)]
-C'(2, 3) ~
4
"v/~A'( 2, 3)[(J2)+ (/4)]
4
A'(4, 5)[(14) + (/6)1
X/~-~[-C'(2, 3) V~--21- c'(6, 5) v'~ (/6>
6
X//~A'( 4, 5)[(/4> + (]6>]
I
4
(/2> + C'(4,
3
5
6
X/~[C'(4, S) A < j 4 ) - C ' ( 6 , s)x l/---~II(]6>1 7 - C ' ( 6 7) "~~ ( / 6 > ,
7 8
X/~-[-C'(6, 7) x/~ (16>1
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