Hyperfine Interactions 15/16 (1983) 305-308 �9 J.C. Baltzer Scientific Publishing Company

305

MISALIGNMENT OF ORIENTED NUCLEI IN FERROMAGNETS

J. ANDRIESSEN and H. POSTMA

Department of Physics, University of Technology, Lorentzweg 1, NL-2628 CJ DELFT, The Netherlands

Two atomic model ca lcu la t ions are car r ied out in order to understand misalignment of nuclei in ferromagnets sugges- ted by var ious authors. I t is concluded that an un l i ke l y large exchange i n te rac t i on is needed in these model calcu- la t ions to exp la in a substant ia l misalignment.

i . In t roduc t ion

It has been suggested that magnetic moments of nuclei implanted in ferromagnetic metals may not always feel an internal magnetic field which is exactly colinear with the direction of the applied magnetic field, even in the case of full saturation. Possible exper~ mental indication of such misalignments may have been obtained in several nuclear orientation experiments with Mn, Co, Ta and Ir iso- topes [ i, 2] and in an angular distribution experiment with Nd and W isotopes [3]. Recently Ligthart [4 ] in an NO experiment found evi- dence that nuclear spins of gold isotopes in iron show such a mis- alignment, but this did not occur for Mercury isotopes in the same fields. A further important result is the rather small quadrupole interaction, at least in the cases Au and Ir, which seems to be co- linear with the applied field [5].

In an attempt to explain misalignment, Aharoni [ 6]used a con- tinuum model approach in which he assumes the effect orkginates from magnetostriction in the host, caused by the difference in atomic radii. He implicitly assumes that the magnetic field in the host has a different direction near the impurity. Although he arrives at mis- alignment angles comparable with experiment, the approach suffers from serious short comings, particularly the neglect of exchange effects, which are probably mo~e important than magnetostriction.

This work was started to establish whether it is possible to have misalignment induced in the impurity atom without assuming mis- alignment in the host atoms surrounding the impurity atom. This approach was partly motivated by the absence of any measurable mis- alignment in the case of Mercury isotopes, which have atomic radii comparable with that of gold.

2. Theoret ica l approach

There can be not much doubt that extension of the work of Demangeat [ 7]would be a good candidate for a satisfying description of the effects involved. There are, however, two reasons why in this study a different approach was taken. First it is doubtful whether spin-orbit interaction can be handled as a perturbation in the third transition series. It seems not appropriate to write Au 5d-wave-func- tions as linear combinations of Fe d-wave and s-wave functions. Se- condly a simpler formalism is more suited to study physical effects,

306 J. Andriessen, H. Postma, Misalignment of Oriented Nuclei

such as the occurrence of orbital momentum in a closed shell and the related misalignment of orbital momentum in a cubic environment.

Model of the calculation

In our model the Au atom is considered as an isolated entity surrounded by its eight nearest Fe 2+ neighbours of which the elec- tronic density is represented by spheres (3s, 3d) with spins polar- ized in a direction (8,~) with the z-axis (see figure i). The Au atom is thought to have effectively lost two electrons, which assumption is motivated by the absence of any misalignment in case of Mercury isotopes where the presence of one extra 6s electron is expected to be responsible for the different behaviour. The almost free part of the electrons is not of much importance for the misalignment and will be neglected.

Z

e 2+

x ~ -\y

Fig. i. The model for Au z+ implanted in Iron.

In order to show the way mis- alignment can occur, we take as a first case (A) the ionic view- point of a hole in the 5d shell (Au 2t) and the effects of the crystal field and the applied magnetic field are studied, without any overlap or exchange with the surrounding Fe ions. In case B we study the effects of exchange on a 5d shell of which all orbitals have the same occupation. This must be much more realistic if we take into account the small quadrupole coupling that was found.

~se A.

The HF wavefunctions of Au 2+ were calculated using the MCDF program of Grant et al. [8]. The level difference between j=3/2 and j=5/2 was found to be 12460 cm -I or 1.5 eV. Therefore we can assume the hole is in the j=5/2 shell; thus the j=3/2 shell will be neglected. In the cubic field of the neighbouring ions (figure i) the fourfold degenerate level (F8) will be the level of interest.

The wavefunctions are as follows:

~I = 15/2, 5/2 > + 1 5/2, -3/2> ,

~2 = 15/2' 1/2 > ,

~3 = 15/2' -1/2> and

~4 = ~ '5/2, -5/2 > + ~ '5/2, 3/2> �9 (i)

307

Z Z j

If now we diagonalize the interaction with the magnetic field:

H = g ~bJz,Hz , (2)

for a direction z' in the xz-planemaking an angle (@) with the z- axis, it is straightforward to show the occurrence of a perpendicu- lar components Jx' with angular dependence (sin48) as shown in fig.2.

J. Andriessen, H. Postma, Misalignment of Oriented Nuclei

Fig. 2. Occurrence of misalignment of orbital momentum. The arrows on the circle denote the direction of the Jx' component.

The misalignment angle came out to be between 0 and 15 degrees. However, the calculated orbital field is more than i00 Tesla, which is roughly ten times too large. More seriously, the qua- drupole field comes out to be two orders of magnitude larger than experiment. It is clear that the neighbouring Fe ions average out to a large extent the orbital momentum (quenching) by hybridizing with the 5d or- bitals and thus a spherical zero-order state seems to be much more appropriate.

Case B.

This approach is to use a perturbation expansion to study t~e effect of the exchange interaction with the electrons of the Fe zT ions on the occurrence of orbital momentum in a spherical (or cubic) shell. Because of hybridization with the Fe electrons it is thought to be not very essential how many electrons occupy ~he 5d shell. We take for the zero-order Hamiltonian the isolated Au ion. The per- turbation Hamiltonian is formulated as follows:

f * 1 Hex~(1) = i ~i (I) ~i(2) ~ ~(2)dv2 (3)

J 12

The summation is over all orbitals ~i of the eight Fe z+ ions and is an electronic wavefunction of gold. With the interaction given in eq. 3, the excitations 5ds/2 to kd are considered. The analysis, in the form of Feynman diagrams, is given in fig. 3. Only one member of a family is given. The exchange vertex is represented by an arrow with subscript Fe. The orbital operator is denoted by a vertex with subscript H. and the quadrupole operator by a subscript vertex H .

l q Orbital momentum (colinear with the field) occurs already in first order, whereas quadrupole coupling is different from zero from second order on. The misalignment phenomenon begins in third order because of the expected sin4@ dependence (see fig.2). Misalignment of quadrupole coupling starts in fourth order.

It is not easy to get numerical results because of the unknown strength of the exchange and the form of the proposed hybridization. However, if we take the exchange rather small so that the perturba-

308 J. Andriessen, H. Postma, Misalignment of Oriented Nuclei

5 d

_ _ ~ _ _

5d I \ ~ Hq

Fe ~ kd

I s t order, 2nd order, 3rd order, o rb i t a l o r b i t a l , quadru- misalignment moment pole moment

Fig. 3. Analysis of the occurrence of orbital momen- tum and quadrupole coupling of Au implanted in bcc iron.

tion series is rapidly converging, it is found that the misalignment angle is very small. This is understandable if we realize that a uniformly polarized medium has zero misalignment and a cubic crystal as shown in figure 1 is not much different from such a medium. One has to assume strong exchange effects to have sizable misalignment angles. These unlikely large exchange effects should be the conse- quence of the open-shell character of the 5d shell, mentioned earlier. This is in accord with the absence of misalignment in mercury. The quadrupole coupling estimated with our model is not conflicting with the experiment as it was in case A.

References

[ i] K.S. Krane, B.T. Murdoch, W.A. Steyert, Phys. Rev. Lett. 30 (1973) 321.

[ 2] K.S. Krane, W.A. Steyert, Phys. Rev. C7 (1973) 1555.

[3] I. Ben-Zvi, P. Gilad, G. Goldring, P. Hillman, A. Schwarzschild, Z. Vager, Phys. Rev. Lett. 19 (1967) 373.

[ 4] H.J. Ligthart, 'Nuclear orientation experiments concerning odd-A Gold isotopes', Groningen, 1982.

[ 5] P.D. Johnston, N.J. Stone, J. Phys. C5 (1972) L303.

[6] A. Aharoni, Phys. Rev. B2 (1970) 3794. [7] C. Demangeat, J. Phys. F5 (1975) 1637. [8] I.P. Grant, B.J. McKensey, P.H. Norrington, D.F. Mayers,

N.C. Pyper, Comp. Phys. Comm. 21 (1980) 207.