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DIRECTION OF THE MAGNETIZATION OF NICKEL NEAR TUNGSTEN IMPURITIES
Amikam Aharoni
To cite this version:
Amikam Aharoni. DIRECTION OF THE MAGNETIZATION OF NICKEL NEAR TUNG- STEN IMPURITIES. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-677-C1-678.
�10.1051/jphyscol:19711236�. �jpa-00214064�
L OIS D 'AIMA N TA TION
DIRECTION OF THE MAGNETIZATION OF NICKEL NEAR TUNGSTEN IMPURITIES
AMIKAM AHARONI
Weizmann Institute of Science, Rehovoth, Israel
R6sum6. - On estime la direction du moment magn6tique dans Ni polycristallin, p r b des atomes d'impuret6 de W, mini- misant 1'6nergie impliquk par la mkthode de Ritz. L'angle moyen entre le champ appliqu6 et le champ au noyau de W est proche de 2 7 O pour un champ applique de 1 000 Oe, en concordance avec les conclusions t i r h des Btudes expkrimen- tales sur la distribution angulaire des rayons y 6mis par W implant6 dans Ni.
Abstract. - The direction of the magnetic moment in polycrystalline Ni, in the vicinity of W impurity atoms, is estimated by a Ritz-method minimization of the energy involved. The average angle between the applied field and the field at the W nucleus is about 2 7 O at an applied field of 1 000 Oe, in agreement with the conclusions from experimental study of the angular distribution of the y-rays emitted by W implanted into Ni.
1. Introduction. - Experiments on y-rays emitted by nuclei recoiling into Fe or Ni foils, were inter- preted in terms of large angles between the applied field and the field at the site of the embedded nucleus [I]. This cail agree with measurements of the foil magnetization, only if the direction of the magnetiza- tion in the immediate vicinity of the impurity atoms differs considerably from its direction in most of the foil. The difference was suggested [2] to be due to
~nagnetostrictive anisotropy near the impurity atom.
This view was qualitatively confirmed by small angles found for W in low-magnetostriction permalloy [3].
Theoretically, angles of the correct order were estimated [4] for W or Nd in Fe, but for W in Ni
the model predicted much too large angles at high applied fields, evidently because it did not take into account the change in the Ni magnetic moment near the impurity. This effect is too large to be neglected for W in Ni, where a bound state is formed [5, 61 Analysis of neutron scattering data, assuming sphe- rical symmetry and a continuous material, leads [5,6].
to the graph which is reproduced here in figure 1, marked (< experimental D. But this curve need not be taken into account very accurately, since Comly et al.
[6] say that the curve marked (( theoretical )> in figure 1
is (< virtually indistinguishable )> from the other curve,
for the interpretation of the neutron scattering data.
Therefore, anything near these curves should do as well.
Radiol d i s t a n c e ( % I
FIG. 1. -The magnetic moment of Ni as a function of the distance from a W impurity atom.
2. The model. - In polar coordinates, r, 6, q, whose origin r = 0 is at the W nucleus and whose axes are parallel to the cubic axes of the Ni crystallite, the following functional form is assumed for the direction cosines of the magnetization of Ni, when the field H is applied along a direction defined by the polar angles O,, 9, :
a = cos o(r) cos cp,,
P
= cos o(r) sin q0 , y = sin o(r), for r 2 R , (la) a = D cos o(R) cos q o+
A sin (nr/R) sin (2 6) cos q,
= Q cos w(R) sin 9,
+
A sin (zr/R) sin (2 6) sin q,
y = Q sin o(R)
+
A sin (zr/R) cos (2 0),
for r
<
R.
(lb)Here A, p, q are parameters with respect to which the energy is minimized,
sin o(r) = tanh (qr) cos 8, - sech (qr) sin 6,
,
i2 = sin (pr)/sin (pR) , (2) and R was chosen as twice the distance between Ni nearest neighbours, taking into account the lattice distortion due to the insertion of the larger W atom, namely R = 4.979 8
A.
The magnitude of the magne- tization,MIM, = (a2
+ P2 i- y2)%, (3)
is seen to be I , for r
>
R. For r<
R, a reasonable requirement of the model (1) is that this magnitudeArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711236
C 1 - 678 AMIKAM AHARONl will increase monotonically to 1, as it does in figure 1.
A suficient condition for this is that
This is not the most general necessary condition, but it is convenient to have separate condition for each parameter, and such a choice is permissible in a Ritz method. Note, however, that the saturation state, A = 0, is always included in minimization under the constraint (4).
The magnetocrystalline anisotropy energy, the inter- action with the applied field, the exchange energy, the interaction with the mechanical strains of the lattice, and the magnetostatic self-energy, for the model (1), were calculated by the conventional expres- sions. By minimizing the total energy, A, p, and q were determined. Using these values in (1)-(3), the magnetic moment, M / M , was evaluated. Its average over 9 , 6, 41, and 6, is plotted in figure 1 as a function of r, for two values of H. It is seen that in the region of interest the dependence on H is not significant, and that the approximation to the neutron scattering data is good enough, which justifies the use of (1).
3. Results. - The potential problem for the model (1) has an analytic solution, which enables a calculation of the field B everywhere, once A, p, q are found by minimizing the total energy. In particular, at the W nucleus, r = 0, the angle O between B and the applied field is given by
sin O = a sin 8,/[l
+
2 a cos 0,+
a2]%,
a = 4 A Sin, (5) where Si is the sine integral function. The angle O thus calculated is practically independent of q,. Its dependence on do, which will be reported elsewhere 171, is symmetric with.respect t o 6, = 900, but not with respect to 8, = 450, because of the constraint (4).
The average of this 0, for a random distribution of p,, 6,, is plotted in figure 2. Also plotted in this figure is the deviation of the magnetization from its saturation value. The latter is obtained by averaging
APPLIED F I E L D (OERSTED)
FIG. 2. - The average, for a random distribution of crystal- lographic axes, of the angle @ between the applied field and the field at the site of the W nucleus, compared to the deviation of the average magnetization from its saturation value, in the same
field. The dashed curve represents arcsin (< sin @ > ).
the component of the magnetization in the direction of the applied field for a random distribution of crystallite axes, and taking its average value in a sphere whose diameter is the average distance between impurity atoms. It is seen from the figure that at about 1 000 Oe field, a measurement at the W nucleus should yield large angles, whereas this deviation is already too small to be detected by magnetic measurements in dilute alloys.
References
[I] BEN-ZVI (I.), GILAD (P.), GOLDRING (G.), HILLMAN (P.), [4] AHARONI (A.), Phys. Rev. 1970, B2, 3794.
ScHwARzscHILD and VAGER (Z.), Phys- [5] Low (G. G.), J. Appl. Phys., 1968, 39, 1174.
Rev. Letters, 1967, 19, 373.
[2] AHARONI (A.), Phys. Rev. Letters, 1969, 22, 856. [6] COMLY (J. B.), HOLDEN (T. M.) and Low (G. G.), [3] BEN ZVI (I.), GILAD (P.), GOLDBERG (M. B.), GOLD- J . Phys. C , 1968, 1, 458.
R ~ (G.), G KALISH (R.) and SPEIDEL (K. H.), [7J AHARONI (A.), Proc. Conf. Hyperfine Inter., Reho- Proc. Conf. Hyperfine Inter., Rehovoth, Sept., voth, Sept., 1970 (to be published).
1970 (to be pubhshed).
