PHYSICS

F. M. GAL’PERIN

EFFECTIVE MAGNETIC FIELDS AT ATOMIC NUCLEI OF FERROMAGNETS

(Presented by Academician N. V. Belov, October 7, 1964)

It is known that large effective magnetic fields \(H_{\mathrm{eff}}\) act on the atomic nuclei of ferromagnets and that the latter are usually negative, i.e., antiparallel to the direction of the magnetization vector. In paper \((^{1})\) it is indicated that comparison with experimental data reveals the weakness of the theory, and therefore at present it is expedient to seek empirical relations for calculating \(H_{\mathrm{eff}}\). Such relations are proposed by us below.

First let us consider the pure metals Fe, Co, Ni. The first of the indicated relations has the form

\[ H_{\mathrm{eff}}=-[10^2 m/\mu_B-H_s]\ \text{kOe}, \tag{1} \]

where \(m\) is the atomic magnetic moment, \(\mu_B\) is the Bohr magneton, \(H_s=N_s(\downarrow)H_A(\downarrow)\). Here \(N_s(\downarrow)\) is the number of unpaired (with spins antiparallel to the spin of the \(3d\)-shell) conduction electrons in the \(4s\)-band of the metal, \(H_A(\downarrow)\) is the negative contribution to \(H_{\mathrm{eff}}\) of a \(4s\)-electron (with spin antiparallel to the spin of the \(3d\)-shell) of an isolated atom, calculated in paper \((^{2})\) by the Hartree–Fock method. The first and second terms of (1) give the contributions due to the Fermi contact interaction of the nucleus, respectively, with the inner \(s\)-electrons of the ionic cores of the crystal lattice and with the outer \(4s\)-conduction electrons polarized by the unpaired \(3d\)-electrons of the same cores.* In order to measure directly the contribution \(H_s\), in paper \((^{3})\) the Mössbauer effect was studied on \(^{119}\mathrm{Sn}\) nuclei dissolved in Fe, Co, and Ni. The values of \(H_s/10^5\) Oe thereby obtained are close to the values of the quantity \(p\) introduced by us \((^{4})\), related to the contribution to the moment \(m\) of conduction electrons (see Table 1).

Fig. 1. Concentration dependence of \(H_{\mathrm{eff}}\) at \(^{57}\mathrm{Fe}\) nuclei in Fe—Ni alloys. \(a\) — experimental data; \(b\) and \(c\) — calculated: \(b\) — data for \(m\) from work \((^{12})\) were used, \(c\) — from work \((^{13})\).

Let us find \(N_s=H_s/H_A\) from the experimental data \((^{3})\). The numbers thus obtained

* In paper \((^{2})\) it is assumed that the first contribution is equal to \(-126(m/\mu_B)\) kOe and that, in agreement with Marshall \((^{5})\), \(H_s>0\); therefore in paper \((^{2})\) (in it \(H_s=N_s(\uparrow)H_A(\uparrow)\) for Fe) \(H_A>0\) is used. In all cases considered below, however, we use \(H_A<0\), since, as experiment shows \((^{3,6,7})\), \(H_s<0\). Marshall, who earlier had supposed that \(H_s>0\), taking into account the results of studies \((^{8,9})\) showing that the contributions of conduction electrons to \(m\) and to \(H_{\mathrm{eff}}\) are negative, acknowledges in paper \((^{10})\) that \(H_s<0\).

Table 1

Magnetic fields at the atomic nuclei of tin and ferromagnetic metals

* \(H_A(4s\uparrow)=+1850,\ +1951\) and \(+2053\) koersted for Fe, Co, and Ni, respectively (\(\uparrow\)—spin parallel to the spin of the \(3d\) shell; \(\downarrow\)—spin antiparallel).

are close to \(N_s=0.1|p|\); the latter expression is used by us in calculating the second term for Fe, Co, Ni, and for alloys. The second of the indicated relations has the form:

\[ H_{\mathrm{eff}}=-(m/\mu_B-1.5p)\cdot 10^2\ \text{koersted}, \tag{2} \]

\(H_s/p=150\) (koersted); the same value is possessed by \(H_{\mathrm{eff}}/m\) for the pure ion \((^{1})\). Tables 1 and 2 and Fig. 1 compare the values of \(H_{\mathrm{eff}}\) calculated from (1)—(2) with the experimental values.

We shall now find \(N_s\) from the data of works \((^{6-7,\ 15-18})\). It is seen from Table 3 that, for solutions of gold in iron, the measured values of \(H_s\) are in approximately the ratios \(1:2:3:4\), while the values of \(N_s\) found are close to the following discrete numbers: \(0.5|p|=0.4;\ 0.8;\ 1.2\) and 1.6, i.e., 5, 10, 15, and 20 times larger than in the case of pure iron \((0.1|p|=0.08)\). The calculated values of \(H_s\) corresponding to these discrete \(N_s\) values are also discrete, and the measured values of \(H_s\) are grouped around them; fields at the nuclei of impurity atoms have not yet been found for intermediate values of \(N_s\).

Analogous regularities are observed upon dissolving gold in Co and Ni (see Table 3 and Fig. 2).

Fig. 2. Dependence of the fields \(H_s\) at the nuclei of diamagnetic elements (\(^{198}\mathrm{Au}\), \(^{199}\mathrm{Au}\), \(^{119}\mathrm{Sn}\), etc.) dissolved in Fe, Co, Ni on the number \(N_s\) of unpaired conduction electrons in the \(4s\) band of the solvents. The straight lines were calculated (present work). Points are experimental data: \(a\)—from \((^{15})\); \(b\)—\((^{6})\); \(v\)—\((^{17})\); \(g\)—\((^{7})\); \(d\)—\((^{16})\); \(e\)—\((^{18})\); \(zh\)—\((^{19})\); \(z\)—\((^{3})\).

Thus, the fields considered at the nuclei of diamagnetic elements dissolved in Fe, Co, Ni apparently depend mainly on the solvent metal (in particular, on the quantity \(p\) characterizing it). The fact described recalls the appearance and disappearance of “giant” localized magnetic moments in dilute solid solutions of iron in nonmagnetic elements, observed at quite definite electron concentrations \((^{9})\).

Table 2

Magnetic fields at \(^{57}\mathrm{Fe}\) nuclei in ferromagnetic alloys

Note. \(p=[0,63\lambda^2\sum n_i(r_i-R)/A+\beta]/(1-\beta)\), where \(r_i\) and \(n_i\) are, respectively, the interatomic distance and the number of nearest-neighbor atoms in the coordination sphere of the lattice; \(\lambda\) is the atomic concentration of Fe; \(R=0,13[(3,75-C/2)^2+K(8-C)^2+21]\); here \(C\) is the electron concentration (the number of \(3d\)- and \(4s\)-electrons) of the isolated atom; \(K=2\) for \(C>8\); \(K=1\) for \(C\leq 7\); \(\beta=-1\) for pure Ni; \(\beta=0\) for Fe and Co. The values of the two arbitrary constants, \(K\) and \(\beta\), are given, each of them taking only two values (2 or 1 for \(K\) and \(-1\) or 0 for \(\beta\)). For all cases given in Table 2, \(R=2,733\) Å and \(H_A=-1361\) kOe. In 18 cases the alloys have a b.c.c. structure (), and therefore for them all quantities (except \(\lambda\)) entering the second term of (1) are the same; with constant \(m\), the field \(H_{\mathrm{eff}}\) should vary parabolically (as the square of the concentration \(\lambda\)), which is indeed observed experimentally; see Fig. 1. In 4 cases the alloys have an f.c.c. structure (*).

Table 3

Magnetic fields at the nuclei of atoms of diamagnetic elements in Fe, Co, Ni

In the case of Fe and Co \((r_1 < R)\), the conduction electrons are also in the \(3d\) band (\(s\)—\(d\) hybridization) and give a positive contribution, which is compensated \((\beta = 0)\) by the negative contribution due to antiferromagnetic polarization \((^{2,9})\). In the case of Ni this hybridization is absent (because \(r_1 > R\)), and there is only the above-mentioned negative contribution \((\beta = -1)\), which reduces both \(H_{\mathrm{eff}}\) by \(10^3\) oersted \((H_{\mathrm{eff}} = -190\) instead of \(-90\) oersted, according to (1)—(2)), and the moment
\(m = (N_d - 1 + 0.63 \sum n_i (r_i - R)/\text{\AA})\mu_B\) by \(\beta\mu_B\) (\(0.6\) instead of \(1.6\,\mu_B\) at \(\beta = 0\)). \(N_d\) is the number of unpaired \(3d\)-electrons of the isolated atom. Compression of Ni, under which it turns out that \(r_1 \leqslant R\) \((\beta = 0)\), should lead to a moment of \(1\mu_B\) (instead of the \(0.6\mu_B\) observed experimentally).

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