Cullity B.D. Introduction to Magnetic Materials. Second Edition (2008)

Fig. 5.11 Diffraction by (a) disordered and (b) ordered structures.

and 2 are exactly equal, because planes X and Y are statistically identical when the solution is disordered. Scattered rays 1 and 2 therefore cancel each other, as indicated in the sketch of the scattered wave form, and so do 3 and 4, 5, and 6, etc. There is no 100 reflection from the disordered solution. There is, however, a 200 reflection; this is obtained by increasing the angle u until def ¼ l so that rays 1 and 2, scattered from the (200) planes X and Y, are in phase. In Fig. 5.11b there is perfect order: C atoms occupy only cube corners and D atoms only cube centers. For first-order (n ¼ 1) reflection from (100) planes, scattered rays 1 and 2 are again exactly out of phase. But now their amplitudes differ, because planes X and Y now contain chemically different atoms, with different numbers of electrons per atom and hence different X-ray scattering powers. Therefore, rays 1 and 2 do not cancel but combine to form the wave indicated by the dashed line in the sketch. The ordered solid solution thus produces a 100 reflection. If we examined other reflections, from planes of different Miller indices hkl, we would find other examples of lines which are present in the diffraction pattern of ordered solutions and absent from the pattern of disordered ones. These extra lines are called superlattice lines, and their presence constitutes direct evidence of order.

The detection of order in magnetic systems with neutrons is exactly analogous. We now regard Fig. 5.11a as representing a lattice of chemically identical ions, C ions, say, each with an identical magnetic moment randomly oriented in space. For the same reasons as in the X-ray case, there will be no 100 neutron reflection. In Fig. 5.11b we have magnetic order: the spins on the corner ions are “up,” say, and those on the body-centered ions, “down.” There will now be a 100 neutron superlattice line, because the neutron magnetic scattering is sensitive to the differing directions of the spin moments on adjacent planes.

Before considering a particular example, we must qualify the remarks just made about “up” and “down” spins. No magnetic scattering at all can take place if the spin axes are normal to the reflecting planes, for reasons described by G. E. Bacon [Neutron Diffraction, 2nd ed., Oxford University Press (1962)]. Thus, if “up” and “down” mean

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Fig. 5.15 Projection of the MnF2 unit cell on a (010) face. Only the Mn ions are shown.

in the drawing, have opposite spin. The antiferromagnetic state of MnO has one feature not found in MnF2: The magnetic unit cell differs from the chemical (also called the nuclear) unit cell. Although a unit cell may be chosen in many ways, the choice must meet certain requirements. One is that the “entity” (chemical species, spin direction, etc.) at one corner of the cell be the same as that at all other corners. The unit cell in Fig. 5.16a is the chemical unit cell and has a manganese ion at each corner; it is also the magnetic unit cell above TN, because the spin directions are then random and the manganese ions are, in a magnetic sense, statistically identical. But when magnetic ordering sets in, the spin direction at one corner of the chemical unit cell is opposite to that at the three nearest corners. It is then necessary to choose a magnetic unit cell twice as large along each cube edge, as shown in Fig. 5.16b.

Neutron diffraction has disclosed spin structures in which the spins in alternate layers are not antiparallel but inclined at some angle other than 1808. MnAu2 is an example and

Fig. 5.16 Structure of MnO. (a) Chemical unit cell of Mn and O ions. (b) Chemical and magnetic units cells, Mn ions only.

To conclude this section we will consider what neutron diffraction has revealed concerning the spin structure of some transition metals.

5.3.1Antiferromagnetic

Chromium is antiferromagnetic below 378C and manganese below 95K. Neither has a susceptibility which varies much with temperature and neither obeys a Curie–Weiss law. (Inasmuch as they are both electrical conductors, rather than insulators, we do not expect their behavior to conform closely to a localized-moment, molecular-field theory.) Chromium turns out to have a peculiar magnetic structure known as a incommensurate spin-density wave, in which the magnitude of the spin forms a spatial wave whose wavelength is not an integral number of unit cell edges. Manganese has a complicated crystal structure with 29 atoms per unit cell, and develops a complicated antiferromagnetic structure with moments varying from 0.25 to 1.9mB per Mn atom.

5.3.2Ferromagnetic

For iron, nickel, and cobalt, neutron diffraction shows that the spins on all the atoms are parallel to one another and that the moment per atom is in accord with values deduced from measurements of saturation magnetization. (Furthermore, the diffraction experiments show that each atom has the same moment. This evidence disposes of a suggestion that had been made that a nonintegral moment, such as 0.6mB/atom, was simply an average, resulting from the appropriate mixture of atoms of zero moment and atoms with a moment of one Bohr magneton.) It has even been possible to discover the way in which the magnetization is distributed around the nucleus [C. G. Shull, in Magnetic and Inelastic Scattering of Neutrons by Metals, T. J. Rowland and P. A. Beck, eds, Gordon and Breach (1968)]. In cobalt this distribution is spherically symmetrical. In iron, however, the magnetization is drawn out to some extent along the cube-edge directions of the unit cell; in nickel, it tends to bulge out in the face-diagonal and body-diagonal directions.

5.4RARE EARTHS

The 15 rare earth metals extend from lanthanum La (atomic number 57) to lutetium Lu (71). They are all paramagnetic at room temperature and above. At low temperatures their magnetic behavior is complex. Because almost all the rare earths are antiferromagnetic over at least some range of temperature, it is convenient to give their magnetic properties some brief consideration here.

The rare earths are chemically very similar, and it is therefore difficult to separate them from one another or to obtain them in a pure state. This near identity of chemical behavior is due to the fact that the arrangement of their outer electrons is almost identical. However, the number of electrons in the inner 4f shell varies from 0 to 14 through the series La to Lu, and the magnetic properties are due to this inner, incomplete shell. Because the 4f electrons are so deep in the atom, they are shielded from the crystalline electric field of the surrounding ions; the orbital moment is therefore not quenched, and the total magnetic moment has both orbital and spin components. The total moment can become very large in some of the atoms and ions of the rare earths (see following text).

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The “light” rare earths, lanthanum (La) to europium (Eu), remain paramagnetic down to 91K or below, and then five of the seven become antiferromagnetic. [Promethium (Pm) is not found in nature and has no stable isotope; the properties of the metallic form are unknown.]

Of the eight “heavy” rare earths, six become ferromagnetic at sufficiently low temperatures, and five of these (terbium, Tb, through thulium, Tm) pass through an intermediate antiferromagnetic state before becoming ferromagnetic. Gadolinium (Gd) just misses being ferromagnetic at room temperature; its Curie point is 168C. All six ferromagnetic rare earths have magnetic moments per atom mH exceeding that of iron; if they only retained their ferromagnetism up to room temperature, they might make useful, although expensive, materials. The one with the largest moment is holmium, Ho, which has mH ¼ 10.34mB/ atom, or almost five times that of iron (2.22mB). The rare earth atoms are so heavy, however, that their saturation magnetizations s0 per gram at 0K are not very different from that of iron. For example, we may calculate, by means of Equation 4.27 and the moment per atom given above, that s0 for holmium is 351 emu/g, compared to 221.9 emu/g for iron.

The rare earths and their alloys have provided a rich field for research by neutron diffraction. The spin structures of the antiferromagnetic states include helical and even more complex arrangements. Even the ferromagnetic structures are sometimes unusual. Consider, for example, gadolinium and holmium, which have the same crystal structure (hexagonal close-packed). Ferromagnetic Gd has a simple arrangement of parallel spins, like iron. Antiferromagnetic Ho has a helical spin structure like that of Mn Au2 in Fig. 5.17; the spins in any one hexagonal layer are all parallel, but they progressively rotate about the c-axis from one layer to the next. In the ferromagnetic state below 20K, this spiral spin structure is retained, but added to it is a ferromagnetic component of spins parallel to the c-axis in every layer. (The c-axis is normal to the hexagonal layers.) The resultant of these two components, one parallel and one at right angles to the hexagonal layers, gives ferromagnetic Ho a kind of conical spin arrangement.

5.5ANTIFERROMAGNETIC ALLOYS

Antiferromagnetism is now known to exist in a considerable number of alloys, most of them containing Mn or Cr. It is more common in chemically ordered structures, which exist at simple atomic ratios of one element to the other, like AB or AB2, but it has also been found, surprisingly, in some disordered solid solutions.

An example of antiferromagnetism in an ordered phase has already been given: MnAu2 in Fig. 5.17. In the same alloy system, the phases MnAu and MnAu3 are also antiferromagnetic. Some other antiferromagnetic ordered phases are CrSb, CrSe, FeRh, FePt3, MnSe, MnTe, Mn2As, and NiMn. The spin structure of the latter is interesting. The unit cell is face-centered tetragonal and the (002) planes, normal to the c-axis, are occupied alternately by Ni and Mn atoms. Each (002) layer of atoms, whether all Ni or all Mn, is antiferromagnetic in itself, i.e., half the atoms in one layer have spins pointing in one direction and parallel to the plane of the layer, and the other half have spins pointing in the opposite direction.

Among disordered alloys antiferromagnetism has been observed in Mn-rich Mn–Cu and Mn–Au alloys. They have a face-centered tetragonal structure. All the spins in any one (002) plane are parallel to one another and to the c-axis, but the spins in alternate

PROBLEMS 173

(002) layers point “up” and “down.” Disordered MnCr is also antiferromagnetic. It is body-centered cubic, with the spins on the cell-corner atoms antiparallel to those on the body-centered atoms. In none of these examples is there any chemical ordering. Each lattice site in, for example, the Mn–Cu alloys is occupied by a statistically “average” Mn–Cu atom, and each average atom appears to have a magnetic moment of the same magnitude. This behavior is understandable on the basis of the band theory, which envisages all the 3d and 4s electrons as belonging to a common pool, but not on the basis of a localized-moment theory. If the moments were localized, the various exchange interactions (molecular fields), between Mn–Mn, Mn–Cu, and Cu–Cu atoms, would have different orientations from one unit cell to the next in a disordered alloy, so that it would be difficult to understand how any long-range magnetic order could result.

Finally, it should be noted that the susceptibility–temperature curves of alloys do not usually give evidence for, or against, the existence of antiferromagnetism, because a Curie–Weiss law is not often followed. Neutron diffraction is the only sure test.

PROBLEMS

5.1MnF2 is antiferromagnetic and at high temperatures its Curie constant per mol is 4.10. Its molar susceptibility xM is 0.024 emu/Oe/(g mol) at the Ne´el temperature. Assuming the ideal behavior described in Section 5.2, and assuming all the magnetic

moment of Mn is due to spin, calculate

a.The value of J.

b.The spontaneous magnetization of each sublattice at 0K.

c.The molecular field acting on each sublattice at 0K.

d.The angle a in Fig. 5.5a when a field of 1.2 T is applied perpendicular to the spin axis of a single crystal at 0K.

5.2Show that Equation 5.24 reduces to Equation 5.2 at high temperatures, to Equation 5.18 at TN, and to zero at 0K.

5.3In a body-centered tetragonal arrangement of metal ions (Fig. 5.12), the cell-center ions form the A sublattice and the cell-corner atoms form the B sublattice. If an A ion is to have only B ions as nearest neighbors, find the range of allowable values for c/a.