Some Problems of Phenomenological Theory of Ferro and AntiferromagnetismS. V. Vonsovsky and E. A. Turov Citation: Journal of Applied Physics 30, S9 (1959); doi: 10.1063/1.2185984 View online: http://dx.doi.org/10.1063/1.2185984 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/30/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Water nucleation: A comparison between some phenomenological theories and experiment J. Chem. Phys. 137, 124702 (2012); 10.1063/1.4754662 Nuclear Magnetic Resonance in Ferro- and Antiferromagnets Am. J. Phys. 41, 935 (1973); 10.1119/1.1987430 Some Critical Properties of QuantumMechanical Heisenberg Ferroand Antiferromagnets J. Appl. Phys. 40, 1546 (1969); 10.1063/1.1657755 Linear Heisenberg Model of Ferro and Antiferromagnetism J. Math. Phys. 5, 1091 (1964); 10.1063/1.1704213 Theory of Ferro and Antiferromagnetism by the Method of Green Functions J. Appl. Phys. 34, 1153 (1963); 10.1063/1.1729412

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

138.251.14.35 On: Sat, 20 Dec 2014 09:44:39

OU.RNAL OF APPLIED PHYSICS SUPPLEMENT TO VOL. 30, ~O. 4 APRIL, 1959

Some Problems of Phenomenological Theory of Ferro- and Antiferromagnetism

S. V. VONSOVSKY Ar-.-n E. A. TUROV

Instifute oj .lIe/oJ, Physics, Academy oj Sciences, Sverdlovsk, U.S.s. R .

This paper presents a general review of the modern state of phenomenological theory of ferro- and anti­ferromagn etism and the results of some investigations of the authors in this Geld .

N the quantum theory of ferro- and antiferro­magnetism there a re essential difficulties when at­

empting to build adequate microscopic models. To eceive quantitative or yet only qualitative results one

must make many poorly grounded simplifications, which essen the value of the results of the theory to a great

oxtenl. Therefore it is very useful to t ry to avoid these ~[fficu lties and to create the general phenomenological theory in which there would be a minimum of all

nproved model approximations. This would give a Food chance to fo rmulate those questions that have real pbysical meaning.

Weiss l was the fi rst \vho long ago used th is method in creating his theory of ferromagnetism . This method was eveJoped further in the works of Van Vleck2 and Neel ,3 ho created the theory of molecular field for antiferro­d ferrimagnetism. But the phenomenological theory molecular field has only a limited application (high

emperatures) . The physical nature of molecular field as unknown when creating this theory. Therefore it is tural that the physicists attempted to set up the

tomic theory of ferromagnetism. Ising's classical atomic odel,4 which played a great role in the development of e microscopic treatment of ferromagnetism, should be entioned here. The fact that the nature of the inter­

actions between the atomic magnetic moments still emained unknown was the defect of this model as well

of the theory of Weiss. The physical meaning of this exchange interaction was understood only after the

ork of Frcnkelb and Heisenberg. 6

The principal approximation of the molecular field theory consists in taking into account only the long­

nge magnetic order. Therefore it is only natural that attempts to improve tbe Weiss's theory were made, consideri ng the short-range magnetic order. The neces­sity of this improvement resulted from the sho rt-range character of exchange interaction also.

A number of papers (of Peierls7 and others8.9) were

IPublished, in which the short-range correlation in spin iystems were taken into account using different methods. ~ut in fact these ' ~ improved" treatments in some re-

I P. Weiss, J. ph)'S. radium 6, 661 (lOOn 2 J. H. Van Vlcek, J. Chem. P hys. 9, 85 (1941 ). I L. Ncel, Ann . phys. 3,137 (19-18). 4E. Ising, Z. Phvsik 31 , 253 (1925 ). ']. I. Frenkel, i. Physik 49, 31 (1928). ~W. Heisenberg, Z. Physik 49, 619 (1928). . R. Peieris, Proc. Cambridge Phil. Soc. 32, 477 ( \936). a L. S. Stilhance, Zhur. Ek.sptl. i T co reL Fiz. 9, 4-32 (1939 ).

(I'S. V. Vonsovsk y, Doklady Akad. :\fauk. S.S.S.R . 27, 550 94ll).

95

speets worsened the principal positIOns of the theory because of a greater number of rough mode! approxima­tions. However, the results of molecular field theory, in spite of rough character of approximations, gave good qualitative explanation of ferromagnetic propertics of matter. The solution of this paradox was possible afte r the appearance of exact thermodynamic treatment of ferromagnetism.

The fundamental \ ..... orks of Landau !O on the theory of phase transit ions laid the foundation for the beginning of creation of the consistent thermodynamical theory of condensed mediums. His method permitted the estab­lishment of a base for all the correct resul ts of Weiss 's theory, and the building of the exact thermodynamical theory of magnetic properties of ma Uer. *

The Landau method consists in using the most general ideas of group theory. The explicit form of the expansion of thermodynamic potential (in series by the power of small parameter) near the phase transition points is obtained using the general symmetry considerations. The magnetic state equation can be found from the general conditions of thermodynamic equilibrium. The recent work of Dzialoshinskyll can be mentioned as an example of the great possibilities of this method . The physical explanation of "weak" ferromagnetism, for a long time known in Fe203 and recently found in a number of other compounds, was given in that work. But the thermodynamic theory also possesses some defects. Firstly, it contains some phenomenological con­stants which require detailed information about atomic structure of crystals. Secondly, the resu lts of this theory can be obtained only in the narrow temperature interval near the phase t ransit ion points. This re!:'ults from the fact that only in this temperature range we Can be satisfied with the few first members in the expansion of thermodynamic potential by the powers of small parameters (magnetization). Therefore it was necessary to de\'elop the lheory to be applied to the low-tempera­tu re range. The wave funct ions and energy spectrum of crystal elect rons must be determincd for complete microscopic description of these substances. Howc\'er

10 L. D. Landa.u, Zhur. Eksptl. i Teoret. Fiz. 7, 19 (1937 ). ,. One of the authors~l and Cinsburg2l ftrst applied tbis method to

ferromagnetic phase transitions. The Landau method can be extended [reference 22 and 1. La.ndau, Zhur. Eksptl. i T co rel. Fiz. 7, 1232 (l937)J for the case of magnet ic short.range ord er with the help of statistical thermodynamics theory of the fluctuat ions. The work of Zaitzt!v [ Il. 1\1. Zaitzcv, Zhur . Eksptl. i Teort!t. Fiz . 34, 1302 (1958); tran slation: Soviet Phys . JETP 7, 898 (1958)J Q1USt be ment ioned in connection with this problem also. • II r. E . Dzialoshinsk)' , Zhur. EksPll. i T eo reL 1"iz. 32, 15.H (1957 ) [translation; Sov iet Phrs. J F:TP 5, 1259 (1957)].

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

138.251.14.35 On: Sat, 20 Dec 2014 09:44:39

lOS S. V. VONSOVSKY AND E. A. TUROV

this problem cannot be solved practically. There is one important reason why the macroscopic bodies cannot be in precisely stationary states, besides the mathematical difficulties result ing from great many degrees of freedom. The fact is that the breadth of energy levels, resulting from their quantum nature ["'E~(h/"'t)J, is much greater than the splitting of these levels. Therefore, it is necessary to use the quantum-statistical treatment with the density matrix for precise description of macro­bodies. This method was realized in the above-men­tioned work of Heisenberg. 6 He substituted the t f ue energy spectrum of crystal electrons by statistical mean values-the energetic centers of gravity. However, this method is a very rough approximation of quantum­statistical treatment. It can be used fo r description of the properties of magnetics only in the region near (or more exactly above) the Curie and Keel points, as in the method of molecular fields. Therefore, it is necessary to develop a consistent quantum-statistical theory fo r the wide range of temperatures including the very low ones. There is no such theory at present, however, but it is possible to mention cases when we can use the quantum-mechanical theory for macrobodies. This may be done when calculating the ground state (the lowest energy state) and the nearest weakly excited states. For the latter case the motion of complex macrosystems may be described with high precision as the free gas of quasi­particles or elementary excitations. It is easy to find the energy spectrum and wave functions for this gas.

The method of quasi-particles was used in microscopic ferromagnet ic models with great success. These models, however, possess some defects caused by rough simpli­fications. Tn 1935 Landau and Lifshitsl2 gave the pro­gram for the generalization of the phenomenological theory of ferromagnetism for the range of low tempera­tu res. Herring and Kittel t3 were the first to use this method for creation of the phenomenological theory of spin waves (ferromagnons) . t

2

f erro-, antife rro-, and ferrimagnetics differ from all other substa nces by thei r atomic magnet ic structure. This means that magnetoactive ions in crystals of these substances possess nonzero time average values of spin or orbit magnetic moments ( Sj)t). All these moments in ferromagnetics are parallel. Therefore a spontaneous resulting moment appears in these crystals. In antife rro­magnetics we have the anti parallel arrangement of moments with complete compensation in each ele­mentary cell. This compensation is absent in fcrri­magnctfcs, and therefore in these crystals the nonzero spontaneous difference resulting moment exists. And lastly in paramagnetics the time average moments in all

a L. D. Landau and E. 1\1. Lifsbits, Physik. Z. Sowjetunion 8, 153 (1935); E. M. LifshilS , J. Phys. U.S.S.R. 8, 61 (1942 ).

n C. Herring and C. Kittel , Phys. Rev. 81, 869 (1950). t See also the works of Heller and Kramcrs [G. Heller :'ll\d .rI.

Kramcrs, Proc. Acad. Sci. Amstcrdam 37, 378 (1934)J and DOring [W. Doring, Z. Physik 124, 501 (1947)].

lallice points are equal to zero.t The space distributio of (SJ)t .can be given as the 0 function

I:(Sj),o( r - r '). (!

In Eq. (1) the sum is taken over alllallice points. The function is the analytic expression of discrete nature 0

crystals \vith magnetic atomic structure. But this ki of description is not always necessary as can be see from the Debye's heat capacity theory. The crystal be regarded as a quasi-continuous medium, if We tak into account only such modes of motion in which t discrete lattice structure is not important. This fo the basis for the phenomenological description of rna netics. The term Hquasi" emphasizes the fact that w have the medium with th~ important properties discrete crystal atomic structure (crystal symmet elements, differences in types of lattice points and types of magnetoactive ions, and so on) .

The oscillating character of motions must be tak: into account when selecting the criterion for th degrees of freedom of crystal at which the lattice haves as quasi-continuum. If the characteristic wav length of this mode of mot ions is much greater than th lattice constant, the crystal can be considered as quasi. continuum. A large number of atomic particles simuJ taneously take part in these oscillations. Therefore tb are of cooperative character. The transition from di .. crete to continuous description is expressed mathe .. matically by placing the magnetic moment densit functions M j(r) instead of the ° function · (1). Tho number of these functions is determined by the number of nonequivalent magnetic ions§ in the elementary crystal cell. Each such function M j(r) corresponds to different magnetic sublatt ices in the discrete de­scription .

The functions M i(r ) in the ground state of magnetic are considered as uniform: Mjo. Such uniform Hspread· ing" of magnetic moment over the volume correspondi to completely ordered distribution of the atomic roo-­ments over the magnetic sublattice points. The small fluctuations of magnetic moment densities, .1M j = Mj(r) - M jo, arise at weak perturbations of the system. For solution of concrete problems of theory one must know I he equations of motion for magnetic moment densitiCi (classical or quantum). At first we must find the Hamiltonian X(M ,) II of system as a fun ction of these densities. \Vhen determining the Hamiltonian we must use some results of quantum theory of magnetism and also the experimental data for the crystal and magnetic atomic struClure of the substance:; in question . These data may be taken from x·ray and neutron diffraction experiments.

t It is esscntial to note that averaging is connccted in our case with time and not with physical infinitesimal volume. .

§ The ions, which cannot be transformed into onc another In translation. .

11 For the casc of high temperatures it is tbe thcrmodynamlC potcntial.

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

138.251.14.35 On: Sat, 20 Dec 2014 09:44:39

PHENOMENOLOGICAL THEORY OF FERROMAG NETISM liS

Two types of terms can be expected in this Hamil­tonian. Firs t, the terms, resulti~g from the isotropic I trostatic exchange lI1teractlon. These quantum ~~i determine the very existence of bodies wi th the rna(Detic ordered structure and also the temperature

n,e of their existence. Th en the te rms, caused by the ra iJotropic magnetic (relativistic) interactions. They :termine the orientations of the magnetizations M j in the crystal as well as the behavior of magnetic sub­taD£t:S under different external influences (magnet ic ~Idi elastic stresses, and so on) . The order of magni­Wde ' af exchange terms ('""-' 10--13 erg/ atom) is much maher than of the " magnetic" ones ('" 10--]6 erg/ atom) , Therefore the last men tioned may be considered as Qnall perturbations.

3

Now, let us use the symmetry properties of magnetic ayotals to find the phenomenological Hamil tonian

(M,). The fundamental characteristic of magnetic qys;tal, namely, its magnetic mome~t, is connec~ed "lth microcurrents, and therefore WI th the part icle

ties. Therefore in such crystals we must consider flle symmetry as to the change of t ime sign

(J~ -;:t:R operation '4) . Using this symmetry element we can determine the general criterion for crystals wi th atomic magnetic structure. If the R operation is in itself the symmetry element of the crystal (namely, it fJoes not change its thermodynamic equilibrium state), ~ this crysta l does not possess any magnetic struc­we. In fact, by applying R to magnetizations Mj, we change their signs: R· M j = - M i ' Since the crysta l state does not change under th is operation, the mag­oetim:tions also remain invariable (Mj= - Mj) . There­fore. tbere is only one possibili ty, i.e., that all M {== O, which means that in this case the magnetic structure is abient. If on the contrary, the R operation is no t the symmetry element of the crys tal, then the latter can pos$eSs some atomic magnetic st ructure.

If we know the atomic symmetry of crystal lattice, ~ We can find theoretically all ordered magnetic states permitted by this symmetry. The crystal lattice sym­metry elements must contain the symmetry clements of each such state. ** Using these considerations we can find under what conditions among diffe rent magnetic statCi the phase t ransitions of the fi rst or second order

14 L. D. Landau and E . J\L Lifshits, Electrodynamics of Continu­"UJ'Medinms (Moscow, 1957).

, EAch crystal possesses a definite set of symmetry operations, which transforms it in itseH (rotation axes, rotation-reflection 111$, screw axes, reflection planes, glide planes, and t ranslations). ~ form the so-called space group of symmetry. The R operation !Ililielf' and in different combinations with all others must be :clUded in this group in the case of crystal without magnetic V ruMcture (param'!-gnetics ). Tavge.r and Zaitzey [B . A. Tavger !lnd ... Zaltzev, Zhur. EksplL I Teoret. Flz. 30, S64 (19.)6) ; ~e a detailed group theoretical investigation of crystal magnetic

.. metry. the The number of magnetic symmetry clements cannot exceed fJJn number of atomic symmetry elements. Therefore the magnetic tr metry group usually forms the subgroup of crystal symmetry

take piace.IO"s Sometimes not only one but a great many magnetic states may correspond to the same set of symmetry elements. T hen, if the crystal with definite atomic structure admits at least one of these magnetic states, it admits them all as well.

One must know the explicit form of Hamil tonian JC(M j) and its minimum to find the conditions under which one or the other ordered magnetic state is realized . One should make a number of notes here.

(a) JC(M j) must rema in invariant under all sym­metry transformations of crystal lattice. In this case the magnetic structure should not be considered, because for all symmetry operat ions (including R) of paramag­netic state, all lattice points and axes are transformed into equivalent ones. Therefore each magnetic moment gets in the equivalent lattice point and is oriented along the equivalent axis. Hence there is no reason for the change of JC.

(b) The translations over distances equal to the period of elementary magnetic cell multiplied by whole numbers leave all the M j invariant. T herefore such operations may be excluded. Hence the invariance of X only wi th respec t to all rotation and screw axes, reflec­tion, glide planes, and the R operation is required.

(c) One must know the t ransformation ru les for different M j (the number of which v is determined by elementary cell structure) for all symmetry operations. In other words, it is necessary to know the index permutations P(1,2,· .. ,v) of different M j for all sym­metry operations. tt The t ransformation properties of functions M i include their axial character as well.

(d) The expansion X as a series of powers of M j must contain only even powers because of its invariance with respect to the R operat ion.

The most general form of expansion of X(M j ) IS

given by the formu la

M i a. Mj'fJ __ a"f a! a.Mja. aMi'fJ X=Aa.{3ii' - - --+Ba.fho"'---- ----

M j o Mj,o M fO Mj,o aT., aro

Mja. M j '{3 M j "., M j '"6 + Ca.fh6 ii'i"i'''--------

M j o M j'o 1l1 j " o Mj" ,o

+ ·· ·-i L:MjHj,M_L:M jH. (2) ii'

Here Aa.t/i', Ba.lht/i', and Ca.fJ"f6ii'i"i"', ... are tensors of energy density dimension, independent of Y a , caused by isotropic electrostatic and anisotropic magnetic inter­actions; a"f are length dimension constants of the order of magnitude of lattice parameter, 0', (3, 'Y, o· .. =x, y, z;

16 L. D. Landau and E. M. Li fshits,Stal islical Physics ()'1oscow, 1951).

tt These permutations are caused by the mutual transitions of magnetic ions from one lattice point into another under the sym­metry operat ions. Dzialoshinskyu was the first to point out this fact. Earlier it was wrongly thought that difTerent ~[ j tra nsform only independently from each other under all symmetry opera~ tio'ns. Only identical index permutation was taken into account. All other permutations were not considered, which could lead to loss of important terms in X_

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

138.251.14.35 On: Sat, 20 Dec 2014 09:44:39

125 S . V. VONSOVS KY AND E. A. TUROV

j, j', .. = Jil )2, .. ' J j.; V is the number of magnetic sublattices; Mja-a. components of vectors M j . Summa­tion is made over all indices occuring twice in (2) . The terms with the space derivatives aMja./arfJ describe the deviations from the unifo rm magnetization in the system. In (2) we may be satisfied with the terms of second order only in derivatives in the case of long-wave oscillations. The ratios M ja/ M i for each sublatt ice j a re the directional cosines about the axes a. Nea r the Curie and Neel points the expansion (2) is carried out in fact over small dimensionless parameter Mia./M jO (M jO are the maximum ground-state magnetizations) , The lit! jal M jO may be not small ("-' 1) at low temperatures. In th is case, therefore, t\\'o additional problems of dete rmina­tion of the system ground state ("vacuum") and selec­tion of elementary excitations ar ise. T he magnetic field H iM describes the long-range part of magnetic inter­act ions, w'hich are not included in tensors A, B, C, In general, H/I must be found from the ::'vlaxwe!l's equations.1a \Ve can be satisfied only wi th magneto­statics equations rotH/v = 0 and divH/1 = - 411" divM j, if we neglect the displacement currents. If we take in to account the dipole inte raction in the first approxima­tion, we get the expression for the magnetic field intensity

1 J divM j( r')dr'

H iM. = V' I r-r' l f _M_j_(r_')n_o, I

dS . Ir- r' l

(3)

Here in the first term the integration is carried out over the volume, and in the second term over the external surface of specimen (no is the unit length vector normal to this surface.) T he last term in (2) is the magnetic energy wi th respect to external magnetic field H .

4

:"Jaw let us il!us t rate this theory by two isomorphous rhombohedric crystals Cr,O" and a-Fe,O, (the space group D3d 6) . i n the elementary cell there a re four magnetoactive ions Cr+3 or FeH placed along the [ 111J axis (Fig. 1) . According to neutron diffraction data, the magnetic unit cell coincides with the chemical one in these crystals.

- -~-0-- [/I/J (a)

u, u, I I I I I 3 2

I I [/II] • I

b 0 b 0

I I (b)

'FIG, 1. (a) The unit cell of rhombohednc cl) stal \a-Fe10 3, Cr103); 0, ions Fc+3 or CrTl. (b) a,-the pOints of InterseCtion of lj~ axes v.·ith [ lll J; b, the points of inversion and of intersection of reflection plane for S6 axes with [111].

( 0 ) ( b) ( c )

FiG. 2. Possible types of antiferromagnetic structurcs; (a) Cr (b) Fe20s; (c) u nknown. .

T he following symmetry operations of paramagn~ stateH;,17 must be taken into account when we wan examine the possibility of any of magnetic states

2Ca, 3U2, I) 256) 3iTd, R.

Here C3 are the th ird-order axes parallel to [ 111J; are the second-order axes, perpendicular to [ l11J, tersecting [ 111 J in points " a" [ Fig. l (b)] ; 1 andS. the inver~ion and the sixth-orde r rotation-reflection ' with respect to points "b" on [ 111J axis, respective if d are the glide planes with t ransla tion for one h period along the [ 111J axis and perpendicular to the aXIS .

Neutron diffraction experimentsl8 .19 proved the exii ence of two antiferromagnetic st ructures for Cr [Fig. 2(a)J and o-Fe,03 [ Fig. 2(b)J, respectively. T third structure [ Fig. 2(c)J is possible also. In Fig. there are shown only the mutual magnetic marne orientations in sites of the cell. 1\10st different orien t ions of antiferromagnetic axis!! are possible as w

(for example, along the [ 111 J, normal to [ 111J, etc. The set of symmetry elements of given type of

netic structure must transform the unit cell mome distribution in itself. For instance, if in the cell there the resultant momen t m = LiSj)I~ O, it must r . invariant under all of these operations. In all magnetic states of "a" and "e" types, among allowable operations, there is one complex I · Roper tion . Applying it to m, we get -m, which gives m­because of the invariance req uirements. Therefore in ' to;

and "e" afm (a ntiferromagnetic) structures ferraro netism is impossible. This agrees \ .... ith the observ properties of C r~03. The symmetry requirements allo three possible varieties in the "b"-type states : (1) th - '-6 R. W. G. Wyckoff, The StrltetuTe of Crystals (New 'I 1931).

Jj fnfernalionalle rabellClt ZllT BcsfilllIJllmg fon KrislaUstruklfV

(1935 ), Vol. 1. ) 18 Shull, Shtrauser, and \Vonan, Phys. Rc\,. 83 , 333 (1951 . I~ B. N. Brockhause, J. Chern. Phys . 21 ,961 (1953). Iat tt That is, the axis along which the magnetizations of sub

tices are oriented in both directions.

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

138.251.14.35 On: Sat, 20 Dec 2014 09:44:39

PHENO",IE:-IOLOGICAL THEORY OF FERROMAGNETISM 13S

afm axis coincides with [111 J ;. (2) the afm a xis lies in the iyrometry plane (1 d (ItS Orienta tion In Ud IS dClcr­"uned from the minimum value of Ha,;,iltonian) ; (3) the aim axis cOincides With Oile of the L 2 axes, normal to [IIIJ [Fig. 3(a), (b) , (c)].

In case (I) the symmetry elements arc given by ( ~) "itb Ule exception of R. L et us assume that there is the reiultant moment m;;eO in the elementary cell. 'We can J'fiOlvem into two components : mi l parallel to the [l 11J axil and m,l normal to [11 1J axis. mJ. is nol invariant with respect to the C 3 transformation, and m lJ Lo li z. Therefore the in variance of m=mu+ mj, (wi t h respect IOC, and U,) is possible only if m=O. Thus fcrromag­netiim is impossible in afm structure (1) . In a-Fc:tOa thf. is the case at T:'0250°K.

In (2) the symmetry elements are:

I, iT" and U, [normal to plane of Fig. 3(b)]. (5)

fie operations permit rotations of pairs of moments Ito S. and Sz, Sa toward onc another from the plane of ,. 3(b) . This rotation creates a resultant moment o along the U'!. axis. Therefore ferromagnetism is

&ifble in this case. In a-FeZ03 it is observed as "weak" magnetism in the temperature range from 2S0oK to

SO"K. The structure (3) possesses the symmetry elements

(6)

e elements permit any ro tation of moments Sj (at angles), causing the appearance of a resul tant

ent m~ O, oriented in the qt/, plane. I n a-FeZ03 this ture does not occur.

Now let us consider the isomorphous crystals FeC03,

I.CO" and CoCO,. They are of rhombohedric sym­type also, but they possess two magnetic ions

10+2, Fe+z, or CO+2 in the uni t cell (Fig. 4) . The etry operations of paramagnetic sta te are given as

fore by (4). The U 2 axes intersect the [ 111J "xis at Ints "e" (Fig. 4). The J and 56 must be taken wi th

eipect to "d" points. A translation for half a period along the [l 11J axis corresponds to the ii , glide plane (from lattice point 1 toward 2) . Only one afm structure

[IIIJ [IIIJ

3

4 Ie)

;.3. Th ree possible types of orienta tions of arm axis in structure of Fig: . 2(h) (a-Fe2Da) .

u. u. I 1 1

2 I

I I • • • [III J 1 1

d c d c d I I

FIG. 4. Order of magnetic ions (1 ,2) along [ 111J in crystals rvlnC03, FeCOl , COC03. Here c represents the points of inter­section of Uz axes with [ 111 J, and d the poin ts of inversion andof intersection of reflection plane for S6 axes with [1 t 1].

with three possible types of orientations of afm axis [Fig. 5(a), (b), (c)J is possible in this case. It may be proved by analogy with the foregoing that for structure of Fig. 5 (a) the ferromagnetism is impossible, while for those of Figs. 5(b), (c) it is possible. The former occurs in FeC03) and the latter is observed as " weak" ferro­magnetism in l\,InC03 and COC03.~

The aforementioned theoretical analysis permits one to determine whether or not any magnetic structures are compatible with the given crystal lattice symmetry. In particular we obtained the general conclusion as to the possibility of coexistence of the whole set of magnetic structures.

5

Further invest igat ions of Hamiltonian (2) a re neces­sary to examine the conditions of occurence of any of magnetic states. Let us consider FeC03, ,!\I nCO:!, and CoCO, again.§§ For the symmetry operations (4) the index t ransformat ions of densities M 1 and M2 are :

C 3 : 1 -71,2-72; U 2 :1+=±2 ;

7, S,: 1 -> 1, 2 -> 2; ii,: 1 <=' 2. ( i )

Using (4) , (7) , and axial properties of M j vectors, we find from (2) (OZ axis il [ 1I1J) :iiI:

A A, H JC = - - ,Mj.M j.+- M ,.M2.+--(.1f ,,'+ M 2" )

2M 0- M 02 2lvl 02

H, D +--M"M2,+-(M,~¥f,,-M, .M, y)

.UOZ .U02

C C, + - - (vMj.vMj.)+-(vM,.vM,.) + ...

2.M02 M 02

- }(M,+ M ,)(H '''+2H), (HJf= H ,M+H,M). (8)

I t is easy to prove that Eqs. (8) are invariant for all symmetry operat ions (3) . The terms wi th A and A, describe the exchange in teraction inside a nd between the two magnetic sublattices. The terms with C and C 1

describe the influence of magnetiza tion nonuniformitics.

20 A. S. Borovic-Romanov and \1. P. Orlova, Zhu r. Eksptl. i T (."Oret. Fiz. 31 , 5;9 (1956) [transla tion: Soviet Phys . J ETP ", 531 ( 19;7)].

§§~This is va lid for a- Fc20J also. II H In (8) we neglect all the terms o j the order abovc the second

~v hile in .the terms wit~ r:;.\f ja only the tcrms of isot ropic excha ng~ mteract lOn are taken IOto account.

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

138.251.14.35 On: Sat, 20 Dec 2014 09:44:39

14S S . V . VONSOVSKY AND E. A. TURO V

[II/J [II/J [II/J

c C CTd - 0 -- - U2

2 2 2

(Il ( II ) ( III )

FiG. 5. Three possible types of orientations of afm axis in afm structure of MnC03, an d CoC03•

The terms with Band B\ determine the magnetic aniso tropy inside and between the sublatliccs, re­spectively. The term with D of the magnetic nature also was found first by Dzialoshinsky.ll This t erm is the direct result of the t ransformations (i) . The term with D determines the "weak" ferromagnetism and influences the resonance frequencies and their dependence from external conditions as well. Though this term is quad­rat ic in Mj, it is different for different "uniaxial" crystals.21 For example, in the case of tetragonal sym­metry (,\1 nF 2) this term is not like that in the crystal of rhombohedric system (MnCO,).

The minimization of Hamiltonian (8) is our next step. Using th is operation we can find the relations among the coefficients A, B, . . . . These rela tions qc­te rmine any of the admissible magnetic states of given crystal symmetry. Here we must distinguish two differ­ent cases, namely, the cases of high and low temper­atures.

6

The formula (8) gives the expansion of po tential ¢ in series by the powers of small parameter Jl fa/ M jO for high temperatures near the phase transition from paramagnetic to ordered magnet ic state. 'We introduce two new vectors

(9)

instead of vectors Mj according to Dzialoshinsky,ll to show morc dearly the fm and afm componen ts of magnetic struc ture. Th e m vector characterizes the fm statc while I charac terizes the afm statc. \Ve must also note, tha t the 1 vector determines the afm axis. Using E qs. (9). and omitting the terms with (v M j .= divM j = 0) and with the demagnetization effect of sample surfaces, from (8) we find

a a, b h, ¢ = - ll+-m2+ - I; 2+- m l2+d (I ~m 1l -l llm,)

2 2 2 2 e f

+-I'+-m'- mH. (10) 4 4

Here coefficients a, h, c, .. are simply expressed by

211. E. Dzialoshinsky, Zhur. Ekspll. i Teorct. Fiz. 33 1-l5-l (1957) [translat ion: Soviet Phys. JETP 6,1120 (1958)]. ,

A, B, C, ... from (8) . The terms with e and f are ex change mvanants of the four th order wi th respect to [ and m. In the "main" exchange terms of Eg. (10) fI "

and "a," ~re posi t ive in the paramagnetic state. The:e fore there IS no magnetic order for the minimum value of 1> (m= O, 1=0). We shall get a purely afm structure (m= O, Ir:!'O at H=O), if the coefficient "a" changes i sign and "al" remains> ° at the poin t of phase transi_ ~ion between the paramagnet ic and ordered states and If \ve neglect the term with "d." If we take into accOunt the. term with d, then for the minimum va lue of 4>, besides the purely afm structure, there exist the struc_ tUre with m r:!' O, i.e., the "weak" ferromagnetism of relativistic na ture. T his case is con,sidered in detail by Dzialoshinsky.lI

On the contrary, if at the point of phase transition tha coefficient 'Ia" changes its sign and a,>O in both phases then for the minimum value of cp \ve shall get the f~ state (mr'O, 1=0) of the usual exchange type. In thi' case it is possible to get the admixture of "weak" anti ferromagnetism of relativist ic type a lso, if we take into account the term with d. Indeed, we fi nd the solu­tions : (I) 1=0, m.= m,=O; and (II) 111,=0, 1,=0 and 1.= - (d/ a)m" 1,= (d/ a)m" if we de termine the mini­mum of (10) with given m. In the state (I ) lVe bave ferromag.netism with axis of easy magnetization alon, [111J \vlthout the aamixture of weak antiferromag_ netisffi. In state (II) the axis of easy magnetizat ion liea in the (111) plane, but the magnetizations of sublatticei are rotated at a small angle towards one another. There­fore the afm components along the afm axis perpen­dicular to vector m arise.

This effect comes practically only to one of the causes­of inequality of the spontaneous magnetization with iti nominal valuc. It is necessary to take into account the higher order invariants (with respect to nh and m,,), \vhich determine the magnetic anisotropy in the (111) plane, to lind the orientat ions of vec tors m and I in the same plane.

The investigations of potential (10) extrema deter­mine the character of phase t ransitions among tbe different magnetic states, if \ve take into account some further terms in the expansion of (10) (see reference 11). Let us note only that for the second-order phase tran­sitions it is neccssary to fulfill two condilions : (1) the symmetry group one of phase must be the subgroup of the other phase; (2) the expansion of cp must not contai n any th ird-order terms of small parameter. This is always the case for t ransit ion from paramagnetic state to the magnetic ordered one.!O,12 In this case we have the second-order transition . In other cases more often we have the first-order transitions, which do not require the fulfillment of the above-ment ioned condi~ tions. In particular this occurs in a-Fe~03 at 250 0 K Y

:rhe invariants, containing elastic st ress tensor com· ponents, and also thei r products by the sublattlce magnetization components, can be taken into account in the expansion of cp. The detailed thermodynamic theorY,

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

138.251.14.35 On: Sat, 20 Dec 2014 09:44:39

PHENOMENOLOGICAL THEO RY OF FERROMAG:-IETISM 155

lda!iJ1I,t.c ordered medium in the region of phase points can be created, if we take into (on­the external magnetic field effect. Essential

of phase transition points in magnetic Wt!siances•can take place under the influence of external lIiIi!&E,etJ.c fields and elastic stresses, \vhich can greatly

the magnetic crystal anisotropy. l!l\f""Y problems of thermodynamic theory of magnetic

It!t~:~:!-:,~are investigated In the works of Soviet (temperature dependence of spontaneous

&:\Jietioa1tio'n, curves of the para-process, the influence stress, magnetostriction and spontaneous

displacement of transition point, galvano­and other kinetic effects near the Curie point, important result of these investigations is the of getting the precise theoretical expressions

~fer''''t physical quantities, as for instance, mag­magnctos t riction, etc. It is possible also to

t~elaltiOlJS among the phenomenologic constants, be proved by experimen ts .

terms with the gradients of magnetization in cP taken into account in the considered temperature

This is important for the calculation of lor.g-ra •• ge order fluctuations near the phase

these fluctuations (i.e., magnetic order) are relatively very great. This prob­

discussed.zz

7

us consider the applications of phenomeno-~iI~ .. )ryfor low temperatures. First we shall describe

the general spin-wave method within the limits theory. ground state and eigenfrequencies spectrum of Dt.,turb"d stales of the system, described by

1;:~~~ (2), can be obtained using classic motion P magnetizations Mi(r) , or with the help of P.'! mleu.oo of second quantization.

The claisic motion equations are

(aMi/ al) =y;[MiX H;]. (11)

Hj i£ the effective field, which acts on Mj and is J,de1tennlr.ed from formula":

ax a ( ax ) H=- aM/ dr. a(aMi/ar.) .

(12)

is the magnetomechanic ratio (or morc precisely, the of $pectroscopic splitting) for the j sub lattice,

I~,=.," .I~ whert {3 is Bohr's magneton, 21dt is Plank's lcolos~lnt. and giis the Lande factor.

[~~~.n!C'v~;y. Izvest. Akad. Nauk S.S.S.R. Ser. Fiz.ll , 485 Vonsovsky and J. S. Shur, Ferromagnelism (~ I os-

Eksptl. i Teoret. Fiz. 17,833 (194 i ). Fiz. Nauk 65 , 207 (l958).

H. rshmuchametov, Zhur. Eksptl. i

. Phy,. Soc. (London) 64, 968 (1951 ).

The equi librium positions of vectors M jo are found from cond itions aM j/iJl =O or from the minimum of Hamiltonian (2). Now let us suppose M j = M jo+ 6.M j

and linearize the system of Eqs. (II ) with respect to deflections .6~[ j . The solution of this system usually is carr ied out in the form <lMi~exp[i ("'I+kr) J, where '" is the frequency and k the vector of quasi-momentum. The eigenfrequencies wand the dispersion relations for them w(k) are found if we take the system determinant as being equal to zero.

The method of second quant ization, wh ich is the generaliza tion of the well-known Holstein-Primakoff method,z1 is often more simple to usc, however. This method has a number of advantages over the classic method, because the zero point energy is taken into account. The relaxation processes may be considered. This method is also convenient for its "automatic" characte r~narnely, to get the dispersion relations,:!8 it is enough to find the quadratic form of second quantiza­tion operators.

Using the method of second quantization, the classic pseudovectors M j must be replaced by the operators M/(r), which satisfy the following commutation ru les:!9:

M i : (r)M i·.' (r') - M i' .' (r')M i:(r) =i" iM i ,'0ii·o(r-r'), etc., (13)

where Poj= gil'3. The second quantization operators +b/ and br i , given by formula27 •30

.Uil!/= (!Po j.MjO)iC/ibri++b//i) ,

MiY/= G" iM iO) l(jib,i- +b,ij;) , (14)

M iJ/ = M jo - p, j+b/b/; (/i = [ 1-Poj+bribr}/ 2MiOJ!),

can be int roduced, if its own coordinate system OXj YjZ j

is chosen for each sublatticc j , so that the OZj axis is parallel to the quantization axis of the M j vector (i .e., along classic M jo vector). The mean values of magnetiza­tion deflections are relatively small: {p.j +brjb,J)b = (.6Mjzj )Av«M jO, because of the initial assumption that the system perturbations are weak. Insert ing (14) in (13) , it is easy to see that operators +b, i and b,i agree \vith the Bose commutation relations

It is necessary to pass to one common coordinate system OX V Z , connecled \ .... ith the crystal axes. This transition is made by using the table of di rectional cosines QQfJj,

M jQ' = L a"fJjl'vfjfj/' . , (16)

The whole Hamiltonian of the system can be represented as a series

(1 7)

l7 T. Holstein and H. Primakoff, Pbys. Rev. 58, 1098 (19-W ). uN. N. Bogolubov Lectures of Qllantum Statistics (Kiev, 19-19). "c. Kittel and E. Abrahams, Revs. Modern Phys. 25, 233

([953). JO E. A. Turovand V. G. Shavrov, Papers Inst. Phys. Mel. t\ cad.

Sci . U.S.S.R., Sverdlovsk, No. 20 (1958), p. 101.

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

138.251.14.35 On: Sat, 20 Dec 2014 09:44:39

16S s. v. \. 0 :-I S 0 v SKY AND E. A . T U R 0 V

Here we introduce the Fourier components +hk ' and bki

of +b ri and b/, respectively, and JC n is the It-order term with respect to operators +bk

i and bk i .

The ground state, or the equilibrium values of a,,(1° at T =ooK , can be found by minimization of the zero­order term 5Co (i ndependent from +b;/ and bk

i ) with respect to directional cosines. Th e first-order term 3C] in (17) i5 identically equal to zero fo r a.,o Th erefore the ground state can be found also from the condition X]=O. The term JCz can be presented as the sum of individual elementary excitations-spin waves, afte r re­ducing diagona l form. T his term determines the mag­netic self-oscillation spectrum in fi rst approximation. The higher order te rms in (17) (:lea, etc.) describe the Hcollision" processes among spin waves (ferromagnons, antiferromagnons) . The differen t kinetic phenomena in magnetics ca n be investigated using these terms.

8

The case of un iaxial ferromagnetics with the easiest magneti;;:ation axis along OZ, influenced by external field H of arbit rary magnitude and orientation, may be taken as a concrete example. This problem was in­vestigated in a number of works!3.2i (for the case MoIIH), It is known that for H-LOZ there exi5ts a criti c.al field I-I A, which is de termined by magneti c anisotropy energy, and by which the rotation of vec to r M toward the field is accomplished . The dependence of tield H ..t (= }fA (T)) on T was not taken in to account,13.2i however, when calculating the temperature change of the magnetization curves (at H <H A). This dependence 11 .-1. (T) is a result of the temperature change of the equilibrium position of :Mo, i.e., from the temperature dependence of anisotropy constant. Therefore, the following expressions for functions M/I (T ) were found31

for H -LOZ.

11 [ ~ «T)IJ .104 1/(1')= .11 0 - 1+0.06- - ;

If A .1040 [

[ ~ <1' I]

.1041/ (1')=.110 1-0.06.1040( 1 ) ;

(I is the exchange interaction parameter) . Therefore the theory predicts the increase of mag­

netization (al H ~ 1I.4 T) with the rise of temperature in the unsaturated ferromagnetic:;. Such effect is obseryed in common polycrystallic samples as \Yell. However, the ri se of magnetization in the latter case is caused not only by the facilitation of rotation processes, but also by the displacement processes of domain boundaries ill ferro­magnetics.

31 E. :\, Turov and U. P. rrchin , !zvcst. ,\kad. Nauk 5.S.S.I{ . Sec. Fiz. 22, 11 68 (1958).

9

Now . let us consider the ferrimagnetics with tw magneuc sublalticcs, the magnetizations of which in th ground state are anti parallel and are not equal to each other: MIO~.tf 20 . T he resultant magneti;;:ation is no zero and is equal to I'J lo-M :!O and directed along th field H at T=ooK. The spin wave spectru m is given b

, (k) O.'I = [( 'I + ,,)'- 'I,'J'± ('1- , ,j (18

if we do not consider the magnet ic te rm:; in Hamiltonia (isotropic ferrimagnetics) . Here

~;

2,;=-.104. (AI,+C;jk')- ( -I);~;ll , • }o '

(j = 1,2) ,

(19

A 12, Cii, are the phenomenological constants of exchang interaction . The ferromagnetic and exchange resonan frequencies may be found from (1.8) at k = O. One ca easily see that the spin wave energy (18) is in linear dependence from k (at Il = 0) under the condition

(20)

This fact is characteristic for the afm state. Form~l (20) permits the existence of the resultant magnetiza. tion, i.e., ferromagn etism as w"cll. Indeed, from (20) i follows ,

l1~ M~o = .U1o- .Mto = - .M"o,

~

(21)

where l1~=~ (~ I -~'), ~=i(PI+~')' and M o=HMI + .11 20) . Thereiore, on the one hand the compensated spin anti ferromagnetism occu rs because NISI = lV~~'!, according to (20) (.\"; and S; arc the numbers of m~­netic ions and thei r spins in sublattices, respectively) On the other hand, the noncompensated "orbi tal" ferromagnetism appears, caused by the different g factors of ions in different sublatLiccs. T his may resul from the existence of different ions with gl;;eg2 in th sublatLices. T he gj factors nevertheless can be different in the case of identical ions, if these ions occupy' nonequivalent sites with diffe rent type of nearest neigh~ borhood in the lattice. ,Yc obtain

for tilC spin wave energy of th is 'lorbital" ferromagnetism [u'ing Egs. (18)- (20) at CII =C,,]. Here![::; ( .'.~/p)HB' II r: is the effective exchange field, and J is the constant defined by exchange coeOicicl1ts . I l~ and Cii , from Eq. (19). 'fhe temperature and tit:ld depcndence of mag· nctizat ion can be calculated by using lhe spectrum (22).

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

138.251.14.35 On: Sat, 20 Dec 2014 09:44:39

PHENOMENOLOGICAL THEORY OF FERROMAGNETISM 175

his dependence is (under condition Kr»u.t:.I'H EHJI)

M(T,H)~M.(T)+x(T) ·H. (23)

is the spontaneous magnetization, which is

I"H g(KT)2]

12Mol (24)

d X is the susceptibility (analogous to XII in antiferro­gnetics), which is equal to

(25)

herefore, the spontaneous magnetization in this case ollows the P law, and not the T~ law as in ordinary erro- and ferrimagnetics. ~~ It should be noted that the

rt of magnetization, which depends upon the tem-

1'2(KT)'(t:.1' ) t:.MT~---- -fIe-4H ,

12J' I'

anges its sign at the field 1I ~ t(t:.1'11')1I E.

10

Now let us consider the case of antiferromagnetism. e atomic theory of anti ferromagnetism is based upon

e supposition that there exists an ordered ("check-ed") alternation of "right" and "left" ion spin COffi­

nents in the crystal lattice points. The entirely defi­te spin orientation is ascribed to each ion. This is in ntradiction with quantum mechanics, because the dividual ions cannot possess definite spin components,

wing to exchange interaction. This difficulty does not x.is,t in phenomenological theory, in wh ich one uses only e symmetry properties of afm crystals. Only one natural afm axis (trigonal, tetragonal, or

ex-agonal) exists in the uniaxial afm crystals. If we mit the higher order terms, \vhich describe the ani­HOPY in the base plane, then the Hamiltonian will be

the same for all these crystals, with the exception of the mns with D [see Eq. (8)J, which may be different. For

nstance, this term has the form (DIMo')(M\.M\ , - M,.M, ,) in tetragonal crystals (MnF2 type and the ke). This expression is not the same as the term with D n (8) for the rhombohedric crystal. The terms with D ust be taken into account in some cases, though they ve no great influence upon the general character of gnetic propert ies of uniaxial antiferromagnetics. For

the details of calculations the readers are referred to the original papers.31- 33 Here we shall mention briefly only

11"11" The T2 law for spontaneous magnetization of ferrites, dis­vered in the work by Vonsovsky and Seidev [So V. Vonsovsky

'(nd U. M. Seidev, Lwest. Akao. Nauk S.S.S.R. Ser. Fiz. 18,319 19S4)] is valid in that case (N lSl=N:S2 and glr'gZ). ~ M. I. Kaganov and V. M. Tsukernik, Zhur. Eksptl. i Teoret.

F~. 34, 106 (1958) [tmnsIation, Soviet Phys. JETP 7, 73 (1958)]. SS;QL. Neel, Ann. pbys. 8, 232 (1936); Izvest. Akad. Nauk .. S.R. Se .. Fiz. 21, 896 (1957).

some results connected with the consideration of the term with D. Let the external field 1I be parallel to the OZ axis. Then the afm axis 4. will be normal to the OZ axis at the fields greater than the threshold value (1I?: H o) . If the term with D is considered, there appear the magnetization components Ml~2Mo (1I vI H ) per­pendicular to the OZ axis (and H), besides the z com­ponents of magnetization M:= ~i\.1 lz+M2z= 2Mo(H/ H E) proportional to external field H. Here HD~ (DI Mo) is the "Dzialoshinsky field ." The magnetization Ml in the first approximation is independent of the external field and is of the same relativistic nature as the "weak" ferromagnetism in a-Fe203 (see Sec. 6) . Let us note that in general M1«M 1l , since Ho»H D, because lIo is determined by the mean geometrical value of ex­change and magnetic energies and H D is determined only by magnetic energy. The spin wave spectrum and the afm resonance frequencies were determined by Turov and Irchin. 31 The splitting in this spin wave spectrum remains even at H = 0, if we take into account the long-range dipole-dipole magnetic interaction. How­ever, this splitting is relatively small because the con­stant "magnetic addition" is always small in comparison with the exchange interaction term.

The term with D influences the resonance frequencies in a different way for rhombohedric and tetragonal crystals. In the former case hV=J.I..[Ho2_HD2]i±J.l..H, and in the latter h,~ 1'[ (Ho±H)' - H v2JI. The energy gaps are different in the various branches of anti ferro­magnon spectrum. The resonance frequencies have differen t dependencies from the magnetic field in the state 41.H at H II OZ and Hl.OZ.

The temperature dependencies of magnetic suscepti­bility and spin heat capacity were calculated31 for various orientations and magnitudes of magnetic field, using the spin wave spectrum. The anisotropy of these dependencies was determined by Turov.34 Some of these deductions are in good qualitative agreement with the results of the experiments of Handel and othersY' The theoretical conclusion as to the relation between the parts (dependent on temperature) of perpendicular and parallel susceptibilities of uniaxial antiferromagnetics is of interest. This dependence has the following form : LlX zl1 = - 4Llx/=aT2. *** The present experimental data, unfortunately, are insufficient for verification of this quantitative conclusion of spin-wave theory.

II

Finally, let us cite the resu lts of the calculations of spin-wave spectrum, of resonance frequencies, and of temperature dependence of magnetization for "weak" ferro magnetics, like a-Fe203 and lVfnC03 for low tem-

l4 E. A. Turov, Zhur. Eksptl. i Teoret. Fiz. 34, 1009 (1958) [translation : Soviet Phys. 7, 796 (1958)].

35 Van den Handel, Gijeman, and Poulis, Physica 18, 862 (1952 ). ... Here ~x = x(1')- x(O) ; the lower index indicates the orienta ­

tion of magnetizing field H, and the upper index, the mutual orientation of 4. and H.

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

138.251.14.35 On: Sat, 20 Dec 2014 09:44:39

18S S. V. VONSOVSKY AND E. A. TUROV

peratures.36 Here two branches of spin~wave spectrum were found also

. (k)(l)~[u'(H+H D)H+l'k']I, (26)

.(k)'" ~ (~'[Ho'+ H D(H D+ H)]+ l'k'} I . (27)

Here all symbols are the same as in Sec. 9. At k=O the resonance frequencies are

hOl~~[(H+H D)lJ]l, (28)

(29)

The frequency v, in (28) corresponds to the usual fm resonance, when the easy magnetization axis of tht uniaxial crystal lies in the base plane. The formula (28) in contrast with that of KitteP7 contains the Dzialoshinsky field H D instead of the magnetic (uni­axial) anisotropy field H A, _ To explain the experimental data fo r fm resonance in a -Fc203,38 it is necessary to take into account, perhaps, the magnetic anisot ropy in the base plane.39 The second resonance frequency (29) lies in the shorter wavelength region . This frequency corresponds to the usual afm resonance. As fa r as we know, this resonance was not observed in a·Fe20~ as yet.

For the temperature in terval J..I.H o«KT«"eN (e.\' is the ~eel point), using the spectrum (26) and (27), we find that

M (T,IJ ) = M .(T)+x (T) · IJ = xU') (Il D+ II). (30)

Here M, equal to

is the spontaneous magnet ization, which is

Ill) M, (T) ~2Mo-(I-aP)

lIe (31 )

and X the susceptibility (analogous to Xl of antiferro­magnetics), which is

(32)

Here a is the constant. The coefficient a in (31) must be substituted by la, if the th resholrl field H o is so great that ~}[o»KT»~H, p.H D .

Thus the spontaneous moment of "weak" ferro· magnetism follows the T2 law instead of Lhe TI 1;;\\' . ..- like the "orbital" ferromagnetism (see Sec. 9) . Borovic­Romanov and Orlova21 obtained the two· term formula analogous to (3 1) for ,lnCO, and CoC03• Unfortu­nately, one can speak only about the qualitative

n E. A. Turov, Zhur. Eksptl. i Teorct. Fiz. (to he Jluhlished ). 37 C. Killel , Phys .• Rev. 73 ,155 (1948). 18 Kumagai , Abc, Dno, Hayashi, Shimaua, anu Jwanaga, Phys.

Rc\·. 99, 1116 (l955). 19 :,\1. Shimizu, J. Phys. Soc. Japan 11, lOiS (1956).

agreement between theory and experiment at present ").feasurements in the lower temperature range for rno detailed verification of this theory should be made Howcvcr this checking is reasonable only if there is phase transition from the "weak" ferromagnetic sta t.o pure antiferromagnetic one (as in a ·FezOa) in these substances.

It is interesting to note that lithe Dzialoshinsky field" /I v) which is equal to lv! ~/x) may be determined inde. pendently, measuring M. and X and comparing the above value of H v with II v found from resonance experiments.

The comparison of fo rmula (32) for susceptibility 01 weak ferromagnetics with the respective formula for Xl, the usual in case of antifcrromagnctics in ~ .l H state shows that they are similar. This fact may help ~ d ist inguish easily the weak ferromagnetism of the Htransverse" type from the ('longitudinal orbital" one (see Sec. 9). The Jinear dispersion law for spinwave energy is realized for all ((weak" ferromagnelics as we as for the antife rromagnetics: , (k) ~ I k (at KT»Mio where !:lEo is the energy gap for the spin-wave excita­tion) . Therefore the spin heat capacity must be pro-; portional to T 3. The formulas for this heat capacity are the same as those found in the case of anti.ferro~ magnetism .34

12

In the conclusion of our repOrL we must note that the phenomenological theory can be applied with consider able success not only to a great number of problems 0

magnetic phenomena) but to the solid state physics as whole as well. \\'e cannot mention, of CQurse) all th problems in this repor t : the phenomenological treat­ments, the problems of interaction of magnetic an elastic properties of ferro- and antiferromagnetics,4.o. the problems of interaction between spin waves a conducting electrons in transition metals (s-d-exchang modeP2.JO) , etc.) may be noted here. Th e great ad vantages of strict thermodynamical treatment for tb explanation of many imporLant problems of magnet! phenomena stated in the foregoing are now qui evident .

III E .. \. 'IuTO ..... and C. P. [rchin, Fiz. ~[cta.l. i :'letalloved. Abd. Xauk S.S.S.R., Ura.l Fi lial 3, 15 (1956).

~! :'\1. I. Kaganov and V. :\r. Tsukernik, Zhur. Eksptl . i Teoret. fiz . 34, 1610 (1958) [tra.n~lation: Soviel Phys. 7, 110i ; Achieser. Bariachtar, and Pclctmiosky, Zhur. EksPl l. i Tcorct. Fiz. 34, 228 (1958) ; A. 1. .-\chieser aod L. .\. Shishkin ibid. 34, 1267 (1958 [tr:;~slation : Soviet Phys. JETP 7, 8~5 (958)]. V ~ S. V. Voo!',Ovsky, J. Phys. U.S.S.R. 10, 468 (1946); S. .

Vonso\'sky and E. A. 'l"urov, Zhur. Eksptl. i Teoret. Fiz. 24,419 (1953).

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

138.251.14.35 On: Sat, 20 Dec 2014 09:44:39