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On the exchange interaction in magnetic crystals
Physica
IV, no 3
Maart
ON THE EXCHANGE
INTERACTION CRYSTALS
1937
IN MAGNETIC
by W. OPECHOWSKI Instituut
voor
Theoretische
Natuurkunde
der Rijks-IJniversiteit,
Leiden
Summary A simple method, due to K r a m e r s, of approximate evaluation of the partition function (Zustandssumme) in the case when kT > 1 J 1 (J = the exchange integral), but pH/kT arbitrary, is explained and applied to discuss some questions related to ferromagnetism (J > 0) and to the ‘adiabatic demagnetization experiments at low temperatures (J < 0).
Introdzlction. The physical model used in the present paper to discuss some magnetic properties of crystals, is that intfoduced by H e,i s e n b e r g 1) in his first paper on ferromagnetism, and can be briefly characterized as follows : the crystalline lattice is built up of atoms each having only one s-electron which is not in a closed shell *) ; only these electrons have a part in the exchange interaction; the magnetic interaction between their spins is neglected; each atom interacts exclusively with its nearest equidistant neighbours. Starting from this model H e i s e n b e r g derived, as is well known a formula for the magnetization of a ferromagnetic crystal as a function of the external magnetic field. and the temperature, and showed that the positive exchange integral (J > 0) is a necessary condition for the ferromagnetism. The calculations of H e i s e nb e r g, however, are based on the assumption that the energy states of the crystal with the same resultant spin are distributed about their mean value according to a Gaussian curve. On this assumption, the dependence of the magnetization on the crystal structure involves only the number of nearest neighbours of any given atom, i.e. the “co-ordination number”, and not their arrangement. Later; from much more rigorous calculations of B 1 o c h “) which are valid, however, only for sufficiently low temperatures it appeared that H ei se n b er g’s formula is for this range of *) Cf., however,
the foot-note
on p. 196.
-
181 -
182
W.
0l;ECHOWSKI
temperature certainly not even approximately correct. Moreover, B 1 o c h showed that the three-dimensionality of the crystal lattice is essential for the ferromagnetism, thelinear and the two-dimensional lattice being never ferromagnetic, and, thus, that the knowledge only of the co-ordination number does not suffice. Recently K r a m e’r s “) proposed another mathematical treatment of the H e i s e n b e r g model of ferromagnetism and this treatment yields for low and high temperatures respectively the results of B 1 o c h and H e i s e n b e r g. This new method differs essentially from that of H e i s e n b e r g or. B 1 o c h in that it consists in an approximate evaluation of the partition function without explicit calculation of the energy levels, of the crystal. In the same paper K r am e r s briefly indicated still another way of evaluation of the partition function, valid only for high temperatures (kT > J), but for arbitrary magnetic field strength and for any sign of the exchange integral, and calling for no simplifying assumption in the calculation itself. This treatment will be the object of the present paper. In $ 1 a detailed presentation of the method is given and calculations are made for the close-packed hexagonal and face-centred cubic (i.e., close-packed cubic) structures. The method gives essentially the free energy in the form of an expansion in descending powers.of kT/ J, the coefficients being functions of the magnetic field (more exactly: of pH/kT). Consequently the magnetization is obtained in the same form and in 5 2 it .will be shown that this result, to the terms quadratic in J/kT, is exactly equivalent to the H e i s e n b e r g implicit formula, the limitations on the validity of which appear thereby more clearly; further terms in the expansion depend generally not only on the co-ordination number, but also on the arrangement of atoms in the crystal. All calculations and results of $$ 1 and 2 are written so as to be valid for both signs of the exchange integral. In 5 3 the case J > 0 is chosen and the question of the C u r i e point is briefly discussed. In 5 4 is analysed the case J < 0, that of the “anti-ferromagnetic” crystal, where a diminishing of the magnetization results from the exchange interaction. This is of some interest for the discussion of the adiabatic demagnetization experiments at low temperatures, and, though most of these experiments’ were done with crystals in which our simple assumption are certainly not fulfilled (the magnetic
ON
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183
interaction playing a preponderant role *) - not to mention other no less important circumstances), still some formulae and some figures related to the anti-ferromagnetic case will be given, since no quantum theory treatment of this case has yet been presented before. There is only the case of the experiments with ‘crystalline cerium fluoride for which one coulq hope for various reasons that the treatment given in this paper IS not entirely inadequate; however, no unambigous discussion appeared possible, as will be seen in 5 4. S $ 1. We write the formula free energy and the partition
which gives the relation between the function, in the following way: e-FN/RT _spur e-
- !iJ x [bd where the components P a u 1 i matrices:
+ 11= - $J x
(bkbJ)
-
+JNf
of a ck vector operator are represented
(4 by the
and where the sum is extended over all pairs of adjacent atoms in the crystal, and, thus, consists - if we neglect the effect of the surface - of Nf terms, 2f being the c&ordination number. For the second part of the energy operator one has the expression - pH r, ah”’ (4 where the summation extends over the N atoms of the crystal; k is the magnetic moment of each atom and H the magnetic field which is supposed to point in the direction of the z axis. We now replace the variables T and H by new dimensionless variables which we shall denote, throughout the rest of this paper by T and a, in the following way: *) Professor v a n V 1 e c k has kindly informed soon publish in the Physical Rev&u some articles closely investigated.
us that he and his collaborators in which the magnet.ic interaction
will is
184
W.
OPECHOWSKI
kT
>T
-pH +a, J
J
thus, T is no longer the absolute temperature, but a quantity proportional to it, With this new notation eq. (1) becomes e-FNIT = spur e-W,
F being here equal to F/J in the old notation,
(6)
and
E = Et-J+ EH
(7)
where EiJ = -
4 s (CT&
(8)
EH = - a ; op (9) In passing from (2) to (8) we have dropped the term (- Nf J/2), which changes F only by an additive constant, equal to (- f/2), and is of no meaning for our calculations and results. Our problem is now to calculate F, since knowing F as a function of T and a, one obtains immediately by differentiations the magnetization crand the entropy S both per atom and in units corresponding to the choice (5) of independent variables : aF s,-!E O=-aa aT * To this end, we proceed, after K r a m e r s 3), as follows. Since Ed commutes with Ed, we may write e-W = e-~oiT e-qiIT = e-qg/T e-~olT (11) We now introduce the following notation: if P is any operator, then p = spur (Pe-‘H”) (12) spur t?-‘HIT and we shall call p the average value of P. This will be the only meaning of the expression “average value” used in the present paper. Owing to this way of averaging, our results will be valid for any magnetic field strength. Thus, from (1 l), we have: spur e-c/T = e-cc-/T spur emcHIT
(13)
Now we put e-co,/T
where G is a certain function expanded, as follows :
=
e-NGIT
(14
of T and ‘a which can be formally
ON
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CRYSTALS
185
(15) and the A,‘s are given by eq. (20) below. Equations and (15) yield, thus, F = -
(16)
the A,‘s, we expand the right hand member e--NW
=
g
(-
n=O and putting
m A” + X n=iF’
$ In spur e+HiT
In order to calculate of (14):
(6), (13), (14)
1)”
WG" n! T"
(17)
-
(15) in (17), we obtain
e-NGIT= 1-s ++$ .(
+$ + . ..)+
PAI (
+zT+p+
A2
2 . . . .
)
-
. . . .
Now, because of the linearity of the definition (12), we have 1 T eT%iT = 7 --AL + ---EO ,TT3 + *-** l! T 2!T2 and, thus, according to (14), the coefficients sions (18) and (19) must be equal; hence A,, = coefficient
of N in (-
(18)
(19)
of (1 /T)” in the expan-
2 l)“+l -$,
(1~ = 1, 2. . . .)
(20)
Equations (16) and (20) show that, for sufficiently great T, i.e., the thermal energy much greater than the exchange energy, the free energy will be known, if one calculates the average values of the lOWeSt powers Of the Operator Eg, and the value of the first term on the right side of (16). This term is evaluated immediately, since;from (9), one has: spur
e-‘H
=
spur eazo~‘/T = spur R=l,
n 2...N
e4’#IT = k=l,
n
spur eacrt’/T
(21)
2...N
where the last equality is a consequence of the fact that the spur of a “product matrix” is equal to the product of the spurs of the factor matrices. Now (22) spur eoa&T = e4T + e-dT
186
W. OPECHOWSKI
and, thus, from (21), one obtains: spur e-%/T = (&T + e-@)N.
We shall now calculate
(23)
ET. We have from (8)
-G L - 4 Nf(ckq), (24) since, by symmetry, each term in the sum in eq. (8) has the same average value. Now *),
(25) ax’
0:)/T
spur e
spur [(&J@
=: (&T + e--dT)N-2
+ @+)
spur [&lojd
,+@+#‘WT]
and hence, if, for brevity,
ea(uP+ub-J
= 0
= (eJT _ e-Q/T)2
.
we put tanh (a/T) = T,
(26)
(CrkbJ = 72.
(27)
eq. (25) becomes: Thus, from (24) and (27),
(28)
,=-iNf+’
and this, with regard to (20), gives
(29)
A, =--if?
We calculate
further 2:
3 = ii@ WQ)~= *[ffl (wo)~ + 2fN (2f - 1) (ckd (‘Wn) + fN (fN - 4f + 1) (bkol)
+ (%dl
(30)
The last equality needs comment. Squaring the sum Z (o& we obtain (Nf)2 t erms which we divide into three classes, the terms belonging to the same class all having, by symmetry, the same average value. These are .the classes of terms having respectively two, three and four unequal indices (Where the contrary is not explicitly stated, different subscripts of the ok’s relate, everywhere throughout this paper, to different atoms). As is easily seen, the *) 27 means summation
over
all m’s for which
m # k, C.
ON
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INTERACTION
IN
MAGNETIC
three classes contain respectively fN, 2fN (2f -2(2f - 1) - l] t erms. Hence eq. (30). Now, from the identity (6pJJ2 = 3 - 2 (GpsJ) and from eq. (27), we obtain Further eq. (27))
(c&2 = 3 - 2;L we have, by an argument similar
CRYSTALS
187
1) and fN [fN -
(31)
(32) to that which leads to
(wd b&J = 72 (33) To find the average value of the terms of the third kind, it is sufficient to observe that, by definition (12), the average value of any two operators which act on different electrons, is equal to the product of the average values of each of these operators. Hence, by eq. (27), (w) (%%) = T4 (34) Substitution of (32), (33) and (34) in (30), and rearrangement, gives z=
?[3
+ 43 (f-
1) + r* (1 -4f)]
+ Qp.
(35)
Hence, by eq. (201, A,=--$[3+4ra(f-I)
+r”(l-4f)].
(36)
Thus, from (16), (23), (29) and (36), we obtain for the free energy, in this approximation, the expression : F =--Tln(e~Tfe-U’r)-&$-$[3+47’(f-
l)+~*(l-4f)]
(37)
We may remark here that the knowledge of & and;; suffices for the derivation of the H e i s e n b e r g formula mentioned in the introduction. We postpone the discussion of topics related to this formula to $5 2 and 3, but we should observe now that our calculation of G and 3 depends on the kind of crystal lattice only through f. Now, that is no longer the case with the average values of higher powers’of Q. In general the result will here also depend on the arrangement of the neighbours atoms about the given atom. To calculate 3 and 2, as we shall now do, we must, thus, explicitly state with what crystal lattice we have to deal. We choose for this purpose the close-packed hexagonal lattice (f = 6) and the results will appear to hold for the face-centred cubic lattice too.
188
W. OPECHOWSKI
To find 2 we must evaluate the third power of the sum C (I& and, doing so, we get (Nf)3 terms .of eight different classes, all terms of each class having the same average value, Counting the number of terms in each class and calculating the average values of their representative terms, one obtains the results which are given in Table I. TABLE Representantive term *)
I
-4verage ..sl..n . S.UI
Number
of terms
(f = 6)
It is now necessary to say something about the way in which the data of Table I are obtained. We first consider the average values of the third column of Table I. The average values [3] and [8] follow immediately from eq. (27) and (32) with the aid of the remark which has yielded eq. (34), and the average values [l] and [2] from eq. (31), (27) and (32). We shall now formulate a general rule by which the average values [5], [6] and [7] can be readily found, and which will be of great use later, in the case of calculation of ET Let (fil%)
@3%)
* * * * (%m--1%)
(38)
be a product which does not split in groups of factors acting on different electrons, and which contains no two equal factors; moreover, contrary to the convention used in this paper, it may here well be that By = n, for i # k. Now, if the product (38) does contain at least one index n, which is not equal to any of the other 1~~‘s (i # k) in (38), then bn,%J
(%3%4)
- * - * b~m-*~“2m)
=
9
(39)
where p is the number of such atoms that the index of each of them occurs in (38) an odd number of times. The premissae of this rule express actually only the sufficient conditions that in an expression such as (38), the terms which contain o@)‘s or &‘)‘s or both of them, have all the average value zero; in other words - that the average value of the whole product be equal to that of the one term which contains only &)‘s. l)
For brevity,
we write
down
only
the subscripts
of the uk’s.
ON
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189
CRYSTALS
The term [4] in Table I obviously does not fulfil these conditions; here the ~(4’s and &‘)‘s do play a role. The calculation of the average value is, however, still very easy and, using the well-known relations between the P au 1 i matrices, one obtains the value given in Table I. The calculation of the numbers of terms in each class is, of course, a purely arithmetical one and presents few interesting features. It canbe done generally in several different ways, which yields a good check on the results. This circumstance has no importance in the present case (2) where the number of classes is small, but gives much help in the case of3 where there are twenty-three classes, and the counting of the number of terms becomes rather complicated. One will find below (eq. (40)) th e numbers of terms in each of the eight classes written down in such a way as to make clear the simplest manner in which they can be counted. We denote by L [ ] where the representative term of a class is to be put in [ 1, the number of terms of this class. Further we understand by L’[(kZ) (Zwz)] the number of such (AZ) (&)-terms for which atoms k and m are neighbours, and by L”(kZ) (Zm) - th e number of all other @I) (Zm)-terms. We have then: = fN
w431 Lrw2 W)l .
= 3L[(kZ) (Zm)]
LrWI2b41
= 3L[(kZ) (m)]
L[(W W) (41 = L'iI(k4 (41
Lll(k4 v4 WI = w4 b41 Pf - 2) L[(kz) (zm)@%)I = Q{L’[(kz) (zm)]2(2f -2) + L”[(W (W] Wf - 1)) LlI(k4 (z4 (+)I = 3 V’[ (kz) (WI [fN - Wf - 4 - 31 + + L”[(kZ) (Zm)] [fN -
L[(W b.4 &)I =
fN(fN-
1) (fN-2)
(40)
2(2f - 1) - (2f - 2) -
- W(W
(W
211 (Ml+
+ u-w (W WI + LCWP4 b41 + + LW) (I4 h91>
where (cf. p. 187, first line) L[(kZ)2] = fN L[(kZ) (Zm)] = L’[(kZ) (Zm)] + L”[(kZ) (Zm)] = 2fN (2f -
1).
190
W.
OPtiCHOWSKI
Equations (40) hold for an arbitrary crystal lattice without regard to the arrangement of the neighbour atoms about a given atom. The above mentioned dependence on this arrangement lies in L’[(kZ) (Zm)] which is not uniquely determined by the co-ordination number. For example, in the case of the hexagonal layer-structure (i.e., a two-dimensional lattice) L’[(kZ) (Zm)] = 4Nf and in the case of the simple cubic lattice L’[(kZ) (Zm)] = 0, both structures having the same co-ordination number 6. For the close-packed hexagonal lattice and for the face-centred cubic lattice L’[(kZ) (Zm)] is the same and equal to 8Nf, and, hence, L”[(kZ) (Zm)] = .14Nf; these values must be substituted in (40) to give the numbers of the last column of Table I. Thus, from Table I and eq. (20), A3 = -&)[9 + 2763-931 r4 + 646~~1 (41) The results of the calculation of 2 which are valid for both closepacked hexagonal and face-centered cubic lattices, are given in Table II. The calculation of average values of the terms is very easy and does not involve any new considerations beyond these used in the case of z The counting of the numbers of terms in the classes, on the other hand, though it presents, of course, no essential difficulties, becomes cumbersome. TABLE
II Number
Average
-
value
fN
of terms
(1 = 6)
1
-
-
-
(fW’
8: -92
4
-2
3
48 1320 2808 1404 -8880 -4440 7752 960 -1056 27120 1980 -38640 132 29400 -123912 -70320 294456 -120126
264 132 -414
6
32 880
-
2808 14.52 -11916 6755
132 -138
1
ON
From A,=
THE
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CRYSTALS
1-91
Table II and eq. (20),
-A[51
+ 95443-69688~~
+ 120156~6-60063~B)
(42)
The calculation of the average values of higher powers of Eg would be very complicated, since the number of classes of terms grows very quickly with the power of Eg. In relation to this point, it should be added that, with the increasing power of Ed, the sort of term in which indices of many different atoms are present, becomes more and more frequent (this may be illustrated by the comparison of Tables I and II). Thus, if one should try to approach the C u r i e point from high temperatures, the terms of this sort would play a preponderant role in the calculation (another question is whether such a calculation, is possible, and whether the expansion in powers of l/T here used, is convergent) ; indeed, from the work of B 1 o c h 2), and, in particular, from that of K r a m e r s 9), it seems that such long chains of atoms which stretch across large regions of the crystal, are decisive for its properties in the neighbourhood of the C u r i e point. Substitution of (23), (29), (36), (41) and (42) in (16) yields finally for the free energy, in the case of close-packed hexagonal and facecentred cubic lattices (f = 6),
a/T+ e-W) - 33 - ?4T (3 + 20 P - 23 TV)-
F=-Tln(e
+(51
&
(9 + 276 3 -
+9544?---69688~~+
93 1 ~~ + 646 +‘) -
120156~6-60063?)-.
..
(43)
Other thermodynamical functions, such as the entropy S and the total energy E, are immediately obtained from the free energy. One has from (43), with the aid of (IO), S=ln2-$--$---2-.... +lncosha--
+
T
T
where
B, = $ (3 + i0 P - 23 TV) B,=+(9+276P-93L4+646+) Ba ‘= G2 (51 + 9544 ? -
69688 -r4 + 120156 $’ -
60063 ?)
192
W.
OPECHOWSKI
C,=6(1--72) C4 = 3 (1 -3) (10-2372) C6 = (1 r,?) (138 - 931 72 + 969 TV) C,=)(l-?)(2386-348441++90117~~-60063P). Since E = F + TS, one obtains for the energy; in the case of the absence of the external magnetic field,
E=-;f-~+-;+ Equations
_....
(44) and (45) will be used in 5 4.
(45)
,
$2. We are now able to obtain, by a simple differentiation, expression for the magnetization of a crystal, ferromagnetic “anti-ferromagnetic”. From (37) and (lo), one has o=T+
d1 --I
T(l--P)j+
T
[f(f-
--4f)‘1
‘1 + id1
T2
the or . (46)
This formula gives explicitly the dependence of the magnetization on the temperature and the magnetic field. We shah now show that it is exactly equivalent to the implicit formula of H e i s e n b e r g which in our notation is ‘) : c = tanh
(47)
To this end, we choose as independent terms qudratic in l/T, we have then
variables
r~ and T. To the
(48) where c&T) = /CT(1 - GJ) c&J) = fo (1 - 0.y [-- 1 + 4 (1 + 2/p] From equations
(48) and (26),
We now expand Taylor series:’ f=
the right
argtanhcr-(?+
hand member
$1~(argtanho)
of this equation +
++($+...)2$(argtanhcr)= argtanha----
c3
1
1-G
T
in a
.... =
1
+-....
(51)
ON
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Substitution
INTERACTION
IN
MAGNETIC
of (49) in (51), and rearrangement,
+
arg tanho---
fc
f~(~-2)
T
2T2
19.3
CRYSTALS
yields
_.
(52)
*-’
which gives immediately the formula (47) of H e i s e n b e r g. The above derivation shows that the argument of tanh in the H e i s e n b e r g formula is actually an expansion in l/T, its coefficients being polynomials in C, since if one starts from the infinite expansion 00 ~a2n+d4
O=T+Z
, (ak(7) is a polynomial
P
n=l
in T.of degree k)
instead of from (46), one may re-write equations (52) in a quite general form, as follows: OQ
T=G-zl
~CZn+l(d
, (ck(b) is a polynomial
P
*=1
F = arg tanh G =krO arg tanh
+
&
.[
O”
c-
X
-
i
G-
O”
(48), (50), (51) and
in cr of degree k)
(48a)
CZn+l(d
X
P
n=l
z qlk ?I=1
(46a)
-$
>
I
(arg tanh CJ)
(514
P2n-l(4
n=l
P
’
(p,(c) is a polynomial
in d of degree k) (52a)
For example, including the (1 ,/T)3-term, one has for the magnetization, obviously, an expression of the form
o=T+
+ 41 --If
‘w-3f
[f - 1) + HI --f)~l
T
+
T2
+ T(l - 3 (fo +
f23
+
f4T4)
(53)
T3
where fo, fi, f4, are coefficients depending only on the crystal lattice, and eq. (53) can, in this way, be shown equivalent to the following one : a +T fc
cs= tanh
;
-fc+BfD3+
P
+ (fo+2f2-f3b + (fi--f2 + Ff3b3 f (fa,+ 3f2-5f3)$ + T3
1
... .
It should be emphasized that the H e i s e n b e r g formula (47) is valid for both signs of the exchange integral; and, thus, gives a sort Physica
IV
13
194
W. OPEC’HOWSKI
of justification of the “molecular field” hypothesis of W e i s s not only for the case of ferromagnetism, but for that of anti-ferromagnetism too. For the further discussion we still need the expression for the magnetic susceptibility x. Since X=
PFI aa Lo
--
one obtains, from (43), for the close-packed centered cubic lattice, lT=lfT
5+2!i!+lE+lE+....
hexagonal
and the face-
(54)
To the term quadratic in l/T, one can, of course, write down a formula for XT which is valid for any crystal lattice: XT=
1 +f+@-) T
T2
5 3. We now consider a ferromagnetic crystal. We thus assume the exchange integral to be positive, and hence T > 0 (cf. eq. (5)). It is, obviously, not possible to get any exact idea about the C u r i e point with the treatment given here (cf. the remarks on p. 19 1). All one can do is make arather arbitrary extrapolation from the domain of high temperatures where our treatment holds. Defining the C u r i e point, as usual, as that temperature for which the paramagnetic susceptibility becomes infinite, and observing that the expression for ~/XT is in every finite approximation a polynomial in l/T, one can look for real positive roots of this polynomial and, if such roots exist, interpret the smallest one as the C u r i e temperature. From (55), - 1 -1-_f_+t T T2 XT and, hence, one has for the C u r i e temperature the expression given by Heisenberg: T, =
2
1-m. This equation, if it were sufficiently accurate, would yield as a necessary condition for ferromagnetism 1 2 4. From the work of B 1 o c h “) it is known, however, that this condition is certainly not
ON
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195
a necessary one. Taking, for the simple cubic lattice and for the diamond lattice (f = 3 and 2, respectively), one term more in the expansion of ~/XT in powers of 1/T, one sees directly that such a condition derived from (56) is quite illusory. From Table I, with L’[(KZ) (Zm)] = 0 for [both lattices, one obtains, namely, for the ( 1/T)3-term in this expansion the values (-2/T3) and (-4/3T3). Thus, for the simple cubic lattice: L,l-$+3-L XT and for the diamond lattice: 1 -Ml-++&&, XT each of these polynomials obviously
T2
having a positive
$4. In the case of anti-ferromagnetism, with, we must put T < 0 in all formulae J<
T3
real root *) .
which will now be dealt of $9 1 and 2, since here
0.
So far as we know, there is only one case in which the existence of a negative “molecular field”, not masked by other effects has been experimentally stated ‘with certainty. This is the case of tysonite (Ce + Nd + Pr + La, F3), the F a r a d a y effect of which was investigated by’ B ecquer el, de H aas and van den H a n d e 1”). It appears from their experiments that in tysonite the paramagnetic rotation in the direction of the optical axis can be represented in the whole range of liquid helium temperatures roughly speaking - by tanh (pH’/kT) where H’ is the external magnetic field diminished by a term proportional to the rotation itself; this term corresponds obviously to a negative “molecular field”. Moreover, it was stated that practically only the cerium ions were responsible for the paramagnetic rotation. On the other hand, there exist for cerium fluoride the adiabatic demagnetization Wiersma and Kramers6). experiments of de Ha as, Now the form of the dependence of the paramagnetic rotation on H’/T, tangens hyperbolicus, shows that all cerium ions are here, at liquid helium temperatures, in doubly degenerated state. One may *) In the case of a face centred cubic crysta1, or a close-packed hexagonal one, we have for the C u r i e temperature, taking successively into account in the expression for ~/XT obtained from (54) the polynomials of first, second and third degree in I/T, 6.0, 4.7 and 4.3 respectively. The polynomial of fourth degree has no positive real root.
196
W. OPECHOWSKI
thus assume the applicability of the treatment given in the present paper to the case of cerium fluoride, the isotropy of the operator 3 (cT~R~~) being the only considerable simplification *) . The investigations of 0 f t e d a 17) have established that the cerium ions in cerium fluoride form very nearly a closed-packed hexagonal lattice and that is the reason why the calculations of $ 1 were done for this lattice. One may thus directly compare the theoretical curve (cf. Fig. 1) representing the dependence of XT on 1T) and given by eq. ($0 _________
-- _____
-----------
wY?ot.torhkt?---
---------
t
LA----20
SO
40
50
I
Fig. 1. The product of susceptibility and temperature as a function of the temperature in different approximations. y. and T are dimensionless variables which correspond respectively to JxJp’ (x is taken here per atom) and kT/J in ordinary units, J being negative. Curves I, II, III and IV are given by eq. (54), with T < 0, where, on the right side, terms are taken respectively to the first, second, third and fourth power in l/T inclusive. Curve V is obtained by a somewhat arbitrary extrapolation, adding on the right side of eq. (54) a (l/T)‘-term with the coefficient equal to 2400. The best approximation curve is drawn thick in the part which is quite safe.
(54), where T is negative, with the corresponding experimental one in Fig. 6 of the paper of de Ha as, W i er sm a and K r a*) An isotropic exchange interaction in a cry&al in which all atoms are in doubly degenerated states, can be written in.the form const. Z (cr~oi), even when the electrons which are not in closed shells, do possess an orbital angular momentum 0).
ON
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EXCHANGE
INTERACTION
IN
MAGNETIC
CRYSTALS
197
m e r s. It appears from this comparison that the value of the exchange integral can be so chosen as to give a fairly good agreement between experiment and theory (it must be remembered that our T correspond to kT/J in the ordinary notation). The other arbitrary constant, i.e., the (average) magnetic moment p of the doubly degenerated state of the cerium ion (p is contained implicitly in the x of the formula (54)), cannot be determined, for the experimental
Fig. 2. Entropy as a function of the temperature for different magnetic field values. S and T correspond respectively to S/k and kTlJ in ordinary units. Curves 1, 2, 3, are given by eq. (44), with T < 0, in which a = pH/J is taken respectively equal to 0, 7.4, 13.0. The extrapolated part is dotted.
values of XT are only relative. However, a rough value of p can be obtained from the data on paramagnetic rotation: since here pll = 1.3 (p,, is the component of lo in the direction of the optical axis, expressed in number of B o h r magnetons) and pLI seems to be much smaller than one B o h r magneton, p is probably a little smaller than one B o h r magneton. Such an approximative value of lo and the value of J, chosen as explained above (one must take,
198
W. OPjXHOWSKI
namely, k/J = -r 5), can now be put in the formula (44) for the entropy. Thus, in the range of temperatures where (44) is sufficiently exact, the final temperature reached after an adiabatic demagnetization can immediately be determined graphically from a figure such as Fig. 2, provided that the initial temperature and magnetic field are given. Unfortunately, however, the final temperatures appear to lie actually in the domain for which our expression for XT and for the entropy are no longer valid, and no reasonable extrapolation can be made *).
0c _--- ---------
_ ----l wmpt~~torM=~~
-0.6-
-6 I -1.0
I O-RI10
I 20
I 30
I 40
I 50
60
Fig. 3. Energy as a function of the temperature, in the case of no magnetic field. E and T correspond respectively to El J and kT/ J in ordjnary units. E is given by eq. (45), with T < 0.
Perhaps it should still be remarked that this failure is essentially connected with the fact that J must be chosen comparatively large. This fact is a little astonishing, for k/J = - 5 corresponds to A = 1.2 in the C u r i e-W e i s s formula, while it is found from the analysis 5) 8) of paramagnetic rotation data on tysonite that A,, M 0.3 only, which yields a much smaller J. This disagreement may be due to a very strong anisotropy of the exchange interaction in tysonite or in cerium fluoride. *) In the experiments of de H a as, Wiersma and Kramer.s, the initial temperature was about 1.4’, and the initial magnetic field 27600 gauss. In Fig. 2, Curve 2 corresponds to this field, if p = 0.8 and k/J = -5, thus, 5 unities on the axis of abscissae being taken equal to 1”.
ON
THE
EXCHANGE
INTERACTION
IN
MAGNETIC
CRYSTALS
Figures I,2 and 3 refer to the general anti-ferromagnetic are all drawn in our dimensionless variables.
199 case and
I am indebted to Professor K r a m e r s for the suggestion of this’ problem and for his help during the work. To him is due the derivation of the H e i se n b e r g formula in $ 2. Received
January
27th
1937.
REFERENCES 1) W. H e i s e n b e r g, 2. Phys. 40, 619, 1928. The formula for the magnetization of a ferromagnetic crystal is given on p. 630, eq. (22). 2) F. B 1 o c h, Z. Phys. 61, 206, 1930. 3) H. A. K r a m e r s, Rapports et communications issus du Laboratoire Kamerlingh Onnes, pr6sent6s au 7e Congres International du Froid, la HayeAmsterdam, juin 1936, No. 29; or, Commun. Kameriingh Ormes Lab., Leiden, Suppl. No. 83. The method discussed in the present paper is indicated there on p. 14.. 4) J. Becquerel, W. J.de Haas and J.van den Handel, Physical,383, 1933-1934. 5) W.J.de Haas, E.C. Wiersma and H.A. Kramers, Physical, I, 19331934. 6) ,H. A. Kramers, Physical, 182, 1933-1934. 7) I. 0 f t e d a 1, Z. phys. Chem. B, 13, 190, 1931. 8) J.H. van Vieck andM. H. Hebb, Phys. Rev.46, 17, 1934.
IV, no 3
Maart
ON THE EXCHANGE
INTERACTION CRYSTALS
1937
IN MAGNETIC
by W. OPECHOWSKI Instituut
voor
Theoretische
Natuurkunde
der Rijks-IJniversiteit,
Leiden
Summary A simple method, due to K r a m e r s, of approximate evaluation of the partition function (Zustandssumme) in the case when kT > 1 J 1 (J = the exchange integral), but pH/kT arbitrary, is explained and applied to discuss some questions related to ferromagnetism (J > 0) and to the ‘adiabatic demagnetization experiments at low temperatures (J < 0).
Introdzlction. The physical model used in the present paper to discuss some magnetic properties of crystals, is that intfoduced by H e,i s e n b e r g 1) in his first paper on ferromagnetism, and can be briefly characterized as follows : the crystalline lattice is built up of atoms each having only one s-electron which is not in a closed shell *) ; only these electrons have a part in the exchange interaction; the magnetic interaction between their spins is neglected; each atom interacts exclusively with its nearest equidistant neighbours. Starting from this model H e i s e n b e r g derived, as is well known a formula for the magnetization of a ferromagnetic crystal as a function of the external magnetic field. and the temperature, and showed that the positive exchange integral (J > 0) is a necessary condition for the ferromagnetism. The calculations of H e i s e nb e r g, however, are based on the assumption that the energy states of the crystal with the same resultant spin are distributed about their mean value according to a Gaussian curve. On this assumption, the dependence of the magnetization on the crystal structure involves only the number of nearest neighbours of any given atom, i.e. the “co-ordination number”, and not their arrangement. Later; from much more rigorous calculations of B 1 o c h “) which are valid, however, only for sufficiently low temperatures it appeared that H ei se n b er g’s formula is for this range of *) Cf., however,
the foot-note
on p. 196.
-
181 -
182
W.
0l;ECHOWSKI
temperature certainly not even approximately correct. Moreover, B 1 o c h showed that the three-dimensionality of the crystal lattice is essential for the ferromagnetism, thelinear and the two-dimensional lattice being never ferromagnetic, and, thus, that the knowledge only of the co-ordination number does not suffice. Recently K r a m e’r s “) proposed another mathematical treatment of the H e i s e n b e r g model of ferromagnetism and this treatment yields for low and high temperatures respectively the results of B 1 o c h and H e i s e n b e r g. This new method differs essentially from that of H e i s e n b e r g or. B 1 o c h in that it consists in an approximate evaluation of the partition function without explicit calculation of the energy levels, of the crystal. In the same paper K r am e r s briefly indicated still another way of evaluation of the partition function, valid only for high temperatures (kT > J), but for arbitrary magnetic field strength and for any sign of the exchange integral, and calling for no simplifying assumption in the calculation itself. This treatment will be the object of the present paper. In $ 1 a detailed presentation of the method is given and calculations are made for the close-packed hexagonal and face-centred cubic (i.e., close-packed cubic) structures. The method gives essentially the free energy in the form of an expansion in descending powers.of kT/ J, the coefficients being functions of the magnetic field (more exactly: of pH/kT). Consequently the magnetization is obtained in the same form and in 5 2 it .will be shown that this result, to the terms quadratic in J/kT, is exactly equivalent to the H e i s e n b e r g implicit formula, the limitations on the validity of which appear thereby more clearly; further terms in the expansion depend generally not only on the co-ordination number, but also on the arrangement of atoms in the crystal. All calculations and results of $$ 1 and 2 are written so as to be valid for both signs of the exchange integral. In 5 3 the case J > 0 is chosen and the question of the C u r i e point is briefly discussed. In 5 4 is analysed the case J < 0, that of the “anti-ferromagnetic” crystal, where a diminishing of the magnetization results from the exchange interaction. This is of some interest for the discussion of the adiabatic demagnetization experiments at low temperatures, and, though most of these experiments’ were done with crystals in which our simple assumption are certainly not fulfilled (the magnetic
ON
THE
EXCHANGE
INTERACTION
IN
MAGNETIC
CRYSTALS
183
interaction playing a preponderant role *) - not to mention other no less important circumstances), still some formulae and some figures related to the anti-ferromagnetic case will be given, since no quantum theory treatment of this case has yet been presented before. There is only the case of the experiments with ‘crystalline cerium fluoride for which one coulq hope for various reasons that the treatment given in this paper IS not entirely inadequate; however, no unambigous discussion appeared possible, as will be seen in 5 4. S $ 1. We write the formula free energy and the partition
which gives the relation between the function, in the following way: e-FN/RT _spur e-
- !iJ x [bd where the components P a u 1 i matrices:
+ 11= - $J x
(bkbJ)
-
+JNf
of a ck vector operator are represented
(4 by the
and where the sum is extended over all pairs of adjacent atoms in the crystal, and, thus, consists - if we neglect the effect of the surface - of Nf terms, 2f being the c&ordination number. For the second part of the energy operator one has the expression - pH r, ah”’ (4 where the summation extends over the N atoms of the crystal; k is the magnetic moment of each atom and H the magnetic field which is supposed to point in the direction of the z axis. We now replace the variables T and H by new dimensionless variables which we shall denote, throughout the rest of this paper by T and a, in the following way: *) Professor v a n V 1 e c k has kindly informed soon publish in the Physical Rev&u some articles closely investigated.
us that he and his collaborators in which the magnet.ic interaction
will is
184
W.
OPECHOWSKI
kT
>T
-pH +a, J
J
thus, T is no longer the absolute temperature, but a quantity proportional to it, With this new notation eq. (1) becomes e-FNIT = spur e-W,
F being here equal to F/J in the old notation,
(6)
and
E = Et-J+ EH
(7)
where EiJ = -
4 s (CT&
(8)
EH = - a ; op (9) In passing from (2) to (8) we have dropped the term (- Nf J/2), which changes F only by an additive constant, equal to (- f/2), and is of no meaning for our calculations and results. Our problem is now to calculate F, since knowing F as a function of T and a, one obtains immediately by differentiations the magnetization crand the entropy S both per atom and in units corresponding to the choice (5) of independent variables : aF s,-!E O=-aa aT * To this end, we proceed, after K r a m e r s 3), as follows. Since Ed commutes with Ed, we may write e-W = e-~oiT e-qiIT = e-qg/T e-~olT (11) We now introduce the following notation: if P is any operator, then p = spur (Pe-‘H”) (12) spur t?-‘HIT and we shall call p the average value of P. This will be the only meaning of the expression “average value” used in the present paper. Owing to this way of averaging, our results will be valid for any magnetic field strength. Thus, from (1 l), we have: spur e-c/T = e-cc-/T spur emcHIT
(13)
Now we put e-co,/T
where G is a certain function expanded, as follows :
=
e-NGIT
(14
of T and ‘a which can be formally
ON
THE
EXCHANGE
INTERACTION
IN
MAGNETIC
CRYSTALS
185
(15) and the A,‘s are given by eq. (20) below. Equations and (15) yield, thus, F = -
(16)
the A,‘s, we expand the right hand member e--NW
=
g
(-
n=O and putting
m A” + X n=iF’
$ In spur e+HiT
In order to calculate of (14):
(6), (13), (14)
1)”
WG" n! T"
(17)
-
(15) in (17), we obtain
e-NGIT= 1-s ++$ .(
+$ + . ..)+
PAI (
+zT+p+
A2
2 . . . .
)
-
. . . .
Now, because of the linearity of the definition (12), we have 1 T eT%iT = 7 --AL + ---EO ,TT3 + *-** l! T 2!T2 and, thus, according to (14), the coefficients sions (18) and (19) must be equal; hence A,, = coefficient
of N in (-
(18)
(19)
of (1 /T)” in the expan-
2 l)“+l -$,
(1~ = 1, 2. . . .)
(20)
Equations (16) and (20) show that, for sufficiently great T, i.e., the thermal energy much greater than the exchange energy, the free energy will be known, if one calculates the average values of the lOWeSt powers Of the Operator Eg, and the value of the first term on the right side of (16). This term is evaluated immediately, since;from (9), one has: spur
e-‘H
=
spur eazo~‘/T = spur R=l,
n 2...N
e4’#IT = k=l,
n
spur eacrt’/T
(21)
2...N
where the last equality is a consequence of the fact that the spur of a “product matrix” is equal to the product of the spurs of the factor matrices. Now (22) spur eoa&T = e4T + e-dT
186
W. OPECHOWSKI
and, thus, from (21), one obtains: spur e-%/T = (&T + e-@)N.
We shall now calculate
(23)
ET. We have from (8)
-G L - 4 Nf(ckq), (24) since, by symmetry, each term in the sum in eq. (8) has the same average value. Now *),
(25) ax’
0:)/T
spur e
spur [(&J@
=: (&T + e--dT)N-2
+ @+)
spur [&lojd
,+@+#‘WT]
and hence, if, for brevity,
ea(uP+ub-J
= 0
= (eJT _ e-Q/T)2
.
we put tanh (a/T) = T,
(26)
(CrkbJ = 72.
(27)
eq. (25) becomes: Thus, from (24) and (27),
(28)
,=-iNf+’
and this, with regard to (20), gives
(29)
A, =--if?
We calculate
further 2:
3 = ii@ WQ)~= *[ffl (wo)~ + 2fN (2f - 1) (ckd (‘Wn) + fN (fN - 4f + 1) (bkol)
+ (%dl
(30)
The last equality needs comment. Squaring the sum Z (o& we obtain (Nf)2 t erms which we divide into three classes, the terms belonging to the same class all having, by symmetry, the same average value. These are .the classes of terms having respectively two, three and four unequal indices (Where the contrary is not explicitly stated, different subscripts of the ok’s relate, everywhere throughout this paper, to different atoms). As is easily seen, the *) 27 means summation
over
all m’s for which
m # k, C.
ON
THE
EXCHANGE
INTERACTION
IN
MAGNETIC
three classes contain respectively fN, 2fN (2f -2(2f - 1) - l] t erms. Hence eq. (30). Now, from the identity (6pJJ2 = 3 - 2 (GpsJ) and from eq. (27), we obtain Further eq. (27))
(c&2 = 3 - 2;L we have, by an argument similar
CRYSTALS
187
1) and fN [fN -
(31)
(32) to that which leads to
(wd b&J = 72 (33) To find the average value of the terms of the third kind, it is sufficient to observe that, by definition (12), the average value of any two operators which act on different electrons, is equal to the product of the average values of each of these operators. Hence, by eq. (27), (w) (%%) = T4 (34) Substitution of (32), (33) and (34) in (30), and rearrangement, gives z=
?[3
+ 43 (f-
1) + r* (1 -4f)]
+ Qp.
(35)
Hence, by eq. (201, A,=--$[3+4ra(f-I)
+r”(l-4f)].
(36)
Thus, from (16), (23), (29) and (36), we obtain for the free energy, in this approximation, the expression : F =--Tln(e~Tfe-U’r)-&$-$[3+47’(f-
l)+~*(l-4f)]
(37)
We may remark here that the knowledge of & and;; suffices for the derivation of the H e i s e n b e r g formula mentioned in the introduction. We postpone the discussion of topics related to this formula to $5 2 and 3, but we should observe now that our calculation of G and 3 depends on the kind of crystal lattice only through f. Now, that is no longer the case with the average values of higher powers’of Q. In general the result will here also depend on the arrangement of the neighbours atoms about the given atom. To calculate 3 and 2, as we shall now do, we must, thus, explicitly state with what crystal lattice we have to deal. We choose for this purpose the close-packed hexagonal lattice (f = 6) and the results will appear to hold for the face-centred cubic lattice too.
188
W. OPECHOWSKI
To find 2 we must evaluate the third power of the sum C (I& and, doing so, we get (Nf)3 terms .of eight different classes, all terms of each class having the same average value, Counting the number of terms in each class and calculating the average values of their representative terms, one obtains the results which are given in Table I. TABLE Representantive term *)
I
-4verage ..sl..n . S.UI
Number
of terms
(f = 6)
It is now necessary to say something about the way in which the data of Table I are obtained. We first consider the average values of the third column of Table I. The average values [3] and [8] follow immediately from eq. (27) and (32) with the aid of the remark which has yielded eq. (34), and the average values [l] and [2] from eq. (31), (27) and (32). We shall now formulate a general rule by which the average values [5], [6] and [7] can be readily found, and which will be of great use later, in the case of calculation of ET Let (fil%)
@3%)
* * * * (%m--1%)
(38)
be a product which does not split in groups of factors acting on different electrons, and which contains no two equal factors; moreover, contrary to the convention used in this paper, it may here well be that By = n, for i # k. Now, if the product (38) does contain at least one index n, which is not equal to any of the other 1~~‘s (i # k) in (38), then bn,%J
(%3%4)
- * - * b~m-*~“2m)
=
9
(39)
where p is the number of such atoms that the index of each of them occurs in (38) an odd number of times. The premissae of this rule express actually only the sufficient conditions that in an expression such as (38), the terms which contain o@)‘s or &‘)‘s or both of them, have all the average value zero; in other words - that the average value of the whole product be equal to that of the one term which contains only &)‘s. l)
For brevity,
we write
down
only
the subscripts
of the uk’s.
ON
THE
EXCHANGE
INTERACTION
IN
MAGNETIC
189
CRYSTALS
The term [4] in Table I obviously does not fulfil these conditions; here the ~(4’s and &‘)‘s do play a role. The calculation of the average value is, however, still very easy and, using the well-known relations between the P au 1 i matrices, one obtains the value given in Table I. The calculation of the numbers of terms in each class is, of course, a purely arithmetical one and presents few interesting features. It canbe done generally in several different ways, which yields a good check on the results. This circumstance has no importance in the present case (2) where the number of classes is small, but gives much help in the case of3 where there are twenty-three classes, and the counting of the number of terms becomes rather complicated. One will find below (eq. (40)) th e numbers of terms in each of the eight classes written down in such a way as to make clear the simplest manner in which they can be counted. We denote by L [ ] where the representative term of a class is to be put in [ 1, the number of terms of this class. Further we understand by L’[(kZ) (Zwz)] the number of such (AZ) (&)-terms for which atoms k and m are neighbours, and by L”(kZ) (Zm) - th e number of all other @I) (Zm)-terms. We have then: = fN
w431 Lrw2 W)l .
= 3L[(kZ) (Zm)]
LrWI2b41
= 3L[(kZ) (m)]
L[(W W) (41 = L'iI(k4 (41
Lll(k4 v4 WI = w4 b41 Pf - 2) L[(kz) (zm)@%)I = Q{L’[(kz) (zm)]2(2f -2) + L”[(W (W] Wf - 1)) LlI(k4 (z4 (+)I = 3 V’[ (kz) (WI [fN - Wf - 4 - 31 + + L”[(kZ) (Zm)] [fN -
L[(W b.4 &)I =
fN(fN-
1) (fN-2)
(40)
2(2f - 1) - (2f - 2) -
- W(W
(W
211 (Ml+
+ u-w (W WI + LCWP4 b41 + + LW) (I4 h91>
where (cf. p. 187, first line) L[(kZ)2] = fN L[(kZ) (Zm)] = L’[(kZ) (Zm)] + L”[(kZ) (Zm)] = 2fN (2f -
1).
190
W.
OPtiCHOWSKI
Equations (40) hold for an arbitrary crystal lattice without regard to the arrangement of the neighbour atoms about a given atom. The above mentioned dependence on this arrangement lies in L’[(kZ) (Zm)] which is not uniquely determined by the co-ordination number. For example, in the case of the hexagonal layer-structure (i.e., a two-dimensional lattice) L’[(kZ) (Zm)] = 4Nf and in the case of the simple cubic lattice L’[(kZ) (Zm)] = 0, both structures having the same co-ordination number 6. For the close-packed hexagonal lattice and for the face-centred cubic lattice L’[(kZ) (Zm)] is the same and equal to 8Nf, and, hence, L”[(kZ) (Zm)] = .14Nf; these values must be substituted in (40) to give the numbers of the last column of Table I. Thus, from Table I and eq. (20), A3 = -&)[9 + 2763-931 r4 + 646~~1 (41) The results of the calculation of 2 which are valid for both closepacked hexagonal and face-centered cubic lattices, are given in Table II. The calculation of average values of the terms is very easy and does not involve any new considerations beyond these used in the case of z The counting of the numbers of terms in the classes, on the other hand, though it presents, of course, no essential difficulties, becomes cumbersome. TABLE
II Number
Average
-
value
fN
of terms
(1 = 6)
1
-
-
-
(fW’
8: -92
4
-2
3
48 1320 2808 1404 -8880 -4440 7752 960 -1056 27120 1980 -38640 132 29400 -123912 -70320 294456 -120126
264 132 -414
6
32 880
-
2808 14.52 -11916 6755
132 -138
1
ON
From A,=
THE
EXCHANGE
INTERACTION
IN
MAGNETIC
CRYSTALS
1-91
Table II and eq. (20),
-A[51
+ 95443-69688~~
+ 120156~6-60063~B)
(42)
The calculation of the average values of higher powers of Eg would be very complicated, since the number of classes of terms grows very quickly with the power of Eg. In relation to this point, it should be added that, with the increasing power of Ed, the sort of term in which indices of many different atoms are present, becomes more and more frequent (this may be illustrated by the comparison of Tables I and II). Thus, if one should try to approach the C u r i e point from high temperatures, the terms of this sort would play a preponderant role in the calculation (another question is whether such a calculation, is possible, and whether the expansion in powers of l/T here used, is convergent) ; indeed, from the work of B 1 o c h 2), and, in particular, from that of K r a m e r s 9), it seems that such long chains of atoms which stretch across large regions of the crystal, are decisive for its properties in the neighbourhood of the C u r i e point. Substitution of (23), (29), (36), (41) and (42) in (16) yields finally for the free energy, in the case of close-packed hexagonal and facecentred cubic lattices (f = 6),
a/T+ e-W) - 33 - ?4T (3 + 20 P - 23 TV)-
F=-Tln(e
+(51
&
(9 + 276 3 -
+9544?---69688~~+
93 1 ~~ + 646 +‘) -
120156~6-60063?)-.
..
(43)
Other thermodynamical functions, such as the entropy S and the total energy E, are immediately obtained from the free energy. One has from (43), with the aid of (IO), S=ln2-$--$---2-.... +lncosha--
+
T
T
where
B, = $ (3 + i0 P - 23 TV) B,=+(9+276P-93L4+646+) Ba ‘= G2 (51 + 9544 ? -
69688 -r4 + 120156 $’ -
60063 ?)
192
W.
OPECHOWSKI
C,=6(1--72) C4 = 3 (1 -3) (10-2372) C6 = (1 r,?) (138 - 931 72 + 969 TV) C,=)(l-?)(2386-348441++90117~~-60063P). Since E = F + TS, one obtains for the energy; in the case of the absence of the external magnetic field,
E=-;f-~+-;+ Equations
_....
(44) and (45) will be used in 5 4.
(45)
,
$2. We are now able to obtain, by a simple differentiation, expression for the magnetization of a crystal, ferromagnetic “anti-ferromagnetic”. From (37) and (lo), one has o=T+
d1 --I
T(l--P)j+
T
[f(f-
--4f)‘1
‘1 + id1
T2
the or . (46)
This formula gives explicitly the dependence of the magnetization on the temperature and the magnetic field. We shah now show that it is exactly equivalent to the implicit formula of H e i s e n b e r g which in our notation is ‘) : c = tanh
(47)
To this end, we choose as independent terms qudratic in l/T, we have then
variables
r~ and T. To the
(48) where c&T) = /CT(1 - GJ) c&J) = fo (1 - 0.y [-- 1 + 4 (1 + 2/p] From equations
(48) and (26),
We now expand Taylor series:’ f=
the right
argtanhcr-(?+
hand member
$1~(argtanho)
of this equation +
++($+...)2$(argtanhcr)= argtanha----
c3
1
1-G
T
in a
.... =
1
+-....
(51)
ON
THE
EXCHANGE
Substitution
INTERACTION
IN
MAGNETIC
of (49) in (51), and rearrangement,
+
arg tanho---
fc
f~(~-2)
T
2T2
19.3
CRYSTALS
yields
_.
(52)
*-’
which gives immediately the formula (47) of H e i s e n b e r g. The above derivation shows that the argument of tanh in the H e i s e n b e r g formula is actually an expansion in l/T, its coefficients being polynomials in C, since if one starts from the infinite expansion 00 ~a2n+d4
O=T+Z
, (ak(7) is a polynomial
P
n=l
in T.of degree k)
instead of from (46), one may re-write equations (52) in a quite general form, as follows: OQ
T=G-zl
~CZn+l(d
, (ck(b) is a polynomial
P
*=1
F = arg tanh G =krO arg tanh
+
&
.[
O”
c-
X
-
i
G-
O”
(48), (50), (51) and
in cr of degree k)
(48a)
CZn+l(d
X
P
n=l
z qlk ?I=1
(46a)
-$
>
I
(arg tanh CJ)
(514
P2n-l(4
n=l
P
’
(p,(c) is a polynomial
in d of degree k) (52a)
For example, including the (1 ,/T)3-term, one has for the magnetization, obviously, an expression of the form
o=T+
+ 41 --If
‘w-3f
[f - 1) + HI --f)~l
T
+
T2
+ T(l - 3 (fo +
f23
+
f4T4)
(53)
T3
where fo, fi, f4, are coefficients depending only on the crystal lattice, and eq. (53) can, in this way, be shown equivalent to the following one : a +T fc
cs= tanh
;
-fc+BfD3+
P
+ (fo+2f2-f3b + (fi--f2 + Ff3b3 f (fa,+ 3f2-5f3)$ + T3
1
... .
It should be emphasized that the H e i s e n b e r g formula (47) is valid for both signs of the exchange integral; and, thus, gives a sort Physica
IV
13
194
W. OPEC’HOWSKI
of justification of the “molecular field” hypothesis of W e i s s not only for the case of ferromagnetism, but for that of anti-ferromagnetism too. For the further discussion we still need the expression for the magnetic susceptibility x. Since X=
PFI aa Lo
--
one obtains, from (43), for the close-packed centered cubic lattice, lT=lfT
5+2!i!+lE+lE+....
hexagonal
and the face-
(54)
To the term quadratic in l/T, one can, of course, write down a formula for XT which is valid for any crystal lattice: XT=
1 +f+@-) T
T2
5 3. We now consider a ferromagnetic crystal. We thus assume the exchange integral to be positive, and hence T > 0 (cf. eq. (5)). It is, obviously, not possible to get any exact idea about the C u r i e point with the treatment given here (cf. the remarks on p. 19 1). All one can do is make arather arbitrary extrapolation from the domain of high temperatures where our treatment holds. Defining the C u r i e point, as usual, as that temperature for which the paramagnetic susceptibility becomes infinite, and observing that the expression for ~/XT is in every finite approximation a polynomial in l/T, one can look for real positive roots of this polynomial and, if such roots exist, interpret the smallest one as the C u r i e temperature. From (55), - 1 -1-_f_+t T T2 XT and, hence, one has for the C u r i e temperature the expression given by Heisenberg: T, =
2
1-m. This equation, if it were sufficiently accurate, would yield as a necessary condition for ferromagnetism 1 2 4. From the work of B 1 o c h “) it is known, however, that this condition is certainly not
ON
THE
EXCHANGE
INTERACTION
IN
MAGNETIC
CRYSTALS
195
a necessary one. Taking, for the simple cubic lattice and for the diamond lattice (f = 3 and 2, respectively), one term more in the expansion of ~/XT in powers of 1/T, one sees directly that such a condition derived from (56) is quite illusory. From Table I, with L’[(KZ) (Zm)] = 0 for [both lattices, one obtains, namely, for the ( 1/T)3-term in this expansion the values (-2/T3) and (-4/3T3). Thus, for the simple cubic lattice: L,l-$+3-L XT and for the diamond lattice: 1 -Ml-++&&, XT each of these polynomials obviously
T2
having a positive
$4. In the case of anti-ferromagnetism, with, we must put T < 0 in all formulae J<
T3
real root *) .
which will now be dealt of $9 1 and 2, since here
0.
So far as we know, there is only one case in which the existence of a negative “molecular field”, not masked by other effects has been experimentally stated ‘with certainty. This is the case of tysonite (Ce + Nd + Pr + La, F3), the F a r a d a y effect of which was investigated by’ B ecquer el, de H aas and van den H a n d e 1”). It appears from their experiments that in tysonite the paramagnetic rotation in the direction of the optical axis can be represented in the whole range of liquid helium temperatures roughly speaking - by tanh (pH’/kT) where H’ is the external magnetic field diminished by a term proportional to the rotation itself; this term corresponds obviously to a negative “molecular field”. Moreover, it was stated that practically only the cerium ions were responsible for the paramagnetic rotation. On the other hand, there exist for cerium fluoride the adiabatic demagnetization Wiersma and Kramers6). experiments of de Ha as, Now the form of the dependence of the paramagnetic rotation on H’/T, tangens hyperbolicus, shows that all cerium ions are here, at liquid helium temperatures, in doubly degenerated state. One may *) In the case of a face centred cubic crysta1, or a close-packed hexagonal one, we have for the C u r i e temperature, taking successively into account in the expression for ~/XT obtained from (54) the polynomials of first, second and third degree in I/T, 6.0, 4.7 and 4.3 respectively. The polynomial of fourth degree has no positive real root.
196
W. OPECHOWSKI
thus assume the applicability of the treatment given in the present paper to the case of cerium fluoride, the isotropy of the operator 3 (cT~R~~) being the only considerable simplification *) . The investigations of 0 f t e d a 17) have established that the cerium ions in cerium fluoride form very nearly a closed-packed hexagonal lattice and that is the reason why the calculations of $ 1 were done for this lattice. One may thus directly compare the theoretical curve (cf. Fig. 1) representing the dependence of XT on 1T) and given by eq. ($0 _________
-- _____
-----------
wY?ot.torhkt?---
---------
t
LA----20
SO
40
50
I
Fig. 1. The product of susceptibility and temperature as a function of the temperature in different approximations. y. and T are dimensionless variables which correspond respectively to JxJp’ (x is taken here per atom) and kT/J in ordinary units, J being negative. Curves I, II, III and IV are given by eq. (54), with T < 0, where, on the right side, terms are taken respectively to the first, second, third and fourth power in l/T inclusive. Curve V is obtained by a somewhat arbitrary extrapolation, adding on the right side of eq. (54) a (l/T)‘-term with the coefficient equal to 2400. The best approximation curve is drawn thick in the part which is quite safe.
(54), where T is negative, with the corresponding experimental one in Fig. 6 of the paper of de Ha as, W i er sm a and K r a*) An isotropic exchange interaction in a cry&al in which all atoms are in doubly degenerated states, can be written in.the form const. Z (cr~oi), even when the electrons which are not in closed shells, do possess an orbital angular momentum 0).
ON
THE
EXCHANGE
INTERACTION
IN
MAGNETIC
CRYSTALS
197
m e r s. It appears from this comparison that the value of the exchange integral can be so chosen as to give a fairly good agreement between experiment and theory (it must be remembered that our T correspond to kT/J in the ordinary notation). The other arbitrary constant, i.e., the (average) magnetic moment p of the doubly degenerated state of the cerium ion (p is contained implicitly in the x of the formula (54)), cannot be determined, for the experimental
Fig. 2. Entropy as a function of the temperature for different magnetic field values. S and T correspond respectively to S/k and kTlJ in ordinary units. Curves 1, 2, 3, are given by eq. (44), with T < 0, in which a = pH/J is taken respectively equal to 0, 7.4, 13.0. The extrapolated part is dotted.
values of XT are only relative. However, a rough value of p can be obtained from the data on paramagnetic rotation: since here pll = 1.3 (p,, is the component of lo in the direction of the optical axis, expressed in number of B o h r magnetons) and pLI seems to be much smaller than one B o h r magneton, p is probably a little smaller than one B o h r magneton. Such an approximative value of lo and the value of J, chosen as explained above (one must take,
198
W. OPjXHOWSKI
namely, k/J = -r 5), can now be put in the formula (44) for the entropy. Thus, in the range of temperatures where (44) is sufficiently exact, the final temperature reached after an adiabatic demagnetization can immediately be determined graphically from a figure such as Fig. 2, provided that the initial temperature and magnetic field are given. Unfortunately, however, the final temperatures appear to lie actually in the domain for which our expression for XT and for the entropy are no longer valid, and no reasonable extrapolation can be made *).
0c _--- ---------
_ ----l wmpt~~torM=~~
-0.6-
-6 I -1.0
I O-RI10
I 20
I 30
I 40
I 50
60
Fig. 3. Energy as a function of the temperature, in the case of no magnetic field. E and T correspond respectively to El J and kT/ J in ordjnary units. E is given by eq. (45), with T < 0.
Perhaps it should still be remarked that this failure is essentially connected with the fact that J must be chosen comparatively large. This fact is a little astonishing, for k/J = - 5 corresponds to A = 1.2 in the C u r i e-W e i s s formula, while it is found from the analysis 5) 8) of paramagnetic rotation data on tysonite that A,, M 0.3 only, which yields a much smaller J. This disagreement may be due to a very strong anisotropy of the exchange interaction in tysonite or in cerium fluoride. *) In the experiments of de H a as, Wiersma and Kramer.s, the initial temperature was about 1.4’, and the initial magnetic field 27600 gauss. In Fig. 2, Curve 2 corresponds to this field, if p = 0.8 and k/J = -5, thus, 5 unities on the axis of abscissae being taken equal to 1”.
ON
THE
EXCHANGE
INTERACTION
IN
MAGNETIC
CRYSTALS
Figures I,2 and 3 refer to the general anti-ferromagnetic are all drawn in our dimensionless variables.
199 case and
I am indebted to Professor K r a m e r s for the suggestion of this’ problem and for his help during the work. To him is due the derivation of the H e i se n b e r g formula in $ 2. Received
January
27th
1937.
REFERENCES 1) W. H e i s e n b e r g, 2. Phys. 40, 619, 1928. The formula for the magnetization of a ferromagnetic crystal is given on p. 630, eq. (22). 2) F. B 1 o c h, Z. Phys. 61, 206, 1930. 3) H. A. K r a m e r s, Rapports et communications issus du Laboratoire Kamerlingh Onnes, pr6sent6s au 7e Congres International du Froid, la HayeAmsterdam, juin 1936, No. 29; or, Commun. Kameriingh Ormes Lab., Leiden, Suppl. No. 83. The method discussed in the present paper is indicated there on p. 14.. 4) J. Becquerel, W. J.de Haas and J.van den Handel, Physical,383, 1933-1934. 5) W.J.de Haas, E.C. Wiersma and H.A. Kramers, Physical, I, 19331934. 6) ,H. A. Kramers, Physical, 182, 1933-1934. 7) I. 0 f t e d a 1, Z. phys. Chem. B, 13, 190, 1931. 8) J.H. van Vieck andM. H. Hebb, Phys. Rev.46, 17, 1934.