- Email: [email protected]
The antiferromagneticism of CuCl2 · 2H2O
J. Phys.
Chem. Solids
Pergamon
Press 1958. Vol. 7. pp. 159-164.
THE ANTIFERROMAGNETICISM w.
OF CuCl,
.2H,O
MARSHALL*
Atomic Energy Research Establishment, (Received
8 April
Harwell, Be&s.,
England
1958)
Abstract-A simple explanation is given of the unusual behavior of CuCl, * 2Hz0, which has a NCel temperature of 4*33”K but a broad maximum in the magnetic susceptibility at about 4*8”K. the strongest interaction exists only The explanation is based upon the fact that in CuCl, * 260 in linear chains, and hence it is not as effective in determining the long-range order as it is in determining the short-range order.
CuClz * 2H20 is an antiferromagnet with a NCel temperature of 4-33”K.(l) The specific heat has been measured by FRIEDBERG,@) who found a curve of typical shape but with a rather larger “tail” than is usual above the NCel temperature. The magnetic susceptibility has been measured by GORTER et al., (3)who found a normal behavior below the NCel temperature; but above the NCel temperature the susceptibility continues to rise with temperature and goes through a broad maximum at about 46°K. GORTERalso points out that the magnitude of the susceptibility is rather smaller than one would expect. The purpose of the present paper is to suggest a simple explanation for these three departures from the usual behavior of antiferromagnets. The physics of the explanation can be summarized very easily after studying the structure of the crystal. CuCls .2H20 is a crystal of the rhombic bi-pyramidal class with a unit cell, shown in Fig. 1, of dimensions 7.38, 8.04, and 3.72 A for a, b, and c, respectively. In the unit cell there are dCU
FIG. 1. The unit cell of &Cl,
* 2H,O. Only copper atoms and oxygen atoms are shown.
* Present address : Department of California, Berkeley 4, Calif.
of Physics, University
two copper atoms at (0, 0, 0) and (4, 4, 0); four oxygen atoms at (0, O-25, 0), (0, -0.25, 0), (0.5, O-75, 0), (0.5, O-25, 0); four chlorine atoms at (0.25, 0, O-37), (-0.25, 0, -O-37), (0.25, 0.5, The hydrogen positions 0*37), (0.75, o-5, -0.37). are unknown. From Fig. 1 we therefore see that each copper atom has two nearest neighbors, vertically above and below along the c-axis and four next-nearest neighbors in the a-6 plane. From nuclear-magnetic-resonance studies, POULIS and HARDEMAN (1)conclude that all the electron spins in an u-b plane are parallel to one another and antiparallel to the electrons in the u-b planes vertically above and below. The magnetic studies of GORTER et al. support this conclusion and show that the “easy direction” for the spins is along the a axis. This arrangement of spins is shown in Fig. 1. We now postulate that this spin arrangement is due to an antiferromagnetic coupling between the nearest neighbors along the short c-axis and a ferromagnetic coupling between the next-nearest neighbors in the a-b plane. This is certainly the most obvious coupling scheme which would give rise to the observed spin arrangements. Actually the following argument would not be affected if there were additional ferromagnetic couplings in the a-b plane, but it is essential that the antiferromagnetic coupling from one plane to another should exist only between the nearest neighbors along the c-axis. Thus, for example, it is essential for the argument that any coupling between two copper atoms such as (0, 0, 0) and (4, 4, 1) should be negligible.
159
W. MARSHALL
160
Let the antiferromagnetic coupling along the c-axis be of magnitude J and the ferromagnetic coupling in the a-b plane be of magnitudeJ’. Then we can understand the behavior of CuCls * 2HsO if we accept the second postulate that
J’
(1)
The explicit calculations to be described later give a best fit with a value of 0.165 for the ratio J’/J. We now notice that the NCel temperature is chiefly determined by the magnitude of J’ because the stronger antiferromagnetic coupling exists in only one dimension. Thus, for example, if J’ were zero, we would be left with a set of unconnected linear chains running along the c-axis, and hence T, would be zero because a linear chain cannot support long-range order no matter how strong the antiferromagnetic coupling is. Hence if J’ is small T, is also small. Now consider the short-range order in the crystal. Roughly speaking, this is determined by the Boltzmann factors exp (-J/kT) for vertical pairs and exp(-J’/kT) for horizontal pairs. Since T, is small (roughly m J’/k) even at this NCel temperature, the first factor, which becomes exp( -J/kTc), can be very small, i.e. there can be a great deal of short-range antiferromagnetic order along the c-axis. Thus the spins will still be “locked in” antiparallel and so the susceptibility will continue to rise with temperature until thermal agitation gradually breaks down this coupling and the susceptibility will then go through a broad maximum. This simple explanation accounts for the other unusual features of CuCls * 2HsO also. Because there is an exceptionally large amount of shortrange order along the c-axis, the tail to the specificheat wave must be larger, and the absolute magnitude of the susceptibility must be smaller, than we would normally expect at any particular value of T/T,. For explicit calculations we use the simple Ising model and solve the order-disorder problem by using the Bethe-Peierls method. We divide the lattice into two sub-lattices, p and m, each one consisting of alternate a-b planes. Thus each atom has two nearest neighbors vertically above and below along the c-axis which are on the other sublattice and to which it is coupled antiferromagnetically, and four next-nearest neighbors lying in the same a-b plane and hence the same sub-lattice to
which it is coupled ferromagnetically. For complete order all the spins on the p sub-lattice are + and all those on the m sub-lattice are -; in this case we say the spins are all “right” and any deviations we call “wrong”. In the usual way we must consider a cluster of atoms consisting of a central atom and the surrounding shell of atoms to which it is coupled. Because the outside atoms will have different effects on the shell atoms, depending upon whether the latter lie in the same a-b plane or not, i.e. on the same sub-lattice or not, we must use two long-range-order parameters for the cluster. Furthermore, in the presence of a magnetic field clusters centered on a p sub-lattice atom are not equivalent to those centered on an m sub-lattice atom, so that we must consider these two types of cluster separately and use a pair of long-rangeorder parameters for each. First consider a cluster centered on a p sublattice atom. The relative probability of any arrangement of spins over this cluster of seven atoms can be written down immediately by using the following factors :
t =exp(-J/kT)forevery(++)or(--) pair along the c-axis (i.e. between sublattices) r = exp( -J’/kT) for every (+ -) pair in the u-b plane (i.e. in the same sublattice) s = exp(--2&Y/kT) if the central atom is
I
’ (2)
t-1
Al for each nearest neighbor
in the shell which is (+), i.e. “wrong” $1 for each next-nearest neighbor in the shell which is (-), i.e. “wrong”. Here hr and 41 are long-range-order parameters introduced in the usual way. Then we can sum over all those distributions with the central atom (+), i.e. “right” to give the total relative probability of this atom being “right” as R, = (l+t~l)z(l+r+1)4. (3) the sub-script p being introduced to signify this refers to a p sub-lattice atom. Similarly the total relative probability of the central atom being (-) i.e. “wrong” is w, =
S(t+h)v++ly.
(4)
THE
ANTIFERROMAGNETICISM
Then, if Pp and Mp are the real probabilities of a spin on the p sub-lattice being (+) and (-) respectively, Pp = R,I(%+
W,),
MP = I%#%+
IQ-,).
(5)
But from the rules given by equation (2) we can also write down an alternative expression for Pp by asking for the probability that one of the shell atoms belonging to p be (+). This probability is the sum of two terms: the probability the center atom is (+) times the probability the shell atom is (+) given that the former is (+), plus the probability that the center atom is (-) times the probability the shell atom is (+) given that the former is (-), i.e. Pp = P,/(I ++)+M&++r)
(6)
161
2H,O
is w?n =
(t+h2)2(y++2>4.
(11)
Hence from this cluster we obtain the real probabilities that a spin of the m sub-lattice be (-) or (+) as Mm or Pm respectively where Mm = %I(&+
Wm);
Pm = WmI(&n+ Wm). (12)
An alternative expression the same way as (6), is Mm =
for Mm, derived in just
M~/(1+~~2)+PmTI(y+~2),
(13)
that is Mm~z/(l+~~2)=Pml(~+~2).
(14)
Finally from this cluster we can write down the probability that a p sub-lattice spin be (+) as
that is, J’&l(l ++I)
= M~l(r+&).
(7)
Similarly from this same cluster we can write down the probability that an m sub-lattice spin be (-)
“wrong”,
OF CuCI,.
as Mm =Pp/(l+t~~)+Mpt/(t+hl).
(8)
Now consider a cluster centered on an atom of the m sub-lattice. We can write down the relative probability of any arrangements of spins over this cluster using the factors: t = exp( -J/KT) for every (+ +) or (--) pair along the c-axis (i.e. between sub- \ lattices) I = exp(--J’/KT) for every (+ -) pair in the u-b plane (i.e. in the same sublattice) s = exp( -2pH/kT) if the central atom is }(9) (-) hs for each nearest neighbor in the shell which is (-), i.e. “wrong” 4s for each nearest neighbor in the shell ) which is (+), i.e. “wrong”. hs and $2 are also long-range-order parameters. The total relative probability that the center atom be (-), i.e. “right”, is &
=s(l+
and the total relative
fh2)2(1+r42)4
probability
it be (+),
(10)
i.e.
Pp =Mm/(l
+t~z)+Pmt/(t+~z).
(15)
From these expressions we must find four consistency equations to solve for the four parameters /\I, 41, X2,+2. Two such equations are given already by equations (7) and (14) ; two more come from equating the expressions for Pp given by (5) and (15) and by equating those for Mm given by (8) and (12). The problem is now solved in principle. These four equations are solved for the unknowns hl, $1, hs, 952and then we can write down the magnetic moment as M=&Np[P,-Mp+Pm-Mm]
= N/.L[P,-Mm]. (16)
The problem is greatly simplified because we need only the susceptibility in small fields H. We therefore expand S=l--E E= 2pHJkT
(17)
is small. We first solve the equations for E equal to zero and then compute the corrections to first order when E is small but non-zero. When E is zero, the equations are satisfied identically by Al = A2 = A,
(W q51=42=+.
162
W.
MARSHALL
Hence Rp=Rm=R=(1+tX)2(1+r~)4,
w,= w,= w= (t+~)2(~++)4,
(19)
and (7) becomes $(y+$)R Equating
= (I +4w.
(20)
(8) to (12) gives /\(t+X)R = (l+th)W.
(21)
From (20) and (21) we deduce l$s(l+th)+C$r(l--hz)--h(t+h)
=o.
where t, is the critical value of t. As a becomes small, t, tends to zero, i.e. T, tends to zero. This confirms the qualitative argument given earlier that T, is chiefly dependent on the value of J’ and goes to zero as J’ goes to zero. Above the critical value of t the only solution is the disordered solution given by equation (23) ; below t, equations (20) and (22) must be solved simultaneously for X and #, using numerical methods. Assuming this to be done, we now solve the equations in the more general case when E is not zero. Working throughout just to first order in C, we put
(22)
x1 =h+crC,
#Jl=$+&%
A2 =h+c2r,
$2 =4+d2%
(31)
From this equation we notice that $ and h become unity together and one solution is always the disordered solution h=l=$.
(23)
From an examination of (20) and (22) we deduce that below a certain critical value of T there exists another solution with X and 4 unequal and smaller than one. This is the ordered solution. We determine the critical point in the following way. Let
where X and 4 are the solutions for zero E. Solving for cl, cs, dl, and d2, a great deal of tedious algebra gives eventually ~~(1+~4)(r+$)
= -dl(r+26+42),
@(1+4)(r+$)
= &(~+2++rC2), (32)
BX(l-t2) A+ (l+tq(t+q
h=l--61,
+=1-G&
(24)
where 61 and 82 are small. Then (22) gives 1+r sa = -4, 1+t
c~t(t+2X+th2)(1+2th+P)
s =
(l--t)u+r)
2
(l+t)(zr-1$1.
where we have introduced
The solutions of these equations or
+
B=l-
(1+tq are 61 and Sz zero
Y= l-t/2, which is an equation Define a by
46
4dg
(26)
(1+m
Gq
4dzr
44
+
+
In terms of these quantities equation (16) is given by
(33)
r++’
the susceptibility
from
(27)
to give the critical point. which, solving (32) and (33) for (A+@, J’=aJ
then
’
h(1 +th)s(t+h)s
=
A=l+
whereas (20) gives
’
X(l+tX)yt+X)s
=
Ah(l-t2) B+ (l+th)(t+h)
(25)
-czt(t+2X+tP)(1+2th+As)
(28)
7 = ta
(29)
tp = 1- &/2,
(30)
and (27) becomes
’ =
4pw
h(t+X)(l+tX)
kT
(1+2tX+Xs)2
x
gives
THE
In the disordered simplifies to
ANTIFERROMAGNETICISM
region this expression
$N Tc 2t -’ = 2kT, T l--2t(l--r)
for x
(36)
This expression has been plotted, for various values of u, against T/T, in Fig. 2 using the relation obtained from equation (2) that T
log tc
I,._\
OF
CuCla.
163
2Ha0
seem to show an abrupt change slightly above T,. It seem likely that this is not a real effect. According to GORTER the susceptibility maximum occurs at about 4-S”. Now from equation (36) it is possible to plot the position of the maximum as a function of a. Determining CC by this method gives ct=O.165.
(3%)
We can also compute the specific heat and compare to experiment. We find eventually C,/Nk = 2t[log t]2 x
I) ’ (39) which is plotted for various values of Min Fig. 3. 2aJ
\ o%J
a x 0430
By 0.05
i
3 ::-m
0
0.75 ,
1.00 /,
1.25
1.50
1.7
.oo
FIG. 2. The magnetic susceptibility of C&l, * 2H,O for various values of CL.The heavy curve is the experimental curve as determined with a single crystal. The dashed curve is the experimental value deduced from powder measurements.
and for each value of c(solving equation (30) to give tc. From Fig. 2 we see that the susceptibility has a broad maximum above the NCel temperature for all values of CCsmaller than 0.21. We also show the susceptibility below the NCel temperature which is obtained by solving for h and 4 numerically and substituting into equation (35). From the experimental curve given in Fig. 2, we see that a M O-165 gives a good fit between theory and experiment. The precise value of u to take depends upon which of the two experimental curves is chosen. The theory shows that there is an abrupt change of slope in the susceptibility curves at the NCel temperature, whereas both the experimental curves
0 0.50
0.75
VOO
I.25
1.50
?-75
2co
T/T, FXG. 3. The specific heat of CuCI,
- 2H,O.
We see from this figure that the agreement with experiment is very poor, but this is only to be expected because we have used the Isirg model to describe the interactions, and this is a very crude model especially for calculating the specific heat. We can see this very easily by computirg the partition function at high temperatures and evaluating the susceptibility and the specific heat. The leading terms of a series in powers of l/kT are CO/R = - l 3@ KZTZN
(40)
where S, is the total x component of spin and .@ is the Hamiltonian. Now ciis just N/4, and so the leading term for the susceptibility is the same for Ising and Heisenberg models. But Zs for the Heisenberg model is exactly three times that for the Ising model. At high temperatures therefore, because we have used the Ismg model, we should
164
W.
MARSHALL
expect to get the correct result for x but a result only one-third of the experimental specific heat. Furthermore because the total amount of entropy associated with the specific-heat curve is the same on either model, i.e.
m-+T cw
c
J 0
= R log 2,
1
it follows that, since the Ising model gives low values for C, at high temperatures, it must give high values at low temperatures. From Fig. 3 it can be seen that this is just the behavior found. From this argument we may expect to get reasonable agreement with the magnetic-susceptibility curve, but we must not expect to get any agreement with the specific-heat curve. The difference between theory and experiment in Fig. 3 is therefore only to be expected. In the near future it is planned to reconsider this problem using the correct Hamiltonian, and then we would expect good agreement with experiment for both susceptibility and specific heat.
Acknowledgement-1 would like to thank Professor C. J. GORTER for bringing this problem to my notice and for his interest in the calculation. Note added in proof: Since this paper was written, Dr. H. M. Gijsman has kindly informed me of some details of the measurements of xa made by him and VAN DEN HANDEL. These measurements were briefly reported by VAN DERMAREL et al.@) From their measurements VAN DEN HANDEL and GIJSMAN conclude that the maximum in xa for small applied fields probably occurs at a temperature a little higher than 5.O”K. (Unfortunately this temperature range cannot be examined carefully.) If we take the position of the maximum to be about 5.2”K instead of the 4.8”K deduced by VAN DER MAREL et al.t3) and use this result to determine a, we obtain a result of OL= .138. I am grateful to Dr. GIJSMAN for communicating this information before pubbcation and for a correspondence on this point.
REFERENCES 1. POULIS N. J. and HARDEMANG. E. G. Physica 18, 207 (1952). 2. FRIEDBERGS. A. Physica 18, 714 (1952). 3. VAN DER MAREL L. C., VAN DEN BROCK, WASSCHER J. D. and GORTERC. J. Physica 21, 685 (1955).
Chem. Solids
Pergamon
Press 1958. Vol. 7. pp. 159-164.
THE ANTIFERROMAGNETICISM w.
OF CuCl,
.2H,O
MARSHALL*
Atomic Energy Research Establishment, (Received
8 April
Harwell, Be&s.,
England
1958)
Abstract-A simple explanation is given of the unusual behavior of CuCl, * 2Hz0, which has a NCel temperature of 4*33”K but a broad maximum in the magnetic susceptibility at about 4*8”K. the strongest interaction exists only The explanation is based upon the fact that in CuCl, * 260 in linear chains, and hence it is not as effective in determining the long-range order as it is in determining the short-range order.
CuClz * 2H20 is an antiferromagnet with a NCel temperature of 4-33”K.(l) The specific heat has been measured by FRIEDBERG,@) who found a curve of typical shape but with a rather larger “tail” than is usual above the NCel temperature. The magnetic susceptibility has been measured by GORTER et al., (3)who found a normal behavior below the NCel temperature; but above the NCel temperature the susceptibility continues to rise with temperature and goes through a broad maximum at about 46°K. GORTERalso points out that the magnitude of the susceptibility is rather smaller than one would expect. The purpose of the present paper is to suggest a simple explanation for these three departures from the usual behavior of antiferromagnets. The physics of the explanation can be summarized very easily after studying the structure of the crystal. CuCls .2H20 is a crystal of the rhombic bi-pyramidal class with a unit cell, shown in Fig. 1, of dimensions 7.38, 8.04, and 3.72 A for a, b, and c, respectively. In the unit cell there are dCU
FIG. 1. The unit cell of &Cl,
* 2H,O. Only copper atoms and oxygen atoms are shown.
* Present address : Department of California, Berkeley 4, Calif.
of Physics, University
two copper atoms at (0, 0, 0) and (4, 4, 0); four oxygen atoms at (0, O-25, 0), (0, -0.25, 0), (0.5, O-75, 0), (0.5, O-25, 0); four chlorine atoms at (0.25, 0, O-37), (-0.25, 0, -O-37), (0.25, 0.5, The hydrogen positions 0*37), (0.75, o-5, -0.37). are unknown. From Fig. 1 we therefore see that each copper atom has two nearest neighbors, vertically above and below along the c-axis and four next-nearest neighbors in the a-6 plane. From nuclear-magnetic-resonance studies, POULIS and HARDEMAN (1)conclude that all the electron spins in an u-b plane are parallel to one another and antiparallel to the electrons in the u-b planes vertically above and below. The magnetic studies of GORTER et al. support this conclusion and show that the “easy direction” for the spins is along the a axis. This arrangement of spins is shown in Fig. 1. We now postulate that this spin arrangement is due to an antiferromagnetic coupling between the nearest neighbors along the short c-axis and a ferromagnetic coupling between the next-nearest neighbors in the a-b plane. This is certainly the most obvious coupling scheme which would give rise to the observed spin arrangements. Actually the following argument would not be affected if there were additional ferromagnetic couplings in the a-b plane, but it is essential that the antiferromagnetic coupling from one plane to another should exist only between the nearest neighbors along the c-axis. Thus, for example, it is essential for the argument that any coupling between two copper atoms such as (0, 0, 0) and (4, 4, 1) should be negligible.
159
W. MARSHALL
160
Let the antiferromagnetic coupling along the c-axis be of magnitude J and the ferromagnetic coupling in the a-b plane be of magnitudeJ’. Then we can understand the behavior of CuCls * 2HsO if we accept the second postulate that
J’
(1)
The explicit calculations to be described later give a best fit with a value of 0.165 for the ratio J’/J. We now notice that the NCel temperature is chiefly determined by the magnitude of J’ because the stronger antiferromagnetic coupling exists in only one dimension. Thus, for example, if J’ were zero, we would be left with a set of unconnected linear chains running along the c-axis, and hence T, would be zero because a linear chain cannot support long-range order no matter how strong the antiferromagnetic coupling is. Hence if J’ is small T, is also small. Now consider the short-range order in the crystal. Roughly speaking, this is determined by the Boltzmann factors exp (-J/kT) for vertical pairs and exp(-J’/kT) for horizontal pairs. Since T, is small (roughly m J’/k) even at this NCel temperature, the first factor, which becomes exp( -J/kTc), can be very small, i.e. there can be a great deal of short-range antiferromagnetic order along the c-axis. Thus the spins will still be “locked in” antiparallel and so the susceptibility will continue to rise with temperature until thermal agitation gradually breaks down this coupling and the susceptibility will then go through a broad maximum. This simple explanation accounts for the other unusual features of CuCls * 2HsO also. Because there is an exceptionally large amount of shortrange order along the c-axis, the tail to the specificheat wave must be larger, and the absolute magnitude of the susceptibility must be smaller, than we would normally expect at any particular value of T/T,. For explicit calculations we use the simple Ising model and solve the order-disorder problem by using the Bethe-Peierls method. We divide the lattice into two sub-lattices, p and m, each one consisting of alternate a-b planes. Thus each atom has two nearest neighbors vertically above and below along the c-axis which are on the other sublattice and to which it is coupled antiferromagnetically, and four next-nearest neighbors lying in the same a-b plane and hence the same sub-lattice to
which it is coupled ferromagnetically. For complete order all the spins on the p sub-lattice are + and all those on the m sub-lattice are -; in this case we say the spins are all “right” and any deviations we call “wrong”. In the usual way we must consider a cluster of atoms consisting of a central atom and the surrounding shell of atoms to which it is coupled. Because the outside atoms will have different effects on the shell atoms, depending upon whether the latter lie in the same a-b plane or not, i.e. on the same sub-lattice or not, we must use two long-range-order parameters for the cluster. Furthermore, in the presence of a magnetic field clusters centered on a p sub-lattice atom are not equivalent to those centered on an m sub-lattice atom, so that we must consider these two types of cluster separately and use a pair of long-rangeorder parameters for each. First consider a cluster centered on a p sublattice atom. The relative probability of any arrangement of spins over this cluster of seven atoms can be written down immediately by using the following factors :
t =exp(-J/kT)forevery(++)or(--) pair along the c-axis (i.e. between sublattices) r = exp( -J’/kT) for every (+ -) pair in the u-b plane (i.e. in the same sublattice) s = exp(--2&Y/kT) if the central atom is
I
’ (2)
t-1
Al for each nearest neighbor
in the shell which is (+), i.e. “wrong” $1 for each next-nearest neighbor in the shell which is (-), i.e. “wrong”. Here hr and 41 are long-range-order parameters introduced in the usual way. Then we can sum over all those distributions with the central atom (+), i.e. “right” to give the total relative probability of this atom being “right” as R, = (l+t~l)z(l+r+1)4. (3) the sub-script p being introduced to signify this refers to a p sub-lattice atom. Similarly the total relative probability of the central atom being (-) i.e. “wrong” is w, =
S(t+h)v++ly.
(4)
THE
ANTIFERROMAGNETICISM
Then, if Pp and Mp are the real probabilities of a spin on the p sub-lattice being (+) and (-) respectively, Pp = R,I(%+
W,),
MP = I%#%+
IQ-,).
(5)
But from the rules given by equation (2) we can also write down an alternative expression for Pp by asking for the probability that one of the shell atoms belonging to p be (+). This probability is the sum of two terms: the probability the center atom is (+) times the probability the shell atom is (+) given that the former is (+), plus the probability that the center atom is (-) times the probability the shell atom is (+) given that the former is (-), i.e. Pp = P,/(I ++)+M&++r)
(6)
161
2H,O
is w?n =
(t+h2)2(y++2>4.
(11)
Hence from this cluster we obtain the real probabilities that a spin of the m sub-lattice be (-) or (+) as Mm or Pm respectively where Mm = %I(&+
Wm);
Pm = WmI(&n+ Wm). (12)
An alternative expression the same way as (6), is Mm =
for Mm, derived in just
M~/(1+~~2)+PmTI(y+~2),
(13)
that is Mm~z/(l+~~2)=Pml(~+~2).
(14)
Finally from this cluster we can write down the probability that a p sub-lattice spin be (+) as
that is, J’&l(l ++I)
= M~l(r+&).
(7)
Similarly from this same cluster we can write down the probability that an m sub-lattice spin be (-)
“wrong”,
OF CuCI,.
as Mm =Pp/(l+t~~)+Mpt/(t+hl).
(8)
Now consider a cluster centered on an atom of the m sub-lattice. We can write down the relative probability of any arrangements of spins over this cluster using the factors: t = exp( -J/KT) for every (+ +) or (--) pair along the c-axis (i.e. between sub- \ lattices) I = exp(--J’/KT) for every (+ -) pair in the u-b plane (i.e. in the same sublattice) s = exp( -2pH/kT) if the central atom is }(9) (-) hs for each nearest neighbor in the shell which is (-), i.e. “wrong” 4s for each nearest neighbor in the shell ) which is (+), i.e. “wrong”. hs and $2 are also long-range-order parameters. The total relative probability that the center atom be (-), i.e. “right”, is &
=s(l+
and the total relative
fh2)2(1+r42)4
probability
it be (+),
(10)
i.e.
Pp =Mm/(l
+t~z)+Pmt/(t+~z).
(15)
From these expressions we must find four consistency equations to solve for the four parameters /\I, 41, X2,+2. Two such equations are given already by equations (7) and (14) ; two more come from equating the expressions for Pp given by (5) and (15) and by equating those for Mm given by (8) and (12). The problem is now solved in principle. These four equations are solved for the unknowns hl, $1, hs, 952and then we can write down the magnetic moment as M=&Np[P,-Mp+Pm-Mm]
= N/.L[P,-Mm]. (16)
The problem is greatly simplified because we need only the susceptibility in small fields H. We therefore expand S=l--E E= 2pHJkT
(17)
is small. We first solve the equations for E equal to zero and then compute the corrections to first order when E is small but non-zero. When E is zero, the equations are satisfied identically by Al = A2 = A,
(W q51=42=+.
162
W.
MARSHALL
Hence Rp=Rm=R=(1+tX)2(1+r~)4,
w,= w,= w= (t+~)2(~++)4,
(19)
and (7) becomes $(y+$)R Equating
= (I +4w.
(20)
(8) to (12) gives /\(t+X)R = (l+th)W.
(21)
From (20) and (21) we deduce l$s(l+th)+C$r(l--hz)--h(t+h)
=o.
where t, is the critical value of t. As a becomes small, t, tends to zero, i.e. T, tends to zero. This confirms the qualitative argument given earlier that T, is chiefly dependent on the value of J’ and goes to zero as J’ goes to zero. Above the critical value of t the only solution is the disordered solution given by equation (23) ; below t, equations (20) and (22) must be solved simultaneously for X and #, using numerical methods. Assuming this to be done, we now solve the equations in the more general case when E is not zero. Working throughout just to first order in C, we put
(22)
x1 =h+crC,
#Jl=$+&%
A2 =h+c2r,
$2 =4+d2%
(31)
From this equation we notice that $ and h become unity together and one solution is always the disordered solution h=l=$.
(23)
From an examination of (20) and (22) we deduce that below a certain critical value of T there exists another solution with X and 4 unequal and smaller than one. This is the ordered solution. We determine the critical point in the following way. Let
where X and 4 are the solutions for zero E. Solving for cl, cs, dl, and d2, a great deal of tedious algebra gives eventually ~~(1+~4)(r+$)
= -dl(r+26+42),
@(1+4)(r+$)
= &(~+2++rC2), (32)
BX(l-t2) A+ (l+tq(t+q
h=l--61,
+=1-G&
(24)
where 61 and 82 are small. Then (22) gives 1+r sa = -4, 1+t
c~t(t+2X+th2)(1+2th+P)
s =
(l--t)u+r)
2
(l+t)(zr-1$1.
where we have introduced
The solutions of these equations or
+
B=l-
(1+tq are 61 and Sz zero
Y= l-t/2, which is an equation Define a by
46
4dg
(26)
(1+m
Gq
4dzr
44
+
+
In terms of these quantities equation (16) is given by
(33)
r++’
the susceptibility
from
(27)
to give the critical point. which, solving (32) and (33) for (A+@, J’=aJ
then
’
h(1 +th)s(t+h)s
=
A=l+
whereas (20) gives
’
X(l+tX)yt+X)s
=
Ah(l-t2) B+ (l+th)(t+h)
(25)
-czt(t+2X+tP)(1+2th+As)
(28)
7 = ta
(29)
tp = 1- &/2,
(30)
and (27) becomes
’ =
4pw
h(t+X)(l+tX)
kT
(1+2tX+Xs)2
x
gives
THE
In the disordered simplifies to
ANTIFERROMAGNETICISM
region this expression
$N Tc 2t -’ = 2kT, T l--2t(l--r)
for x
(36)
This expression has been plotted, for various values of u, against T/T, in Fig. 2 using the relation obtained from equation (2) that T
log tc
I,._\
OF
CuCla.
163
2Ha0
seem to show an abrupt change slightly above T,. It seem likely that this is not a real effect. According to GORTER the susceptibility maximum occurs at about 4-S”. Now from equation (36) it is possible to plot the position of the maximum as a function of a. Determining CC by this method gives ct=O.165.
(3%)
We can also compute the specific heat and compare to experiment. We find eventually C,/Nk = 2t[log t]2 x
I) ’ (39) which is plotted for various values of Min Fig. 3. 2aJ
\ o%J
a x 0430
By 0.05
i
3 ::-m
0
0.75 ,
1.00 /,
1.25
1.50
1.7
.oo
FIG. 2. The magnetic susceptibility of C&l, * 2H,O for various values of CL.The heavy curve is the experimental curve as determined with a single crystal. The dashed curve is the experimental value deduced from powder measurements.
and for each value of c(solving equation (30) to give tc. From Fig. 2 we see that the susceptibility has a broad maximum above the NCel temperature for all values of CCsmaller than 0.21. We also show the susceptibility below the NCel temperature which is obtained by solving for h and 4 numerically and substituting into equation (35). From the experimental curve given in Fig. 2, we see that a M O-165 gives a good fit between theory and experiment. The precise value of u to take depends upon which of the two experimental curves is chosen. The theory shows that there is an abrupt change of slope in the susceptibility curves at the NCel temperature, whereas both the experimental curves
0 0.50
0.75
VOO
I.25
1.50
?-75
2co
T/T, FXG. 3. The specific heat of CuCI,
- 2H,O.
We see from this figure that the agreement with experiment is very poor, but this is only to be expected because we have used the Isirg model to describe the interactions, and this is a very crude model especially for calculating the specific heat. We can see this very easily by computirg the partition function at high temperatures and evaluating the susceptibility and the specific heat. The leading terms of a series in powers of l/kT are CO/R = - l 3@ KZTZN
(40)
where S, is the total x component of spin and .@ is the Hamiltonian. Now ciis just N/4, and so the leading term for the susceptibility is the same for Ising and Heisenberg models. But Zs for the Heisenberg model is exactly three times that for the Ising model. At high temperatures therefore, because we have used the Ismg model, we should
164
W.
MARSHALL
expect to get the correct result for x but a result only one-third of the experimental specific heat. Furthermore because the total amount of entropy associated with the specific-heat curve is the same on either model, i.e.
m-+T cw
c
J 0
= R log 2,
1
it follows that, since the Ising model gives low values for C, at high temperatures, it must give high values at low temperatures. From Fig. 3 it can be seen that this is just the behavior found. From this argument we may expect to get reasonable agreement with the magnetic-susceptibility curve, but we must not expect to get any agreement with the specific-heat curve. The difference between theory and experiment in Fig. 3 is therefore only to be expected. In the near future it is planned to reconsider this problem using the correct Hamiltonian, and then we would expect good agreement with experiment for both susceptibility and specific heat.
Acknowledgement-1 would like to thank Professor C. J. GORTER for bringing this problem to my notice and for his interest in the calculation. Note added in proof: Since this paper was written, Dr. H. M. Gijsman has kindly informed me of some details of the measurements of xa made by him and VAN DEN HANDEL. These measurements were briefly reported by VAN DERMAREL et al.@) From their measurements VAN DEN HANDEL and GIJSMAN conclude that the maximum in xa for small applied fields probably occurs at a temperature a little higher than 5.O”K. (Unfortunately this temperature range cannot be examined carefully.) If we take the position of the maximum to be about 5.2”K instead of the 4.8”K deduced by VAN DER MAREL et al.t3) and use this result to determine a, we obtain a result of OL= .138. I am grateful to Dr. GIJSMAN for communicating this information before pubbcation and for a correspondence on this point.
REFERENCES 1. POULIS N. J. and HARDEMANG. E. G. Physica 18, 207 (1952). 2. FRIEDBERGS. A. Physica 18, 714 (1952). 3. VAN DER MAREL L. C., VAN DEN BROCK, WASSCHER J. D. and GORTERC. J. Physica 21, 685 (1955).