The temperature dependence of the spontaneous magnetization in an antiferromagnetic single crystal

Poulis, N. J. Hardeman, G. E. G. 1953

Physica XIX 391-396

THE TEMPERATURE DEPENDENCE OF THE SPONTANEOUS MAGNETIZATION IN AN ANTIFERROMAGNETIC SINGLE CRYSTAL by N. J. POULIS

and G. E. G. HARDEMAN

Synopsis The time average magnetic moment of the Cu-ions in an antiferromagnetic single crystal of CuC1,.2H,O is measured using the nuclear magnetic resonance method. This average moment is plotted as a function of the temperature. The curve obtained differs considerably from that calculated on the basis of the theory of N C e 1 l) and of that of V a n V 1 e c k *) even if the square of the magnetic moment is plotted against the temperature. It approaches the temperature axis vertically. Measurements were carried out in weak (1640 D) and strong magnetic fields (10490 0).

1. Introduction. As described in earlier papers the proton magnetic resonance line in a single crystal containing paramagnetic ions is split into a number of lines 3). Experiments on single crystals of CuSO,.SH,O and CuC1,.2H,O show that each water of crystallization causes four resonance lines. This splitting of the proton magnetic resonance line is due to the local magnetic field caused by the Cuions at the positions of the protons. When H, is the component of the local field at the position of the proton parallel to the constant magnetic field H,, the resonance frequency can be found at : 27~ = y(H, + HA>

(1)

where y is the magnetogyric ratio for the considered protons. It is found that the position of the resonance lines depends 1”) the position of the different protons in the unit cell, 2”) direction of the external magnetic field H, to the crystal axes, the temperature of the crystal. The value of H,, and therefore the shift of the resonance lines, be calculated using the formula: H, = PO’ C Nib-) c

W-)/~” -

391 -

-

Gc.~o)/~3>~

on: the 3”) can (2)

392

N. j. POULIS

AND

G. E. G. HARDEMAN

,iic is the time average magnetic moment of the cth Cu-ion, ,u,, that of the proton and r the distance of the cth Cu-ion to the considered proton. Thus Hd is equal to p, multiplied by a geometrical factor. Proton magnetic resonance measurements on rhombic single crystals of CuCl,.2HzO showed that this salt behaves antiferromagnetically at liquid helium temperatures. From the proton magnetic resonance measurements we can conclude that in the paramagnetic state the Cu-ion has its magnetic moment reversed many times during a time of the order of the transversal relaxation time t,. The time average of the magnetic ‘moment is, for each Cu-ion, approximately parallel to the external magnetic field H,. This time average magnetic moment in the paramagnetic state follows the law of Curie-Weiss. Just above the NCel point, however, we found that & = f(T) d eviates strongly from this law. It may be remarked incidentally that the intensity of the resonance peaks becomes very strong in this temperature region. It is found that within a few thousandths of a degree a long range antiferromagnetic order sets in. Earlier measurements show that the antiferromagnetic single crystals of CuC1,.2H,O can be divided in two magnetic sub-lattices A’ and A”. It appears that in the antiferromagnetic state the magnetic moments of the Cu-ions reverse many times during a time of the order of t,, just as in the paramagnetic state. The time averaged magnetic moment for ions of lattice A’ is now, ,however, directed along the +a-axis, while for those of lattice A” the time averaged magnetic moment is directed along the -u-axis. It is, however, impossible to draw conclusions about the momentary spin directions from the nuclear magnetic resonance experiments. As all the resonance diagrams for the antiferromagnetic CuC1,.2H,O show sharp peaks, one can conclude that the time average magnetic moment s, for each of the Cu-ions is exactly the same (except for the sign) and is independent of the position of the Cu-ion in the crystal. The spontaneous magnetization of one of the sub-lattices, e.g. A’, is thus N/2.,&, where N is the number of ions in the whole crystal. This spontaneous magnetization depends only upon the temperature and not upon the magnetic field strength. 2. Experimental

magnetic

method. From

resonance

earlier experiments on the proton in antiferromagnetic CuC1,.2H,O we derived

MAGNETIZATION

-.

IN

AN

ANTIFERROMAGNETIC

CRYSTAL

393

the dependence of F, on temperature. These measurements were not accurate as they were carried out on a number of crystals at different times, only five points of the curve being measured. In the introduction (formula 2) we mentioned the splitting of the resonance lines to be proportional to,& and to a geometrical factor. We assumed that the geometrical factor is independent on the temperature in the temperature range of liquid helium. We measured the splitting of the resonance lines as a function of the temperature for two orientations of the magnetic field, via. for H, parallel to the a- and to the b-axis of the crystal. To detect the nuclear resonance the oscillator method is used. The c-axis of the crystal is parallel to the axis of the coil. The measurements were carried out in a constant field of

00

0.0

-

T

LO

20

30

40

50%

Fig. 1. The splitting A between the extreme resonance lines as a function of the temperature T, p is a geometrical factor. A Curve for Ho parallel to the u-axis, 0 curve for H,, parallel to the b-axis. The dashed line shows the theoretical curve from formula 3 normalised at T = 0°K and T = T,“K for the curve with Ho parallel to the u-axis.

394

N. J. PO ULIS AND G. E. G. HARDEMAN

1640 0, while a few measurements

were carried out in a field of

10490 0.

3. Results and discussions. The results of the measurements are shown in fig. 1. Along the ordinate the distance between the extreme resonance lines when the magnetic field is along the a- and b-axis is plotted. As these two curves differ only by the geometrical factor, there must be a multiplication factor independent of the temperature to make these curves coincide. It proves that this factor is constant within 1o/o over the whole temperature region. The splitting of the resonance lines for magnetic fields along the a- and b-axis for T = 0°K can be extrapolated from both curves. The value of the magnetic moment calculated from the formulae in 3) is of the order of 1 Bohr magneton. Many experiments were carried out in the temperature range from 4.O”K to 4.35”K, to find how the curve approaches the temperature axis. It proves that the experimental curves approach perpendicularly to a NCel temperature of 4.336”K for both

8

MHz2

1

Fig. 2. A2 as a function of the temperature ? for H, parallel to the u-axis. The dashed line shows I;,, normalised at T = 0°K and T = TN”K.

MAGNETIZATION

IN

AN

ANTIFERROMAGNETIC

CRYSTAL

395

directions of the magnetic field. V a n V 1 e c k “) as well as N C e 11) deduce a relation for the spontaneous magnetization I,, of one sublattice of an antiferromagnetic crystal as a function of temperature : ISA’ = M.fh

(CI,,,/T).

(3)

The theoretical curve Is,, = f(T) should be influenced very little by relatively small fields as used by us. Identifying IsA, with. the extrapolation of our measured curve to T = 0°K this theoretical curve is given in fig. 1 as a dotted line.The curves differ considerably. The most striking difference between the curves is the behaviour near the NCel temperature which is shown more clearly in fig. 2 where ,,Gzis plotted as a function of T. For the ferromagnetic Ni a similar deviation occurs just below the Curie temperature. According to N 6 e 1 this deviation can be accounted for by considering fluctuations.

‘O8 :6 -

4

-

2

-

A-ho -I

1 Fig. 3. A extrapolated

-T.

2

4

6

do = f(T) plotted on double logarithmic scale. in fig. 1 for T = 0°K. The slope of the two lines from 1°K to 3.5”K is about 4.

8

IO K”

3, is the value lefthand drawn

396

MAGNETIZATION

IN

AN

ANTIFERROMAGNETIC

CRYSTAL

From T, to a temperature of about 2°K the curves have different shapes. The experimental and the theoretical curve coincide within the limits of accuracy in the temperature range from 0°K to 3°K if one uses in formula (3) a Neel temperature of 5.O”K instead of the measured value of 4.336”K. In fig. 3 we plotted the difference between the splitting at a temperature T and that obtained by extrapolation to T = 0°K on a double logarithmic scale. It is found that in the temperature range from 0°K to 3.5”K the spontaneous magnetization (N time-average magnetic moment of the Cu-ions) can equally well be represented by : I, -

I,,

N T4.

(4)

This result is in contradiction with results obtained from the spin wave theory by H u 1 t h C n “) and K u b o 5). They deduce for the spontaneous magnetization at the lowest temperature a dependence given by : I,-II,,wT2,

(5)

where I,, is the spontaneous magnetization at T = 0°K. The experiments in a magnetic field of 10490 0 are carried out in this field along the u-axis of the crystal. The curve obtained has the same general shape as that for weak fields. It must be mentioned, however, that the shift of the NCel temperature is somewhat larger than one should expect theoretically. Our thanks are due to Prof. C. J. G o r t e r for his suggestions and criticism. This work is part of the research programme of the “Stichting voor Fundamenteel Onderzoek op het Gebied van de Materiel’ and was made possible by a financial support from the “Stichting voor Zuiver Wetenschappelijk Onderzoek”. Received

I) 2) 3)

4) 5)

14-2-53.

N C e I, L., Ann. Physique (I I) :% (1948) 137. \’ a II \‘I e c k, J. H., J. them. Phys. !) (1941) 85. P o u 1 is, N. J., and H a r d c m a II, G. E. G., Commun. Kamcrlingh Onyxes Lab., Leiden, Nos. 287.1 and 2883; Physica 111 (1952) 201 and 315. P o u Ii s N. J., H r7 rd e m a n, G. E. G., and B ii 1 g e r, B., Commun. No. 288:; Physica 18 (1952) 429. P o u I i s, N. J., Thesis Leiden, 1952. H u I t h 6 n, L., Proc. kon. Wet. Amsterdam :I!) (1936) 190. K u II o, R., Phys. Rev. 87 (2) (1952) 568.