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The magnetic susceptibility and relaxation of a MnCl2.4H2O single crystal in the paramagnetic and antiferromagnetic states
Lasheen, M . A . Van den Broek, J. Gorter, C. J. 1958
Physica X X I V 1061-1075
THE MAGNETIC SUSCEPTIBILITY AND RELAXATION OF A MnCI~.4H~O S I N G L E CRYSTAL IN T H E P A R A M A G N E T I C AND A N T I F E R R O M A G N E T I C STATES by M. A. LASHEEN *), J. VAN DEN BROEK and C. J. GORTER Communication No. 312b from the Kamerlingh Onnes Laboratorium, Leiden, Nederland Synopsis
The magnetic susceptibility of a MnCls. 4H~O single crystal has been measured parallel to its preferred (c-axis) and perpendicular (b-axis) directions of magnetization in the t e m p e r a t u r e regions of liquid hydrogen and liquid helium. I t was found to follow a Curie-Weiss law Z = C/(T + A). The susceptibility has the same value in the two directions a t higher temperatures down to about 1.7°K where it becomes anisotropic. I t passes through a m a x i m u m in the b-axis direction and decreases suddenly in the c-axis direction at the same t e m p e r a t u r e 1.62°K, the N6el temperature, which is the same as t h a t found b y F r i e d b e r g and W a s s c h e r in the specific heat determination. Its behaviour in the antiferromagnetic state is in general agreement with the molecular field theory. The dependence of the susceptibility on the magnetic field was studied and c o m p a r e d with the extension of the N6el molecular field theory b y G o r t e r and Mrs. V a n Peski-Tinbergen. Relaxation measurements have been carried out on the crystals with each of the two axes parallel to the field a t liquid helium temperatures and in alternating fields at frequencies below 1135 Hz. I n the liquid helium region at temperatures above the l a m b d a point of liquid helium the experimental dispersion and absorption curves deviate from the C a s i m i r and D u P r 6 theoretical curves due to the limited heat conduction in the crystal. The agreement between the experimental and the CasimirDu Pr$ theoretical curves below t h a t temperature is good. The relaxation parameters are independent of the field strength and v a r y w i t h the inverse 4th power of t e m p e r a t u r e for each of the two axes. Values of the specific h e a t of the spin system are inversely proportional to T~ above 2°K. Below t h a t t e m p e r a t u r e the specific heat is increasing more r a p i d l y as the t e m p e r a t u r e approaches the N6el temperature. This is in agreement with t h e direct specific heat determination.
1. Introduction.
The present work was carried out in order to study the s u s c e p t i b i l i t y a n d t h e r e l a x a t i o n i n t h e s i n g l e c r y s t a l of M n C 1 2 . 4 H ~ 0 w h i c h changes from the paramagnetic to the antiferromagnetic state in the region of t h e l i q u i d h e l i u m t e m p e r a t u r e s . I t is k n o w n t h a t o n e of t h e m a r k e d *) Now at Physics Department, Faculty of Science, Alexandria University, Alexandria, Egypt, United Arab Republic. --
1061
--
1062
M . A . LASHEEN, ~. VAN DEN BROEK AND C. J. GORTER
phenomena in a non-cubic antiferromagnetic crystal is the high anisotropy which decreases rapidly when the N~el temperature is approached. The susceptibility was thus measured in the preferred direction (c-axis) and in one perpendicular to it which is taken in the b-axis direction. The measurements were carried out using a Hartshorn mutual inductance bridge whose details were described in a paper by V a n d e r M a r e l and two of usl). In this bridge the magnetic substance is subjected to a magnetic field of the form H = Hc + h exp(2~ivt) where He is a constant magnetic field produced by a direct current in a large solenoid and h is an alternating magnetic field produced by an alternating current in the primary coils of the cryostat. The primary coil was adjusted so as to have h in the direction of the field He. The magnetic moment acquired b y the substance is thus given by M = M e + m exp(2~ivt).
The static susceptibility is given by X0---- (OMe/OHe)T while the complex susceptibility at the frequency concerned is = m/h ---- Z' -- ix"
where X' and X" are the real and imaginary parts of the complex differential susceptibility. This last definition is only valid if h is taken small compared to the change in Hc producing an appreciable change in z(Hc). The frequency interval used ranges from 2 to 1135 Hz. The field h is about 5 oersted, which is sufficiently small. The values of the constant magnetic field used vary from 0 to 4500 oersted. 2. Theoretical ]ormulae. a) S pi n-1 a t t i c e r e 1a x a t i o n. One of the theoretical descriptions of the spin-lattice relaxation is due to C a s i m i r and D u P r 6 8) who assumed that the system of magnetic dipoles which is called the spin system is in internal thermal equilibrium. According to their theoretical formulae the dispersion (Z') curves are Debije curves added to a constant adiabatic susceptibility while the absorption (if) curves have the Debije shape : Z'/Xo = F/(1 + p2vg) + 1 -- F (1) X"[Xo = Fpv[(1 + p2v2)
(2)
where, if Curie's law is satisfied to a good approximation F = Cn~l(b + CH~);
(3)
b is the constant of the specific heat of the spin system at constant magnetic moment b/T 2 and C the Curie constant. The relaxation parameter p = 2m, ,~ being the relaxation time.
SUSCEPTIBILITY AND RELAXATION OF MnC12.4H20 CRYSTALS 1063 In the dispersion and absorption diagrams 7.'/~o and if/Z0 respectively are plotted versus 10log v. If formulae (1) and (2) are obeyed the following conditions are fulfilled: 1) at X'/Xo = 1 -- F/2 the dispersion curve has a point of inflexion, 2) at this point the slope S = --d(x'/Xo)/d(l°logv ) = 1.1513 F, 3) at the same frequency the absorption curve has a maximum of magnitude h = X"/Xo = F/2, 4) the dimensionless widths 6(1) and ~(r) which are respectively 2 10log (v(m)/vt (1)) and 2 lOlog(vt(r)/v(m)) are theoretically equal and equal to the C.D. value of 1.1438; r(m) is the frequency where the ~bsorption X"/Xo has its maximum value and vl(r) and vt(l ) are the frequencies at which Z"/Zo has half the value of that maximum; vj(r) > vt(1). (See fig. 1). O.5~ vo=~b, 0.4
,I/"\
0.3
0.2
V"
O.
Xld
J
-t.5
-,
T
t-
,
vt
I
-o.~
o
d~
',
&
o[,
I .
,Is
t
"x
0.8
½F
0.6
0.4
0 -4.$
-~
log vJvo
-o:,
o
Fig. 1. Absorption and dispersion curves according to Casimir and Du Pr6's theory. F = 0.8. b) D e t e r m i n a t i o n constant magnetic
of t h e s p e c i f i c h e a t of t h e s p i n s y s t e m at m o m e n t . From the thermodynamic reasoning it
1064
M . A . LASHEEN, J. VAN DEN BROEK AND C. J. GORTER
is easy 8)to deduce the relation ZadlZo = CMICH
(4)
where Xaa is the adiabatic susceptibility, CM is the specific heat of the spin system at constant magnetic moment and CH is the specific heat at constant field. The relation between CM and Cn is given b y CH
=
CM --
T (OM/OT)H (OH/OT)M.
The relation between the magnetic moment M, the susceptibility Z and the field H in the paramagnetic region is M = zH. Hence (OM/OT)~ = H dx/dT and (OH/OT)M = -- (M/z2) d z / d r = -- (H/z) dx/dT. Therefore CH --'~ CM 2F (TH2/z)(dz/dT) 2
and by substituting in (4) l--F----
Zad
X0
CM CM + (THZ/z)(dx/dT)2
or
l/(1 - - F ) =
1 + (TH2/CMz)(dz/dT) ~ = l +
(T3H2/bz)(d~,/dT) ~ ' (5)
if CM = biT 2. Equation (5) can be used for the calculation of b as will be seen later. If Curie-Weiss' law is followed and if CM = biT 2, equation (5) can be reduced to Hz 'F = (6) (b/C) (T + A)S/T 8 + H 2 from which it is clearly seen that F must be expected to be reduced on lowering the temperature. If between M and H a relation exists of the form M = X(0)H + ½~H3, then (OM/OT)H = H(dz(O)/dT + ½n2do~/dT) and (OM/OH)f = z(O) + ~'H2. From this a modified relation (5) is obtained: 1
- - = 1 -- F
T H ~ {dx(0)/dT + ½n 9 d~/dT}~
1 + - -
CM
x(O) +
(Sa)
oLH2
This relation will be used later on to estimate the effect of a possible relaxation in the antiferromagnetic region. 3. Measurement o[ the susceptibility, a) c-axis. A fairly large well formed crystal weighing 3.894 g was chosen from a large number of crystals
SUSCEPTIBILITY AND RELAXATION OF
MnC12.4H20
CRYSTALS
1065
prepared from the solution of MnCls.4H~O in water and placed with its c-axis, the direction of preferred magnetization, in the direction of the field. This was done by embedding the base of the crystal in a layer of paraffin wax in a tube and placing the tube inside the primary coil in the cryostat. Although the crystal was placed with great care so that the c-axis would be in the field direction, yet it m a y be that it is tilted by a small angle which m a y amount to a few degrees. 0.30
iii i o,
J
,:
,.o
,°
°K
2.O
]Fig. 2. S u s c e p t i b i l i t y of a M n C I 2 . 4 H 2 0 single c r y s t a l along t h e c-axis a t f r e q u e n c y v = 227 Hz. A, (D H c = 0 ® He = 2 2 5 0 O e H e = 3375 Oe [] H c = 4500 Oe.
Several series of measurements of the susceptibility were carried out in the temperature regions of liquid hydrogen and liquid helium with the constant field He = 0, 1125, 2250, 3375 and 4500 Oe at a frequency v = 227 Hz. Frequencies of 6.55, 62.5, 119 and 640 Hz were also used with He = 0 to check the results and the agreement was found satisfactory. From the values of the measurements taken in the region of liquid hydrogen and those obtained at the relatively high temperatures of the liquid helium region the susceptibility was found to obey a Curie-Weiss law according to the formula ;~ = C/(T + A) where A = 1.86°K. This A was reported earlier by G i j s m a n 4) to be equal to 1.5°K. Since the susceptibihty ;~ is only measured in the arbitrary units of the bridge, it appears useful to plot in the graphs x/C which has the dimension Physica XXIV
1066
M. A. LASHEEN, J. VAN DEN BROEK AND C. J. GORTER
of a (temperature) -1 and which b y definition approaches T -1 at very high temperatures. The N6el temperature for He = 0, at which the susceptibility decreases suddenly was found to be 1.62°K which is exactly the same as that found from the specific heat measurements b y F r i e d b e r g and W a s s c h e r S ) . It was found to shift to a lower value when a constant field Ho is used. The shift which is accompanied b y higher maxima of Z is increasing with He, in agreement with P o u l i s and G i j s m a n ' s 4) results. The results of the measurements are summarized in fig. 2. A comparison with theory will be given in the next section. b) b-axis. Another • well formed crystal weighing 1.915 g was chosen from the prepared set of crystals. This was taken to .be smaller than the one used in the c-axis measurements in order to suit the size of the tube without any need for grinding the crystal which would produce some difficulty in • identifying the required direction. The smaller size of the crystal helped also, as will be seen later, in decreasing to some extent the effect of the heat of magnetization on the dispersion and absorption curves at temperatures higher than the lambda point of liquid helium. The crystal was placed with its b-axis, which is perpendicular to the direction of preferred magnetization, in the direction of the field b y the same w a y as in the c-axis. The susceptibility was then measured at a series of temperatures in the liquid hydrogen and liquid helium regions with He = 0, 1125, 2250, 3375 and 4500 Oe at a frequency v = 227 Hz. As a check for the results at He = 0, frequencies of 36, 119 and 640 Hz were used at some selected temperatures. • The susceptibility was found, using measurements in the liquid hydrogen and in the high temperatures of the liquid helium region, to obey CurieWeiss' law % = C / ( T + A) withA = 1.79°K which was reported b y G i j s m a n to be equal to 2.4°K. The susceptibility passes through .a maximum at the N&I temperature of 1.62°K which is the same value found for the c-axis and from the specific heat measurements 5). The shift of the N6el temperature to a lower value, when a constant magnetic field is applied, exists also in this case. The maximum is becoming higher and the shift is increasing with the increase of the field. The shift for the same constant field is smaller in this case than the corresponding one in the c-axis. This is all in agreement with reference 4). The results of the measurements are shown in fig. 3.
4. Comparison with molecular/ield theory. On the basis of N6el's molecular field model Mrs. V a n P e s k i - T i n b e r g e n 0) and one of us have calculated the magnetic moment M of an antiferroma'gnetic substance as a function of the magnetic field He and the temperature T. Above the N6el point they predict a susceptibility following Curie-Weiss' law X = C/(T + A) with A = (Ax -- Dx)(S + 1)/3Sk~ where Ax and D~ are the interaction constants.
SUSCEPTIBILITY AND RELAXATIONOF MnCb.. 4H20 CRYSTALS 1067 The N4el temperature is given b y TN = (Az + Dz)(S + 1)/3Sk, k being Boltzmann's constant. Below that temperature they find, when the field is along the preferred axis of magnetization, a non-hnear relation b e t w e e n the reduced magnetic moment m = M x / # z and the reduced magnetic field 0.270
deq I.
0.265
¢
I
0.255
~.2
T
I1.6
t4
t.8
Fig. 3. Susceptibility of a MnCls.4HsO single crystal along v = 227 Hz. G, A
He = 0 H e = 3375 Oe
the
°K
2.0
b-axis at frequency
H e = 2250 O e [ ] H o = 4500 Oe.
®
h = #J-I~/(A~ + D=) (#~ is the magnetic moment per ion). For small values of h this relation can be approximated b y a series expansion: m =
al h + 1 + al~,
2a2~/(1 -- ax) + a8 h s ... (I + al~,) 4
(7)
Here ax, as, a3 are the first coefficients of the series expansion in h of Brillouin's magnetization function Bs(h/t) at the point where B s ( h / t ) = h. ? = (A~ -- D~)/(A~ + D~) = A/TN; the reduced temperature t---= kT/(A~ + D~).
1068
M . A . LASHEEN, J. VAN DEN BROEK AND C. J. GORTER
Taking A-----1.86°K, TN = 1.62°K," S-----5/2 and g = 2 the theoretical relation M = z(0)
+
...
(Ta)
following from equation (7) has been calculated. At H = 0 the parallel susceptibility Sit = X(0) is thus obtained. The perpendicular susceptibility X± following from the model is a constant which is approximately equal to XII(TI~), accepting that there is no large anisotropy in g, A and D. 0.3
0.2
! / I /
o.1
/1111} I
a~
Fig. 4. Susceptibility of a MnC12.4H~O single crystal. C) values along the c-axis & values along the b-axis - theoretical values.
This state of affairs is summarized and compared with experiment in fig. 4, where all the quantities have been divided by Curie's constant C. There is seen to be qualitative agreement; at 4°K even quantitative agreement. The apparent deviation of •± in the antiferromagnetic state from a temperature independent value m a y be explained by a slightly tilted orientation of the crystal. For the c-axis crystal the susceptibility ~ H e ) obtained from the experiment at 227 Hz was found to contain a term proportional to He I. The proportionality coefficient ~ calculated from that has been compared in fig. 5 with the theoretical ~ defined by eq. (Ta). The theoretical ~ has also been
SUSCEPTIBILITY AND RELAXATION OF M n C 1 2 . 4 H 2 0
CRYSTALS
1069
obtained from graphical constructions following ref. 6) ; the results from this are in good agreement with the calculations, from which it can be concluded that higher powers of h do not notably influence our results. It must be kept in mind, however, that there can be other reasons for quadratic effects in •(He). As a matter of fact, from measurements on powdered MnC12.4H20, which will be discussed in a separate paper, it was concluded that there occurs a relaxation phenomenon at frequencies of the order of 3 Hz. As there are no measurements available enabling us to make sure if this occurs in the single crystals, we cannot exclude the possibility that the measurements at 227 Hz give the adiabatic X(He) in stead of its isothermal value. The possible effect of this has been estimated b y means of equation (5a) at a few temperatures using the specific heat data from ref. 5) and is given also in fig. 5. For a more complete discussion on these matters we refer to the paper on the powder measurements. 6O
o 4O
20
0
Fig. 5. - --(D
T
.~,..
05
!
OK
',.S
TN
T h e coefficient ~ in M : X(0)H + ~/-/3. t h e o r e t i c a l results eq. (Ta) e x p e r i m e n t a l results e x p e r i m e n t a l values c o r r e c t e d for r e l a x a t i o n .
5. Relaxation. a)c-axis. The relaxation parameters at any given temperature and at different constant fields were measured from a set of dispersion and absorption curves. They were measured at different temperatures in the liquid helium region above the N~el temperature. No relaxation was detected in the available range of frequencies in the liquid hydrogen region of temperatures, while the data below the N6el temperature did not permit a conclusion about the existence of small relaxation effects. It was observed that the value of F is decreasing with the decrease in temperature and this is according to expectation (see eq. (6)). The dispersion and absorption curves measured at temperatures above the lambda point of liquid helium do not agree with the description given b y C a s i m i r and D u P r 6 2) as they are considerably flatter and broader than
1070
M . A . L A S H E E N , J. VAN D E N B R O E K AND C . J .
GORTER
the theoretical ones. This is a phenomenon which has already been found earlier in concentrated paramagnetic salts x). It may be ascribed to the limited heat conduction in the crystal. The heat of magnetization has to pass not only the usual bottleneck between the spin system and the crystal lattice, but in series with this is a heat resistance in the lattice itself, which differs for different volume elements. The result is a lengthening of the relaxation parameter for the inner parts of the crystal (cf.7), s)). The quantitative description of this phenomenon is not yet satisfactory. It may however be observed that the outer parts of the crystal which have good contact with the surrounding helium bath, will have the shortest relaxation time. So from the corresponding high frequency part of the dispersion and absorption curves it is possible to obtain an estimate of this relaxation parameter. In the case of the c-axis it was even better than this. A second incomplete peak was clearly observed at high frequencies in the absorption curves measured at 3.988°K and 3.010°K, which give results that are much better in agreement with those below the lambda point. At the same frequency for each of the two temperatures the dispersion curves have their TABLE I The relaxation parameters pdisp and pabe as functions of He and T. (The numbers, given in brackets have been obtained by extrapolation). ~0~) (ms) \ pdl~p pabs p=bs (2nd peak) pdlsp p=bs p~bs (2nd peak) pdisp pabs
450
900
1125
1575
2250
.3375
75 (23.7) 4.45 405 119 15.9 98 79.4
77 (16.8) 5.60 485 104 15.9 78 84 100 126
103 (21.1) 5.60 635 345 25.1 80 71
26.6 4.00
63 (23.7) 4.45 305 168 12.5 (40) 79.4
pdlsp psbs
pdlsp 155
• pa,bs
pdisp psbs pdisp p~bs pdlsp pabs pdtsp p~bs
pdlsp pabs pdlsp pabs pdlsp p~bs
4500
T(°K) direction
i
17.8 22.4
17.8 20.0
(40) 65
20.0 19.3 25. I 28.2 71 67 89 89 118 118 126
28.2 22.1
60 65
3.988
c-axis
3.010
84 75 150 112 188 155
2.106
31.6 27.9 71 79 63 67 92 92 139
4.000
157 159 156 159
1.780
1.955 1.813
3.012 2.084 2.014 1.901
1.672
b-axis
SUSCEPTIBILITY AND RELAXATION OF
MnClz.4HsO CRYSTALS
1071
steepest points which were used for determination of the corresponding F and S values. The abrupt change of the shape of the relaxation curves at the lambda point may indicate an abrupt increase of the thermal contact between the crystal and the liquid. Considering the situation below the lambda temperature one must not forget the well-known boundary layer effect, which was first described by K a p i t z a 9). The order of magnitude of the relaxation 2.S
c-axis
2.0
b-axis
o
®
T-~a ~
T.
1
Q
\ ,T-4.,
1.0 \
02 log Pabs
IC I
IocJT0.2
0.4
2.5
I
0 IogT0.2
0.6
0.4
0.6
b-axis
c-axis
2.0
log Pobs
I '5 ~0 ~
I
o.s_j_~
I
2
I 4
I lOkOc
I 0.5
Hf I b
J
I
2
4
i lOkO~t
Fig. 6. Upper diagrams: log pabs V s log T at H c = 2250 Oe (c-axis) and at H e = 4 5 0 0 O e (b-axis). (D normal procedure A second peak
Lower diagrams: log pabs v s log He at 2 . 1 0 6 ° K (c-axis) and at T = 2 . 0 8 4 ° K (b-axis).
time resulting from this heat resistance and the heat capacity of the sample can be estimated from the geometrical surface area of the crystal and numerical data from B e e n a k k e r e . a . 10) and F r i e d b e r g 5). T h i s , turns out to be of the order of 1/5 s below TI~ and 0.01 s between TI~ and the lambda point. A s , ~-- 1/5 s means # ~-~ 1 s and as this is definitely longer than the measured relaxation parameters, it may be concluded that the effective area is much larger than the geometrical one, which might indicate that the He II penetrates into cracks in the crystal. This would also explain why the
1072
M.A.
LASI-IEEN, J. VAN D E N B R O E K A N D C. J. G O R T E R
TABLE II The shape parameters as functions of He and T. Oe)
450
l.O
2h/F
.80 S/1.1513F
900
1125
.44 1.0 .48 l.O
.30 .87 .42
2.19
(~(r)/1.1438
8(I)/I. 1438
1.12
1575
2250
3375
.42 .96 .37 1.0
.41 1.0 .34 .94
.38 1.0 .40 .90
,29
(I.0) (1.0) .24
.70 .48 .64
.68 .38 .83 .69 .69
2.18 1.16 2.49 1.27 1.04
1.91 l,ll 2.44
1.95
2.25
.80
.86
.93 1.05
.24 .74 .26 .82
.91 1.0
.83' .80 .78
2.14 1.27
2.82 1.66 1.00
1.00 1.05 1.05
.89
.43
.65 .54
S/I.1513F
1.70 1.0 ~(r)/I.1438
.68 l.O .82 l.O .85 .81
c-axis
3.988 3.988 (2nd peak) 3.010 2. 106 1.955 1.813 3.988 3.988 (2nd peak) 3.010 3.010 (2nd peak) 2.106 1.955 1.813
.56 .52 .85 .81 .61 .55 1.91 1.17 1.0 1.29 1.46
.69 .79
4.000 3.012
.91
.93 1.0 .85 .81 .69
2.084 2.014 1.901 1.780 1.672
.46
.56 .70 1.11 .83 .70 .73 .86
4.000 3.012 2.084 2.014 1.901 1.780 1.672
1.77 1.54 l.Ol 1.09 1.29 1.22
4.000 3.012 2.084 2.014 1.901 1.780 1.672
1.03
1.79 1.08
1.27
l.O
3.988 3.988 (2nd peak) 3.010 2.106 1.955 1.813
.68
(I.81)
~(])/1.1438
direction
3.988 3,988 (2nd peak)
1.56
2h/F
T(°K)
4500
1,01 1.0
(.90)
(1.00)
(1.65)
4.000 3.012 2.084 2.014
b-axis
SUSCEPTIBILITY AND RELAXATION OF
MnC12.4H20 CRYSTALS
1073
internal heat conduction in the crystal becomes better, as is proved by the smaller deviations from C.D. theory. T h e r e l a x a t i o n p a r a m e t e r s . Values of the relaxation parameters Pdisp and pabs are given at different temperatures in table I. At the two temperatures higher than the lambda point of liquid helium values obtained from the normal procedure, which are affected by the crystal heat conductivity are given together with the values acquired from the second peaks of the absorption curves and the corresponding steepest point of the dispersion curves. Log Pabs has been plotted as a function of log T at the constant field Hc = 2250 Oe in the left hand side up diagram of fig. 6. This diagram shows that p varies with T -4.2. In the left hand side down diagram of fig. 6 log Pabs has also been plotted as a function of the constant field He at T = 2.106°K. The plot is a straight line parallel to the Ho-axis which indicates that the relaxation parameter is independent of the value of the constant field applied. T h e s h a p e p a r a m e t e r s . Values of each of the shape parameters divided by its theoretical (C.D.) value, that is 2h/F, S/1.1513 F, ~(r)/1.1438 and 6(1)/1.1438 are given in table II. Each of these ratios should be equal to unity if the dispersion and absorption curves follow the C.D. formulae. The agreement is good a t temperatures below the lambda point; the best is at 2.106°K which is just below that temperature. The deviation of the dispersion and absorption curves measured at both &988°K and 3.010°K, due to the heat effect, can easily be seen from the deviations of their shape parameters from the theoretical values. The shape parameters given from the second absorption peak and the corresponding steep point on the absorption curves at these two temperatures are in good agreement with the C.D. values as shown in table II. b) b-axi s. Several sets of dispersion and absorption curves were measured at different values of the constant field at different temperatures in the liquid helium region above the N6el temperature. As in the c-axis no relaxation was found in the available range of frequencies below the N6el temperature and in the liquid hydrogen region of temperatures. The value of F was, also in this case, found to decrease with the decrease of the temperature as is expected. It was stated above that a crystal, smaller than that used in the c-axis measurements, was chosen in order to decrease the effect of the heat of magnetisation at temperatures higher than the lambda point of liquid helium. This really helps in decreasing the deviation of the dispersion and absorption curves from the theoretical formulae at these temperatures as can be clearly seen from table II. Although the dispersion curves were steeper and the absorption curves were narrower than those measured in the c-axis at the corresponding temperatures, yet the relaxation parameters measured from them were not in agreement with those measured at tem-
1074
~t. A. LASHEEI~, J . VAN DEN BROEK AND C. J. GORTER
peratures below t h e lambda point. No second peaks, resembling those found in t h e c-axis measurements, were observed on the absorption curves and it m a y be that the two have merged. The relaxation parameters. Values of the relaxation parameters pdisp and Pabs at different temperatures and different constant fields are given in table I. Log Pabs is plotted as a function of log T at the constant field He = 4500 oersted in the right hand side up diagram of fig. 6. From this diagram it is found that p varies with T-4.1 which is nearly the same as that found for the c-axis, pabs has also been plotted as a function of the constant field He at T = 2.084°K in the right hand side clown diagram of fig. 6. This shows that the relaxation parameter is independent of the applied field strength. T h e s h a p e ' p a r a m e t e r s . Values of 2h/F, S/1.1513 F, ~(r)/1.1438 and 6(1)]1.1438 the shape parameters divided b y their theoretical values are given in table II. It can be seen from these values that the agreement between the measured and the C.D. values is quite good at temperatures below the larnbda point, this being much better at temperatures close to the lambda point than at lower temperatures. This deviation at lower temperatures m a y be due to the difficulty of measuring some of these shape parameters due to the decrease in the value of F as stated above. At temperatures higher than the lambda point there is a deviation from the theoretical values although it is less than that in the case of the c-axis due to the use of a small crystal.
6. Determination of the speci[ic heat. Equation (5) was used for the •determination of b, the constant of the specific heat of the spin system at constant magnetic moment. This was carried out b y first p l o t t i n g 1/(1 -- F) for different field strengths at the same temperature versus H ~ (= Ho 2) and determining the slope of the resulting straight line. This slope gives the value (T3/bZ)(dx/dT) 9'. The value of the susceptibility Z is to be determined from the experimental curve at the corresponding temperature T. For the determination of dx/dT great care must be taken as it is in the second power in the formula and so any small error would lead to a highly deviated value of b. For this reason the slope was carefully determined from the experimental curve at different temperatures and t h e n plotted versus the temperature. The curve which should be smooth helped in noticing and correcting any deviation in the measured value of the slope at any temperature. Then b y substituting for all the known values, the value of b is calculated. The value of b found b y this w a y is expressed in the arbitrary units of the bridge.The value of b/C calculated b y dividing the value of b b y the measured value of C gives the ratio of the real value o~ b and the Curie constant. Values of b/C at different temperatures for both c and b crystal axes are shown in table III. Above 2°K, for each of the two axes, the value of b/C is constant and equal to 19.0 × 106 Oe 2. This value is in agreement with
S U S C E P T I B I L I T Y A N D R E L A X A T I O N OF
MnC19..4H20
CRYSTALS
1075
T A B L E III
blC as
a function of T
c-axls
T(°K) [ b/C(kOe~) [ T(°K) 3.988 3.795 3.606 3.360 3.199 3.010 2.800 2.608 2.429 2.314
19.1 19.0 19.0 18.9 19.0 18.9 19.0 19.0 19.0 19.1
2.192 2.106 1.993 1.955 1.914 1.813
" b-axis
[blC(kOe') 19.2 19.8 20.3 21.1 21.4 22.4
=(oK) [ b/C(kOe2) ] T(°K) 4.000 3.884 3.713 3.517 3.295 3.146 3.012 2.991 2.861 2.690
'
19.1 19.0 19.0 19.0 19.0 19.0 18.8 18.8 18.9 19.0
2.546 2.413 2.243 " 2.105 2.084 2.014 1.979 1.901 1.780 1.672
I
b/C(kOe") 19.0 19.0 19.0 19.0 19.0 19.0 19.1 19.8 23.2 40.6
the values 19.5 × 108 and 19.8 × 10e Oe2 found at high temperatures by T e u n i s s e n and G o r t e r 11) and Starrl2). Below 2°K the value of b/C is increasing as is expected from the specific heat measurements by F r i e d b e r g and W a s s c h e r 5) indicating that the specific heat is no more given by b/T 2. The authors are indebted to Dr. L. C. V a n d e r M a r e l for his cooperation and valuable discussions and to the technical staff of the laboratory for their help with the experimental preparations. One of us (M.A.L.), who was working at the Kamerlingh Onnes Laboratory for one year, expresses his gratitude to members of the academic staff of the laboratory for the cooperation and assistance he received during his stay. He also appreciates the leave granted to him by Alexandria University. Received 11-9-58. REFERENCES 1) V a n d e r M a r e l , L. C., V a n d e n B r o e k , J. and G o r t e r , C. J., Commun. Kamerlingh Onnes Lab., Leiden No. 306a; Physica 2 3 (1957) 361. 2) C a s i m i r , H. B. G. and D u P r 6 , F. K., Commun. Suppl. No. 85a; Physica 5 (1938) 507. 3) G o r t e r , C. J., Paramagnetic relaxation (Elsevier, A m s t e r d a m 194~/). 4) G i j s m a n , H. M., Thesis Leiden (1958). 5) F r i e d b e r g , S. A. and W a s s c h e r , J. D., C o m m u n No. 293c; Physica 19 (1953) 1072. 6) G o r t e r , C. J. and T i n e k e V a n P e s k i - T i n b e r g e n , Commun. Suppl. No. ll0b; Physica R2
(1956) 273. 7) 8) 9) I0)
E i s e n s t e i n , J., Phys. Rev. 84 (1951) 548. V a n d e r M a r e l , L. C., Thesis Leiden (1958), p. 23 et seq. K a p i t z a , P. L., J. Phys. U.S.S.R. 4 [1941) 181. B e e n a k k e r , J. J. M., T a c o n i s , K. W., L y n t o n , E. A., D o k o u p i l , Z. and V a n S o e s t , G., Commun. 289a; Physica 18 (1952) 433. ll) T e u n i s s e n , P. and G e t t e r , C. J., Physica 7 (1940) 33. 12) S t a r r , C., Phys. Rev. 60 (1941} 241.
Physica X X I V 1061-1075
THE MAGNETIC SUSCEPTIBILITY AND RELAXATION OF A MnCI~.4H~O S I N G L E CRYSTAL IN T H E P A R A M A G N E T I C AND A N T I F E R R O M A G N E T I C STATES by M. A. LASHEEN *), J. VAN DEN BROEK and C. J. GORTER Communication No. 312b from the Kamerlingh Onnes Laboratorium, Leiden, Nederland Synopsis
The magnetic susceptibility of a MnCls. 4H~O single crystal has been measured parallel to its preferred (c-axis) and perpendicular (b-axis) directions of magnetization in the t e m p e r a t u r e regions of liquid hydrogen and liquid helium. I t was found to follow a Curie-Weiss law Z = C/(T + A). The susceptibility has the same value in the two directions a t higher temperatures down to about 1.7°K where it becomes anisotropic. I t passes through a m a x i m u m in the b-axis direction and decreases suddenly in the c-axis direction at the same t e m p e r a t u r e 1.62°K, the N6el temperature, which is the same as t h a t found b y F r i e d b e r g and W a s s c h e r in the specific heat determination. Its behaviour in the antiferromagnetic state is in general agreement with the molecular field theory. The dependence of the susceptibility on the magnetic field was studied and c o m p a r e d with the extension of the N6el molecular field theory b y G o r t e r and Mrs. V a n Peski-Tinbergen. Relaxation measurements have been carried out on the crystals with each of the two axes parallel to the field a t liquid helium temperatures and in alternating fields at frequencies below 1135 Hz. I n the liquid helium region at temperatures above the l a m b d a point of liquid helium the experimental dispersion and absorption curves deviate from the C a s i m i r and D u P r 6 theoretical curves due to the limited heat conduction in the crystal. The agreement between the experimental and the CasimirDu Pr$ theoretical curves below t h a t temperature is good. The relaxation parameters are independent of the field strength and v a r y w i t h the inverse 4th power of t e m p e r a t u r e for each of the two axes. Values of the specific h e a t of the spin system are inversely proportional to T~ above 2°K. Below t h a t t e m p e r a t u r e the specific heat is increasing more r a p i d l y as the t e m p e r a t u r e approaches the N6el temperature. This is in agreement with t h e direct specific heat determination.
1. Introduction.
The present work was carried out in order to study the s u s c e p t i b i l i t y a n d t h e r e l a x a t i o n i n t h e s i n g l e c r y s t a l of M n C 1 2 . 4 H ~ 0 w h i c h changes from the paramagnetic to the antiferromagnetic state in the region of t h e l i q u i d h e l i u m t e m p e r a t u r e s . I t is k n o w n t h a t o n e of t h e m a r k e d *) Now at Physics Department, Faculty of Science, Alexandria University, Alexandria, Egypt, United Arab Republic. --
1061
--
1062
M . A . LASHEEN, ~. VAN DEN BROEK AND C. J. GORTER
phenomena in a non-cubic antiferromagnetic crystal is the high anisotropy which decreases rapidly when the N~el temperature is approached. The susceptibility was thus measured in the preferred direction (c-axis) and in one perpendicular to it which is taken in the b-axis direction. The measurements were carried out using a Hartshorn mutual inductance bridge whose details were described in a paper by V a n d e r M a r e l and two of usl). In this bridge the magnetic substance is subjected to a magnetic field of the form H = Hc + h exp(2~ivt) where He is a constant magnetic field produced by a direct current in a large solenoid and h is an alternating magnetic field produced by an alternating current in the primary coils of the cryostat. The primary coil was adjusted so as to have h in the direction of the field He. The magnetic moment acquired b y the substance is thus given by M = M e + m exp(2~ivt).
The static susceptibility is given by X0---- (OMe/OHe)T while the complex susceptibility at the frequency concerned is = m/h ---- Z' -- ix"
where X' and X" are the real and imaginary parts of the complex differential susceptibility. This last definition is only valid if h is taken small compared to the change in Hc producing an appreciable change in z(Hc). The frequency interval used ranges from 2 to 1135 Hz. The field h is about 5 oersted, which is sufficiently small. The values of the constant magnetic field used vary from 0 to 4500 oersted. 2. Theoretical ]ormulae. a) S pi n-1 a t t i c e r e 1a x a t i o n. One of the theoretical descriptions of the spin-lattice relaxation is due to C a s i m i r and D u P r 6 8) who assumed that the system of magnetic dipoles which is called the spin system is in internal thermal equilibrium. According to their theoretical formulae the dispersion (Z') curves are Debije curves added to a constant adiabatic susceptibility while the absorption (if) curves have the Debije shape : Z'/Xo = F/(1 + p2vg) + 1 -- F (1) X"[Xo = Fpv[(1 + p2v2)
(2)
where, if Curie's law is satisfied to a good approximation F = Cn~l(b + CH~);
(3)
b is the constant of the specific heat of the spin system at constant magnetic moment b/T 2 and C the Curie constant. The relaxation parameter p = 2m, ,~ being the relaxation time.
SUSCEPTIBILITY AND RELAXATION OF MnC12.4H20 CRYSTALS 1063 In the dispersion and absorption diagrams 7.'/~o and if/Z0 respectively are plotted versus 10log v. If formulae (1) and (2) are obeyed the following conditions are fulfilled: 1) at X'/Xo = 1 -- F/2 the dispersion curve has a point of inflexion, 2) at this point the slope S = --d(x'/Xo)/d(l°logv ) = 1.1513 F, 3) at the same frequency the absorption curve has a maximum of magnitude h = X"/Xo = F/2, 4) the dimensionless widths 6(1) and ~(r) which are respectively 2 10log (v(m)/vt (1)) and 2 lOlog(vt(r)/v(m)) are theoretically equal and equal to the C.D. value of 1.1438; r(m) is the frequency where the ~bsorption X"/Xo has its maximum value and vl(r) and vt(l ) are the frequencies at which Z"/Zo has half the value of that maximum; vj(r) > vt(1). (See fig. 1). O.5~ vo=~b, 0.4
,I/"\
0.3
0.2
V"
O.
Xld
J
-t.5
-,
T
t-
,
vt
I
-o.~
o
d~
',
&
o[,
I .
,Is
t
"x
0.8
½F
0.6
0.4
0 -4.$
-~
log vJvo
-o:,
o
Fig. 1. Absorption and dispersion curves according to Casimir and Du Pr6's theory. F = 0.8. b) D e t e r m i n a t i o n constant magnetic
of t h e s p e c i f i c h e a t of t h e s p i n s y s t e m at m o m e n t . From the thermodynamic reasoning it
1064
M . A . LASHEEN, J. VAN DEN BROEK AND C. J. GORTER
is easy 8)to deduce the relation ZadlZo = CMICH
(4)
where Xaa is the adiabatic susceptibility, CM is the specific heat of the spin system at constant magnetic moment and CH is the specific heat at constant field. The relation between CM and Cn is given b y CH
=
CM --
T (OM/OT)H (OH/OT)M.
The relation between the magnetic moment M, the susceptibility Z and the field H in the paramagnetic region is M = zH. Hence (OM/OT)~ = H dx/dT and (OH/OT)M = -- (M/z2) d z / d r = -- (H/z) dx/dT. Therefore CH --'~ CM 2F (TH2/z)(dz/dT) 2
and by substituting in (4) l--F----
Zad
X0
CM CM + (THZ/z)(dx/dT)2
or
l/(1 - - F ) =
1 + (TH2/CMz)(dz/dT) ~ = l +
(T3H2/bz)(d~,/dT) ~ ' (5)
if CM = biT 2. Equation (5) can be used for the calculation of b as will be seen later. If Curie-Weiss' law is followed and if CM = biT 2, equation (5) can be reduced to Hz 'F = (6) (b/C) (T + A)S/T 8 + H 2 from which it is clearly seen that F must be expected to be reduced on lowering the temperature. If between M and H a relation exists of the form M = X(0)H + ½~H3, then (OM/OT)H = H(dz(O)/dT + ½n2do~/dT) and (OM/OH)f = z(O) + ~'H2. From this a modified relation (5) is obtained: 1
- - = 1 -- F
T H ~ {dx(0)/dT + ½n 9 d~/dT}~
1 + - -
CM
x(O) +
(Sa)
oLH2
This relation will be used later on to estimate the effect of a possible relaxation in the antiferromagnetic region. 3. Measurement o[ the susceptibility, a) c-axis. A fairly large well formed crystal weighing 3.894 g was chosen from a large number of crystals
SUSCEPTIBILITY AND RELAXATION OF
MnC12.4H20
CRYSTALS
1065
prepared from the solution of MnCls.4H~O in water and placed with its c-axis, the direction of preferred magnetization, in the direction of the field. This was done by embedding the base of the crystal in a layer of paraffin wax in a tube and placing the tube inside the primary coil in the cryostat. Although the crystal was placed with great care so that the c-axis would be in the field direction, yet it m a y be that it is tilted by a small angle which m a y amount to a few degrees. 0.30
iii i o,
J
,:
,.o
,°
°K
2.O
]Fig. 2. S u s c e p t i b i l i t y of a M n C I 2 . 4 H 2 0 single c r y s t a l along t h e c-axis a t f r e q u e n c y v = 227 Hz. A, (D H c = 0 ® He = 2 2 5 0 O e H e = 3375 Oe [] H c = 4500 Oe.
Several series of measurements of the susceptibility were carried out in the temperature regions of liquid hydrogen and liquid helium with the constant field He = 0, 1125, 2250, 3375 and 4500 Oe at a frequency v = 227 Hz. Frequencies of 6.55, 62.5, 119 and 640 Hz were also used with He = 0 to check the results and the agreement was found satisfactory. From the values of the measurements taken in the region of liquid hydrogen and those obtained at the relatively high temperatures of the liquid helium region the susceptibility was found to obey a Curie-Weiss law according to the formula ;~ = C/(T + A) where A = 1.86°K. This A was reported earlier by G i j s m a n 4) to be equal to 1.5°K. Since the susceptibihty ;~ is only measured in the arbitrary units of the bridge, it appears useful to plot in the graphs x/C which has the dimension Physica XXIV
1066
M. A. LASHEEN, J. VAN DEN BROEK AND C. J. GORTER
of a (temperature) -1 and which b y definition approaches T -1 at very high temperatures. The N6el temperature for He = 0, at which the susceptibility decreases suddenly was found to be 1.62°K which is exactly the same as that found from the specific heat measurements b y F r i e d b e r g and W a s s c h e r S ) . It was found to shift to a lower value when a constant field Ho is used. The shift which is accompanied b y higher maxima of Z is increasing with He, in agreement with P o u l i s and G i j s m a n ' s 4) results. The results of the measurements are summarized in fig. 2. A comparison with theory will be given in the next section. b) b-axis. Another • well formed crystal weighing 1.915 g was chosen from the prepared set of crystals. This was taken to .be smaller than the one used in the c-axis measurements in order to suit the size of the tube without any need for grinding the crystal which would produce some difficulty in • identifying the required direction. The smaller size of the crystal helped also, as will be seen later, in decreasing to some extent the effect of the heat of magnetization on the dispersion and absorption curves at temperatures higher than the lambda point of liquid helium. The crystal was placed with its b-axis, which is perpendicular to the direction of preferred magnetization, in the direction of the field b y the same w a y as in the c-axis. The susceptibility was then measured at a series of temperatures in the liquid hydrogen and liquid helium regions with He = 0, 1125, 2250, 3375 and 4500 Oe at a frequency v = 227 Hz. As a check for the results at He = 0, frequencies of 36, 119 and 640 Hz were used at some selected temperatures. • The susceptibility was found, using measurements in the liquid hydrogen and in the high temperatures of the liquid helium region, to obey CurieWeiss' law % = C / ( T + A) withA = 1.79°K which was reported b y G i j s m a n to be equal to 2.4°K. The susceptibility passes through .a maximum at the N&I temperature of 1.62°K which is the same value found for the c-axis and from the specific heat measurements 5). The shift of the N6el temperature to a lower value, when a constant magnetic field is applied, exists also in this case. The maximum is becoming higher and the shift is increasing with the increase of the field. The shift for the same constant field is smaller in this case than the corresponding one in the c-axis. This is all in agreement with reference 4). The results of the measurements are shown in fig. 3.
4. Comparison with molecular/ield theory. On the basis of N6el's molecular field model Mrs. V a n P e s k i - T i n b e r g e n 0) and one of us have calculated the magnetic moment M of an antiferroma'gnetic substance as a function of the magnetic field He and the temperature T. Above the N6el point they predict a susceptibility following Curie-Weiss' law X = C/(T + A) with A = (Ax -- Dx)(S + 1)/3Sk~ where Ax and D~ are the interaction constants.
SUSCEPTIBILITY AND RELAXATIONOF MnCb.. 4H20 CRYSTALS 1067 The N4el temperature is given b y TN = (Az + Dz)(S + 1)/3Sk, k being Boltzmann's constant. Below that temperature they find, when the field is along the preferred axis of magnetization, a non-hnear relation b e t w e e n the reduced magnetic moment m = M x / # z and the reduced magnetic field 0.270
deq I.
0.265
¢
I
0.255
~.2
T
I1.6
t4
t.8
Fig. 3. Susceptibility of a MnCls.4HsO single crystal along v = 227 Hz. G, A
He = 0 H e = 3375 Oe
the
°K
2.0
b-axis at frequency
H e = 2250 O e [ ] H o = 4500 Oe.
®
h = #J-I~/(A~ + D=) (#~ is the magnetic moment per ion). For small values of h this relation can be approximated b y a series expansion: m =
al h + 1 + al~,
2a2~/(1 -- ax) + a8 h s ... (I + al~,) 4
(7)
Here ax, as, a3 are the first coefficients of the series expansion in h of Brillouin's magnetization function Bs(h/t) at the point where B s ( h / t ) = h. ? = (A~ -- D~)/(A~ + D~) = A/TN; the reduced temperature t---= kT/(A~ + D~).
1068
M . A . LASHEEN, J. VAN DEN BROEK AND C. J. GORTER
Taking A-----1.86°K, TN = 1.62°K," S-----5/2 and g = 2 the theoretical relation M = z(0)
+
...
(Ta)
following from equation (7) has been calculated. At H = 0 the parallel susceptibility Sit = X(0) is thus obtained. The perpendicular susceptibility X± following from the model is a constant which is approximately equal to XII(TI~), accepting that there is no large anisotropy in g, A and D. 0.3
0.2
! / I /
o.1
/1111} I
a~
Fig. 4. Susceptibility of a MnC12.4H~O single crystal. C) values along the c-axis & values along the b-axis - theoretical values.
This state of affairs is summarized and compared with experiment in fig. 4, where all the quantities have been divided by Curie's constant C. There is seen to be qualitative agreement; at 4°K even quantitative agreement. The apparent deviation of •± in the antiferromagnetic state from a temperature independent value m a y be explained by a slightly tilted orientation of the crystal. For the c-axis crystal the susceptibility ~ H e ) obtained from the experiment at 227 Hz was found to contain a term proportional to He I. The proportionality coefficient ~ calculated from that has been compared in fig. 5 with the theoretical ~ defined by eq. (Ta). The theoretical ~ has also been
SUSCEPTIBILITY AND RELAXATION OF M n C 1 2 . 4 H 2 0
CRYSTALS
1069
obtained from graphical constructions following ref. 6) ; the results from this are in good agreement with the calculations, from which it can be concluded that higher powers of h do not notably influence our results. It must be kept in mind, however, that there can be other reasons for quadratic effects in •(He). As a matter of fact, from measurements on powdered MnC12.4H20, which will be discussed in a separate paper, it was concluded that there occurs a relaxation phenomenon at frequencies of the order of 3 Hz. As there are no measurements available enabling us to make sure if this occurs in the single crystals, we cannot exclude the possibility that the measurements at 227 Hz give the adiabatic X(He) in stead of its isothermal value. The possible effect of this has been estimated b y means of equation (5a) at a few temperatures using the specific heat data from ref. 5) and is given also in fig. 5. For a more complete discussion on these matters we refer to the paper on the powder measurements. 6O
o 4O
20
0
Fig. 5. - --(D
T
.~,..
05
!
OK
',.S
TN
T h e coefficient ~ in M : X(0)H + ~/-/3. t h e o r e t i c a l results eq. (Ta) e x p e r i m e n t a l results e x p e r i m e n t a l values c o r r e c t e d for r e l a x a t i o n .
5. Relaxation. a)c-axis. The relaxation parameters at any given temperature and at different constant fields were measured from a set of dispersion and absorption curves. They were measured at different temperatures in the liquid helium region above the N~el temperature. No relaxation was detected in the available range of frequencies in the liquid hydrogen region of temperatures, while the data below the N6el temperature did not permit a conclusion about the existence of small relaxation effects. It was observed that the value of F is decreasing with the decrease in temperature and this is according to expectation (see eq. (6)). The dispersion and absorption curves measured at temperatures above the lambda point of liquid helium do not agree with the description given b y C a s i m i r and D u P r 6 2) as they are considerably flatter and broader than
1070
M . A . L A S H E E N , J. VAN D E N B R O E K AND C . J .
GORTER
the theoretical ones. This is a phenomenon which has already been found earlier in concentrated paramagnetic salts x). It may be ascribed to the limited heat conduction in the crystal. The heat of magnetization has to pass not only the usual bottleneck between the spin system and the crystal lattice, but in series with this is a heat resistance in the lattice itself, which differs for different volume elements. The result is a lengthening of the relaxation parameter for the inner parts of the crystal (cf.7), s)). The quantitative description of this phenomenon is not yet satisfactory. It may however be observed that the outer parts of the crystal which have good contact with the surrounding helium bath, will have the shortest relaxation time. So from the corresponding high frequency part of the dispersion and absorption curves it is possible to obtain an estimate of this relaxation parameter. In the case of the c-axis it was even better than this. A second incomplete peak was clearly observed at high frequencies in the absorption curves measured at 3.988°K and 3.010°K, which give results that are much better in agreement with those below the lambda point. At the same frequency for each of the two temperatures the dispersion curves have their TABLE I The relaxation parameters pdisp and pabe as functions of He and T. (The numbers, given in brackets have been obtained by extrapolation). ~0~) (ms) \ pdl~p pabs p=bs (2nd peak) pdlsp p=bs p~bs (2nd peak) pdisp pabs
450
900
1125
1575
2250
.3375
75 (23.7) 4.45 405 119 15.9 98 79.4
77 (16.8) 5.60 485 104 15.9 78 84 100 126
103 (21.1) 5.60 635 345 25.1 80 71
26.6 4.00
63 (23.7) 4.45 305 168 12.5 (40) 79.4
pdlsp psbs
pdlsp 155
• pa,bs
pdisp psbs pdisp p~bs pdlsp pabs pdtsp p~bs
pdlsp pabs pdlsp pabs pdlsp p~bs
4500
T(°K) direction
i
17.8 22.4
17.8 20.0
(40) 65
20.0 19.3 25. I 28.2 71 67 89 89 118 118 126
28.2 22.1
60 65
3.988
c-axis
3.010
84 75 150 112 188 155
2.106
31.6 27.9 71 79 63 67 92 92 139
4.000
157 159 156 159
1.780
1.955 1.813
3.012 2.084 2.014 1.901
1.672
b-axis
SUSCEPTIBILITY AND RELAXATION OF
MnClz.4HsO CRYSTALS
1071
steepest points which were used for determination of the corresponding F and S values. The abrupt change of the shape of the relaxation curves at the lambda point may indicate an abrupt increase of the thermal contact between the crystal and the liquid. Considering the situation below the lambda temperature one must not forget the well-known boundary layer effect, which was first described by K a p i t z a 9). The order of magnitude of the relaxation 2.S
c-axis
2.0
b-axis
o
®
T-~a ~
T.
1
Q
\ ,T-4.,
1.0 \
02 log Pabs
IC I
IocJT0.2
0.4
2.5
I
0 IogT0.2
0.6
0.4
0.6
b-axis
c-axis
2.0
log Pobs
I '5 ~0 ~
I
o.s_j_~
I
2
I 4
I lOkOc
I 0.5
Hf I b
J
I
2
4
i lOkO~t
Fig. 6. Upper diagrams: log pabs V s log T at H c = 2250 Oe (c-axis) and at H e = 4 5 0 0 O e (b-axis). (D normal procedure A second peak
Lower diagrams: log pabs v s log He at 2 . 1 0 6 ° K (c-axis) and at T = 2 . 0 8 4 ° K (b-axis).
time resulting from this heat resistance and the heat capacity of the sample can be estimated from the geometrical surface area of the crystal and numerical data from B e e n a k k e r e . a . 10) and F r i e d b e r g 5). T h i s , turns out to be of the order of 1/5 s below TI~ and 0.01 s between TI~ and the lambda point. A s , ~-- 1/5 s means # ~-~ 1 s and as this is definitely longer than the measured relaxation parameters, it may be concluded that the effective area is much larger than the geometrical one, which might indicate that the He II penetrates into cracks in the crystal. This would also explain why the
1072
M.A.
LASI-IEEN, J. VAN D E N B R O E K A N D C. J. G O R T E R
TABLE II The shape parameters as functions of He and T. Oe)
450
l.O
2h/F
.80 S/1.1513F
900
1125
.44 1.0 .48 l.O
.30 .87 .42
2.19
(~(r)/1.1438
8(I)/I. 1438
1.12
1575
2250
3375
.42 .96 .37 1.0
.41 1.0 .34 .94
.38 1.0 .40 .90
,29
(I.0) (1.0) .24
.70 .48 .64
.68 .38 .83 .69 .69
2.18 1.16 2.49 1.27 1.04
1.91 l,ll 2.44
1.95
2.25
.80
.86
.93 1.05
.24 .74 .26 .82
.91 1.0
.83' .80 .78
2.14 1.27
2.82 1.66 1.00
1.00 1.05 1.05
.89
.43
.65 .54
S/I.1513F
1.70 1.0 ~(r)/I.1438
.68 l.O .82 l.O .85 .81
c-axis
3.988 3.988 (2nd peak) 3.010 2. 106 1.955 1.813 3.988 3.988 (2nd peak) 3.010 3.010 (2nd peak) 2.106 1.955 1.813
.56 .52 .85 .81 .61 .55 1.91 1.17 1.0 1.29 1.46
.69 .79
4.000 3.012
.91
.93 1.0 .85 .81 .69
2.084 2.014 1.901 1.780 1.672
.46
.56 .70 1.11 .83 .70 .73 .86
4.000 3.012 2.084 2.014 1.901 1.780 1.672
1.77 1.54 l.Ol 1.09 1.29 1.22
4.000 3.012 2.084 2.014 1.901 1.780 1.672
1.03
1.79 1.08
1.27
l.O
3.988 3.988 (2nd peak) 3.010 2.106 1.955 1.813
.68
(I.81)
~(])/1.1438
direction
3.988 3,988 (2nd peak)
1.56
2h/F
T(°K)
4500
1,01 1.0
(.90)
(1.00)
(1.65)
4.000 3.012 2.084 2.014
b-axis
SUSCEPTIBILITY AND RELAXATION OF
MnC12.4H20 CRYSTALS
1073
internal heat conduction in the crystal becomes better, as is proved by the smaller deviations from C.D. theory. T h e r e l a x a t i o n p a r a m e t e r s . Values of the relaxation parameters Pdisp and pabs are given at different temperatures in table I. At the two temperatures higher than the lambda point of liquid helium values obtained from the normal procedure, which are affected by the crystal heat conductivity are given together with the values acquired from the second peaks of the absorption curves and the corresponding steepest point of the dispersion curves. Log Pabs has been plotted as a function of log T at the constant field Hc = 2250 Oe in the left hand side up diagram of fig. 6. This diagram shows that p varies with T -4.2. In the left hand side down diagram of fig. 6 log Pabs has also been plotted as a function of the constant field He at T = 2.106°K. The plot is a straight line parallel to the Ho-axis which indicates that the relaxation parameter is independent of the value of the constant field applied. T h e s h a p e p a r a m e t e r s . Values of each of the shape parameters divided by its theoretical (C.D.) value, that is 2h/F, S/1.1513 F, ~(r)/1.1438 and 6(1)/1.1438 are given in table II. Each of these ratios should be equal to unity if the dispersion and absorption curves follow the C.D. formulae. The agreement is good a t temperatures below the lambda point; the best is at 2.106°K which is just below that temperature. The deviation of the dispersion and absorption curves measured at both &988°K and 3.010°K, due to the heat effect, can easily be seen from the deviations of their shape parameters from the theoretical values. The shape parameters given from the second absorption peak and the corresponding steep point on the absorption curves at these two temperatures are in good agreement with the C.D. values as shown in table II. b) b-axi s. Several sets of dispersion and absorption curves were measured at different values of the constant field at different temperatures in the liquid helium region above the N6el temperature. As in the c-axis no relaxation was found in the available range of frequencies below the N6el temperature and in the liquid hydrogen region of temperatures. The value of F was, also in this case, found to decrease with the decrease of the temperature as is expected. It was stated above that a crystal, smaller than that used in the c-axis measurements, was chosen in order to decrease the effect of the heat of magnetisation at temperatures higher than the lambda point of liquid helium. This really helps in decreasing the deviation of the dispersion and absorption curves from the theoretical formulae at these temperatures as can be clearly seen from table II. Although the dispersion curves were steeper and the absorption curves were narrower than those measured in the c-axis at the corresponding temperatures, yet the relaxation parameters measured from them were not in agreement with those measured at tem-
1074
~t. A. LASHEEI~, J . VAN DEN BROEK AND C. J. GORTER
peratures below t h e lambda point. No second peaks, resembling those found in t h e c-axis measurements, were observed on the absorption curves and it m a y be that the two have merged. The relaxation parameters. Values of the relaxation parameters pdisp and Pabs at different temperatures and different constant fields are given in table I. Log Pabs is plotted as a function of log T at the constant field He = 4500 oersted in the right hand side up diagram of fig. 6. From this diagram it is found that p varies with T-4.1 which is nearly the same as that found for the c-axis, pabs has also been plotted as a function of the constant field He at T = 2.084°K in the right hand side clown diagram of fig. 6. This shows that the relaxation parameter is independent of the applied field strength. T h e s h a p e ' p a r a m e t e r s . Values of 2h/F, S/1.1513 F, ~(r)/1.1438 and 6(1)]1.1438 the shape parameters divided b y their theoretical values are given in table II. It can be seen from these values that the agreement between the measured and the C.D. values is quite good at temperatures below the larnbda point, this being much better at temperatures close to the lambda point than at lower temperatures. This deviation at lower temperatures m a y be due to the difficulty of measuring some of these shape parameters due to the decrease in the value of F as stated above. At temperatures higher than the lambda point there is a deviation from the theoretical values although it is less than that in the case of the c-axis due to the use of a small crystal.
6. Determination of the speci[ic heat. Equation (5) was used for the •determination of b, the constant of the specific heat of the spin system at constant magnetic moment. This was carried out b y first p l o t t i n g 1/(1 -- F) for different field strengths at the same temperature versus H ~ (= Ho 2) and determining the slope of the resulting straight line. This slope gives the value (T3/bZ)(dx/dT) 9'. The value of the susceptibility Z is to be determined from the experimental curve at the corresponding temperature T. For the determination of dx/dT great care must be taken as it is in the second power in the formula and so any small error would lead to a highly deviated value of b. For this reason the slope was carefully determined from the experimental curve at different temperatures and t h e n plotted versus the temperature. The curve which should be smooth helped in noticing and correcting any deviation in the measured value of the slope at any temperature. Then b y substituting for all the known values, the value of b is calculated. The value of b found b y this w a y is expressed in the arbitrary units of the bridge.The value of b/C calculated b y dividing the value of b b y the measured value of C gives the ratio of the real value o~ b and the Curie constant. Values of b/C at different temperatures for both c and b crystal axes are shown in table III. Above 2°K, for each of the two axes, the value of b/C is constant and equal to 19.0 × 106 Oe 2. This value is in agreement with
S U S C E P T I B I L I T Y A N D R E L A X A T I O N OF
MnC19..4H20
CRYSTALS
1075
T A B L E III
blC as
a function of T
c-axls
T(°K) [ b/C(kOe~) [ T(°K) 3.988 3.795 3.606 3.360 3.199 3.010 2.800 2.608 2.429 2.314
19.1 19.0 19.0 18.9 19.0 18.9 19.0 19.0 19.0 19.1
2.192 2.106 1.993 1.955 1.914 1.813
" b-axis
[blC(kOe') 19.2 19.8 20.3 21.1 21.4 22.4
=(oK) [ b/C(kOe2) ] T(°K) 4.000 3.884 3.713 3.517 3.295 3.146 3.012 2.991 2.861 2.690
'
19.1 19.0 19.0 19.0 19.0 19.0 18.8 18.8 18.9 19.0
2.546 2.413 2.243 " 2.105 2.084 2.014 1.979 1.901 1.780 1.672
I
b/C(kOe") 19.0 19.0 19.0 19.0 19.0 19.0 19.1 19.8 23.2 40.6
the values 19.5 × 108 and 19.8 × 10e Oe2 found at high temperatures by T e u n i s s e n and G o r t e r 11) and Starrl2). Below 2°K the value of b/C is increasing as is expected from the specific heat measurements by F r i e d b e r g and W a s s c h e r 5) indicating that the specific heat is no more given by b/T 2. The authors are indebted to Dr. L. C. V a n d e r M a r e l for his cooperation and valuable discussions and to the technical staff of the laboratory for their help with the experimental preparations. One of us (M.A.L.), who was working at the Kamerlingh Onnes Laboratory for one year, expresses his gratitude to members of the academic staff of the laboratory for the cooperation and assistance he received during his stay. He also appreciates the leave granted to him by Alexandria University. Received 11-9-58. REFERENCES 1) V a n d e r M a r e l , L. C., V a n d e n B r o e k , J. and G o r t e r , C. J., Commun. Kamerlingh Onnes Lab., Leiden No. 306a; Physica 2 3 (1957) 361. 2) C a s i m i r , H. B. G. and D u P r 6 , F. K., Commun. Suppl. No. 85a; Physica 5 (1938) 507. 3) G o r t e r , C. J., Paramagnetic relaxation (Elsevier, A m s t e r d a m 194~/). 4) G i j s m a n , H. M., Thesis Leiden (1958). 5) F r i e d b e r g , S. A. and W a s s c h e r , J. D., C o m m u n No. 293c; Physica 19 (1953) 1072. 6) G o r t e r , C. J. and T i n e k e V a n P e s k i - T i n b e r g e n , Commun. Suppl. No. ll0b; Physica R2
(1956) 273. 7) 8) 9) I0)
E i s e n s t e i n , J., Phys. Rev. 84 (1951) 548. V a n d e r M a r e l , L. C., Thesis Leiden (1958), p. 23 et seq. K a p i t z a , P. L., J. Phys. U.S.S.R. 4 [1941) 181. B e e n a k k e r , J. J. M., T a c o n i s , K. W., L y n t o n , E. A., D o k o u p i l , Z. and V a n S o e s t , G., Commun. 289a; Physica 18 (1952) 433. ll) T e u n i s s e n , P. and G e t t e r , C. J., Physica 7 (1940) 33. 12) S t a r r , C., Phys. Rev. 60 (1941} 241.