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ordering lead to a direct determination of the so-called gap exponent A. This is ... obtained directly from this method and that derived from critical exponents a ...
Physica l12B (1982) 227-231 North-Holland Publishing Company

227

DIRECT E X P E R I M E N T A L D E T E R M I N A T I O N OF THE GAP E X P O N E N T A IN TWO THREE-DIMENSIONAL HEISENBERG FERROMAGNETS W.L.C. R U T I ' E N * , L.J. D E H A A S and J.C. V E R S T E L L E Kamerlingh Onnes Laboratory, State University Leiden, Leiden, The Netherlands

Received 22 July 1981 Measurements of the longitudinal adiabatic and isothermal susceptibility in the critical temperature regime of magnetic ordering lead to a direct determination of the so-calledgap exponent A.This is illustrated for two three-dimensional Heisenberg ferromagnets Rb2CuBr4"2H20and (NH4)2CuBr4"2H20.Within experimental error there is complete agreement between A obtained directly from this method and that derived from critical exponents a and/3 (with the aid of scaling relations), This implies that, in general, ot and fl can be determined by measuring (with only one experimental technique) A and 3'.

1. Introduction It is generally believed that the gap exponents A " ( n = 1,2 . . . . ), defined as successive field derivatives of the Gibbs free energy, are only of academic interest as direct experimental determination of A" does not seem possible [1]. Indirectly, however, employing scaling relations, A 1 (henceforth to be called A) has been calculated for numerous magnetic systems [2]. It is of interest to have a more direct experimental access to A. We will show that it is possible to obtain A by measuring the magnetic field dependence of the macroscopic adiabatic susceptibility Xad in the critical temperature regime above To. The adiabatic susceptibility X~d is obtained from measurements of the frequency dependence of the macroscopic magnetic susceptibility.

2. Experimental method For the accurate measurement of the frequency dependent macroscopic susceptibility a Twin-T bridge with a L - C combination in one of its branches has been used; this bridge is very * Present address: ENT-department (KNO), University Hospital, 10 Rijnsburgerweg, Leiden, The Netherlands. 0378-4363/82/0000-0000/$02.75 © 1982 North-Holland

suitable in the frequency range above about 100 kHz. Details of the bridge and accompanying set-up are given elsewhere [3]. Main characteristics are: frequency range between 100 kHz and 70 MHz, static field range between 0 and 4 0 k O e , temperature range between 1.2 and 300 K. In frequency units the accuracy of the real part A" of the susceptibility is 0.15 Hz, that of the imaginary part X" amounts to 0.5 Hz. In CGS units, the former value of 0.15 Hz corresponds to an accuracy of 0.3x 10-Vcm3 (at a measuring frequency of 16 MHz).

3. Compounds Single crystals of two three-dimensional ferromagnetic Heisenberg compounds, Rb2CuBr4. 2H20 and (NH4)2CuBr4"2H20, were studied. The symmetry of the crystals is tetragonai, space group P42/mnm. These are very well studied compounds (see ref. [3] and references therein). Especially for the Rb compound (henceforth abbreviated as " R b " ) critical exponents c~,/3 and Y are known very precisely as well as Tc ( T c = l . 8 7 4 - + 0 . 0 0 2 K for Rb and Tc=1.837-+ 0.002 K for NH4; for details about finding Tc see ref. [3]). Single crystals were grown by slow evapora-

228

W.L.C. Rutten et al. / Gap exponents in two ferromagnets

tion of a saturated solution of the constituents at a temperature of about 4°C. Orientations along the four-fold c axis and along the perpendicular a axes were performed by recording of Laue diffraction patterns. Samples were ellipsoidally shaped in order to account for demagnetization effects [3].

AM(t, H) = - f ~b(t', 11). AH(t - t') d t ' ,

(2)

0

with @ the relaxation tensor. X(u, H) is related to the time dependent part ~b of ~b by

X(v, H) = ~b(O,H) - 2~-vi f e-i2~"~b(/', 11) dt', 0

4. Relaxation phenomena

(3) For a thorough understanding of the adiabatic susceptibility experiments insight into the spinspin and spin-lattice relaxation phenomena of electron spin systems is necessary. Experimentally the real and imaginary part, X' and X", of the macroscopic susceptibility X = - X ' - i x " , are studied as functions of frequency, static field, crystal orientation and temperature. The susceptibility X relates the temporal variations of the macroscopic magnetization M(t) to variations of the field 11(t). The field 11(0 may be composed of two contributions: a steady field 11 and a highfrequency field 11h.~.(t). These two fields may be oriented parallel or perpendicular to each other. Depending on this orientation different processes occur in magnetic systems. The parallel orientation pertains to relaxation processes and the perpendicular one to resonance phenomena. We shall be concerned with relaxation processes.

where ~b(t, H ) = ~b(t, H ) - ~b(oo,11),

(4)

T

tb(~, 11) :

~m 1 f ~b(t',H)dt'. 0

The relaxation tensor 4~ describes the time evolution of the magnetization macroscopically. When this relaxation tensor decays exponentially with a relaxation time ~'(H)=-p(11)/21r, then ~b(t, H ) = th(0, H ) e -t/~(m ,

(5)

and one finds by Fourier-transforming (3) the well-known D e b y e relation

x(,~, H) = x'(~, H) + x(0, 11)- x'(~, 11). 1 + ipu

5. Relaxation tensor and susceptibility

or, writing out the real and imaginary parts of

In a linear response approximation the susceptibility tensor X(u, It) is defined by

AM(t, 11) = Re[x(u, H).A11 ei2"'],

(6)

x(., H), x(,~, 11) : x'(~, 1 1 ) - ix"(~, 11),

(1) x'(~, 11) = x'( ~, hr) +

where AM is the variation of the macroscopic magnetization in response to an oscillating field All(t) = Re All exp(i21rut). More generally, still in a linear response approximation, one has

x(0,11) - x'(~, 11) 1 + p2,,:

pu X"(~, 11) = IX(0, 11) - X'( ~, 11)1 1 + p:u2"

,

(7)

(8)

A plot of X" versus X' (mrgand or Cole-Cole

W.L.C. Rutten et al. / Gap exponents in two ferromagnets

diagram) yields a semicircle with its centre on the dispersion axis at X'(~) + ½(~(0, H ) - X'(oo)). There are two limiting cases: For pv>> 1, X'(v, H ) = X( oo, H ) and X"(v, H ) = O. For pv 1, X'(V, H ) = X(0, H ) and X"(V, H ) = 0. At the top of the semicircle, where pv = 1, one has X"(v, H ) = ½IX(0, H ) - X ' e o, n ) ] .

229

because X " ( v , H ) = 0 in the frequency region where X'(v, H ) = X~a. Making use of the thermodynamical relations

[1] (3M~ 2

(lO)

xo(Cu - CM) = T \ a T / u ' _/8M'x 2

6. Spin and lattice systems The relations above are not specific for relaxation in the spin system only. Also energy exchange between the spin system and lattice system, which occurs in non-zero static fields, can be described by analogous relaxation processes and characterized by a spin-lattice relaxation rate p~l provided, of course, that the decay of the relaxation function is exponential. If the exchange of energy between spin and lattice systems proceeds much slower than the exchange within the spin system itself, the spin system can be considered as isolated from the lattice in a frequency region v ~ p~l. In that case two wellseparated relaxation processes take place according to (with X'(Oo,H ) taken as zero) X(V, I t ) = Xo

-

X.a +

l + ip~w

(11)

CH(Xo - Xad) = 1 {\ 0- "/~ ] /' H '~

(9)

X.d

l + ipsw '

where we have substituted X(0,0)=X0 and X(0, H ) = X~d. When p~t ~ p~s, a plot of X" versus X' shows two subsequent semicircles (fig. 1). Even when the relaxation processes do not show exponential decay X,d is very well defined,

and restricting ourselves to such low fields that the relation M = XoH holds, one can derive the relation X.__~a= 1 Xo 1 + H21H]I2 '

(12)

Rblla

_

I~o /

-I

/

o'~//J 0

I

142>. I

I

o ~.o -

,

5 I

--

I

lOxlO 80e 2 I

I

Rb IIc

+ N H ¢ II c

*/"

O'~--~i b o o

£,ad

Xo

X'

Fig. 1. Argand diagram of complex susceptibility X with frequency v as parameter.

0

H2

lO

20x106

Oe2

Fig. 2. Plots of 0¢o/)¢*a- 1) versus H 2 to illustrate the linear relationship .a) Rblla: T = 20.4 K(O), T = 14 K ( × ) .b) Rbl]c ; T = 4.24 K (O); NI-hllc: T = 4.23 K ( + ) .

230

W.L.C. Rutten et al. / Gap exponents in two ferromagnets

in which

where G ' and G" are differentials with respect to E A2/"/.

H~/2 =

C~xo (Cn)n=oXo T(OXo/OT) 2= T(Oxo/OT) 2'

(13)

CM and Cu are the magnetic specific heats for constant magnetization and constant field, respectively. H m can be determined from (Xo/X~1) versus H 2 plots. Some examples are given in figs. 2a and b for t e m p e r a t u r e s above the critical t e m p e r a t u r e region.

We now m a k e the assumption that M varies linearly with H, M = G ' - e*2H, which condition is experimentally fulfilled in the critical region for small fields. We then also have G " - 1 (both G' and G" for T > T ~ and small H ) . One can now identify the relations (15) with the critical dependence C , - e -u, x 0 - e - L M ~ e B and M ~ H1/~(e = 0) to yield -a=al-2,

7. The gap exponent A and the critical exponents a , / 3 , y and 8 Extension of the t e m p e r a t u r e range of Xad" m e a s u r e m e n t s into the critical regime gives an opportunity to determine the gap exponent A from a 0C0/Xad--1)/H 2 versus e(-----T / T c - 1 ) plot, in the limit H ~ 0 . W e recapitulate the definitions, as given by Stanley [1], of the gap exponent A and the critical exponents t~, /3, and 6, according to scaling theory. The gap exponents, A", are defined by the successive field derivatives of the Gibbs free energy G, taken at H = 0, e ~ 0 +, as

G" =- ( ~ ' ~ - e-a"G '"-', \OH ] T,H=O

(14)

In scaling theory one assumes G to be of a generalized h o m o g e n e o u s form in e and H so that G(e, H ) = A-1G(A~*e, A~uH), where at, an and h are arbitrary p a r a m e t e r s [1]. With the choice A = e -~/~', we can write G as G(e, H ) = e~G(1, e*2H). For the T and H derivatives of G then follows (for small fields):

I~OG

S =-t-ff~)H___. ~ e X l - l G ,

C . = k OT2 ]u ~ ex~-zG,

[02G~ Xo = kOH2]r

ea~+2.~G,,'

(15)

- y = A~ + 2A2, (16) /3 = A1 q- A2 ' 6 = --~t2/(/~l q'- /~2) ;

(the value for 6 results by taking the limit e -0 0). In the same manner as for the critical exponents above, one finds A" = A = -A:. Inserting these values in the expression found for Hm(13) and expressing 3.2 as combination of a , / 3 and % thus yields H i t z - e a = e 2-~-0

=

e/~+~'

=

e (1/2X~'-'~)+I

-

(17)

8. Results T h e experimental results for H m versus e are presented in figs. 3a and b. In table I the results A = 1.65 _ 0.05 (Rb) and A = 1.67 _ 0.07 (NH4), obtained from figs. 3a and b, are c o m p a r e d to the values of A calculated from the experimental values of y, ct and/3. For Rb the agreement is very good. Especially for N I L m o r e accurate experimental data are necessary. For completeness sake it is mentioned that because of demagnetization effects the external field H has to be corrected according to Hi = H + (pJMol)eDM while M = xoH(pJMol) (M: magnetization, Mol: molecular mass, p~: density, eD: demagnetization factor). This corrected Hi is employed in the calculation of the experimental results.

231

W.L.C. Rutten et al. I G a p exponents in two ferromagnets 1

I r IIIII r

I

I I lllll r

P

I I liIi'.,-

Oe-1

NH4//c

E -1

10

10-~ _

E-1,67

a 1(5 2

_lb I l i m

i

--1

{--~©

lo-3 ~= T- 1_837 ~-T i

i

i ihl~ll

i

10-2

10-3

I

"~ ~

i ~lJlll~

1(5 4

J I i

10-3

lO-1

10-2

10-~

1

Fig. 3. Log-log plot of the inverse characteristic field HT/~ = O(o/Xae - 1)l/2/Hi versus ~ to determine the gap exponent A. a) A = 1.67 for NI--h and b) A = 1.65 for Rb.

Table I Comparison of gap exponent A values

Rb NIL

1.26 _+0.02 1.25-+0.02

-0.041 -+ 0.014

0.370 ± 0.007 0.375-+0.008

1.65 -+ 0.05 1.67-+0.07

1.672 ± 0.02

1.63 -+ 0.02 1.625_+0.01

1.63 ± 0.01

a) This research; b) ref. [4]; o refs. [4, 5], m e a n values.

It is worth mentioning that, once A and 7 are obtained from experiment, application of (17) could serve to yield fl and ~. Thus, measuring X0 for H = 0 and Xad for H # 0 yields a rather complete set of critical exponents, obtained with only one experimental technique.

Acknowledgements Critical reading of the manuscript by Prof. N.J. Poulis is gratefully acknowledged.

References [1] H. Stanley, Introduction to Phase Transitions and Critical P h e n o m e n a (Oxford University Press, London, 1971). [2] L.J. de Jongh and A.R. Miedema, Advances in Physics 23 (1974) 1. [3] W.L.C. Rutten, Static and D y n a m i c Behaviour of some Ferromagnets, Thesis, Leiden (1979). [4] E. V61u, J.P. R e n a r d and B. Lecuyer, Phys. Rev, B14 (1976) 5088. [5] T.O. Klaassen, Thesis, Leiden (1973).