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EFFECT OF A FIELD ON THE
THERMODYNAMICS OF THE RANDOMLY DILUTE
1D HEISENBERG FERROMAGNET
A. Rettori, M. Pini
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Suppl6ment au no 12, Tome 49, decembre 1988
EFFECT OF A
FIELD ON THE THERMODYNAMICS OF THE RANDOMLY
DILUTE
1DHEISENBERG FERROMAGNET
A. Rettori (I) and M. G . Pini (2)
(I) Istituto di Fisica del17UniversitiL di Siena, 53100 Siena, ed Unit& del GNSM-CISM, 50125 Firenze, Italy (2) Istituto Elettronica Quantistica CNR, 50127 Firenze, Italy
Abstract.
-
The thermodynamics of the dilute 1D Heisenberg ferromagnet in a field is investigated. The quantum free spin-wave theory is used to calculate the specific heat and analytical results are obtained for low temperatures and low concentration of impurities. The bulk, surface and size contributions t o the free energy are evaluated by the classicaltransfer matrix method and analytical results for the surface one are given at low and high temperatures.
Impure 1D magnetic systems have ~eceived con- for h = 0. Any thermodynamic property of the dilute siderable interest in recent years [l-lo]. If the non- system is obtained summing over all the segments: e.g. magnetic impurities are distributed randomly, divid- the specific heat per site is C/N = (1
-
c ) ~ CcTC,, ing the chain into non-interacting segments, the freeenergy of the dilute system can be evaluated summing
cT
= CK, [(PEK, 12) cosech (PEK, /2)12.
(4)In the quantum case, the Si are spin operators. At low temperatures one can transform to Bose operators and diagonalize the quadratic Hamiltonian [2, 81. In this free spin-wave theory the free energy of a segment with T
2
2 spins is (KB= 1,P
= 1/T, j = J S ~ , h =~ ~ B H S ) :
over all the segments: F = ~ T ~ F T ( r = 17%
...,
N) In figure 1 we report the numerical results for the tem- weighted by their probability of occurrence PT = perature dependence of C/N. At very low and c + 1,N (1
-
c ) ~ cT, where N is the total number of sites and c the concentration of magnetic ones. For high c, the1
Fig. 1. - Quantum free spin-wave contribution to the spe- cific heat per site of the dilute Heisenberg FM versus T* =
T / J S for c = 0.9. Full line: h = 0; dashed-double-dotted:
h = 0.5; dashea-dotted: h = 1. Insert: comparison with analytical expression (5a, b) (full lines).
classical Heisenberg model was found to explain the zero-field properties of the real quasi-lD compound
it can be evaluated in closed form using the method of steepest descents [2, 41.
c=O 9
C/N = exp (-PhlS) { A -t
(CD3),NMnCCul-,CIS [8-91, but not those of high-
-
- .dilution samples of CsMn,Mg,-,Br3 [lo], for which quantum effects dominate.
In this paper we study the effect of a magnetic field
on the thermodynamic properties of the dilute 1D
5
/U
Heisenberg ferromagnet (FM). We consider both the quantum case at low temperatures and the classical
one, for a single segment with T spins the Hamiltonian o o I
is ( J
>
0) :0 1 2
where EK, are the energies of the harmonic modes
+B ( T ~ / T ) ~ / ~ exp (-To/T)lI3
K T =7rp,,.;p=O,1
,...,
T - 1 ) : x [I+ 2 ( ~ o / x ) ~ (h/j)2+
( T O / X ) ~ (h/jI4])
( 5 4 EK,/JS = 2 (1-
cos KT)+
(h/j) (3)In order t o ensure the convergency of F,, any zero- B = 27r1I2 (1
-
c ) ~/
(27In
(l/c)),
energy mode must be excluded from the summation in To = (2714) x2 ln2 (llc) (j/S).
(2). This implies that for a segment of length T one (5b)
has t o sum over T modes for h
#
0, i.e. one more than TO = [27r2/1n (llc) ( ~ j l ~ ) ] 'I3C8 - 1418 JOURNAL DE PHYSIQUE
In the classical spin approximation, the transfer ma- trix method [6, 7, ll] can be applied to calculate the partition function of a segment of length r :
where
An are the eigenvalues of the transfer integral
equation. For the Heisenderg chain [ll], y5, (2) are the eigenfunctions andWe devote particular care to the separation of bulk (B), surface (SF) and size (SZ) effects on the free en- 'ergy of the dilute chain.
Analytical expressions for the surface contribution f s ~ can be obtained using for Xo the low and high expan- sions and for a$ :
x [(sinh (a
+
ph/2))/
(a+
ph/2)I2.
(9) The latter is obtained using for the lowest eigenfunc- tion the trial form y5T (x) =A
eux [ll]. Analytic ap- proximations for a in the low and high T limits are:where L (2) is the Langevin function
The finite-size contribution to the free energy has to be calculated numerically. Note that fsz= 0 for h = 0
since an+ = 0 owing t o the symmetry of the eigen- functions.
In figure 2 we report the different contributions to the susceptibility as a function of temperature for c = 0.9. For this high value of c, the three contributions are of the same order. The more rapid increase of X S F with
respect t o X B at low
T
is due t o the greater freedomFig. 2.
-
Full lines: ciassical transfer mittrix results for the total (T), bulk (B), surface (SF) and size (SZ) contribu- tions to the susceptibility per site verstis T* = T / J S ~ forj = 1, h = 0.01 and c = 0.9. Dashed lines: low and high temperature expansions (Eq. (lla, b)) for X S F .
in the fluctuations of the surface spins. The low and. high T expansions for X S F
correctly reproduce the numerical da-ta. The size con- tribution Xsz presents a behaviour si.milar to that ob- served for the dilute classical planar model in a field
[GI.
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