CAPÍTULO 2: MARCO TEÓRICO
2.3 EXPERIENCIAS REGIONALES Y NACIONALES
2.3.2 EXPERIENCIAS REGIONALES
Low dim ensional m agnetic system s have been w idely studied in recent years, both theoretically and experim entally. T here are m any interesting ground states and phenom ena unique to low dim ensional m agnets such as the spin P eierls transition (see section 2.2.5.2.1). Further interest has arisen due to H aldane’s^ prediction that there will be an energy gap betw een the singlet ground state and the first excited state for integer spin 1 -dim ensional H eisenberg antiferrom agnets, but not for h a lf integer spin chains. L ow dim ensional m agnetism is a general term that m ust be clarified. Low dim ensional character originates from the dim ensionality o f the interacting spins; i.e. zero- (clusters), one- (chains), tw o- (planes) or three-dim ensional. The
H am iltonian equation for the interaction o f any tw o spins Sj and
S-
on coordinatesx,y,z can be w ritten as:
H = - 2 J [aS jS "-hb(Sj S^-HSfS"')]
i > j ' ^ ^ ^
W hen the constants a and b are equal to 1, the interaction is w holly isotropic and three-dim ensional w hich is m ost suitably described by the H eisenberg m odel. I f a = 1 and b = 0 then the interaction is linear and described by the Ising m odel or if a ^ 0 and b = 1, then the interaction is planar and can be represented by the X Y model. It is know n from the theorem s o f M erm in and W agner^ that one and two dim ensional systems can not sustain long range m agnetie order at finite tem peratures; there m ust be a small but finite interchain or interplane interaction to induce such order.
2.2.5.1 Dimer Model
The sim plest type o f m agnetic exchange is betw een two m agnetic centres w ith no influence from any other. This is not only possible in discrete m olecules but also within chains. The H am iltonian for a dim er is
H = - 2 J S S
1 J
W hen a p air o f S = l/2 ions interact antiferrom agnetically, the interaction gives a singlet ground state and a triplet excited state. The splitting betw een these states m ust be large, o f the order o f kT in order to produce a dim erised state. M ajum dar
and G o sh '7 th eoretically predicted that in S = l/2 antiferrom agnetie chains there are
com peting nearest neighbour (J^) and next nearest neighbour (J2) interactions w hich
are in a d im erised state for ^ 1/2. H aldane^, h o w ev er, arg u ed th a t a
spontaneously dim erised state exists for a w ider range o f J2/J1.
B leaney and B ow ers deriv ed an equation^ to describe th e b eh avio ur o f hydrated copper acetate as dim ers:
2Ng\^
1
k T 3 + e x p ( - 2 J / k T)
w here N is the num ber o f dim ers in the sam ple and J is the exchange interaction betw een n eare st n eig h b o u r m agnetic m om ents. The dim erised state w as later confirm ed in Sr^^Cu2^0^^ by M atsuda and K atsum ata^, w ho obtained the same expression by an alysing E S R results betw een the singlet and trip let states. The susceptibility o f a dim erised state falls rapidly to zero as T - 0 . C opper acetate is still regarded as a good m odel o f a dim er system . The susceptibility o f copper acetate m onohydrate passes through a m axim um near room tem perature and then decreases rapidly as the tem perature decreases to becom e zero at about 5OK. The assum ption th at this system interacts as a d im er system w as borne out by the structure ^0.
2.2.5.2 One Dimensional Magnetic Systems
One dim ensional m agnetic system s have been studied theoretically since B ethe in 1 9 3 0 k It is som etim es possible to exactly calculate the ground state o f these system s and to predict the therm odynam ie properties such as susceptibility and heat capacity. These system s can be described like any other m agnetic system , by the linear Ising, planar X Y or three dim ensional H eisenberg models. H owever, the Ising and X Y m odels, because o f their anisotropy, describe the susceptibility in term s o f
%ll and
X .
A s all the sam ples in this thesis are pow ders, it is not possible to resolveM ost theoretical w ork on one dim ensional system s centres on com pletely isolated chains w hile experim ental w ork inevitably has longer range interactions. There is no closed form solution for the one dim ensional Heisenberg m odel, but using
approxim ations, m odels for estim ates have been found 1 2 - 1 7 B onner and Fisher 1^
calculated the energy levels and eigenvectors o f short chains and rings o f S = l/2 ions and w ere able to predict the therm al and m agnetic properties for varying anisotropy and m agnetic fields in these system s. The results for 11- and 12- m em bered chains o f antiferromagnetically exchange coupled S = l/2 ions were used to extrapolate to the infinite len gth chain. T hese resu lts w ere later taken by H alT ^ w ho fitted the theoretical results obtained by Bonner20 to a quadratic/cubic function.
A + B x ' + Cx^ '1
1 + Dx-t-Ex^ + Fx^
w here x = -L-L, A = 0 .2 5 , 3 = 0 .1 4 9 9 5 , C = 0 .30094, D = 1 .9 8 6 2 , E = 0 .688 54 and kT
F=6.0626. The m axim um in reduced susceptibility calculated from this expression
kT Y |J|
occurs at — 1. 297 and — —— = 0.07337. The expression is valid for T>J/2k.
|J|
In this ex p ressio n , the m a g n itu d e o f the m ax im um o f the an tiferro m ag n etic susceptibility is determ ined by the g value for a given J. A discrepancy in this part o f the fit m ay im ply that the given m odel is inappropriate. The susceptibility o f the one dim ensional linear chain approaches a non-zero value as T ^ 0 , in contrast to the dim er model. This is due to the absence o f an energy gap in the continuum o f energy levels.
2.2.5.2
.1 Spin Peierls
The spin Peierls transition w as theoretically predicted in the 1970’s. It is an unusual effect o b serv ed in a lim ited n u m b er o f one dim en sio n al H eisen b erg S = l/2 antiferrom agnets. In the spin Peierls m odel, a uniform quasi-one-dim ensional S = l/2
H eisenberg antiferrom agnetie chain is coupled to three dim ensional lattice vibrations, w hich causes the chain to distort into pairs as a function o f tem perature. A bove the transition, the m agnetie behaviour is th at o f a one dim ensional antiferrom agnet, below , the dim érisation o f th e spin lattice system leads to alternatin g exchange constants. There is a gap in the excitation spectrum betw een the non m agnetic
sin glet g ro un d state and the n ex t energy level. The su scep tib ility decreases
exponentially tow ards zero w hich is a direct result o f the energy gap in the m agnetic excitation spectrum contrasting to a finite value at low tem perature found for the uniform linear antiferrom agnetie chain. The susceptibility in a dim erised spin-Peierls state can be fitted to a m odel proposed by Bulaevskii:
X(T) = q (T ) f - 2 [ l + 8(T)]Jp(T) I
w here
a
is the exchange ratio J2/Jj.2.2.5.2.2 Alternating Chain Model
The B onner-Fisher m odel applies to a regular one dim ensional chain but w hen J ^ a J , as in the figure 2.5, the B o nn er-F ish er m odel is replaced by the H eisenberg alternating chain model.
a S
J
oJ
Figure 2.5 Exchange param eters in an alternating chain
The H am iltonian for an antiferromagnetie H eisenberg alternating chain is
N / 2
i = l
w here J is the exchange integral betw een a spin and its right neighbour and a J is the exchange integral betw een the sam e spin and its left neighbour. The alternation
param eter, a , is independent o f tem perature and has values 0 < a < 1. W hen a = 1,
the m odel reduces to the linear chain m odel and w hen a = 0 the m odel reduces to that
o f m agnetically isolated dim ers. H alP ^ repeated calculations done by D uffy and Barr22 for the reduced susceptibility o f short alternating rings o f up to ten S = l/2 ions for a = 0 , 0.1, 0.2, 0.3, 0.4, 0.6, 0.8 and 1.0 using the cluster approach and derived the expression given below for the m agnetic susceptibility o f an S = l/2 antiferromagnetie alternating Heisenberg chain
Xu =
' ' N g V e W A + B x + Cx"kT 1 + D x + E x^ + Fx^
the values for the param eters