Spin Correlations in the Paramagnetic PhaseJ. Hubbard Citation: Journal of Applied Physics 42, 1390 (1971); doi: 10.1063/1.1660262 View online: http://dx.doi.org/10.1063/1.1660262 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/42/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Paramagnetic spin correlations and spin dynamics in doped manganites as the precursors of their magneticordering J. Appl. Phys. 105, 07D707 (2009); 10.1063/1.3054370 Highenergy spindensitywave correlated fluctuations in paramagnetic Cr+5 at.% V J. Appl. Phys. 73, 6454 (1993); 10.1063/1.352631 Some Consequences of the Non Mixing Behaviour of the Four Spin Correllation Function in HeisenbergParamagnets AIP Conf. Proc. 18, 713 (1974); 10.1063/1.3141806 Nuclear Magnetic Resonance in Paramagnetic Solutions. A Correlation between Contact Shifts andNuclear Spin Coupling Constants J. Chem. Phys. 54, 3238 (1971); 10.1063/1.1675323 Spin Correlations in Paramagnetic Nickel J. Appl. Phys. 38, 1247 (1967); 10.1063/1.1709563

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JOURNAL OF APPLIED PHYSICS VOLUME 42. NUMBER 4 IS MARCH 1971

Spin Correlations in the Paramagnetic Phase

J. HUBBARD*

Brown University, Providence, Rhode Island 02912

A method for calculating approximately the time-dependent spin-correlation functions in the paramagnetic phase of a Heisenberg ferromagnet is sketched. The predicted inelastic neutron scattering line shapes and widths at high temperatures and in the critical region (where dynamic scaling is predicted) are described.

I. INTRODUCTION

The appearance in recent years of much inelastic neutron scattering datal on magnetic materials in the paramagnetic phase above and near the transition temperature has stimulated theoretical interest in the spin dynamics in this phase. A number of theoretical approaches have been tried.2 The results obtained from a particular class of methods applicable to Heisenberg spin systems will be reviewed here. These methods are to be found in the work of Resibois and DeLeener,3 Kawasaki,4 Wegner,5 and Hubbard and Blume.6

Fortunately these rather diverse approaches all lead to the same fundamental equation [Eq. (14) J, and there­fore to the same predictions. In Sec. II, a rather concise derivation of Eq. (14) (adapted from Kawasaki's discussion) is given and in the remaining sections some results obtained by solution of (14) are described.

II. THE BASIC EQUATION

The general theory of neutron scattering7 gives in the paramagnetic case a differential cross section of the form

(d2ul dOdw) €X (k'lk) 11iw1[1-exp( -,8Iiw) JI xJ'q(w) (1)

for the inelastic scattering of a neutron from an in­cident state k to a final state k' with wave vector change q=k-k' and energy loss liw=fi2(k2-k'2)/2M (M = neutron mass). Here,8= llkBT, kB=Boltzmann's constant, T= absolute temperature,

(2)

i2 essentially the q-dependent susceptibility and Fq(w) is the Fourier transform of the relaxation shape function Fq(t) defined as

Fq(t) = {S_q', Sq·(t) I I I S_q', Sq'). (3)

lA, B(t) I stands for the Kubo relaxation functionS

{A,B(t)l= j(J (e).HAe-).HB(t»dX, (4) o

where H is the Hamiltonian,9 and Sq(t) is a Fourier component of the spin densitylo:

Sq(t) =N-l ~ Si(t) exp(iq·Ri ). (5)

obtainable from the neutron measurement is embodied in Fq(w) or equivalently in Fq(t). The general theory of Morill shows that Fq satisfies the generalized Langevin equation

Fq(t) = - [kq(t-t')Fq(t')dt', (6) o

where kq(t) is a memory function given by

kq(t) = I i-q',j/(t) II I S_q', Sq'l

~IS_q', S/(t) I/IS-q', Sq'), (7)

fq(t) is an operator12 closely similar to Sq(t); the second half of (7) follows from making the approxima­tion frSq •

From (5) and the usual Heisenberg Hamiltonian

one finds

H=-! ~J;jS;·Sj, i,i

(8)

(9)

where J q = ~;Jij exp[iq· (R;-Rj )]. Substituting (9) into (7) and using (2) gives

kq(t) = (1/xq) ~ Jq.Jq" I Sq""S-q_q,,lI- Sq"lIS_q_q,,"', g'q"

Sq,x(t) Sq_q,lI(t) - Sq,lI(t) Sq_q."'(t) I. (to)

To evaluate the right-hand side of (10) we make a further approximation. Consider the term ISq":z:S-q-q,,lI, Sq,"(t) Sq_q,lI(t) I. If one had instead the correlation

. function (Sq""S_q-q,,IISq,"'(t) Sq_q,1I(t», a natural de­coupling approximation would be

(Sq""S_q_q,,lISq,"(t) Sq_q.II(t) )

~(Sq":z:Sq,,,(t) > (S_q_q"lISq-q,lI(t) ). (11)

However, one can show (see Appendix) that in the paramagnetic regime (11) implies (to an accuracy of one or two percent) a similar decoupling for the re­laxation function:

{Sq""S-q_q,,lI, Sq,"(t) Sq_q,lI(t) }

"""",a-l{ Sq"'" Sq,"(t») I S_q_q,,1I, Sq_q,1I(t) I ~,a-l~q" ,-q'Xq'Xq-q,Fq, (t) Fq_q, (t), (12)

i where the last line follows from the definitions (2), All the information concerning the spin dynamics (3), the fact that {Sq,,"Sq,"(t)} =0 unless q'+q"=O

1390

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SPIN CORRELATIONS IN THE PARAMAGNETIC PHASE 1391

and I S-l, Sqz(t) } = I S_q', Sq·(t)} in the paramagnetic regime. Carrying out corresponding approximations on the other terms on the right of (10) leads to the final app roximation

kq(t)~(2/{3Xq) L (Jq.-Jq-q.)Jq• q'

1.6 r---...---r--,---,---,---.---, .........

1.4

1.2

XXq'Xq-q.Fq.(t)Fq_q.(t). (13) 1.0

I t should be noted that the errors involved in the ap­proximations at Eqs. (7) and (12) to some extent cancel.

Finally, substituting (13) into (6) gives the basic equation yielded by all the methods cited in Sec. I, namely

Fq(t) = - (2/{3xq) L (Jq·-Jq-q·)Jq'Xq'Xq-q· q'

x [Fq.(t-t')Fq_q.(t-t')Fq(t')dt'. (14) o

III. THE HIGH TEMPERATURE LIMIT

In the limit T -'HX>, (3-4J, one has Xq= S(S+ 1) {3/3N, and (14) reduces to

Fq(t) = - (2/3)S(S+I)N-I L (Jq.-Jq-q.)Jq• q'

x [Fq.(t-t')Fq-q.(t-t')Fq(t')dt'. (15) o

0.8 (s f (0) shtl)

0.7 tS(S+1)

0.6 (sf(o) S~lt) • -tSIS+ I)

0.5

0.4

0.3

0.2

0.1

--- -:'-=--.::..-.-. 0

-0.1 0 0.8 1.6 2.4 3.2 4.0 4.8

.. FIG. 1. Autocorrelation function and nearest-neighbor cor­

relation function at T= 0> for a simple cubic lattice with nearest­neighbor interactions. The heavy curves represent the result obtained by solving Eq. (14), the dots and circles Windsor's computer simulation results. The dashed curves are simpler approximations (see text) . .,. is a reduced time variable (see text).

OB

0.6

0.4

0.2

00 0.4 0.8 1.2 1.6 2.0 2.4 2.8

W

FIG. 2. The Fourier transform of the autocorrelation function at T= 0> shown in Fig. 1. The heavy line gives the solution ob­tained from (14), the circles are Windsor's simulation results. The dashed curve is a simpler approximation (see text). w is measured in units of 2J[S(S+l) Jl/2/1i.

This equation may be solved numerically by integrating forward in time beginning with the initial condition Fq(O) = 1 obtained from (3). Such calculations have been carried out for the case of a simple cubic lattice with nearest-neighbor interactions6 and some of the results are shown in Figs. 1-4.

Figure 1 shows the autocorrelation function (S;'S;'(t) and nearest-neighbor correlation function calculated from the solution of (15) (heavy lines). These results are compared with those obtained from Windsor's computer simulation calculation for the same modeP3 (circles and dots) with which they show very good agreement. The dashed curves represent simpler approximations for the autocorrelation function obtained by other methods.3,6 T=2Jt[S(S+I)]1/2/1i is a reduced time, where J is the nearest-neighbor inter­action.

Figure 2 shows the Fourier transform of the auto­correlation function (the heavy line) again compared with Windsor's results and a simpler approximation.6

This curve has fairly characteristic shape and should give the spectrum of neutrons inelastically scattered through large angles from a polycrystalline material at high temperature. In fact Windsor's neutron datal for RbMnF3 in this case gives a curve of this shape.

Figure 3 shows the correlation functions (S_qZSqz(t) for several q and Fig. 4 the corresponding Fourier transforms, the dots and circles in Fig. 3 again giving Windsor's results. For small q one can show that (15) has a solution of the form

(16)

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1392 J. HUBBARD

08

FIG. 3. The correlation function (S_q'Sq'(t)) at T= '" calculated from (14) (heavy lines) compared with Windsor's simulation results (dots and circles). r=2J[S(S+l)]!/2/1l.

i.e., an ordinary diffusive form giving a Lorentzian line for Fq(w). The solution of (15) gives a diffusion constant D in close agreement with Mori and Kawasaki's result.14

For larger q, Fq(w) has a Gaussian-like shape and for the largest q, a plateau form reflecting the tendency to oscillation in the corresponding F q(t).

IV. INTERMEDIATE TEMPERATURES IN THE PARAMAGNETIC PHAS~

On going to lower temperatures it is necessary to de­termine the Xq entering into Eq. (14). To do this, we use the identity8 {S_q', S/} = -i([S-qz, Sqz]). Using (2), (3), and (9) this identity gives on evaluating the commutator

XqFq(O) = -i([S_qZ, Sqz])

= -22: (Jq,-Jq_q,) (S_qZSq') ql

~-2rrl 2: (Jq,-Jq-q,hq' (17) ql

the last line following by using (S_q'Sqz)"-'{l-I{Sq', S_qZI"-'{l-IXq which is good to a few percent in the paramagnetic phase (see Appendix). On the other hand, (14) gives

XqFq(O) = -2rrl 2: (Jq,-Jq-q,)Jq'Xq'Xq-q" (18) ql

For (14) to be consistent with the sum rule (17) one must have

for each q. These constitute a set of equations for the Xq which may be easily verified to have the solution

(20)

for any A. Thus the Xq are determined within the param­eter X which must be a function of T. Equation (20) is of course approximate having been derived from the approximate Eq. (14).

To determine the temperature dependence of X, one

may use the relation

q

(21)

i.e., (22)

q

This gives with (20) a formula for Xq equivalent to the spherical model result15 and a relation between A and T which will be referred to as the "spherical model (SM) temperature scale." Alternatively one may notice that Xq=O= (>,-JO)-1 is just the ordinary susceptibility x, so

l/(A-Jo) =x (23)

and may determine X as a function of T by substituting a Pade approximantl6 for X as a function of T on the right-hand side of (23). This will be called the "Pade temperature scale" and will be better than the SM scale in the critical region. Using (20), Eq. (14) may be integrated numerically for various X corresponding to various temperature (according to the SM or Pade scales). The results of some such calculations for a simple cubic lattice with nearest-neighbor ferromagnetic interactions are shown in Figs. 5 and 6.

Figure 5 shows the line shapes Fq(w) for several qat various temperatures T/Tc (Tc is the critical temper­ature at which Xq=O diverges, i.e., X=Jq=o). The shapes are on the whole much the same as at high temperatures except for stronger side peaking at q= (t t, t) and a change in shape at T= Tc for small q. The widths of the lines change with temperature, however. In Fig. 6

2.0 r--I,---r----,---r-----r---,--..,--r----.

1.8

1.6

1.2

0.4 0.8 1.2 1.6 2.0 2.4 2.8 W

FIG. 4. fo. (w) calculated from (14) at T = "'. w is measured in units of 2J[S(S+l)]I/2/1i.

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SPIN CORRELATIONS IN THE PARAMAGNETIC PHASE 1393

the half-widths at half-height ~q have been plotted as a function of T for various q.l7 For large q, the width increases with decreasing T, accelerating as T~Te. For small q the width initially decreases in conformity with the fact that for small q one has ~q <X Dq2 and a diffusion constant D which decreases as T ~Te. How­ever the width has a minimum at some temperature above Te and thereafter rapidly increases again as T~Te, the total increase being about a factor 2 be­tween the minimum and Te.

V. THE CRITICAL REGION: DYNAMIC SCALING

The calculations at small q for T near Te discussed in Sec. IV are in the critical region. In this region the

FIG. 5. The shapes of P.(",) calculated from (14) at various q and temperatures for a simple cubic ferromagnet with nearest­neighbor interaction. The vertical scales are arbitrary, the horizontal ",-scale is in units of 2J[S(S+ 1) J1I2/1i, and the temperatures are according to the Pade scale (see text).

solution of (14) has interesting properties and in fact predicts dynamic scaling.ls

For small q one may expand J q in powers of q

which substituted in (20) gives for small q

Xq<X 1/(q2+ K2) ,

(24)

(25)

i.e., an Ornstein-Zernicke form where K is the (temper­ature dependent) inverse correlation length. As T~Tc one has K~O and Xq becomes very large for small q (critical fluctuations). In this case, for small q the q' sum in (13) is dominated by the small q' terms (because of the divergence of Xq'Xq-q' for small q').

q

(HO)

(Ht)

(too) (Ho)

(foo) (Hi) (HO)

(too)

TlTe

TlTe

, iD

, iD

lao , 0

5.0

I I 10.0 5.0

LO~ N .... ., ""

3'0 0 io ' L~" "~o PADE SCALE

I 0 I 0 II II~ sltLE 3.0 2.0 L5 1.0

FIG. 6. The half-width at half-height 8. of P.(61) as a function of temperature calculated from (14) for the cubic ferromagnet with nearest neighbor interaction. p.= J._o/X (see text) and the SM and Pade temperature scales are given.

For small q one may therefore substitute the forms (24), (25), in (14). Furthermore the q' sum may be replaced by an integral dq' extending over the whole of q' space giving in the critical regime

Fq(t) = -U2(q2+K2) It dt' f dq' q2-'22q

'2q

' o q +K

r 4.

3.0

2.0

XFq, (t-t') Fq_q, (t- t') Fq(t') , (26)

• •

• x

x <tOO) POINTS

• <Hi) POINTS

• x

•

1.0ko ---'--'O:;';.5.----''----;,.!;:;O---'--......J.,.5

K/q

FIG. 7. Approximations to the universal function g(x) of Eq. (30) obtained from the solution of (14) by plotting 8./uq6l2

against K/q for small q.

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1394 J. HUBBARD

0.8

0.6

0.2

2 4 w/trql

6 8

FIG. 8. The universal functionJ(x, Tf)/J(O, '1) of Eq. (29) for several Tf obtained from the solution of (14) by plotting Fq(w)/Fq(O) against W/trq5/2.

where CT is a constant depending upon the details of the interaction and lattice.

It can now be verified that the solution of (26) is

(27)

where j( u, 1J) is a universal function satisfying the equation

d 1" f 1-2x·e a- j(u, 71) = - (1+712

) du' dx u 0 ~+~

Xj [(u-u')x5/2, 71/XJj (u-u') / x-e /5/2, _71_) / x-e /

Xj(U,71) (28)

and the initial condition j(O, 71) = 1, where e is an arbitrary unit vector. From (27) one has

(29)

wherej(w,71) is the Fourier transform ofj(u, 71). Thus the shape of the line in the critical region is determined solely by the ratio q/ K independent of interaction, lattice, etc. The width Oq in this region is from (29) given by

(30)

where g is another universal function. At T = Tc, K = 0, and Oq= uq5/2g(0) giving the 5/2 law for the ferro­magnet at the critical point. For q«K one can show that g(K/q)~d(K/q)I/2 giving oq=uq2dKI/2, i.e., a diffusion constant D=dKI/2 scaling as KI/2 as expected.I8

In Fig. 7, Oq/ q5/2 is plotted against K/ q for two small values of q taken from the calculations described in

Sec. IV to obtain the shape of the function g(x). It is seen to be a minimum at q~K and to have infinite slope at K = 0 [as can be shown from (28)]. A similar result has been obtained by Resibois and Piette.

In Fig. 8, the function Fq(w)/Fq(O) has been plotted against w/uq5/2 for several values of K/q from the results of the calculations described in Sec. IV to give the shape of the functionj(u, 1/). For the larger K/q, the curve is Lorentzian, but at K=O, i.e., T=Tc, has a distinctly non-Lorentzian shape which agrees fairly well with the shape observed at T= Tc in neutron measurements of Fe and Ni,19 and with the calculation of Wegner.5

Application of (14) to the antiferromagnet in the critical region has proved less successful. Correct scaling laws are predicted but the observed three-peaked line shape at T= TN is not obtained.

APPENDIX

To demonstrate the near equivalence of (11) and (12) in the paramagnetic phase denote the Fourier transforms of (Sq"zS-q_q"ySq,z(t) Sq_q,v(t) ), (Sq"ZSq,z(t), and (S_q_q"YSq_q,lI(t) by SI(W) , S2(W) , and Sa(w) and the transforms of the corresponding re­laxation functions [Sq"ZS_q_q"Y' Sq,z(t) Sq_q,lI(t) J, etc., by RI(W) , R2(w), and Ra(w). Then general theory gives the relations

Si(W) =hw/[I-exp( -(3hw) JRi(W) i= 1,2,3 (AI)

and (11) is equivalent on Fourier transformation to

SleW) = f S2(W') Sa(w-w')dw'.

Then (AI) and (A2) together give

RI(W) = JR2(w')Ra(w-w')

( 1-exp ( - (3liw ) liw'

X liw l-exp(-{3liw')

(A2)

X h(w-w') )dw' (A3) l-exp[ -(3h(w-w')J .

The Ri(w), Si(W) are only largefor hw ;5J[zS(S+ 1) JI/2 in the paramagnetic phase, while in this phase {3<{3c'" [JzS(S+l)J-I, where {3c is the critical (3, J the inter­action, and z the number of near neighbors. Thus hw{3<hw{3c',-{zS(S+1)J-I/2«1. If one expands the quantity in the large parentheses in (A3) one finds

l-exp( -(31iw) hw' h(w-w')

liw l-exp(-{3liw') l-exp[ -{3h(w-w')J

= (3-I[1 + (l/12) {32fi2w' (w-w') + ... J, (A4)

where the correction term is ",1/12zS(S+1) ;51/70. Neglecting this term, which introduces an error of

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SPIN CORRELATIONS IN THE PARAMAGNETIC PHASE 1395

only a percent or so, in (A3) gives

Rl(W) =(3-1fR2(w')Ra(w-w')dw' (AS)

which is the Fourier transform of (12).

* On leave from A.E.R.E., Harwell, En&land. 1 C. G. Windsor, G. A. Briggs, and M. Kestigan, J. Phys. C 1,

940 (1968); C. G. Windsor, Proc. Phys. Soc. (London) 87, 501 (1966); 89,825 (1966); 91,353 (1967); D. Cribier and B. Jacrot, in Inelastic Scattering of Neutrons (International Atomic Energy Agency, Vienna, 1963), Vol. 2, p. 309; M. F. Collins and R. Nathans, J. Appl. Phys. 36, 1092 (1965); N. S. Satya Murthy, in Inelastic Scattering of Neutrons (International Atomic Energy Agency, Vienna, 1965), Vol. 1 p. 433; Phys. Lett. 26A, 108 (1968); see also Ref. 19.

2 P. G. deGennes, J. Phys. Chern. Solids 4, 223 (1968); H. S. Bennett and P. C. Martin, Phys. Rev. 138, A608 (1965); M. F. Collins and W. Marshall, Proc. Phys. Soc. (London) 92, 367 (1967); H. S. Bennett, Phys. Rev. 174, 629 (1968); 176, 650 (1968); R. A. Tahir-Kheli and D. G. McFadden, Phys. Rev. 182, 604 (1969); see also Refs. 3-7 and 14.

3 P. Rcsibois and M. DeLeener, Phys. Rev. 152, 305 (1968); 152,318 (1968).

4 K. Kawasaki, J. Phys. Chern. Solids 28, 1277 (1967); Progr. Theoret. Phys. (Kyoto) 39, 285 (1968).

• F. Wegner, Z. Physik 216, 433 (1968). 6 M. Blume and J. Hubbard, Phys. Rev. B 1, ,~!lLi (1970). 7 W. Marshall and R. D. Lowde, Rept. Pro)l;r. l'hys. 31, i05

(1968l. 8 R. Kubo, J. Phys. Soc. Japan 12, 570 (1957). 9 The additional term - (A ) (B ) wiII vanish in all cases con­

sidered here. 10 N is the number of lattice sites, Ri and Si are the Jlosition

and spin vectors of site i, and q is any wave vector in the tirst Bril­louin zone. All q sums are over this zone.

11 H. Mori, Progr. Theoret. Phys. 33,423 ()965). 12fq(t) ",exp[(l-P)iLt]Sq in Mori's notation. 13 C. G. Windsor, in Inelastic Scattering of Neutrons (Inter­

national Atomic Energy Agency, Vienna, 1968), Vol. 2, p. 83. 14 H. Mori and K. Kawasaki, Progr. Theoret. Phys. (Kyoto)

27, 529 (1962). 15 T. Berlin and M. Kac, Phys. Rev. 86, 821 (1952). 16 R. G. Bowers and M. E. Woolf, Phys. Rev. 177,917 (1969). 17 ,.. '" J q _ o/>-' is a convenient measure of >-. which runs from 0 to 1

as T passes from '" to Te• 18 B. 1. Halperin and P. C. Hohenberg, Phys. Rev. Lett. 19,

700 (1967). 19 M. F. Collins, V. J. Minkiewicz, R. Nathans, L. Passell,

and G. Shirane, Phys. Rev. 179,417 (1969); V. J. Minkiewicz, M. F. Collins, R. Nathans, and G. Shirane, Phys. Rev. 182, 624 (1969).

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