^ecliniccil ^2ote 92©. 327

SIGNIFICANCE of DIMENSIONALITY in the SPIN

WAVE THEORY of FERROMAGNETISM

BY

HENRY UNRUH, JR.

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical ^ote 327

ISSUED April 19, 1966

SIGNIFICANCE OF DIMENSIONALITY IN THE SPIN WAVE THEORY

OF FERROMAGNETISM

BY

HENRY UNRUH, JR.

Radio Standards Laboratory

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington D.C. 20402

Price: 20*

Contents

Abstract

5. References

in

1

1. Introduction 1

2. Calculations 2

3. Conclusions * 3

4. Acknowledgments 13

14

The Significance of Dimensionality in the Spin Wave Theory

of Ferromagnetism

Henry Unruh*

National Bureau of Standards, Boulder, Colorado

The significance of dimensionality in the magnetic

properties of a single domain ferromagnetic system is

determined utilizing a new approach to the spin wave theory

and a few plausible assumptions. If conditions are such

as to produce spontaneous magnetization in a finite temperature range for a specific dimensionality then under the

same conditions there exists a finite temperature range

for every dimensionality in which spontaneous magnetization occurs.

Key Words: Spin Waves, Linear chain, ferromagnetism.

1. Introduction

It has been claimed on the basis of the spin wave theory that

spontaneous magnetization for one and two dimensional systems cannot

occur in contrast to the three dimensional system^. In this regard,

the ferromagnetic thin film has been the subject of both theoretical

and experimental-^* 4 investigation. The experimental results so far

appear inconclusive. In this paper the dependence of the spontaneous

magnetization on dimensionality is re-examined, paying particular

attention to the role played by anisotropics such as that due to an

externally applied field. The results are applicable in all dimensionalities

if the low temperature limit is taken, and further, are applicable for

unrestricted temperature in the one dimensional case if one makes a few plausible assumptions.

^Permanent address: Dept. of Physics, Wichita State University,

Wichita, Kansas.
2. Calculations

In what follows the Hamiltonian operator-* is assumed for a system of

N spins to be given by

N

X=Xi ~ gPS- £s (1)

j=l J

with

3Cl = - J ZZgj * & (2)

J z

where J is the exchange integral (effective only between nearest

neighbors), H is the field intensity and includes contributions from

crystalline anisotropics, pis the Bohr magneton, g is the spectroscopic splitting factor and Si is the operator for the spin in the j position. The sum over z shall include only those spins nearest

neighbor to j. The Heitler- London model of well localized spins is

assumed. Only the case of spin /2is considered.

For the operator (1) there exist eigenstates (in any dimensionality)

with q spins directed downward, which have the form

4

J (N,q)= ),a(m1

, m2 *-* L a

. . . mn ) a , ... p ...p ...a., (3) 4 L m, m N

i l q mj, m?

. . . m = 1

where the sum is so restricted that two or more inversions cannot be

located in the same position. The c^s and p's represent the up and

down states respectively while the subscript determines the position

of the spin in the lattice. The eigenvalues of ( 1 ) belonging to (3) may

/ 7 Nz j conveniently be written D » ', aside from the unimportant constant -—— ,

4

E(k k . .. k ) = 2J Y (1 - cosk ) - %gfH(N - 2q) (4) v = 1
in which it is assumed that H determines the preferred direction for

the spin. The k are restricted to certain values which are given

exactly for any N and for < q < V2N in the one dimensional case, and

approximately for q < < N in any dimension by Bethe.

From (4) one may obtain the magnetic moment according to

M(N,T,H)=/cT-^- (5)

where k. is Boltzmann's constant, T is the absolute temperature and

Z is the partition function.

The evaluation of (5) has been accomplished by approximate

7 methods , usually with H = 0, and gave the results

M(N, T, 0) = VagpNil - const T ) (6)

for low T in the three dimensional case, and M(N, T, 0)—- - °° for any T

in the one and two dimensional cases. • These results are the basis for

the statement that only the three dimensional system can be spontaneously ferromagnetic.

However, it appears that an improved value for M(N, T, H) which

yields a different relationship between dimensionality and the magnetization can be obtained by separating out a part of Z which can be

done exactly as is shown in the following steps.

Consider the operator S defined by

N

s

± =E(sx

±sy

) j=l J

x y where S . and S . are the x and y components of S . Then

J J ~j
N N

S

_

i|j(N, q) = ^ Tj a ( ml> m2> ••; mq>

f= 1 m,12, m„ . . . m =1

q

X cl a_ . . . 6 . . . (3 . . . (3 . . . a

I 2 f mj m N

that is --the operation by S" increases the total number of inversions byone, and a(m,12, m • • • m ) is independent of the location of the new inverq

sion, (3 , so introduced. It is understood that the sums are so restricted

that two or more inversions cannot be located at the same point. Each

repeated application of S introduces one more such inversion. It is now

convenient to rewrite the eigenfunction, iji(N,q), in the form i(j (N, r) in

which h represents the number of inversions whose locations are independent of the a(m • • m ), i. e. the indices which determine the location

of the spins in the lattice do not appear in the a(m. " • • m ), and there are r

1 q

remaining ones so that q = h + r. What has just been established then is

that

S>h

(N,r) = ^h+1

(N,r).

In fact, if we suppose that we start with a state having h = so that all of

the indices locating the inversions appear explicitly in a(m • • • m ) we see

1 r that

iMN.r) = (S")

h

^ (N,r). n

T o At this point it would be convenient if one had established the correctness of the following two statements:

1. Those inversions, and only those, introduced by the S" operator,

i. e. , all those inversions whose locations have no effect on the a(m • • •

)

coefficients, each have an associated k value of zero.
2. That S Ui__ , (N, r) = • T N-2r

The author believes that he has accomplished these proofs in general

for the one dimensional case and is publishing them separately in

o National Bureau of Standards Technical Note No. 328, since they are

rather lengthy.

For the time being we prove and illustrate the use of the two

statements for the case of one inversion in a linear chain and indicate

some simple interpretations that may be made concerning the physical

system.

For a linear chain with one inversion the a(m) coefficients are

5

well known so that the eigenstate may be expressed by

N . k

.

j = 1 L J

where k is restricted according to

k =——— , I = 0, 1, 2 .. . N-l«

In the notation previously introduced we would write the eigenstate as

\\i. (N, 0) if k = and \\> (N, 1 ) if k £ 0. Of course, for k = the location

1 o

of the inversion has no effect on the coefficient, which is an illustration

of statement one above.

Making use of the fact that

N

and that

S

+ * (N,l)=( 2) e

ikj

) a « a . j=l * J a = N

s

2 = s"s

+ +s

z + (s

z

) 2

we see that
S

2

ib (N, 1) = y^ (N, 1), where y =

1/2{N-2)[ 1/2 (N-2)+ 1]

o o

2 so that \\> (N, 1 ) is in fact an eigenstate of S , which we expect since

[s

2

,^] = 0.

2

Recalling that [S", S ] = it is then clear that

S^N, 1) = Y ^(N. 1)

Repeated operation by S~ yields

S%

h

(N, 1) = Y \(N, 1)

provided h< N-2. If we try to carry this process beyond h = N-2

there results,

S

2

S" +N_ 2(N,1) = N N

S S- [ (N-2)! ] 2 Z e

"

J f\ P

2

• • • ?£.! «i Pi+1 • • ' Pj ' •

• 1=1" j = l ^3

?N

2

N

"k- S [(N-l)!] ( Yj ^

3

) Pi P ?

• • • Pisj =0 ' which illustrates statej =1

two above.

It is now clear that the interpretation that may be placed on

\\i

, (N, 1) is that it represents a state having the same magnitude of

spin, i, e. , the same magnitude of magnetization, as the state

represented by \\) (N, 1) but is tipped at an angle with respect to it. These results just obtained for the case that started out with

just one inversion present hold equally well for the two and three

dimensional cases. But rather than proceed further with this restricted

case the general validity of the two statements is assumed and the
partition function is calculated. Accordingly, the final results obtained

at the end depend on the demonstration of the correctness of the statements when many inversions associated with k ^ are present but

should be valid if only one is present, i. e. , they should be valid in

the limit of low temperatures.

Making use of the fact that [JC T »S ] = 0, there results

3CA (H, r) = 3C T (S") ^ (N, r) = 2J £ (1 - cosk U (N, r) In I o — v h

v=l

in which, using statement 1, none of the k's are zero and in which,

using statement 2, 0<h<N-2r. Thus it is seen that the h inversions

do not contribute to the eigenvalue of 3C T

° We now have

r -s 3C^h

(N, r) =J 2J }] (1 - cosk^) - #g|3H(N -2r -2h) \ i|j (N, r) v = 1 ^

so that the partition function is

VzN N - 2r r Z= Z Z E exp J-—,

J -2J Yj (1 " cosk ) r= k. . . . k h=

1 r kT v=l

^g(3HN - gPHr - gPHh ! j

or

Z = e

gJHN

2/cT

gPH

kT

Z'(H) + e

g(3HN

2/cT

1 - e

g£H

kT -1

Z'(-H) (7)

where %N r - n Z'(H)= 2 Z exPl"^ 2J Z (1 - cosk

y

)+ gpHr |

r=0 k_. . . k ^ L v=l J

J (8)

in which none of the k's are zero.
At this point it may be noted that when H is replaced by -H, Z is left

unaltered so that M(N, T, -H) simply becomes -M(N, T, H) as is expected

on physical grounds.

We may now observe that for sufficiently low T the terms for r ^ can be neglected, so that

^Z'lH^/(-Hhl

in all dimensionalities. This is because all k =0 modes are excluded

from Z

1 so that for r> 1

lim 1 T-0kT ZjJ^j (1 - cos k ) + gpHr

v=l

V

oo

for any H. Then for finite H,

lim

T— Z—e 2/cT

so according to (5)

M(N, 0,H) = Vzg&N

which shows no dependence on dimensionality. On the other hand, for

finite but very low T (so that Z' (H)— 1) there results

M(N, T, O) = O,

which again shows no dependence on dimensionality. These results are

made more explicit in the following paragraphs.
Expanding the exponentials in powers of -

that appear in

Z we obtain

Z = (N+ 1)Q -20,+ o 1 N(N + 1)(N+ 2) (3N -f 6N+ 2 ) 24 o

" 12 U

l

+ iN±ii Q _I Q

2

w

2 3 3 kT

4 3

N_ AN_ N_

48 I 40 8

i£',4Yq .i^43

6 15

t ^.-LV

y o 8 ^24 6 6 45 J

\

(9)

8^6 2 3 y 2 1 2 \, 2 3^3

+ <N±ii Q. - Q 24 4 60 w

5j^ kT J

' o • •

in which

V^N r Q* = Z Z r exp{-7^ Z ( x " coskj

r=0 k,...k L v = l

1 r

v

(10)

Thus for small H (the criterion for "small" is given below)

M(N,T,H) „ |(N^l)(N^2)Qo -l(3N t 6 + |)Q^(l t i)Q

2

- 3

-lQ

3 g(?H

M (N+ 1)Q - 2Q,

o 1 kT

(11)
in which M = /^gPN. For finite temperatures, it is clear that the

Q's are all finite and, therefore, the magnetization is finite or zero

in any dimensionality if the applied field is respectively finite or

zero. This result shows that there is no essential dependence of

magnetization on dimensionality in the sense that the condition required

to make it vanish in any dimension is sufficient to make it vanish in

all dimensions, and the condition required to make it finite in any

dimension is sufficient to make it finite in all dimensions. The actual

value of the relative magnetization will, of course, depend on the

dimensionality in the finite case.

2J

If one assumes T small enough to insure —- » 1 so that on the

average r « N, implying a temperature much lower than the Curie

temperature, then it can be seen from equation (10) that

NQ »Q A 1* 1

Making use of this condition and also the usual situation of N » 1,

equation (9) becomes

Z ^NQ Wg^HN x 2

1 /g(3HN ^

4

24 ^ kT ) 1920 ^ kT / J

(12)

which shows that the criterion for small H is that ° m— « 1. kT

Under these conditions

M(N, T, H) _ 1 gpHN

M kT (13)

That is, for sufficiently small H the magnetization is independent of

the strength of the interaction between the spins.

On the other hand, it can be seen that for large & — and for kT

10
small enough T to insure on the average r << N the second term of

equation (7) may be neglected compared to the first. In this case,

M(N, T,H) JCT 81nZ'(H) _2_-

M M 9H " N

o o •-(-^).

C. L "J or, approximating the sum on k by an integral * ' and neglecting

the last term,

2

m(n,j,h.) Bl __p_ r

kD.ir ,i

§

7t _^"

o (2ir) ^ 2tt L "g"

dk (14)

where p = 1, 2tt, 4it according as D = 1, 2, 3. D is the dimensionality

of the single domain sample, and G is the length of the linear chain or

is the length of an edge in the two or three dimensional cases measured

in units of lattice spacing, i.e., G =-N.

Equation (14) should be used with some caution. It must be

gSHN remembered that -— >>1 is required. This ordinarily does not kT }

place much restriction on H in the three dimensional case unless one

is dealing with small numbers of particles. Also in the three dimensional

case it is customary to evaluate the integral for the case H = and with

2tt the lower limit zero instead of-—, since there is not much contribution

to the integral near the origin. This procedure gives results for

M(N, T, 0)/M ~1 while equation (13) predicts an answer of zero. This

discrepancy is due to the fact that (14) is a very crude approximation to

kT—^r—— for very small H, i.e., H<<——-. However, it should be oH g pN

emphasized that there is no disagreement with the numerical results

of (14) that are usually quoted since for such results it is assumed that

11
at least a very small H exists —and once this is granted, (14) is very

nearly independent of H over an enormous range and nearly equals the

values obtained from (14) for the case H = 0.

In the two dimensional case the mathematical reduction of (14) can

be carried further, yielding, after integration,

M(N, T, H)

= 1 Mo

gpH 4tt 2

J

1 - /cT NkT - e e

-1

(15)

A wide range of physical interest is included if the parameters are

restricted tothe intervals

10 °K > T> 10°K,

4 lOOe >H>.1 Oe , 9 N > 1 .

Such restrictions insure gP >> 1 so that (15) is valid provided the

K. X resulting value for M(N, T, H)/M is near one. Such restrictions also

make "kT— y> Equation (15) then, for very large N, reduces to

M(N, T, H) . /cT . /kT \

T7 = 1 -~a—zrlTli . M 4ttJ VgPH J

12
3. Conclusions

It has been established, with the help of several statements that

are not proved herein, that the behavior of a magnetic system has no

essential dependence on dimensionality. Equation (13) shows that for

sufficiently low fields the relative magnetization is independent of the

interaction between spins. For high fields the results, summarized

in equation (14), are of the same form as previously obtained. The

important difference is that the existance of a finite lower limit on the

integral has been demonstrated, which, together with a finite H assures

one of a finite temperature range in which ferromagnetism will occur.

It appears that there exist a range of parameters, J, N, and H

(for low T) that produce results between those of equations (13) and

(14) that would be of experimental interest. The possibility of the

numerical reduction of the formulas in these cases is currently under

investigation.

4. Acknowledgments

The author is indebted to Mr. John Dalke under whose sponsorship

several months were spent at the National Bureau of Standards, and

also to several staff members, particularly Dr. Robert L. Peterson,

of the Bureau for technical assistance.

13
5. References

1 F. Bloch, Z. Physik61_, 206, (1930).

2 M. J. Klein and R. S. Smith, Phys, Rev. 81_, 378, (1951).

3 C. A. Neugebauer, Phys. Rev. 116 , 1441, (1959).

4 E. C. Crittenden, Jr. and R. W. Hoffman, Rev. of Mod. Phys,

25, 310, (1953).

5 J. Van Kranendonk and J. H. Van Vleck, Rev. of Mod. Phys,

30, 1, (1958).

6 H. A. Bethe, Z. Physik 71_, 205, (1931).

7 A. Sommerfeld and H. Bethe, Handbuch der Physik (Verlag Julius

Springer, Berlin, 1933) Vol. 24, Part 2, 604-618.

8 H. Unruh, Jr. , NBS Tech. Note !N[o. 328.

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