Recent developments in the theory of antiferromagnetism

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Submitted on 1 Jan 1951

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Recent developments in the theory of

antiferromagnetism

J.H. van Vleck

To cite this version:

RECENT DEVELOPMENTS IN THE THEORY OF ANTIFERROMAGNETISM

By J. H. VAN

_VLECK,

Harvard

_{University, Cambridge (Massachusetts).}

Sommaire. 2014 Deux

températures caractéristiques Tc et 03B8 jouent un role fondamental dans la théorie de l’antiferromagnétisme. Tc est la température à _laquelle la _{susceptibilité} est maximum et au-dessous de laquelle l’ordre antiferromagnétique s’établit. 03B8 est la constante de la formule

_{(~ =}

C/T+03B8)

pour la susceptibilité au-dessus du point de Curie. Si l’on tient compte seulement du couplage entre les _premiers voisins, Tc et 03B8 doivent être les mêmes. En réalité, 03B8 est beaucoup plus grand que Tc. Néel a _{montré que} ce _comportement résulte de l’interaction entre les deuxièmes voisins. Si ce _couplage

est assez grand, l’état de plus basse _énergie est celui dans _lequel les _spins des deuxièmes voisins sont

antiparallèles, bien que les spins des _premiers voisins soient, en _{partie, parallèles.} La théorie de P. Anderson, qui analyse l’ordre _{antiferromagnétique} pour des types varies de réseaux, est _présentée

avec _quelque détail. Le travail de Shull et Smart sur la diffraction de neutrons est aussi discuté. Ces auteurs montrent que dans MnO, NiO, MnF2, etc., l’ordre est tel que les spins des deuxièmes voisins sont effectivement _{antiparallèles.} Ce paradoxe, que le couplage est plus fort entre les deuxièmes qu’entre

les _{premiers voisins,} est nettement _{expliqué par} la théorie du " super-échange " (échange indirect)

développée par Anderson. Nous présentons les détails _{mathématiques} de cette théorie sous une forme

un peu différente de celle de la publication originale.

JOURNAL 1951,

It is well known that the

_exchange

eff ects of

quantum

mechanics

_{couple together}

the

_spins

of

different atoms

_by

a

_potentiel

of the form

Here

_S/

is the

_spin

vector of atom

_j,

measured

in

_multiples

of

h

_{and - 1Ajk}

is the

_exchange

integral.

The

_{antiferromagnetic}

case is that in

which the energy is a minimum with

antiparallel

alignment,

and in which the constant

_Ajk

is

conse-quantly

positive.

The

_theory

of

_{antiferromagnetism}

is dominated

_by

the work of Néel

_[1].

He

_pioneered

in

_developing

the

_concept

of an

_{antiferromagnetic}

medium,

with a Curie

_point

above which there is

disorder,

but below which there is a

_staggered

order

of the

_spins.

In section I we will discuss the

_theory

of

anti-ferromagnetism

for

_simple

and

_{body-centered}

cubic lattices. In section 2 we will

_present

Anderson’s

extension of the

_theory

to face-centered cubic

lattices,

the richest case

_{experimentally.}

Section 3

is devoted to the relation between the

suscepti-bility

at T = o and at the Curie

_point.

Section

₄

summarizes the

_interesting

evidence on

antiferro-magnetism

provided

by

the neutron diffraction

data of Shull and collaborators. These data show that

_coupling

between next-to nearest

_neighbors

is

_inordinately

_important.

In section 5 an outline

is

_given

of a calculation

_by

Anderson which shows that

_{superexchange}

does indeed accentuate the role of next-to-nearest

_neighbors.

Mathematical details of this calculation are

_presented

in section 6. A

digression

is made in

_{section 7}

to discuss

_briefly

the

_importance

of the

_perturbing

effect of excited

states for other

_{magnetic phenomena}

_besides

super-exchange,

notably anisotropy.

In the

_concluding

section 8 we summarize some of the limitations of

the

_{existing theory}

of

_{antiferromagnetism.}

1.

Theory

for

_Simple

and

_{Body-Centered}

Cubic Lattices. - The

simplest

type

of

antiferro-magnetic

medium is one in which the

_crystal

can

be divited into two

_sublattices,

_say

_{A, B,}

such that if the

_spins

of one sublattice

_point

east, and if those

of the other sublattice

_point

west, the

_spins

of atoms

which are nearest

_neighbors

are

_{always antiparallel.}

The

_simple

and

_{body-centered,}

but not the face-centered cubic lattice fulfill this criterion. The

body-centered

case is illustrated in the upper

_part

of

_figure

i.

The

_theory

of

_{antiferromagnetism}

is characte-rized

_by

the

_simplicity

of its bold outlines. The first

_{approximation}

is to assume that interatomic interaction can be

_{represented by}

a molecular

field, with,

of course, a constant of

_{proportionality}

having

the

_{opposite sign}

from that in the Weiss

ferromagnetic

case. The molecular field

approxi-mation is at least as

_good

as the first

approximation

of the

_Heisenberg

model,

and

_{mathematically gives}

the same formalism as does the

_Ising

model

pro-vided mean value

_{simplifications}

are used in the

latter.

If an

_{atom i belonging}

to sublattice A is

_coupled

only

to its nearest

_neighbors;

_supposed

_equidistant

and

_equivalent,

its

_exchange

_potentiel

is

where the sum is over the

_nerghbors

of

_j.

The

molecular field

approximation

is to consider the

effective

_{potential acting}

on this atom as

Here MH is the mean

_{magnetization}

_{per unit volume}

of sublattice

B, p

is the Bohr

_{and g}

4

is the Landé factor. The

_{negative sign}

of the

gyromagnetic

ratio is of no real

_import

for our

cal-culation,

and may be

incorporated by taking g

as

negative,

as the final formulas involve

_only

_squares

of q. We choose the definition of the constant of

_{proportionality}

y in such a _way that

_positive

y

corresponds

to

_{antiferromagnetism, negative}

to

ferromagnetism. Obviously

(2)

reduces to

₍₃₎

if

we

_replace

_S~;

_by

its mean value and if we

take ~

where z denotes the number of nearest

_neighbors,

and the number of atoms of sublattice B _per unit volume. However

₍₄₎

need not

_necessarily

be the

_optimum

choice of _y in the molecular field

model,

inasmuch as a somewhat different choice

of _{y may}

_{ultimately give}

some allowance for the effect of

_higher

order

_{perturbations.}

The

calcula-tions of Peter Weiss

_[2]

with the Bethe-Peierls method indicate that the best values of y differ

considerably

from

_(-~4).

Néel

_[3]

has

_recognized

the

_importance

of

generali-zing

the

_theory

to include the influence of an atom’s

next-to-nearest

_neighbors.

When the effect of interactions with next-to-nearest as well as nearest

neighbors

is

included,

the molecular field

_acting

on an atom of sublattice A becomes

and the

_{cotresponding}

field for an atom of sublat-tice B is

Behavior above the Curie

_point.

- Above the Curie

temperature,

the

_only

mean moment

_possessed

by

either sublattice is that induced

_by,

and so

parallel

to, the

_applied

field H. We assume

throu-ghout

that there are no relevant

_crystalline

Starck effects

_interfering

with the free orientation of the

spin.

At

_ordinary

field

_strengths,

saturation effects

are

_negligible,

and the mean moment of an atom is a

linear function

20132013"

of the effective field. Here

Nl

and else-where N _{denotes the total number of}

para-magnetic

atoms per unit volume. The

quantum-mechanical value of the Curie constant C is

C = NB2g2S(S + 1).

The mean moments of the two sublattices are this

For

_simplicity,

the

_quantities

_H,

M,

_appearing

in

_{(6A, B)}

are treated as scalars rather than vectors;

this

_procedure

is

_legitimate

since we assume

H,

Mn,

and to be collinear.

_{By adding}

_(6A)

to

_(6B),

and

_solving

for

MA + MB,

we find that

the

suscep-tibility

Z _ MA + MB si 1 I 1 _{Y Z} == H Sl where

Curie

_point.

- The Curie

temperature

is that below which the sublattices A and B both _possess

a mean

magnetic

moment even without a

_field,

ORDERING OF FIRST KIND

ORDERING OF SECOND KIND

Fig. I. - Body-centered lattice.

Ordering of first _{kind; ordering} of second kind.

though

because of cancellation this is not true for their sum. In

_general,

below the Curie

_point

it is

not

_legitimate

to treat the moment as a linear

function of the effective field.

Saturation, however,

becomes

_unimportant

as the Curie

_point

is

neared,

and so

_linearity

can be assumed in

_getting

the

equations

needed to locate the Curie

_point.

Hence

the Curie

_point

is the

_temperature

Tc

at which

the

_pair

of linear

_{equations (6A, 6B)}

admit a solution

with H = o. Thus we have ,

is to be

_{distinguished}

from the actual

_temperature

T,,

below which the ordered structure sets in. Hence

_7B

may be called the , i

true ", ~ ~ paramagnetic ",

or

~ ~ _{ordered " Curie}

point.

When we

_speak

the of Curie

_point

without

_{qualifying adjectives,}

we

_always

mean

Tc

rather than o.

From

₍₈₎

and

₍₉₎

it follows that the absolute value of the ratio of the

_asymptotic

to the true

Curie

_temperature

is

From

₍₁₀₎

_{it would appear that the}

ratios

can

be made

arbitrarily

large

by

taking a sufficiently

close to Y.

Actually

this is not the case, for if the

ratio ’y

exceeds a certain value it is

_{energetically}

y

more favorable to have different structure in which all the

_spins

in sublattice A cease te be

_parallel

to each

_other,

and

_similarly

for lattice B. The alternative

_arrangement

is

_simpler

for the

body-centered than for the

_simple

cubic

lattice,

for the former has the

_property

that it is

_possible

to make the

_spins

of all next-to-nearest

_neighbors

_{antiparallel.}

Consequently,

for the rest of the

_present

section I we

will confine our discussion to the

_{body-centered}

lattice.

The

_arrangement

which is

_{energetically}

most

favorable

if -

is

_large

is shown in the bottom half

7

of

figure

1. Here lattices A and B

_separately

have

a

_staggered

structure. Such an

_ordering

we call

"

of the second kind ". Since it makes all next-to-nearest

_{neighbors antiparallel,}

this kind _of arrange-ment will be realized if the interaction between next-to-nearest

_neighbors

overshadows that between

nearest

_neighbors.

On the other

hand,

this second

kind of

_ordering

is less favorable to the interaction energy of nearest

neighbors

than is that of the first kind. With the second

_type,

_any

_given

atom

has as _many

_parallel

as

_antiparallel

nearest

neigh-bors,

and the _energy is unaffected if all the

_spins

of sublattice B are

systematically

reversed or all rotated

_through

the same

_angle;

in other words there is no _{necessary correlation between} A and B.

To find which

_type

is

_actually

_achieved,

we

examine whether the first or second kind of

_ordering

has the

_higher

Curie

_temperature.

_{(An equivalent}

result can also be obtained

_{by investigating}

which

has the lower

_energy).

With the second

kind,

there is no _energy

_coupling

the

_spins

of the two

sublattices

A, B,

in the absence of an

_applied

_field,

since it is

_just

as

_likely

that nearest

_neighbors

have

their

_{spins parallel}

as

_{antiparallel.}

Hence if H = o

the

_equations

_analogous

to

₍₆

_A,

6

_B)

for the second

variety

of

_ordering

are

where

_M,

are the moments of the two halves

of lattice A. There are, of course,

_analogous

equations ( 11 B)

for sublattice B. The

equa-tions

_{(11 A),}

also

_{( 11 B)}

will be satisfied if

The calculation of the

_{susceptibility}

above the Curie

point

makes no

_{assymption concerning}

the

_type

of order.

_{Equations (6}

A,

_{6 B), (7), (8)}

are still

applicable,

with

M_, =

_{M B1 + M’B2’}

Mn

=

MD1

+MB2

Fig. 2.

°

the distinction between

MAl

and

_M~,~

enters in our

calculation

_only

for _T

_Tc.

Hence with the second kind of

_ordering,

the absolute value of the

ratio of the

_asymptotic

to the true Curie

tempe-rature is

Comparison

of

₍₉₎

and

₍₁₂₎

shows that the second

type

of

_ordering

will

_give

a

_higher

value of

_Tc

than does the first if

_{oc > ~ ,; .}

Hence

_{(9, 10)}

and

_{(12, 13)}

are

_{respectively applicable according}

as a

~

_y and a

> I

_y. The. maximum allowable

value of

,e

is that achieved at IX

_l

_y, viz. The

_dependance

of 0

,,

on g

I

is shown in the _upper

Unfortunately

no data are available on

nntifer-ron1agnetir

simple

or

_{body-centreed}

cubic lattices

to test the

_theory.

Some suhstances for which the

observed values of

,e

are within the limits

L

allowed

_hy

the

_theory

are as follows :

In all these cases the deviations from cubic

_symmetry

are

quite large,

and it is not clear whether it is a

good

approximation

to

_regard

the lattice as

_naturally

decomposible

into two sublattices in the fashion assumed in the

_present

section. In

_{MnP 2}

and

_FeF2,

an atom’s nearest

neighbors

are those

_along

the

i oo

direction;

then come the

_neighbors

at the

body

centers, and

_finally

those

_along

010 and 001

which are

_considerably

further away; our model

applies

if somehow nearest

_neighbors

in one direction

only

don’t count for much.

2. Anderson’s

_Theory

for Face-Centered Cubic Lattice. - There _are

many

antiferromagnetic

materials for which the observed ratios

2013

are

consi-Tc

derably larger

than the maximum value 3 admitted

by

the

_{preceding theory.}

However,

these

subs-tances have face-centered

lattices,

and we shall show that the

_discrepancy

is removed in a

genera-lized model

_appropriate

to the face-centered struc-, ture,

which has been

_{developed by}

P. Anderson

_[6]

’

and which we shall now

_present.

A face-centered lattice can be

_regarded

as

_composed

of four

_simply

cubic sublattices

A, B, C,

D as shown

in

_figure

3. The twelve nearest

_neighbors

of a

_given

atom _{of any}

_given

sublattice are distributed

_equally

among the three other sublattices. The six next-to-nearest’

_neighbors

_belong

to the same sublattice as

the

_given

atom.

_Hence,

if we include

_coupling

te

both nearest and next-to-nearest

_neighbors,

_the

equa-tions

_analagous

to

₍₆

_A,

_B)

which are

_applicable

above the Curie

_point

are

If one uses the first

_{approximation}

of the

Heisen-berg theory,

the values of z to be used in

₍₄₎

and the

analogous

formula for oc, are

_{respectively z}

==

4

and z = 6.

On

_{solving (15 A,}

_B,

_C,

_D)

for

one has

with

It is

_impossible

to find an ordered structure

such that a

_given

atom is

_antiparallel

to all its

neighboring

atoms. As

_regards

_coupling

between nearest

_{neighbors, nothing}

better can be done than

to assume that the

_spins

of two of the

sublattices,

Fig. 3. - Face centered lattice.

Ordering of first kind; improved ordering of first kind; Ordering of second kind.

say

A,

C

point

east, and the other two,

B,

D, west,

as shown in the first

diagram

of

figure

3. A

given

atom is then

_antiparallel

to

two-thirds,

and

_parallel

to one-third of its nearest

_neighbors.

On

_solving

the chain of

_{Equations (15}

A, B, C,

D)

with

As Anderson

_cleverly

notes, it _{is, however,}

_possible

to find another

_arrangement

_(fig.

₃₎

which we call

" improved ordering

of the first kind " as shown in the second

_diagram

of which is

_just

as favorable as

regards

the

_coupling

between nearest

_neighbors,

and less destrimental relative to the interaction between next-to-nearest

_neighbors.

_{We may arrange} the

_spins

in any

given

001 net

plane

in such a _way that in this

plane

these of nearest

_neighbors

are

_{antiparallel,}

those

of next-to-nearest

_neighbors

are

_parallel.

The

adja-cent

_parallel

001 net

planes

then contain some atoms

which are nearest

_neighbors,

but none which are

next-to-nearest

_neighbors

of those in the first

plane.

The atoms in the

_{adjacent planes}

which

are nearest

_neighbors

are

just

as

_apt

to have

_spins

parallel

as

_{antiparallel.}

In other words there is no

correlation between two successive

_planes.

Inter-action between next-to-nearest

_neighbors

intro-duses

_coupling

between two

_{next-to-adj acent}

oo I

planes,

and this interaction can be made as favorable

as

_possible,

i. e.

_{completely antiparallel, by assuming}

that the

_spins

at

_{corresponding}

sites of these two

planes

are

_always

reversed. In other

_words,

all

spins

of net

_planes

1,

5,

9, ... are rotated

through

1 80° as

_compared

with these of

_planes

3, 7, 11; and

similarly

spins

of

_planes

2,

6,

I o are rotated

through

180° relative those of

_4,

6, 12. A

given

spin

is then

_antiparallel

to four

_{neighboring spins}

and

_parallel

to four next-nearest

_spins,

all in the

same o01 i

_plane;

it is uncorrelated with the

spins

of nearest

_neighbors

located in other

_planes,

and is

_antiparallel

to its

_remaining

two next-to-nearest

neighbors,

situated two

_planes

_away. It is thus

antiparallel

to

two-thirds,

and

_parallel

to one-third of its nearest

_neighbors,

as before the

_improvement,

but it is now

_parallel

to

two-thirds,

and

_antiparallel

to one-third of its next-to-nearest

_neighbors

rather

then

_parallel

to all of them. The

effect,

as

_compared

with the

_type

of

_ordering

assumed in

_obtaining

the value of

_Tc

_given

in

_(18),

is

_{formally equivalent}

to

_reducing

oc

_by

a

factor 3

and so, in

_place

of

_(18),

we have a

_higher

Curie

_{point given by}

As

_long

as

_coupling

between next-to-nearest

neighbors

has any

influence,

one should

_always

use

₍₁₉₎

in

_preference

to

_(18).

_However,

if the

interaction between next-to-nearest

_neighbors

becomes

_{sufficiently powerful}

_compared

with that between

_neighbors,

then we have an "

_ordering

of the

second

kind ",

shown in the bottom of

_figure

3, in

which all next-to-nearest

_neighbors

are

_{antiparallel,}

and in which there is no mean correlation between

nearest

_{neighbors. Analogously}

to

_(I

_{I A)}

we then

have

the factor in the denominator now

_{being 8}

rather than

₄

since there are now

_eight

sublattices

_Ai,

A2,

...,

D2

rather than four as in section i. From

_{(20 A)}

it follows that with

_ordering

of the

second

kind,

the Curie

_point

is

Equation (21) gives

a

_higher

Curie

_point

than

_(19),

3 3

as

_long

while

₍₁₉₎

holds for

a 3y.

4 4 1 .

The maximum value of

rn-

is that realized at a

= 3 i ,

-/c 4 1

viz.

The behavior

of 0-

as a f unction

of a

is shown in

TC y

the bottom half of

_figure

2.

If we had overlooked the

_possibility

of

_using

the "

_improved

"

ordering

of the first

_kind,

and

so

_{employed (18)}

rather than

_(19),

the maximum value of

r8

would have been _7, achieved

at 7

~’c V

-Experimental

values

of £ L

for face-centered

antiferromagnetic

materials,

takes

_mainly

from the work of

Bizette,

are as f ollows :

In all cases the ratio

e

falls between the

theore-TC

tical limits 1. o

and 5. o,

_{though barely}

so in the

case of MnO.

3. Relation between

_{Susceptibility}

at T = _o and T =

Tc.

- A characteristic feature of

anti-ferromagnetic

substances is that below the Curie

point

their

_{susceptibilities}

increase

_slowly

with

temperature,

whereas above this

_point

there is

instead a

_decrease,

_governed

_by

a law of the form

_(16).

In

_calculating

the

_{susceptibility}

below the Curie

point,

allowance must be made for the fact that

the

_applied

field is in

_general

not

_parallel

or

perpen-dicular to the direction of

_alignment

of the

elemen-tary

magnets.

The

_{simplest assumption}

to make

is that the

_spins

of all the various sublattices are

all

_aligned

_parallel

or

_antiparallel

to the same

direction. This

direction,

however,

presumably

varies from one domain to

another,

even

_though

the same for all the sublattices within a domain.

The actual

_{susceptibility}

is then

where Z//

and are

_respectively

the

_{susceptibilities}

of a domain for the

_applied

field

_parallel

_and

267

we will omit the details of the calculation of

_z~,,

and Z i ,

as the

_principles

involved therein are

_quite

straightforward.

Bitter and the writer

_[8]

have

calculatted A.¡¡

for a substance classifiable into two

sublattices,

i. e.

essentially

for

ordering

of the

first kind for a

_simple

or

_{body-centered}

cubic lattice. The basic idea is that one takes as the

argument

of the Brillouin function the sum, or

difference,

depending

on the

sublattice,

of the

applied

field H and the molecular

field,

and

_expands

in a

_Taylor’s

series in H

_through

the linear term.

The most

_important

result is

_{that Z,,}

vanishes

at T =

o, and this feature also holds for the other

models which

we have

_presented

in the two

proce-ding

sections,

as it is an

expression

of the fact that

the work

_required

to turn over a

_{spin against}

the

molecular field is

_{infinitely large}

relative to k T

at T = o. The calculation

_{et X 1}

is

_particularly

interesting,

and was first

_{given by}

Neel

[1].

It is

quite elementary

and consists in

_noting

that

_spins

originally

directed east and west are deflected

_slightly

towards the north

if,

say, this is the direction of

the

_applied

field. The amount of the deflection 10 is determined

_by

the

_requirement

that the total

torque

exerted on a

_{spin equal}

zero. The

_torque

consists of thwo terms, that exerted

_by

the

_applied

field,

and that

_arising

from the fact that after

deflec-tion,

originally

antiparallel spins

make

_{angles 2 80}

with each other. There is no

_torque

between

origi-nally

parallel

spins,

since

_they

remain so even

when deflected. Néel and the writer considered

only

the two sublattice

model,

but we have verified

with this kind of calculation that the relation = _I X.yc

is valid for all the

_types

of model which we considered in the two

_preceding

sections. Without even

_going

through

the details of the calculation it is

_fairly

clear

_{that Z,}

is

_independent

of

_temperature,

for

the

_{susceptibility}

is

_proportional

to

MA60,

where M., is the mean

_{magnetization}

of a sublattice. The

torque

exerted

_by

the

_{nearly antiparallel}

sublattices is

_proportional

to the molecular field and hence

to _{MA. Thus 08} is

_inversely

_proportional

to M, and so the

_{temperature dependent}

element in the

calculation,

cancels out . If we

accept

the result that there is no

discontinuity

in the

susceptibility

at the Curie

_point,

one then has

_{Z I -}

Summarizing

the results of the

_preceding

para-graph,

we have

where X0 denotes X

at T = _o. The

prediction

that the

_{susceptibility}

at T = o be two-thirds of

that at T =

Tc

was first made

_by

the writer

_[8],

and holds

_reaonably

well

experimentally,

as the

following

experimental

values show

Anderson, however,

casts doubts on the

theore-tical

_validity

of

₍₂₁₎

_except

for the two sublattice

model considered at the

_beginning

of section i.

In all other cases there are sublattices which are

uncorrelated with each

other,

and there is a

priori

no reason

_why

the molecular fields of the

uncorre-lated sublattices should be

_{aligned along}

the same

direction. Instead

_they

can be assumed at _{say goo}

relative to each

other,

without

_changing

_{the energy} of interaction. The

_{susceptibility}

_is,

_however,

altered.

If,

for

_instance,

there are two sublattices whese molecular fields are made

_{perpendicular}

rather than

_parallel

or

_{antiparallel,}

there ceases

to be a direction for

_{which /}

= o at T = o. One’s

first reaction is that the

_powder

_{susceptibility}

is unaffected because the rotation

_through

_goo

dimi-nishes the

_{susceptibility}

_along

another axis.

Howe-ver, this is not the case, for the

_{susceptibility}

in a

given

direction is not a linear function of the number

of sublattices

_{aligned perpendicular}

to

it,

inasmuch

as the more the

sublattices,

the

_higher

the

_resisting

torque,

and the lower the rotation or

_{susceptibility}

per sublattice. As a result the

_powder

suscepti-bility

is

increased,

and the ratio of the

_{susceptibility}

at T = o to that at T =

Tc

_can become

greater

than

_2/3,

as the

_following

table shows. Here the

third,

fourth,

and fifth columns

_give

_respectively

the sublattices whose

_spins

are

_aligned

_along

the x,

y, and z axes. The sixth column

_gives

_the

suscep-tibilities at T = _o

along

the _x and y _axes, which

are

_equivalent,

while the seventh column

_gives

the

_{susceptibility}

at T = o in the z direction. The

final column

_gives

the ratio of the

_powder

suscep-tibility

at T = o to that at T = Tc.

In no case does the ratio of the

_{susceptibility}

at T = o to that at T =

Tc

exceed

unity,

when account is taken of the

_permitted

_{range of values}

f’l..

h

_dl(’

I

I

3

of J

_appropriate

to each model

_(viz.

>

1, 1 ,

-1

7 2 2 ₄

3

> 3 ,

for the four rows of the

_table).

In

_particular,

4

when a

_{becomes very}

large,

amplying ordering

of I

the second

kind, #

_{approaches 2/3.}

_However,

ZTc

at the critical

_point

of transition from the

_improved

first to second kind of

_ordering,

which seems to be

the

_point

_appropriate

to

lfln 0,

the value of

#

zTe

furnished

_by

the table is o . _97, and

generally speaking,

for reasonable

_magnitudes

of

_a

it

_gives

values of

~=~

nearer

_unity

than

_2/3.

The fact that the

experimental

values are

usually considerably

closer

to

_{2 /3}

than 1. o,

presumably

means that there is

some _energy of

_{anisotropy causing}

all the molecular

fields in a

_given

domain to have the same direction

except

for

_sign.

4.

_Experiments

on Neutron Diffraction.

-The

_{theory developed}

in sections I and 2 does not

decide whether the

_ordering

is of the first or second

kind,

inasmuch as a

_given

value of

T-1

within the

TC

allowed

limits,

can be realized with either kind of

order. In other words in

_figure

2 a

_given

value

of r8

can be achieved with a value

of x

lying

either

c ,

to the left or

_right

of that

_{corresponding}

to the

_peak

of

r8 .

_FeO,

which

has £

_{== 2,9} 3,

requires

a

c c

small

_negative

value

of *

if the

_ordering

is of Y

type

z

.

Experiments

on neutron

diffraction,

howe-ver, show

unequivocably

that the

ordering

is of the

second kind in the materials so far examined.

First it is

_perhaps

well to _say a word or two as

to the nature of these

_experiments.

The inter-ference is detected between the

_{magnetic scattering}

of different atoms in a

_crystal.

For such

measu-rements to be

successful,

it _{is necessary} that the neutron beam be

_highly

monochromatic. An ordi-nary source of slow neutrons will have far too

_great

a

_dispersion

of velocities to

_permit

detection of

the interference. It is therefore necessary to

"

monochromatize "

the beam

_{by taking only}

the neutrons that are scattered at a

_{given angle by}

some

_crystal

not

_necessarily

_magnetic,

which scatters neutrons

_strongly

_(e.g.

_NaCI).

Then this beam

is in turn sent

_through

a second

_crystal

made of the

_{antiferromagnetic}

material to be studied.

Because of the

_great

reduction in

_intensity

because

the

_experiment

is

_essentially

a double

_scattering

one, it is necessary to use an

_{extremely powerful}

neutron source, such as is obtainable

_only

from a

fission

_pile.

The

_experiments

which we shall discuss were

_performed

at Oak

_{Ridge, by}

Shull,

Smart,

and Wollan

_[9].

The monochromatized

neutrons which

_they

used had de

_Broglie

wave-lengths

1 . 06 A and hence velocities

_{approximately}

three times thermal. Such diffraction

_experiments

are not to be confused with

_ordinary

ones on the

paramagnetic

scattering

of

neutrons,

in which the

scattering

is

_mainly

inelastic and

_incoherent,

and for which sources of unusual

_intensity

are not

required.

The

_scattering

of neutrons

_by

short _range nuclear forces is

_comparable

with,

or often somewhat

_greater

than the

_scattering

of neutrons due to the interaction of the neutron

_spin

with the atomic

_magnetic

moment, ‘for which the

_theory

has been

_developed

by Halpern

and Johnson

_[10].

The

_phase

of the

magneticly

scattered

_part

of the wave

depends

on

the orientation of the atomic

_{spin. Consequently}

if there is no correlation between the

_alignments

of the

_spins

of different atoms, i.e. no

_magnetic

ordering,

all interference effects will arise from

purely

nuclear

_scattering.

The first

_experiments

were

_performed

on

_MnO,

_{IVInF2, lVInS03,}

and

_aFe2O3

(haematite).

Of

these,

the ones on MnO are the most

interesting,

as

_they

_comprised

_temperatures

both

below and above the Curie

_point.

It was found that

in MnO

_(a)

well above the Curie

_point

there are no

magnetic

interference effects

_(b)

below the Curie

point

there are additional

_peaks

which have double

the

_spacing

of the unit cell. Therefore the

_scattering

data demonstrate, since

_only

with this

_variety

do

parallel

spins

have a

_{longer period}

than the

lattice,

beyond

a

_peradventure

that the

_{magnetic ordering}

is of the second kind

_{(c f .}

I or

_3).

An additional

feature of the

_experiments

on MnO is that at

tempe-ratures

_moderately

above the Curie

point,

the extra

peaks

do not

_{disappear completely,}

but instead

persist

in a _{very diffuse form. This is} the same

by

liquids,

and indicates that even above the Curie

_point

there is a limited amount of short

range order of the

_{spins, although}

the true

_long

range order has

_disappeared.

It was, in

fact,

the

magnetic

interference

_{corresponding}

to the

_liquid

rather than solid

_type

of order which was first

detected

_historically

in the brilliant work of Shull and Smart. No

_magnetic

diffraction effects have

been detected for

_MnS04

or

_MnF2,

as is to be

expec-ted,

since

_MnSO4

is not

_{antiferromagnetic}

_(except

perhaps

with an

_extremely

low

_Tc)

and since the

experiments

on

_MnF~

have not

_yet

been extended to low

_enough

_temperatures

to embrace the

_vicinity

of the Curie

_point,

which is

_quite

low for this subs-tance

_{(720 K).}

However,

such measurements are

promised

in the near

_future,

and should be

_quite

interesting

since

_{lVlnF 2}

seams to 1)e one of the best

examples

of the

_{rudimentary theory}

of section 1.

Fe203

has a rhombohedral structure rather more

complicated

than the _{cubic arrays which} we have

considered; however,

the

_magnetic

diffraction data reveal

_again

the existence of

_{antiferromagnetic}

ordering essentially

of the second kind.

Shull and collaborators have also detected the

magnetic

interference in

_{MnSe, NiO, FeO,}

and CoO. All of these substances

_belong

to the face-centered cubic

_system.

_They

all show the double lattice

constant in the diffraction

_experiments

and so have

ordering

of the second kind.

From the form factor revealed

_by

the

_magnetic

diffraction

_data,

it is even

_possible

to deduce infor-mation

_concerning

the

_charge

distribution of the d electrons. The distribution thus _{obtained agrees}

only moderately

well with theoretical calculations

by

Dancoff. The

_degree

of accord

is, however,

adequate

since both

_{experimentally}

and

theore-tically high precision

is difficult.

- The

subject

of neutron diffraction in

magnetic

materials is still in its

_infancy,

but it _{should prove}

the method par excellence of

studying magnetic

order. Measurements can be made on

_{ferromagnetic}

rather than

_{antiferromagnetic}

materials. It should

presumably ultimately

be

_possible

to secure diffrac-tion data on

_Fe,041

_ZnOFe,O,,

_etc., substances which

are

_{particularly interesting}

because of

N6el’s,

theory

[2]

of theirs

_{ferrimagnetism}

".

Unfor-tunately,

the nuclear cross sections of Fe and Ni

for neutrons are rather

_high,

so that

_magnetic

effects in substances

_{containing ferromagnetic}

iron

or nickel are rather difficult to

_{perform, especially}

since in the

_{ferromagnetic}

_state, the mean atomic

spin

at saturation is rather small

_(-

o . 3 in

_Ni,

rv i . i in

Fe),

to be contrasted with the "

_spin

only "

value S 5 for the ion MnO etc.

Cobalt is somewhat more favorable because of its

smaller nuclear cross section.

A rather

_interesting

result

_already

_yielded

_by

experiments

on neutron diffraction is that the

magnetic scattering

powers of the Fe and Co atoms

in FeCO are

_equal.

This fact shows that the Fe

and Co aoms share

_equally

their

_unpaired

electrons,

and so have the same mean atomic

_spin.

Such a

behavior is of the

_type

_{suggested by}

Stoner’s

" _{collective electron}

ferromagnetism

",

but it

is,

of course,

possible

to have a model without

conduc-tion bands which still has the same

_spin

on all

atoms.

Analysis

of neutron diffraction can

_yield

_magnetic

information not obtainable in any other way. The diffraction

_experiments,

for

instance,

show

_beyond

a doubt that NiO is

antiferromagnetic,

a conclusion

not

_altogether

clear from

_{existing susceptibility}

measurements since the latter show

_only

a small

increase

_{of Z}

with T and do not extend to

_high

enough

temperatures

to reveal a maximum _{in Z.} A

_{particularly interesting}

result,

obtainable

_only

by

the diffraction

method,

pertains

to the

_alignment

of the molecular field relative to the

_principal

crystallographic

axes. From

_study

of the

scattering

patterns,

Shull and collaborators find that in

_MnO,

MnSe,

and

NiO,

the

_spins

are

_{aligned along}

100 _axes,

whereas in FeO

_they

are directed

_along

111. A

theoretical basis for this difference in behavior is at

_{present wanting,}

as the models which we have

discussed include

_{only coupling}

of the

_ordinary

Heisenberg exchange

type

and so are

_isotropic.

As

the diffraction measurements have so far been made

on

_powders

rather than

_{single crystals, they}

do not

decide whether uncorrelated sublattices have the

spins aligned

along

the same direction or

_merely

along

crystallographically equivalent

axes. For

instance,

in MnO all four sublattices could have their molecular fields

along

100, or

_{alternatively}

say one

_along

I oo, one

_along

010, and two

along

The measurements therefore do not decide between the model used at the

_beginning

of section 3 that gave

zo

= 2

and the models introduced later that

ZTc "

had molecular fields at

_{right angles}

and that

_yielded

values of

~L°

almost

_equal

to

_unity.

In

_FeO,

the

’ATc

molecular fields of uncorrelated sublattices cannot

make

_{right angles}

with each

other,

but instead of

_having

the same

_direction,

can

_{alternatively}

make tetrahedral

angles

108° with each other. The value of "/.0 for the Io8a model does not differ

X?c

materially

from that which we tabulated for the

goO

one.

An

_{interesting development}

included in the latest work at Oak

_Ridge

is a

_study

of the diffraction

in _{the presence of} a

_powerful

constant

_magnetic

field. Such a field can alter the

_{spin alignments}

and so

_change

the interference

_pattern.

Additional information and further tests of the

_theory

are

270

work of Li indicates that in a _very

_powerful

field

the

_type

of

_ordering

_{may be}

_{completely changed.}

5.

_{Superexchange. -}

The evidence from neutron

diffraction shows _very

_clearly

that in the face-centered

_lattices,

the

_coupling

of next-to-nearest

spins

is

_actually

more

_important

than that between

those which are immediate

neighbors.

It is

_generally

agreed

that in substances like

MnO,

the

_coupling

between the

_spins

of Mn atoms, with the

_possible

exception

of nearest

_neighbors,

is via excited states,

with the 0 atom

_playing

the role of an

interme-diary.

Such a _process, which was first

_proposed

_by

Kramers

_[11],

is

_commonly

called

_{superexchange,}

although

the term "indirect

_exchange"

would

_perhaps

be more

_appropriate.

Now the shortest

path

via

an 0 atom between two Mn atoms which are nearest

neighbors

is via two

_segments

which make a

_right

angle

at the 0 atom, whereas that between two Mn

atoms which are next-to-nearest

_neighbors

is a

straight

line

_piercing

an 0 atom, as reference to

figure

3 will show. In

_figure

3, the

_positions

of

the 0 atoms are not

shown,

but

_they

alternate with the Mn atoms, and so _{occupy all the} unmarked

line-interactions. The inordinate

_importance

of the

_coupling

between next-to-nearest

_neighbors

must

mean that somehow the interaction between Mn atoms is

_stronger

when the bonds

_joining

them to

the

_intervening

atom are

collinear,

as in

_Mn-O-Mn,

Mn

I

rather than at

_right

_angles,

as in

Mn--0,

even

though

in the latter structure the

_separation

of

the two Mn atoms is smaller in the ratio

I .

The

V2

problem

is to show that

_collinearity

accentuates

superexchange,

and _{makes up} for the

_handicap

in distance. Anderson

_[12]

has shown

mathema-tically,

in _{another paper from that}

_quoted

earlier,

that this is indeed the case. His calculation has

contributed

_materially

to the

_{understanding}

of the

mechanism

_by

which

_{superexchange}

works.

.

We will first outline his

_theory

_{descriptively,}

and

reserve mathematical details for section 6. The

model which is used is in its

_simplest

form a four electron one. The

_ground

state of this

_system

is

taken as follows : one electron on an Mn atom in a

state say

d.,

one electron an a state, denoted

_by

_d2

of an Mn atom which is a next-to-nearest

_neighbor

of the atom

_containing

_d.,

and two electrons in

identical p orbits of the 0 atom which have the axis of the p dumb-bell coincident with the Mn-Mn

axis. The excited state is one in which one electron

has _gone over from the 0 atom to one of the Mn atoms,

as shown in the

_following

sketch.

The calculation is

_essentially

a determination of the

perturbation

or , contamination "

of the

_ground

state

_by

the admixture of a certain amount of the

excited state.

The

computation

shows that when allowance is made for this

effect,

the _{energy of} the

ground

state

_depends

on the relative

_alignment

of the

_spins

of the states

_d1,

_d’i

_provided

in the excited state, the

_{exchange integral}

is

_appreciable

between electrons p and

_d2,

and

_provided

also the

energy is different in the excited state when the

spins

of electrons

_d1,

_{d i}

are

_parallel

than when

_they

are

_{antiparallel.}

This result is

_probably

clearest when

_displayed

as a formula.

_~~11,

etc., denote wave

functions whose

_significance

is indicated

_by

the sketch. Let p and J denote

respectively

the

one-electron

_hopping

"

(our

,

resonance

_")

and two-electron

_{exchange " integrals}

where H is the Hamiltonian

function,

already

supposed

integrated

over all coordinates other

than those involved in the

_integral.

Then the

perturbation

calculation shows that the

_disturbing

influence of the excited state

_gives

a

_coupling

Here

_{h v~ .~.}

denotes the

_promotion

_{energy in}

_taking

the electron from the p

configuration

to the state

(d,,

in which the

_spins

of

d1

and

_d’1

are

_parallel,

and denotes a similar

_promotion

_energy to

a state

_(d,,

_d’i)2

in which instead these

_spins

are

antiparallel.

At this

_stage,

one is

_tempted

to ask

_why

this rather recondite effect favors next-to-nearest rater than nearest

_neighbors.

The answer is that the _p

dumb-bell which is left with a free

spin

because

of the transfer of an electron to an Mn atom is one

which

_overlaps

the Mn atom which is

diametrically

opposite

from the first Mn atom. One

could,

of course, in

principle,

imagine

the second Mn atom

as one which is a nearest

_neighbor

of the first one.

Such an atom is indicated

_by

the cross above the

diagram.

However,

the interaction will then be

much

smaller,

because the dumb-bell

_shaped

_p

wave functions have most of their

_charge

distri-bution concentrated

_along

the axis of the

dumb-bell,

and so do not

_overlap

_appreciably

_any atom

reached

_by

a bond at

_right

_angles.

_Actually

some

superexchange

between nearest

_neighbors,

weaker

than that between next-to-nearest one, may result from

_{hybridization}

of 2s

_{and 2p}

wave

_functions,

which

perturbations by

excited states in which an electron is raised to a 3

_quantum

orbit of 0 rather than transferred to an Mn atom.

_However,

the

_high

promotional

energy to the 3

_quantum

_states

pre-sumably

makes the latter effect weak. Of course

there are _{other 2p} orbitals on the 0 atom which

point

in other directions than those we

_considered,

but their electrons are unaffected

_by

the

_particular

transfer we are

_considering,

and so remain

_paired,

and thus

_incapable

of

_excerting

_any

_aligning

influences. The

_gist

of Anderson’s model is that the removal of one member of an electron

_pair

in

a

_given

_p state to an Mn atom at one end of the dumb-bell leaves a

_spin

which is free to

_couple

witch the Mn atom at the other end of the dumb-bell.

It

_gives

a _{very clear-cut} reason

_why

the

_coupling

between next to-nearest

_neighbors

should be so

abnormally important.

One

_point

should be

_particularly

noted. The

superexchange

effect will not exist unless the two

frequency

denominators and

_{h v.~ ~}

_entering

in

₍₂₃₎

are different. This

_point

is not

_brought

out in _{Kramers paper, and} means that in the case

that

_{hv~~ -}

_I

_I

_{superexchange}

is an extra order

_higher

than one would

_gather

from

reading

his _paper.

We have assumed that an electron is transferred from the 0 atom to Mn atom i in the sketch. It

is,

of _course,

_equally

reasonable that it be transferred

to atom 2. This

possibility

has the effect of

making

the

_coupling

coefficient

_actually

twice as

_large

as

given

in

_(23).

The

_question

now arises as to whether

₍₂₃₎

_gives

superexchange

of a reasonable order of

_magnitude.

It does. The observed amount of

_{superexchange,}

A N

io cm-i,

can, for

instance,

be obtained

_by

assuming,

Especially

remarkable is the

_prediction

of the

theory

as

_regards

_sign.

At first

_glance,

the fact that

the

_triplet

state of a two-electron atomic

_system

is

lower than the

_singlet

seems to demand that

_always

; h,y~ C

I

in

_{(23). Actually,}

_however,

the

Mn atom has five rather than one d electron before the supernumerary electron is transferred from the

0 atom to it. The state of _{lowest energy} for d5 is a

quintet

(5S),

while that for d6 is a

_{quartet (4D).}

As

a

_result,

when an extra electron is shifted to

_it,

the lowest energy will be achieved when the

_spin

of this

electron is

_antiparallel

to the

_original

_spin

of Mn.

Hence if we select one

_original

electron of Mn and the added electron as a

_{typical pair}

as is done in our

skeleton

_perturbation

calculation,

we _may

_expect

the _energy to be lower when their

_spins

are

anti-parallel

rather than

_parallel.

This conclusion also

holds,

if instead of

_being

Mn,

the

_paramagnetic

ion is any one

having

its

_original

d shell half or more

complete.

On the other

hand,

if the d shell of the

paramagnetic

ion is less than half

_complete,

the

energy should be lowest when the

spin

of the added

electron is

_parallel

to that of the

_original

electrons.

Thus for

_compounds

_having

_paramagnetic

ions

with d shells less than half

_complete,

one should

expect

the factor

[1 - 1

]

in

₍₂₃₎

-

h2v2-- h2

v+-"*

and hence the

_{superexchange}

_coupling

to have

opposite sign

behavior than obtains when the d

shell is half or more

_complete.

This

_prediction

is in

_striking

accord with

_experiment.

CrTe,

whose

_paramagnetic

constituant is the Cr++ ion in

the d"

_{configuration}

is known to be

_{ferromagnetic.}

Granted that CrTe is

_{ferromagnetic,}

the

_theory

predicts

that MnTe should be

_{antiferromagnetic,}

as is indeed observed

experimentally.

Also

MnO,

FeO,

CoO, NiO,

which

_presumably

should behave like

MnTe,

are all found to be

_{antiferromagnetic}

(No

data are available on

_CrO).

Furthermore

_VC’2

and

_CrCl2

are

_{ferromagnetic,}

while

_MnCl2

and

FeCl2

are

_{antiferromagnetic. ---}

another confirmation of

the

_theory.

_(The

Te and Cl atoms have

_respectively

5p

and

_3p

_{rather than 2p orbital like}

0,

but this difference has no

_bearing

on directional

_properties).

The fact that

_actually

MnTe or MnO is

antiferro-magnetic,

and CrTe

_{ferromagnetic,}

rather than vice versa, shows that the

exchange integral

J involved

in . (23)

is

_negative,

i.e. that the Te-Mn or O-Mn

exchange coupling

effects are

_{antiferromagnetic.}

This behavior is

reasonable,

for

_normally

chemical bonds favor closed electron

_pairs.

Another

_interesting

_{consequence of} the

_theory

is that as the

_{electronegativity}

of the anion

decreases,

there should be a decrease in the ionic character

of the

_material,

and hence a

_{larger integral}

_{p in}

₍₂₂₎

and

_(23),

as it becomes easier to transfer an electron

to the

_paramagnetic

cation. The

_{superexchange}

is then

_{correspondingly}

increased. If the interac-tions between nearest

_neighbors

are caused

_mainly

by ordinary exchange,

and those between

next-to-nearest ones

_{by superexchange,}

the

ratio a

should

, i

increase with

decreasing

electronegativity.

Hence the ratio

0

should increase or decrease when the

TC

electronegativity

decreases,

according

as the left

or

_right

side of

_{figure 2}

is

applicable,

i. e.

_according

as the

_ordering

is of the first or second kind.

Actually 0

decreases,

since

MnO, MnS,

MnSe,

a

TC

sequence

decreasingly electronegative,

have

respec-tively

a = 5 . o,

3 . z, 3 . o. Hence even without

TC

the data on neutron

diffraction,

there is indirect

evidence that the

_ordering

is of the second kind.

6. Derivation of

_Equation

_(23).

- The

present