Collective electron ferromagnetism in metals and alloys

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Submitted on 1 Jan 1951

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Collective electron ferromagnetism in metals and alloys

E.C. Stoner

To cite this version:

COLLECTIVE ELECTRON FERROMAGNETISM IN METALS AND ALLOYS

By

E. C. STONER,

Professor of Theoretical

_{Physics, University}

of Leeds

_(England).

Sommaire. - Plusieurs

rapports généraux ayant paru, récemment sur les diverses méthodes

théo-riques pour aborder le problème du _{ferromagnétisme,} on _s’occupe ici seulement de la méthode des bandes, comme elle a été développée à Leeds. On a _{pu étendre les travaux antérieurs} sur les _propriétés

magnétiques des métaux _{(Stoner, depuis 1933),} pour embrasser avec _précision les variations à toutes les _{températures, grace} au calcul des _intégrales de Fermi-Dirac (Mc Dougall et Stoner, 1938). Une théorie détaillée du _{ferromagnétisme} s’ensuivit, avec une _{comparaison préliminaire} entre les résultats

théoriques et _{expérimentaux (Stoner, 1938-1939).} Plus récemment, Wohlfarth _(1949-1950) a donné des _{comparaisons plus étendues,} et aussi des _{développements} ultérieurs de la théorie.

Toute théorie du ferromagnétisme se fonde _{provisoirement} sur des prémisses simplifiées, et il faut contrôler sur _{chaque point} les conclusions par des expériences si l’on veut mieux comprendre ces

phénomènes complexes. Ceci demande, de la part du théoricien, une appréciation critique des travaux

expérimentaux, et souvent une _{analyse soigneuse} et _pénible des résultats. Il faut aussi que

l’expéri-mentateur soit au courant des vues _{théoriques pour pouvoir établir efficacement} le _plan de ses

expé-riences et _présenter convenablement les résultats.

Les prémisses du traitement fondamental du _problème sont que les électrons effectifs (ou les _trous) sont dans une bande de forme _parabolique, que l’énergie d’échange est _{proportionnelle} au carré de l’aimantation et que la distribution des électrons, qui dépend de la _température et du _champ

magné-tique, doit être déterminée _par application de la _statistique Fermi-Dirac. Les _paramètres

caractéris-tiques sont le nombre des électrons effectifs dans la bande _(qui ne _{correspond pas,} en _général, à un

nombre entier de _{spins par atome,} à cause du recouvrement des _{bandes), l’énergie} de _{Fermi, 03B50,} et la valeur de _k03B8’, mesure de _{l’énergie d’échange.} On peut calculer l’aimantation relative, 03B6 (rapport du nombre des spins parallèles au nombre _total), comme fonction de H et de T pour toute valeur de

k03B8’/03B50.

Dans la limite

03B50/k03B8’

~ 0, les courbes _{correspondent à} celles _qu’on obtient dans la théorie de Weiss. Pour

k03B8/03B5’0 > 2/3

une aimantation spontanée survient au-dessous d’une _température

03B8(03B8/03B8’ 1,

sauf dans la limite

classique)

_qui est le point de Curie. La saturation est incomplète même au zéro absolu

pour k03B8’/03B50

0,794. On donne des courbes d’aimantation spontanée au-dessous de 03B8 et de _{susceptibilité} au-dessus de 03B8 pour une série de valeurs

de k03B8’/03B50.

Pour le nickel, avec _{à peu près} o,6 trous par atome dans la bande d, on obtient une coordination

remarquable des résultats _{expérimentaux pour la variation de la susceptibilité} et de l’aimantation

spontanée et aussi pour la chaleur spécifique électronique aux basses et à hautes _{températures.} On ne

doit pas s’attendre à un accord complet, parce que les valeurs de k03B8’ et de 03B50 ne _{sont pas} constantes, mais varient avec la _{température,} à cause de la dilatation _thermique. Une anomalie _apparente, le renversement de la courbure des courbes

(I/X,

T)

à hautes températures, peut s’expliquer par le transfert calculable d’électrons de

la

bande d à la bande s. Les écarts de la forme théorique des courbes _{expérimentales} pour la variation de l’aimantation spontanée avec la _{température peuvent} être attribués provisoirement à ce _{que l’énergie d’échange} n’est _pas strictement proportionnelle au

carré de l’aimantation,

Par un _{developpement} naturel de la théorie, on _{peut traiter,} d’une manière semblable, un _grand

nombre _d’alliages du nickel, et l’accord avec les _expériences est satisfaisant. Pour de grands domaines de _composition de plusieurs alliages, la saturation au zéro absolu est _incomplète, ce _qui arrive aussi, presque certainement, pour le fer. Pour le fer et le cobalt, pourtant, l’approximation d’une bande

parabolique ne _{peut pas être bonne. On est} en train de faire une _analyse des effets des diverses formes de bandes.

On donne une brève indication des travaux en cours sur des problèmes plus fondamentaux relatifs

aux _énergies d’échange et de corrélation, sur l’effet _{magnétocalorique} dans les _champs forts et sur

les propriétés électriques des métaux ferromagnétiques et autres métaux.

JOURNAL PHYSIQUE

1951,

1. Introduction. ---

Although

it has been

sug-gested

that

_participants

in this conference should survey

generally

the

present

position

in the field

on which

they

are

_reporting,

it would serve no

useful purpose for me to cover more

_hurriedly

precisely

the same

_ground

as do the recent

compre-hensive and

_general

reviews of the

_theory

of

ferro-magnetism by

Van Vleck

_[1945,

see also

_{1947, 1949]}

and

by myself [Stoner, 1948].

Further,

one

_aspect

of the

_subject

will be

_{presented by}

Van Vleck himself with far more

_authority

than I could claim. There are two main

_approaches

to the theoretical

interpretation

of the fundamental

_phenomena

of

ferromagnetism,

one

_deriving

from the

Heitler-London treatment of

_molecules,

_starting

from atomic

wave

functions,

the other

_deriving

from the

elec-tron _energy band treatment of

metals,

which has a

close

_analogy

with the Hund-Mulliken treatment of

molecules,

using

molecular orbitals. It is the second

approach

with which I shall

deal,

and in

_particular

with the theoretical work at Leeds on what I have ventured to call ((collective electron

_{ferromagnetism».}

The

_general

outlook

_underlying

the Leeds work is outlined in

_paragraph

2,

but it is

_appropriate

to

indicate

first,

very

briefly,

the historical course of

development.

In an

_early

_paper

_[Stoner,

1933]

it was shown that with the band treatment a

_satisfying

qualitative explanation

could at once be

_given

of the

occurrence of

_non-integral

_magneton

values for the

saturation moments _per atom in the

_{ferromagnetic}

metals,

and of the variation with

_composition

of the

mean moment _per atom in certain

_{ferromagnetic}

alloys.

The

_original

tentative

_{interpretation}

could be made more definite

_only

in the

_light

of later

work on the _energy band structure. A

_precise

calculation of the forms of the _{energy bands in}

ferromagnetic

metals

_presents

_very

_great

diffi-culties,

but it is

_fairly

certain that in nickel a narrow

d

_band,

_{approximately parabolic}

at its _upper

_end,

is

_{overlapped by}

a much wider s band

_(see

_fig.

i

below).

This was shown

semi-quantitatively by

Slater

_{[1936 a]}

on the basis of calculations for copper

by

Krutter

[1935],

so

_confirming

an earlier

brilliant

_{suggestion by}

Mott

_[1935].

Rather

_loosely,

the

_{ferromagnetic properties may ’ be}

attributed to the " holes "

in the d

band,

approximately

o.6 per

atom. The theoretical treatment, of a collective

electron

_type,

_by

Slater

_[1936

a,

b]

of the

ferro-magnetism

of

nickel,

embodied all the essential

ideas,

but the methods of calculation were such

that

the results obtained were

_{presented only}

as

very

rough approximations.

In the

meantime,

applications

of Fermi-Dirac statistics had been made to free electron

suscepti-bility,

both dia-and

_{paramagnetic [Stoner, 1935],}

free electron

_specific

heat

_{[1936 a],}

and

_spin

para-magnetism

[1936 b],

and to the interrelation in metals between electronic

_specific

heat and

_spin

paramagnetism,

both with and without

exchange

interaction

_{[1936 c].}

This led

_naturally

to a

consi-deration of

_{ferromagnetism,}

but it was soon found

that

_rough

_{interpolation}

between the values

_given

for the relevant Fermi-Dirac

_{integrals by}

the

~~

high "

and , low

_temperature

"

series

_expressions

as used in this earlier

_work,

could not

_give

suffi-ciently

precise

or reliable values for the inter-mediate

_temperature

_{range which is of central}

physical

interest.

_{Comprehensive}

evaluations of the

required

Fermi-Dirac functions for the whole

tempe-rature _range were therefore made

_[McDougall

and

Stoner,

it38];

and I should like to state that the

_preparation

of these standard tables would

hardly

have been

undertaken,

and would

_certainly

not have been carried out with such

_{completeness,}

without the collaboration _{of my}

_colleague

Dr J.

lVlc Dougall.

These tables are basic to the

quanti-tative theoretical treatment of the

_properties

of electronic assemblies whether in metals or in stars

[e.

g.

Stoner, i g38

a,

b,

ig3g

a]

and in

_particular

are

essential to a

satisfactory

treatment of collective

electron

_{ferromagnetism.}

This

_subject

was treated

in two papers

[Stoner,

1938

c,

_ig3o

_{b] shortly}

before

the

War,

the first

_dealing

with the

_temperature

variation of intrinsic

_{magnetization}

below and above

the Curie

_point,

the second with the associated energy and

specific

heat.

_Except

for a short paper on the

_paramagnetic

_magneton

numbers

Qf

the

ferromagnetic

metals

_{[Stoner, 1938 d],}

_however,

no detailed

_comparison

with

_experiment

was made.

Work of this

_kind,

like so much of

_immensely

higher

value,

of

_necessity

_stops

in the face of the immediate

_imperatives

of total war, and it has not

been easy, in the

_precarious

_aftermath,

to

_pick

_up

the threads

_again.

But there is now, once more,

for how

_long

none can _say, a little

_opportunity

and a little time for a

university

scientist to _pursue

knowledge

and

_{understanding}

_freely

and as best

he can within his powers; not

_{indifferently}

to

mena-cing

conflicts,

not

_{irresponsibly,}

_ready

to do all that his

_{opportunities}

allow towards

_bringing

about

a saner

_world,

but conscious that this scientific

search,

despite

its

limitations

in _{scope, is}

_yet

a

thing

of value in

intself,

and not

_simply

a means

to a desirable or undesirable material end.

My

own active research interests in

_magnetism

have extended to the behaviour of

_{ferromagnetics}

in low and moderate

fields,

and

_problems

connected with domain structure and with

_{magneto-thermal}

effects have

_occupied

much _{of my available time.} I

_{have, however,}

been fortunate in

_having

as a

research

student,

and

later,

until very

recently,

as a research

assistant,

Dr E. P.

_Wohlfarth,

who,

under my initial

guidance,

has continued the work

on collective electron

_{ferromagnetism.}

Most of

very recent _{papers of} Wohlfarth I do not _propose to enter into the technical details of

the

work,

which are

_readily

available to those who

wish to

_study

_them,

but I want rather to indicate

the

_underlying

outlook and

_principles,

and then to

discuss very

briefly

the relation between the

conse-quences of the theoretical treatment and the expe-rimental

results,

so far as this has been examined.

2. General outlook. - The status of the collec-tive electron treatment as a "

theory "

of

ferro-magnetism

is so liable to

_{misunderstanding}

that I

am

_following

the unusual course of

_describing,

in this

section,

the

_general

outlook

_underlying

the Leeds work.

Theoretical. r- The theoretical

problem

of

ferro-magnetism might perhaps

be stated somewhat as

follows,

with reference to nickel

_purely

for defi-niteness in

_{exemplification :}

on the basis of the wide theoretical scheme of

_quantum

mechanics,

whose

validity,

over the relevant _range, must be

_accepted

as

established,

and

_given

a

fairly

_complete

and

_precise

knowledge

of the structure and

_properties

of free

nickel _atoms, it is to be shown that the lowest energy

state of an

_aggregate

of such atoms is one in which

they

form a

_crystal,

of a certain

_type

_{(face-centred}

cubic)

and lattice

_dimensions,

and whose

_properties

vary with

temperature

in a

_specific

_way; in

parti-cular,

that these

_crystals

are

_{spontaneously}

magne-tized at absolute zero to a

_{degree corresponding}

to

approximately

(but

not

_exactly)

0.6 Bohr

_magnetons

per

atom;

that the

spontaneous

magnetization

decreases in a definite way with

_temperature

to zero

at the Curie

_point;

that above the Curie

_point,

the

aggregate

is

_{paramagnetic,}

the

_{susceptibility varying}

in a

_particular

way with

temperature;

and that

there are thermal and other characteristics

inti-mately

connected with the

_magnetic

characteristics.

It would no doubt be

_possible,

with an

_adequate

supply

of

dots,

to set up

equations

whose

solutions,

it

_might

be

_stated,

would

_{give everything}

that was

sought.

Such a

procedure

would, however,

be

completely

unrevealing

and indeed

useless;

it is

only

necessary to consider the immense

difpiculty

of

_obtaining

a

_rigorous

solution of the

_hydrogen

molecule

_problem

to realise that the

_problem

of

ferromagnetism,

attacked in _{this way, would} be

completely

intractable.

The

_problem

must be

_enormously

delimited.

It _{may be} taken as

_given

that nickel atoms form a

crystal

of known lattice

_spacing

and that it is a

metal,

so that what is known

_generally

about the

electronic structure _{of metals may} be taken as a

basis. No exact calculation of the electronic band structure in nickel has been

made,

even if it could

be

made,

and the

_{simplest assumption}

consistent with available

_knowledge

may be

adopted.

The

interaction _energy with which

_{ferromagnetism}

is associated is

_only

a small

_part

of the

_total,

and is

unlikely

to be of

_importance

in

_determining

the band structure itself. The 1,

_{ferromagnetic}

inter-action " may, moreover, be

represented

by

a

" _{molecular field ". The manner in which this}

originates

as a result of

_exchange

interaction is

known in a

_general

_{way, but}

_rigorous

calculations of

the

_dependence

on

_{magnetization}

_present

formidable

difficulties. It is reasonable to make the tentative

assumption

that the associated _{energy is} propor-tional to the _{square of} the

_{magnetization,}

and to

examine the _consequences to which this leads.

Further,

it must be assumed that the distribution of electrons among available states,

dependent

on

temperature,

molecular

field,

and

_applied

field,

is determined

_by

Fermi-Dirac statistics.

_Finally,

it _may be

assumed,

in accordance with the

expe-rimental evidence on the

_gyromagnetic

_effect,

that

orbital moment

_plays

a minor

_part

in connection

with the

_primary

effects,

the

_important

_part

_{it may}

play

in various

_secondary

effects

_(e.

g.

magneto-crystalline anisotropy) being

left for

_subsequent,

or at least

_separate,

consideration.

The collective electron treatment of

ferromagne-tism,

when a

_parabolic

form is assumed for the

rele-vant electron

_band,

as in the basic

calculations,

may be

_regarded

_formally

as a

_replacement,

in the

_simplest

molecular field treatment

_adapted

to electron

_spins

as carriers of the

magnetic

_moment, of

Maxwell-Boltzmann

_by

Fermi-Dirac statistics. This

_greatly

complicates

the

_{calculations, and, indeed,}

it has been

_suggested

that the elaboration on the statistical

side is

_{disproportionate,}

in view of the

over-simpli-fication of the

_physical

situation in other

_premises

of the treatment.

This, however,

is

_completely

to

misunderstand the

_position.

It is

_just

because the

consequences of a

_rigorous

treatment of the 11

distri-butional "

side of the

_problem

are in detail so

_complex

and often

_{unexpected (although only}

one additional

parameter,

the band

width,

is

_introduced)

that it

is essential to

_keep

all other

_premises

as

simple

as

possible,

so that the

_logical

interconnection between

premises

and consequences shall be clear.

It turns out that the theoretical

scheme,

in its

simplest

form,

_gives

a wide coordination of both the

magnetic

and the associated thermal

_properties

of metallic

ferromagnetics.

Exact

_agreement

with

experiment (that

is for both these sets of

_properties

over the entire

_temperature

_range)

_{is, indeed,}

not

attained;

nor should it

be,

for it is known thet the

premises

are an

_{oversimplification.}

The character of the

_agreement

or

_disagreement

_may,

_however,

give

a valuable

_guide

as to the direction in which

Experimental.

- The aim of theoretical

physics

is to coordinate

_experimental

_results,

and in so

doing,

to ,

_{predict "}

others which

_might

be obtained

under

_prescribable

conditions. On the

experi-mental

side,

in the field of immediate

interest,

although

there are _many

_important

_{gaps, there is} a

wealth of reliable

_experimental

_information,

pro-vided,

in a

_{large degree,}

_by

the brilliant series of

investigations

carried out between the wars at

Strasbourg,

under the

_inspiration

of Pierre

Weiss,

whose name will be honoured as

_long

as there are men who are interested in

_magnetism.

Comparison

of the results of a ~~

_{theory "}

with

experiment

is

_by

no means so trivial or routine a

task as is sometimes

_supposed

In connection with

collective electron

_{ferromagnetism,}

it would be

desirable to have measurements of the

suscepti-bility

over a wide

_temperature

_range above the Curie

_point,

of the

_{magnetization}

at a series of

_high

fields at

_{suitably spaced}

_temperatures

between the Curie

_point

and absolute zero, and of the

specific

heat over the entire

_temperature

_{range for pure}

ferromagnetic

metals and _{for many}

_alloys.

Where extensive results are

available,

_they

are often due to a number of different

_{investigators}

who have made

measurements over different

_temperature

ranges

using

the diverse

_techniques

which are

appro-priate. Nominally

similar materials may differ

significantly

in

_purity,

and with

_alloys

_the

compo-sition, and,

hardly

less

_important,

the

_degree

of

homogeneity,

may be uncertain.

Unsuspected

secon-dary

eff ects may invalidate the

results,

and there

are

_always

the inevitable "

_experimental

errors " of

many kinds.

Again, experimental

results may be

presented,

often

_unavoidably,

in a form far from

convenient for

_comparison

with

_theory.

Essential information may be

unrecorded,

observations may be

_presented

for

_only

a few

arbitrarily

chosen and

badly spaced

temperatures,

or the _{results may be}

~~

economically

"

presented

in terms of one or two

values derived on the basis of some

_imperfect

theory. If,

for

_example,

interest is centred on

the

_{non-linearity}

of the relation

between %

and T X

above the Curie

_point,

it is

_baffling

to be

_{given only}

a Bohr

_magneton

value,

_pil,

_{corresponding}

to the

slope

of the 1, best "

straight

line that can be drawn

through

unrecorded

_{experimental points}

over an

arbitrarily

chosen

_temperature

_range.

The

_{intercomparison}

of

_experimental

results,

their

critical assessment and their reduction to a

conve-nient form calls for care and

_patience

and an

_ability

to

_appreciate

both

_experiment

and

_theory

of a kind

which tends to become

_atrophied

when

_physicists

confine themselves too

_narrowly

to

_imaginarily

closed _{domains proper} to the

_{experimentalist}

and to the theorist.

_Experimental

results must not

always

be taken at their face

value,

but where there

is

_unambiguous

_disagreement

between

_experiment

and

_theory,

it is

_experiment

which is

_right,

for it is the

_experimental

results which are to be

_explained

by

the

_theory.

The

_theory,

however,

is not

neces-sarily

wrong. The

disagreement

may be due to the sometimes enforced

_{oversimplification}

of the

premises

of a

_manageable

deductive

_argument.

It is

_suggested

that this is the

_position

with

_respect

to the collective electron treatment of

ferro-magnetism.

Once the

_{general applicability}

is

esta-blished,

as it has

been,

genuine

disagreement

between

_theory

and

_experiment

becomes much

more

_interesting

and

_exciting

than

_agreement,

for it is

_just

this which is the

_challenge

to the

discovery

of,

and the invention of a method of

treating, important

factors which may have been

insufficiently appreciated

or

_entirely

overlooked. 3. Statement of

_{premises. -}

In this section

the

_premises

of the basic treatment

_[Stoner,.

₁₉₃₈

_c]

are stated as

_simply

as

_possible,

with a few brief

comments.

(i)

The

_system

under consideration is a set

of N electrons in a

_partially

filled electronic _energy

band _{in which the energy}

_density

of states is

given bye

Fig. I. -

Diagrammatic representation of band structure of nickel near the Fermi limit.

where Eo

is the maximum electron _energy when the

electrons are in balanced

_pairs

in the lowest _energy states.

_{Alternatively,}

_{N may be} the number of

~~ _{holes " in}

a band

_{[c f .}

_Stoner,

_{I g36}

_c],

the energy

then

_being

measured from the

_top

of the band. The number N

_corresponds

_{to q}

electrons

_{(or holes)}

per atom, and q is not

_{necessarily integral.}

This

can arise from band

_overlap.

In nickel the value

of q

is

_{approximately}

o.6, this

_{corresponding}

to the number of holes in the d

band,

as illustrated in

figure

I...

The electrons are not to be

_thought

of as "

_{free ";}

the wave functions _{may be very}

_complicated,

incorporating,

for

_example,

some d-like "

charac-teristics. That the band is

_parabolic,

as for free

electrons,

is a

_simplifying

_{assumption (which,}

for

justification);

but the value of E,

(or,

correspon-dingly,

of the effective mass of the

_electrons,

or

holes)

may be

widely

different from that for free electrons.

(ii) Exchange

interaction

_gives

rise to a term

in the _energy

_{expression proportional}

to the _square

of the

_{magnetization,}

so that the _{energy of} an

electron with

_spin

_antiparallel

and

_parallel

to the

magnetization

may be written

where C

is the relative

_{magnetization i. e.}

T,

where M is the

_magnetic

moment of the

_aggregate

of N

electrons),

and 0’ is a

_parameter

with the dimensions of

_temperature,

and which with the

simple "

classical "

treatment would

_correspond

to the Curie

_temperature.

The

_equation

_(3.2)

is

equivalent

to the first

_{approximation given by}

most treatments of the

_exchange

interaction

_problem.

Reference is made later

_(§

₅₎

to the deficiencies of this

_{approximation.}

(iii)

The distribution of electrons in the band as

dependent

on the

_temperature,

molecular

_field,

and

_applied

_{field, H,}

is to be determined

_by

appli-cation of Fermi-Dirac statistics.

This

_completes

the statement of the

_premises.

It follows from

_(iii)

that-the

_expression

for the _energy distribution of electrons in the _{band may be}

_put

in the form

where E is taken to include the whole of the elec-tronic _energy

_except

that

_dependent

on the

magne-tization

_{[as given by}

eq.

(3.2)]

and on the external field. The

_{corresponding}

terms _{make up F,. An}

implicit equation

for the statistical

_parameter,

~,

related to the Gibbs free _{energy per}

_particle,

is

in which the two terms

_give

the numbers of

elec-trons with

_{spins parallel}

and

_antiparallel

to the

resultant

field;

and the

_{corresponding}

total

_magnetic

moment,

M,

and relative

_{magnetization,}

~, are

given

by

where

and

The Stoner-Mc

_Dougall

tables of the Fermi-Dirac functions

_{(I g38) give F3(n), F1(n),}

and derivatives

-1 -1

of

r,

_(-6)

for the _range -

₄

to

_{+ 20}

_{(corresponding,}

2

in the

_simple

case, to a _{range of -}

,

from

approxi-Eo

mately

o.o5 to

_12);

outside the _{range of} the tables

the functions are

_readily

evaluated from series.

The

_{equations (3. ~[)}

and

_{(3 . ~)}

are _{the basic}

equa-tions of the treatment. Their 11 _solution

",

which in

_general

can

_only

be effected

_numerically,

_involves,

formally,

the elimination of the statistical

para-meter -,0. The solutions must

finally

be

expressed

in a form convenient for

_comparison

with

experi-ment,

that

_is,

_essentially,

in tables or curves

_giving

the variation of the

_{magnetization}

with T and H for a sufficient series of values of the interaction and

band width energy

parameters

and Eo. For an

account of the

_manipulation

of the

_equations,

refe-rence must be made to the

_original

_papers. It must

suffice to _{say that the numerical}

work,

even for the

most unfavourable _{ranges, is}

_reasonably

straight-forward,

and that for some

_{problems (e.}

_g. the

variation of

_{susceptibility}

above the Curie

_point)

the

_equations

reduce to much

_simpler

forms which

are dealt with

_readily.

It _{may be noted that in}

the limit

B ( ~c ~ 0

/

--~ o the

_equations

reduce to those of the

_simple

classical treatment.

It _may be useful to include here the

_expression,

corresponding

to

_(3.4)

and

_(3.5),

for the total energy,

E,

which is basic to the treatment of the

energetic

characteristics,

and,

for

_{completeness,}

that for the _{free energy,}

F,

In the absence of

_exchange

interaction and of an external

field,

these

_expressions

reduce to the much

simpler "

standard " forms for the

_{thermodynamic}

functions with Fermi-Dirac

statistics,

which have been treated in detail

_[Stoner,

_ig3g

_a].

4. General character of the results of the

theoretical treatment. -

numerical results have been obtained for the varia-tion with

_temperature

of the

_{magnetic properties}

of the " idealized

_{ferromagnetic}

",

as described

above,

for a series of values of the

specifying

para-meters

and So;

results for the variation of the

electronic energy and

_specific

heat are less

_detailed,

but

_{sufficiently representative.}

It is to be

under-stood that the basic calculations

_apply

to N

elec-trons or holes

_(q

_per

_atom)

in a

_single

band of

stan-dard

_(parabolic)

form. No account is taken of the

possibility

of a

_temperature

variation of _{and E o.}

This almost

_certainly

occurs in actual

_{ferromagnetics,’}

with effects which are

_by

no means

negligible,

as a

concomitant of thermal

_expansion.

Another

impor-tant

effect,

the " transfer "

effect,

has been taken

into account in more refined calculations in

_special

cases, as mentioned below.

The results have been

_presented

in the various papers in the form of numerical

tables,

with a

degree

of accuracy more than

_adequate

as a basis

for a

_comparison

of

_theory

with

_experiment,

but which is essential if various

_{interesting points}

are

to be

_brought

out

_clearly.

It will be sufficient here

to indicate the

_general

character of the results

_by

diagrams,

with a few short comments. It should

be noted that it is

_{usually appropriate}

to

_{plot suitably}

reduced

_quantities,

so that each curve is a "

corres-ponding

state " curve

_(analagous

to the curve

obtained in the

_{Debye theory}

of

_specific

heat,

T

by

plotting

against £

rather than

_against

_T).

0d

Magnetic

characteristics. - A basic relation is

that

between I

and T for electron

_spin

parama-X

gnetism

with zero

_exchange

interaction. This is

shown,

in reduced

form,

in

_figure

2, curve

1,

the

corresponding

"

classical curve "

_being

the

straight-line 1’. The Fermi-Dirac curve

_approaches

the

clas-sical

_straight

line

_{asymptotically. Writing E0}

=

the value of

OF

would be of the order of 10’’ °

for

strictly

free

electrons;

for the holes in a d band it

may be of the order The limit

corres-ponds

to classical statistics.

The effect of

positive exchange

interaction is to

lower the reduced inverse

_{susceptibility}

curve

by

2013’

* If the lowered _curve cuts the T

axis,

the

E0

point

of intersection

_corresponds

to the Curie

tempe-rature, 0. The continuation of the

_{susceptibility}

curves below 0 has no

_{physical significance,}

as

spon-taneous

_{magnetization}

sets in at this

_temperature

and increases as the

_temperature

decreases. The

Curie

_temperature

0 is

_always

less than 0’

_(except

in the classical

_limit),

and a

_{positive exchange}

interaction does not

_necessarily

_give

rise to

ferro-magnetism.

A _{necessary condition} for

ferroma-gnetism (i.

e. the occurrence of

_spontaneous

magne-tization)

is

It _may also be shown that a condition for

_complete

parallelism

of the

_spins

in the

_{spontaneously}

magne-tized state at absolute zero is

Fig. 2. - Relation between reduced inverse susceptibility

and reduced _temperature.

1, 1’, Fermi-Dirac and classical curves for electron _spin paramagnetism, without exchange interaction; 2, 2’, corres-ponding curves with _{positive exchange} interaction, drawn

-1 3

for k 0’ _{== 2} .

E0

In the _{intermediate range} the relative

sponta-neous

_{magnetization}

at oo K

_is.

less than

_unity.

Symbolically

(see

also

_{fig. 5).}

The numerical values in

_{(4.1 )}

and

_{(4.2) depend,}

of course, on the

_particular

form

of band

_assumed,

but the

_possibility

of

_incomplete

saturation at absolute zero for a certain _range of values of the

_exchange

interaction _energy is not

peculiar

to this form.

The variation of

_spontaneous

_{magnetization}

_below,

and of the inverse

_{susceptibility}

above the Curie

point,

6,

is shown in

_figure

3. The relation between

k0’

,

an -,

it may be

noted,

is

_precisely

E0 eo

vertical and horizontal scales are to be

re-labelled k 0’

!

E0

and k0

_{respectively).}

E0

Fig. 3. - Spontaneous magnetization below (falling

curves, scale on _left) and inverse of _{susceptibility} above _(rising

curves, scale on _right) the Curie _point as functions

of k T ,

Eo

(Stoner, 1938 c.)

The reduced curves of

_figure

3

_(and

the numerical

results from which

_they

are

drawn)

are those most

directly

obtainable from the

_{computations,}

but

_they

do not

_bring

out

_clearly

the essential differences in form for different values of

k Eo 0’

1 nor are

_they

E0

convenient for

_comparison

with

_experiment.

A more

illuminating representation

is that shown in

_figure

_4,

Fig. 4. - Reduced magnetization below, and reduced inverse

susceptibility above the Curie _point as functions

of

The numbers on the curves give the values of

2013’ a Tne limiting curves labelled oo _correspond to the use of clas-sical statistics. _{(Stoner, 1938 c.)}

in which the reduced

_{quantities plotted}

are related

to the

_{experimentally}

observable

_specific

magne-tization, 6

(with

the value 60 at absolute

_zero),

and the mass

_{susceptibility, x, by}

The

_spontaneous

_{magnetization}

curves

_(on

the 7

left in

_{figure 4)}

_plotted

against 0 ,

are

_always

lower

with Fermi-Dirac statistics than the

_limiting

curve

(labelled

oo)

with Maxwell-Boltzmann statistics. The variation with

kol

is

_regular,

but not

mono-E0

tonic. The lowest curve is that for the value

of k 0’

Eo k0 Fig. 5. - C0 as a function of so (Stoner, g38 c.)

for which

_{complete parallelism}

of the electron

_spins

is

_just

attained at absolute zero. The inverse

susceptibility

curves are also lower

_(i.

e. the

corres-ponding susceptibilities

are

_higher)

than the

limi-ting

classical curve. The differences between these

curves is not _very

_great

for the

_higher

values of

k0,

E0

Fig. 6. - Reduced inverse susceptibility curves for a _single

band. Values of C0 are _given on the curves. The curve

for is that

for k

0’-

2 -.L

For values

of k0’

greater

for C0 = 1 is that for

k0

==2

_ ’’ , 1

For values

of -

greater

E0 0

than _this, the curves lie between the _~o = i _curve, and the _limiting curve, cl, corresponding to classical statistics.

(After ’Vo1¡lfarth,

but becomes much more _{marked in the range}

of -

,

E0

values

_{[see (4 .1 )}

to

_(4.3)]

for which the relative

magnetization

at absolute

zero, 0,

is less than

unity.

The variation

of C0 with

kef

,

is shown in

tlgure

5. and the reduced inverse

_{susceptibility}

curves for

,

corresponding

to

_{suitably spaced}

Eo

values

_{of C0}

in

figure

6. These curves

_(or

the more

complete

set of numerical values

_{corresponding}

to

them)

are of

_great

value in the

_analysis

of

experi-mental results.

Associated thermal characteristics. - The variation

of _{electronic energy and}

_specific

heat for electrons in a band of standard form when

_spontaneous

magnetization

does not occur

_(i.

e. for a

non-ferro-magnetic metal)

is shown in

_figure

_7.

Fig. 7. - Variation with

temperature of electronic energy and _specific heat.

E, C, energy and _specific heat for N electrons. £0’ maximum electron energy (band width) at absolute zero. _k,

Boltz-mann constant. The full straight lines are _tangents kT

at -

= _o, and the broken lines the _asymptotes to the

E0

curves. _(Stoner, 1938 b.)

The

_limiting

value of the electronic

_specific

heat

f Kt ..

b

for -

-+ o is

given by

E0 o

From measurements in the

_liquid

_{helium range,} where the electronic contribution to the total

specific

heat can be

_{unambiguously}

_separated

from the lattice

_{contribution,}

it is found that the elec-tronic contribution is in fact

_proportional

to the

temperature.

For some metals

_(e.

_g.

Cu,

_Ag)

the

proportionality

factor is in

_good

_agreement

with

that

_{given by (4.5) if E0}

is calculated on _the

assump-tion that the electrons are

_free;

for other metals

(e.

g.

Fe, Ni, Pd, Pt, Ta,

Nb)

the

_experimental

factor is much

_{higher, indicating}

that the _energy

density

of states at the

_top

of the Fermi distribution is much

_greater,

or, if the band is of standard

form,

that the width of the

_{occupied region}

_(or

_unoccupied

region,

if holes at the

_top

of a band are

_involved)

is much narrower.

For a

_{ferromagnetic}

below the Curie

_{temperature, 0,}

the numbers of

_{spins pointing}

in

_opposite

directions

are

_unequal;

the distributional energy differs from

that for a

_{non-ferromagnetic}

_(for

the same N

_{and s o),}

and in addition there is the

_exchange

_energy propor-tional to ~2

_{[c f .}

_{(3 . 6)I.}

Above

0,

the results illus-trated

_{by figure}

_7, are

_applicable.

The calculation

of the variation of _{electronic energy and}

_specific

heat over the

_temperature

_range up to 0 is

trouble-some and has been carried

_through

in detail

_only

k0’ _i

for =

2 .J

_(the

value for which

_parallelism

of

E0

spins

is

_just

attained at absolute

zero).

The results for the

_specific

heat are shown in

_figure

8.

Fig. 8. - Electronic specific heat _{as a} function

for k6

= 2-1 3

= 0.?937

(k0

= 0.4065 .

Eo Eo /

(Stoner, 1939 b.)

The electronic

_specific

heat,

C,

may be

regarded

as made _{up of} a

_{quasi-magnetic}

_part,

_CM,

associated

with the

_temperature

variation of and a

part,

G’E,

associated with the variation in the _energy distribution. The

_part

CE may in turn be considered

as a sum of two

_parts,

_C1

and

_G’2,

the first

corres-ponding

to

_change

of

_temperature

with C

constant,

the second to

_change

_{of C}

with

_temperature.

A

rather

_{unexpected general}

result

_(see

Stoner,

1939 b,

p.

353)

is that for all values

of k0,

in the low

tempe-E0

rature the

_{quasi-magnetic}

contri-bution,

C1W is

_{numerically equal}

to, but of

_opposite

sign

to

_C2.

Thus the

_limiting

low

_temperature

specific

heat is calculable

_directly

from the distri-bution at absolute zero, and is not

_dependent

on

the manner of variation

of C

with

_temperature.

The

_general

result is :

kT

For - o,

= _o, this reduces to

(4.5).

For C0

= _1,

Apart

from the

_complete

results

’J)

,

E0

and the

_limiting

low

_{temperature specific}

heat,

given by (4.6),

values have been

obtained,

effecti-vely

for the whole _range of values

of - or

of

2013 ) ?

E0 Eo /

of the

_change

_{of energy between absolute} zero and the Curie

_point

and of the

_specific

heat

_immediately

below 0

_(and

of the

_component

_{contributions),}

and

of the

_specific

heat

_immediately

above

0,

and so

of the Curie

_point

_{discontinuity.}

The curves

showing

the variation of these

_quantities

with kG

So have many

interesting

features,

but

_they

will not

be discussed

here,

more

_particularly

because

ade-quate

consideration has not

_yet

been

_given

to the

analysis

of the available

_experimental

results in the

light

of them. It must suffice to state that

_although

k0’

the " classical " values are

_approached

for

k e

consi-- Eo ₀

derably

greater

than

_unity,

for -

= 2

" ,

or less

Eo

(the

relevant range for most actual

_{ferromagnetics),}

the _{theoretical values may be}

_widely

different from those

_{given by}

the " classical "

treatment,

and to a

degree offering promise

of far more

_satisfactory

agreement

with

_experimental

results. As is well

known,

a strict

_application

of Maxwell-Boltzmann

statistics

_gives

results which are

_completely

incom-patible

with

_experiment.

In summary, the treatment of

_{ferromagnetism}

on a

collective electron

basis,

using

Fermi-Dirac

statis-tics,

leads to a consistent scheme

_covering

all the

related electronic

_properties,

thermal as well as

magnetic.

As

_compared

with the classical treat-ment an additional

_parameter

is

required,

the band

width,

s, but this is not introduced

arbitrarily,

but is necessitated

_by

the

_physical

situation. The results of the theoretical and

_{computational}

treat-ment are

_broadly

consistent with

_experimental

findings,

but

_owing

to the

_{simplifications}

made in the

_premises,

the scheme should not be

_expected

to

give

complete

and exact

_agreement

with

_experiment.

Rather,

the scheme

_{provides logically}

consistent sets

of results for a

_{complete family}

of "

idealized "

ferromagnetics, by comparison

with which the

interpretation

of the

_significance

of

_experimental

results for actual

_{ferromagnetics}

_{may be}

_enormously

facilitated.

5.

_Applications

to metals and

_alloys.

-5

(i)

General. - In

the two papers summarized

in §§

3 and 4

_(Stoner,

_very brief

general

discussions were

given

of the then available

experimental

results on the

_{ferromagnetic}

elements in relation to the theoretical

_scheme;

and these were

supplemented by

a somewhat fuller consideration

of the

_{susceptibility}

above the Curie

_point

_{(~ g38}

_c~.

Some of the later

_experimental

results are included in a

_subsequent

review article on

_{ferromagnetism}

(Stoner, 1948).

More detailed consideration was

given

to nickel and

_{nickel-copper}

_{alloys [see §}

5

_(ii)]

by

Wohlfarth who has also discussed

b)

nickel-cobalt and related

_{alloys [see §}

5

_(iii)]

and made a

_preliminary

attack

_{[I g4g}

c,

d,

on some of the

_{underlying quantum}

mechanical

problems [see §

5

_(iv)].

Until the advent of the collective electron

treat-ment,

experimental

results had

_usually

been

consi-dered with reference to the

_simple

form of the mole-cular field treatment, as

_adapted

to electron

_spins

as carriers of the

_magnetic

moment. This modified

Weiss-Heisenberg

treatment, as _{it may be}

called,

is

_essentially

characterized,

on the statistical

side,

by

the use of Maxwell-Boltzmann statistics. It

gives,

in a

_simple

_{way, theoretical} curves for the

variation of

_spontaneous

_{magnetization}

_below,

and

of

_{susceptibility}

above the Curie

_point,

_which,

as far as their

_general

form is

concerned,

_appear at first

to be

_reasonably

consistent with the

_experimental

results. Even

_leaving

aside such matters as the

departures

from

_linearity

of the relation

between /

and T above

0,

there are,

however,

serious

quanti-tative

_{discrepancies.}

For

_example,

the number of effective

_spins

_{per atom,} deduced from the

ferro-magnetic

saturation

_{magnetization}

for T - o, differs

greatly

from the

value,

qp, deduced from the

para-magnetic susceptibility.

For

_Fe,

Co and Ni the qj.

values are 2.22, 1.71 and o.60;

respectively; the qp

values are

_{3.37, 3.27}

and o.86. While it is not

suggested

that the _{qp values "} should " be

identical,

no " natural "

_{explanation (i.}

e. no

expla-nation consistent with the

_general

wider theoretical

scheme)

was ever

_given

of

_the

_very

_great

differences

on the basis of the

_{Weiss-Heisenberg}

treatment.

Further,

although

this treatment could account in

an unforced way for the order of

magnitude

of the

"

excess "

_specific

heat

below,

and the sudden decrease in the

_{neighbourhood}

of the Curie

_point,

it could not account _{in any way either for} the

observed excess above the Curie

_point,

or for the electronic

_specific

heat at _very low

_{temperatures.}

It _may be added that the

_quantitative

characte-ristics of the

_{magneto-caloric}

effect

_(which

_imply

a

decrease of the molecular field coefficient over the

range from the Curie

point

to room

_temperature

_by

a factor of I o or

more)

are in

_complete

disaccord with either an assumed

_constancy

of the molecular field

_{coefficient, N Ir,}

or with the modest

_degree

of

variation which

might

be consistent with the

quantum

mechanical

_origin

of the molecular field