Influence of order on magnetic properties

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Influence of order on magnetic properties

R. Smoluchowski

To cite this version:

INFLUENCE OF ORDER ON MAGNETIC PROPERTIES

By

R. SMOLUCHOWSKI.

Sommaire. - Une nouvelle théorie de saturation

magnétique dans les alliages binaires est présentée. Dans cette théorie on considère les fluctuations de concentration _{électronique} dans tous les _groupes

équivalents des atomes. Dans le cas d’un réseau du cube centré ces _groupes contiennent les _premiers et les seconds voisins et la théorie est en accord avec les données _{expérimentales} pour Fe-Co. Pour les

alliages à faces centrées, comme _Fe-Ni, on _emploie un _{groupe contenant les premiers} voisins. Cette théorie permet aussi de calculer l’influence d’ordre sur les _{propriétés magnétiques} comme le

moment de saturation et la _{magnétostriction} et elle est en accord avec les _expériences dans les cas

connus. L’influence d’ordre sur la _température de Curie, sur _{l’anisotropie magnétique,} sur la force

coercitive et sur la _{perméabilité} est aussi discutée. Enfin l’influence de _{propriétés magnétiques} sur

les _phénomènes d’ordre est considérée.

LE _JOURNAL DE PHYSIQUE ET LE RADIUM. 1’OME

_12,

MARS

_1~~1~

PAGE 389.

1. General characteristics. -

a. The

order-disorder

_phenomena.

-

A brief summary of the

important

features of the

_ordering

_phenomena

_may

not be out of

_place

here. In _many

_binary

_alloys,

usually

those which exhibit

_complete

or

_nearly

complete

miscibility,

at

_{particular compositions}

there can exist below a critical

_{temperature, T,.,}

an « ordered » lattice. In the

ordered

lattice each kind of atom

_occupies

a

_specific

kind of lattice site

in the unit cell.

_Ideally,

at

_sufficiently

low

tempe-ratures this «

long

_{range order » should extend}

throughout

each

_{single crystal.}

However,

at low

temperature

the

_ordering

_{process is} too slow and

at

_higher

_temperature

the

_disturbing

thermal

agita-tion too

_large

to allow the ideal condition ever to

be attained. The ordered state in an actual

_single

crystal (or

grain)

should be

_imagined

as

_consisting

of many small volumes within which the order is

very

high

but

_varying

in a discontinuous manner at

the

_boundary

between these volumes. Each of these volumes can be

_thought

of as

_separately

nucleated

_during

the transition from a random to

an ordered solid solution.

_Clearly,

the over-all

degree

of order in a

_crystal

at

_{equilibrium depends}

upon the size of these blocks of

high

order and it

changes

with

_temperature

and with deviation from the stocchiometric

_composition

which

_corresponds

to an ideal

_complete

order.

For some time it was believed that the order-disorder transformation is a

_homogeneous

trans-formation ;

i. e., the two states cannot coexist in

equilibrium. Recently, mounting

evidence

_[2]

_points

to the conclusion that this is not true and that

many, if not

all,

ordering

reactions are

_{heterogeneous}

and similar to the conventional

_phase

transitions. The

_ordering

_{process is very}

_conveniently

described in terms of a

_change

in the number of nearest

neighbors

of each kind of atoms. The

_ordering

usually

leads to a

_preferential

formation of « mixed bonds n, AB rather than AA or BB. The

_opposite

tendency

leads to a

_separation

of an A-rich and

a B-rich

_phase.

This

_concept

of bonds is

_{intu’itively}

and

_{mathematically}

_convenient,

but it should not

be

_assigned

much

_{physical significance}

since we

know that metallic

_bonding

has a much more

compli-cated

_origin

and also that there are

ordering

reac-tions in which the _average number of bonds of each kind does not

_change

at all.

Finally,

it should be

_pointed

out that above the

critical

temperature,

T"

the

_crystal

exists in an

essentially

random state. There is much

_good

evidence that at least near the

_temperature,

_T,,

there is a

_tendency

for atoms A to seek a B-rich

neighborhood,

and vice-versa. This

_tendency,

the

so-called « short _range

order »,

is best described

as a

_general

decrease in the

_probability

of local concentration fluctuations as

_compared

to those in a

purely

random solid solution.

b. Saturation

_{magnetizafion.}

- When

considering

the influence of order on

_magnetic

_properties

_{[ 1 ~,}

it is

Fig. i. -- Saturation magnetization of transition elements

and their _{alloys (after Pauling).}

necessary to know the

_dependence

of saturation

magnetization

on the

_position

of the elements in the

periodic

table and of the

_{interpolated positions}

of their

_{binary alloys.}

This

_dependence

is illustrated in

_{figure i}

where the saturation moment of several

alloy

series is

_{plotted against}

the number of

elec-trons in the combined 3 d

and 4

s shells. The

striking

regularity

of this

_diagram

found an

_early

interpretation

in the work of

_{Pauling [3]}

which can

be

_{expressed, according}

to

_{Shockley [4],}

in terms

of band

_theory

as follows : let us assume that the

3 d band is

_split

into two

_parts,

the

_higher

one

containing

~~.88

and the lower one ,~.I2 electrons. Half of the states in each of these

_parts

_correspond

to electrons which are

_parallel

to each other and

anti-parallel

to the other

_half,

and we

_imagine

them

occupying

separate

bands which we shall denote R

(right)

and L

_(left).

In

_{ferromagnetic}

metals due

to the

_exchange

_interaction,

the R and L bands are

displaced

with

_respect

to each

other,

and thus an

unbalance of R and L

_spins

is

_produced.

Accor-ding

to this

model,

in a

_{ferromagnetic}

metal the

top

of the band

containing,

the R

_spins

_say, is

lower than the bottom of the _upper

_part

of the band

ccntaining

the L

_{spins. Progressing}

from nickel towards

iron,

the electrons are

_gradually

drained off _{from the upper L}

band,

and thus the number cf unbalanced

_spins

reaches a maximum value

of

2.!~4

at about 8.2 _{electrons per} atom. Further reductions of the total number of electrons lower the number of the R

_spins,

and thus the unbalance

is now

_gradually

reduced in accordance with

_figure

1.

This model,

although

rather artificial and without much additional

_support,

_{is very convenient in}

correlating

the

_properties

of the various

_alloys.

Perhaps

a more

_{satisfactory interpretation}

of

figare

i is based on the differences of the 3 d shells

in the various atoms. Electrons in the 3 d band

are far from free

and,

in

fact,

the

_identity

of 3 d shells is to a

_great

extent

_preserved.

In the band

theory,

the

_exchange

interaction tends to

_separate

the R and L

bands,

and this is counteracted

_by

the

increase of the Fermi _energy The actual

separation

of the R and L bands is determined

_by

a balance between these two tendencies which

depends

on _{many factors which in} turn

depend

_upon

the

_position

_among the transition elements. One of the

_important

factors is the fact that the increase of energy due to a transfer of an electron from the L

to the R band is

_greater

the

_greater

is the width of the 3 d band. This width is

_greater

for lower Z

since then the lower

_charge

of the nucleus allows the 3 d

_{shell to}

_expand

more, and thus the

overlap

and interaction between

_neighboring

3 d shells is

greater.

In

_comparison

with the

_change

in the size of the 3 d

_shell,

the interatomic distance remains

practically

constant for the transition elements in

the

_{ferromagnetic}

_{group. This allows} us to

esti-mate how varies within the _group of elements which are of interest here. The situation can be thus

_{qualitatively}

understood in the

_following

_way,

illustrated in

_figure

2. In

nickel,

all vacancies are

in

L,

and the bands are

_widely

_separated

since the 3 d shell and

_{the AEp}

are

_relatively

small. The unbalance of

_spins

is thus

_equal

to the number

of

_missing

electrons. In

cobalt,

the smaller number

of electrons leaves a still

_larger

number of

_unpaired

spins

than in nickel. At an electron

concentra-tion 8.2, however, the

_separation

of the L and R

bands has decreased to such an extent that the

_top

of the R band reaches the

_{highest occupied}

state in

the L band. From here on

_proceeding

to iron and

beyond

it _{towards manganese,} the

_seaparation

of

OIGPLACEMENT OF BANDS ACCORDING TO THE DIRECTION OF SPIN.

Fig. 2. - Schematic

presentation of _partly filled bands

in a _{ferromagnetic} material.

the two halves will further decrease with

increa-sing

and both halves will have

_unoccupied

states. This causes a

_progressive

decrease of the

number of

_{unpaired spins,}

if the

_separation

decreases

linearly

with

Z,

in accordance with

_{experiment. ,}

Fig. 3. - Density of electronic states

in the 3 d band _{of copper (after Slater).}

The above

_reasoning

can be

_put

into a _very

_rough

quantitative

form : let us make the

_assumption

that Slater’s calculations

_[5]

of the 3 _{d band in copper}

apply

qualitatively

to other transition elements

(fig. 3)

when _{the proper}

_change

of the width of

the band is taken into account. In other

words,

the distribution n

_(E)

of electronic states in the

elec-391

trons in a band of a width W

_(in

atomic

_{units) :}

where a

== 2013o.oi5,

b =

0.045,

c = 0.008. If we

consider now the L and R

_parts

of the band and

transfer one electron from one

_part

to the other

then,

as can be

_easily

shown,

the total _energy

changes by :

This formula is valid

_only

if the number of

trans-ferred electrons is small

_compared

to the

_significant

irregularities

of the curve n

_(E).

The width of the 3 d band in other transition elements can be estimated in the

_following

manner :

From Slater’s calculations on _copper we know how

its band width

_changes

with interatomic distance d. One makes the

_{plausible assumption}

that the

diffe-rence

3rd

in the radius of the 3 d shell as

compared

CHANGES OF ENERGY DUE TO ChlANGE OF DIRECTtOH OF

ONE SPIN PER ATOM.

Fig. 4. - The

change of Fermi energy DEF and the exchange shift E per one electron (in atomic units).

to the 3 _{d shell in copper}

_corresponds

to an

_apparent

change

-

2 of the

_equilibrium

interatomic

dis-tance

_do

_{in copper.} It _{appears then}

that,

well within the _{limits of necessary accuracy, the width}

can be

_expressed

in atomic units

_by

where d is the" effective " interatomic distance

equal

to and c has the value 15.6.

The width W calculated in this way is

_probably

overestimated since no account was taken of the

changing density

of states _{per unit volume of the}

shell.

_{Equations (2)}

and

₍₃₎

allow us to

_compute

the

_change

_{of energy due} to a transfer of an electron from one half band to the

_other,

and the result is

plotted

in

_figure

_4.

This _energy, as mentioned

before,

opposes the influence of the

exchange

energy which tends to shift the two half bands with

_respect

to each other. In order to estimate the latter we use the known total number of electrons available in the 3 d shell

_(assuming

about _0.7 electrons in

the~’4~s

band)

and distribute them among the L and R bands so as to obtain the observed saturation

moment. The _{necessary shift} can be then read off

from

_figure

3 with the

_help

of the known calculated width W. Such an estimate can be

made,

of course,

only

for electron concentrations less than 8.3 and

so besides iron and some of its cobalt

_alloys,

_only

a 50 : 5o iron-chromium

_alloy

was used. This

_alloy

is known to be

_magnetic

at an electron concentration

corresponding

to _manganese which itself is

non-magnetic.

In order to _compare this shift with the

change

of Fermi _energy, we divide the calculated

shift

_by

the total number of electrons transferred from one half band to the other

_(i.

e., half the satu-ration

_{moment) obtaining}

_{in this way the energy}

gain

per one transferred electron. The result is

plotted

in

_figure

_4.

For cobalt and for nickel one

knows

_only

that the shift has to be

_larger

than that

Fig. 5. - The molecular field as a function of the distance berween the 3 d-shells (after N6el).

corresponding

to the vacancies in the L band and

thus the curve in

_{figure 4}

for electron concentrations

higher

than 8.3 is a " _{reasonable "}

extrapolation

of its left

_part,

_falling

above the

_{corresponding}

minimum values for cobalt and for nickel. It is

easily

seen that with

_decreasing

atomic number it becomes less and less favorable to

_{produce unpaired}

spins

and that below a certain electron concentration

ferromagnetism

should not be found. That this

crossing

of the two lines occurs near _manganese

which

is, indeed,

a borderline case, is

_partly

fortuitous

although

the order of

_magnitude

of the various

energy differences is not far off from those known from other sources. It is

_important

to note that

in this treatment the

_slope

of curve E in

_figure

₄

was calculated so as to

_satisfy

the observation that the

_drop

of the

_exchange

shift and of the saturation

moment with

_decreasing

_{Z is very}

_{roughly equal}

to the

_change

of Z. _{This may be related} to the

_general

behavior of

_exchange

forces as illustrated in

_figure

5.

We shall consider thus the

_general

trend of the

392

concentration as understandable in terms of one of the models and will

_approximate

it

_by

an idealized

linear

relationship.

The influence of the size of the 3 d shells on the

exchange

forces has been

_frequently

considered.

In

_particular,

Slater

_{[6] plotted}

_{the energy of}

magnetization against

the ratio of the interatomic distance to the diameter of the 3 d shell. Neel

_[7]

used a somewhat different method

and,

from the

point

of view of the influence of

order,

a more

convenient

_{approach by plotting}

the

_{molecularfield,}

as deduced from the

paramagnetic

behavior,

against

the distance between the 3 d shells

_{(fig. 5).}

We

shall use this result in the

following

discussion.

2. Saturation moment of random

_alloys.

--It is clear that since the saturation moment in

alloys

is influenced

_by

order,

the

_relationship

shown

in

_figure

I should not be

interpreted,

as it

_usually

is,

in terms of _{the average electron} concentration but rather in terms of local electron concentration.

Thus we consider the saturation moment of a random

alloy

to be a sum of the saturation moments of the various local concentration fluctuations. The size of the latter are, of course,

_{strongly dependent}

_upon

the

_degree

of order.

Fig. 6. - Calculated and observed saturation moments in iron-cobalt

alloys.

In _{many of the}

_{binary ferromagnetic alloys}

there

are

_changes

of

_crystal

structure and variations of

lattice constant within each

_phase

_{which may make}

a check of the

_theory

difficult. It was thus

_thought

advisable to use a

_system

which is

particularly

simple.

The

_bodycentered

_{range of the iron-cobalt}

alloys

offers an excellent

_{possibility :}

the lattice

constant

_changes

_very

little,

the maximum moment

occurs in that

_alloy,

and the effect of order is known. The next

_question

to decide is the number of atoms

which should be considered as

_forming

a fluctuation

of concentration

_[8].

It seems natural to choose for that purpose a _group of fifteen atoms

_consisting

of an

atom, its

_eight

nearest

_neighbors,

and its six second nearest

_neighbors,

which are

_only

15 per cent further away.

Assuming

perfect

randomness,

the

proba-bility

of a

_given

concentration of iron and cobalt

atoms can be calculated from the

_expression

where n is the total number of atoms _{in the group,}

r the number of iron

_{atoms, q}

the _average

concen-tration of iron atoms in the random

_alloy,

and p = i - q is the concentration of cobalt atoms. The average electronic

density

per atom in a

fluctua-tion is then

_computed

and the

_{corresponding}

contri-bution to the total saturation moment

_assigned

on

the basis of a curve similar to that in

_{figure i}

-Since the

_{experimental points}

on which

_figure

i is based were obtained on

_presumably

random

_alloys,

it is necessary to choose a _{proper relation for local}

concentrations. This is done in the

_following

way : the saturation moment for

_iron,

which is

body-centered,

is 2.22, the

_{corresponding}

value for a

hypothetical

body-centered

cobalt is not known,

and so we use the value _1.go which can be obtained

by extrapolating

the observed saturation moments

for the

_{body-centered phase.}

The local saturation

moments for all

_compositions

in between are

deter-mined

_by

two

_straight

lines of

_slope

one, as

393

of 2.56 at a concentration

8.34.

_Typical

results

indicating

the contributions from various fluctua-tions in a random 50 : 5o

_alloy

are shown in

_{figure 7,}

NUMBER OF IRON ATOMS IN A GROUP OF 15 ATOMS AT 50 Fe SO Co.

Fig. 7. ---. Contributions of various local fluctuations

to the total saturation moment in a random 5o Fe 5o Co alloy.

the total moment

_being

2.38. The results for the whole _range of

_compositions

are

_plotted

and

com-pared

with

_experimental

data in

_figure

6. The

agreement

is within 1.~ per cent, which is

_quite

_good

in view of the

_{highly approximate}

character of the

theory.

It is

_interesting

to note that a similar

calculation based on fluctuations of concentration

in a nine-atomic _group leads to a rather

_strong

disagreement

with the

_experimental

data as illus-trated

_by

a dashed line in

_figure

6.

In iron-nickel

_alloys

a similar calculation can be made for the

_{body-centered phase.}

In the face

cen-tered

_phase

a _group

_containing

thirteen

_atoms,

an

atom and its twelve nearest

_neighbors,

has to be used and it leads to a

similar,

_though

less

_good,

agree-ment with

_experiment,

which is not too

_surprising

in view of the

_{large change}

in lattice constant and other

_{irregularities}

in that

_system.

The

_{general procedure}

for

_interpreting

and

predic-ting

the influence of order on

_{magnetic properties}

is thus follows : for a

_{body-centered}

lattice one

takes a _group of fifteen atoms, for a

face-cen-tered lattice a _group of thirteen atoms, and calcu-lates the

_{corresponding}

electronic concentration. A linear

_dependence

of local saturation moment

upon electron concentration similar to that in

figure

6

_gives

then the contribution to the observed

total moment.

If,

according

to

_{figure 5 (or}

for some

other

_reason),

between certain

_pairs

of atoms there is no

_magnetic

interaction then the contributions of

various fluctuations have to be _{decreased in}

propor-tion to the number of inactive

_neighbors.

3. Influence of order on volume

_properties.

-In the

_previous

sections we have discussed order

phenomena

and the statistical

_{interpretation}

of saturation moments. The

_procedure

outlined above

should allow us to make now a critical

_comparison

with

_experimental

data. It is convenient to consider

separately

the influence of order on the so-called

structure

_independent

and structure

_dependent

magnetic

properties. Among

the first we shall deal

with saturation

_{magnetization,}

Curie

_temperature,

anisotropy,

and

_spontaneous

_{magnetization.}

a. The Saturation

_{magnetizatiorc.}

- The

satu-ration

_{magnetization}

is known to

_change

with order in

FeCo,

FelVi3, Ni3Mn,

CrPt,

and others. In terms

of the

_theory

here

_presented

these differences in

saturation moment in ordered and random

_alloys

should be accountable for

_by

a

_change

in the

fluctua-tions. This seems to be indeed the case :

FeCo. - In the iron-cobalt

system,

at

50 : 5o per cent there occurs a well-known

_ordering

reaction in which each atom of one kind has

_eight

nearest

_neighbors

of the other kind.

_According

to

figure

6 the saturation moment in a random

alloy

is 2.38. In the

_perfectly

ordered

_alloy

there are no fluctuations of

_composition,

and in the

_previously

considered group of fifteen atoms there can be

either seven iron and

_eight

cobalt

atoms or vice versa. These two have an _average electron

concen-tration of

_8.467

and 8.533

_{corresponding}

to

_2.~3

and

_{2.3 7}

saturation moments,

respectively.

The

moment of the ordered

_alloy

is thus

_2.4o

which is about i _per cent

higher

than the moment of the

random

_alloy.

_Experiment

indicates this difference

to be

about 4

per cent

_[9].

An

_important

_{consequence of the above}

calcula-tion is that the local saturation moment in the

neighborhood

of an iron atom in that

_alloy

differs from that near a cobalt atom

_{only by}

about 3 _per cent

while the usual atomic moments differ

_by

about

15

_{per cent.}

This has an

_{interesting bearing}

_upon

the recent work of C. G. Shull

_[10],

who studied the neutron diffraction in these

_alloys.

The dif-fraction of neutrons

_depends

not

_only

on the

_purely

nuclear

_scattering

but _{also upon the interaction}

between the

_magnetic

moment of the neutron and the

_magnetic

moments of the atoms. In the case

of these iron-cobalt

_alloys

diffraction seems to

occur as if there were no

difference,

within the limits

of

_experimental

error, between the

_magnetic

moments of the lattice sites

_{occupied by}

iron and

_by

cobalt

atoms. This result is in

_agreement

with the

_theory

here outlined.

Fe-Ni3’

- This face-centered cubic

alloy

corres-ponds

to "

permalloy " composition

in which

satu-ration moment increases

_by

about 6 _per cent on

ordering [11]. According

to the outlined

procedure

the _{proper size of} the _group of atoms in this case is

of the various fluctuations are

_easily

calculated.

However, it _{is here necessary} to take into account the fact that

iron,

in a face-centered

lattice,

is

non-magnetic,

a fact which agrees with the

_position

of the

_{corresponding point}

for the nearest

_neighbors

on the curve in

_figure

5.

_(In

fact Néel

_suggested

that in

_{body-centered}

iron the interaction between

nearest

_neighbors

_may be _{very small}

_compared

to the interaction between second

_neighbors.)

It

follows thus that the various contributions to the

saturation moment have to decreased in

_proportion

to the iron-iron

_{pairs occurring}

in each

fluctua-tion. The

_resulting

moment is 1.02. For the

ordered

alloy

the situation is much

_{simpler :}

there

are no iron-iron nearest

_{neighbors ( f g. 8)}

and one

Fig. 8. - Ordered face-centered cubic lattice

obtains

_only

two kinds of

_{neighborhoods,}

an iron

atom surrounded

_by

twelve nickels and a nickel

atom surrounded

_{by eight}

nickels and four irons. The

_{corresponding}

electronic concentrations of these

groups

give

the saturation moments

_o-75o

and 1.21,

respectively.

Taking

the _{proper ratio} 1 : 3 of the

frequency

of their occurrence one obtains for the ordered lattice the total moment I.io which is

about

_eight

_per cent

_higher

than the moment 1.02

calculated above for the random

alloy.

This is in fair

_agreement

with the difference of 6 per cent in the

_{experimentally}

measured moment i. 18.

Ni3Mn. -

This

_alloy

is also face-centered cubic

and is

_strongly

_{ferromagnetic}

in the ordered

condi-tion,

having

saturation moment of about o.9 Bohr

magnetons, while

in the random condition it is

_only

weakly ferromagnetic

[12].

In

_considering

this

alloy according

to our

_procedure

it is

_important

to

take into account the sizes of the 3 d shells.

_Figure

5

indicates that a Mn-3In

_pair

will have a

_negative

contribution to

_magnetism,

while a Ni-Ni

_pair

will

have a

_positive

contribution. The Ni-Mn

pair,

on the other

hand,

corresponds

to a

_point

near zero

interaction. If we make a calculation

_analogous

to that for

_FeNi3l

i. e.,

put

the contributions of the Mn-Ni interactions

_equal

to zero, we obtain

for the ordered structure a _{moment o.g} I in

_good

agreement

with

_experiment.

For the random

_alloy

we consider as before the fluctuations of

concen-tration within the _{groups of thirteen} atoms and take

into account their contribution to the total

satu-ration moment,

obtaining

a

_positive

contribution i from the Ni-Ni interaction and an unknown

_negative

contribution from the Mn-Mn interaction. Whether the latter is

_{big enough}

tao 11

_{compensate "}

for most

of the Ni-Ni contributions is difficult to _say

_[9]

but one can

_expect

the saturation moment for the random

_alloy

to be

small,

in accordance with expe-riment. The scatter of the

_experimental

data for a

quenched alloy

is

_quite

_likely

due to an

_imperfect

randomness.

b.

_{Magnetostriction.}

- Numerous

experiments

indicate a

_{large change}

of

_spontaneous

magneto-striction in various

_alloys

on

_ordering.

The

out-standing examples

are

_FeCo,

_Ni3Fe,

_Fe3Al,

and certain Fe-Si

_alloys.

The

_difficulty

in

_treating

these effects

_{theoretically}

is the lack of a

_{good general}

theory

of

_{magnetostriction.}

Becker based his

theory

[13]

on an interaction of

_magnetic

_dipoles

located at lattice sites.

_Although

this

_point

of view is

_certainly

_{superceded by}

more recent

_quantum

mechanical

_{developments,}

it is rather well suited from a

descriptive, qualitative point

of view

[14]

and in

_particular,

it is convenient for the treatment

of the effects of order. It should be

_remembered,

however,

that this

_{theory gives}

at best reasonable values of

_{magnetostriction.}

Since

_{ordering produces}

radical

_{changes only}

in the immediate

_surroundings

of an _atom, we shall consider

_only

interactions of

nearest and second nearest

_neighbors.

The

_{magnetostriction}

calculated,

as described

below,

from first and second

_{neighbors only,}

_appears

to differ

_{only slightly}

from the

_{magnetostriction}

obtained from a summation over all atoms in the

crystal, indicating

that the contributions of the more

distant atoms

_nearly

cancel out. Another

_interesting

conclusion is the fact that on this

_dipole

model the

only

negative

contribution to the free _energy, due to

positive magnetostriction

),,

in a

_body

centered lattice comes from the

_{angular displacement}

of the nearest

neighbors

which remain at fixed distance from the

central atom, within the

_{approximation}

of terms linear in 1B. All other

_{displacements,}

and of course, the strain energy oppose

magnetostriction.

The final formula

_[14]

for a

_{body-centered}

cubic lattice is :

in which N is the number of

_dipoles

_{per cubic}

395

and

between second

neighbors,

respectively.

In the

case of a random

alloy

tL2 = 2.38 _as

computed

previously

in connection with the

study

of the

varia-tion of the saturation moment with order. For an

ordered alloy,

we have to consider the moments

characteristic

of an interaction between

_pairs

of

atoms. For the nearest

neighbors,

a Ni-Fe

_pair,

we have on the basis of their electron

concen-tration

=

2.!~0,

while for the second nearest

neighbors

we have either 2.22 or _i.go

_depending

upon whether the central atom is iron or cobalt.

The average is = 2.06.

Substituting

these values in

(5)

we obtain :

which has to be

_compared

with an

_{experimentally}

observed factor I .40.

The difference is not

surpri-sing

in view of the _very

_simple

theoretical

assump-tions and also in view of the

_difficulty

in

_obtaining

perfect

order and

_complete

disorder

_{experimentally.}

The latter conclusion has been confirmed

_by

means

of neutron diff raction since

_X-rays

are not suitable

in that case. Since

_{magnetostriction}

data were

available for a

₄₅

Co-55 Fe

_alloy,

the calculation was

made also for that

_composition.

The ordered

lattice was considered as a

_perfectly

ordered 5o : 50

lattice with 10 _per cent of the cobalt atoms

_randomly

displaced

by

iron atoms. The calculated ratio is .6o as

_compared

to an

_experimental

factor i.3o

which,

for similar reasons as

_before,

can be considered

as a

_satisfactory

_agreement.

Fe-Si. - A

large

eff ect of order on

magneto-striction has been

_recently

observed

_by

Carr

_{[ 15],}

who,

in his

_study

of

_single

_crystals

of various Fe-Si

alloys, compared

_{values for annealed and}

quen-ched

_crystals.

_{It appears that around 11 I atomic} per cent silicon the

_{magnetostriction}

in the cubic

direction

is about two times

_larger

in the

_quenched

alloy

than in the annealed

_condition.

_Although

the _{structure and}

properties

of these

_alloys

are still

not well

_understood,

_{it seems}

plausible

that the

effect is due to an

_ordering

reaction which is

_supposed

to occur _{at 25}

per cent silicon. A

_comparison

with

theory

is

_complicated

_{due to the known}

rapid change

of

_{mechanical and,}

presumably,

elastic

_properties

With

_composition

_{and with heat}

treatment.

_Thus,

not

_only

p-

but also

_perhaps

_{G may be affected}

although

the _{energy of}

_ordering

_{is small}

_compared

to _the

energy

of

_binding

_{(approximately}

_{heat of}

sublimation).

Changes

of

_{magnetostriction}

on

orde-lilig have

been

_recently

observed in

_FeNi3

and

_Fe3xl

bY

_Goldman

_[161

_{who obtained}

an increase of

about

Ioo _{per cent in the former.}

temperafure.

- There _{are a} few _cases

In

which

_{the Curie temperature}

for an ordered and

a disordered

lattice

of the same

_composition

is

known, and it is,

therefore, interesting

to see whether

t a effect

_{could be}

interpreted

in a

_simple

manner.

Since _{there is} no such

_simple

_{relation between the}

Curie

_temperature

_and

composition

’as there is for

the

_saturation

moment, we have to use another

approach.

This is based on the

_admittedly

_uncertain

assumption

that the

_{exchange integral}

_{to which the} Curie

_{temperature according}

_to

modern theories is

proportional (the

constant of

_{proportionality}

being

determined

_by

the

_type

of

_lattice)

_{can be}

represented

in a

_{binary A-B}

alloy

as a sum of contributions

and

_{If ilj3}

_{of the}

_individual

pairs

of atoms. This is

_analogous

_{to the}

simple

11

bond "

inter-pretation

of the total

_binding

_energy of a

_crystal

which is _{made up of}

_positive

contributions

and

_Vun

from each

_pair

_{of atoms. A}

stronger

bond

_implies,

_thus,

_a

lower free energy.

If any two atoms A and B are

_{interchanged,}

_then

the total

_binding

_{energy in a}

_crystal

changes by

a

multiple

of &

while the _average

_{exchange integral}

changes by

a

multiple

of

--

’=’

In an ordered

_alloy

V > o and the free _energy is

lower when there are more _{A-B bonds. If the}

corresponding

magnetic

interactions are such that

~

= _o, then the _state

or order would have no influence on the Curie

_{temperature. For J}

_> o on the

con-trary,

the order would

_{promote strong}

_exchange

interaction and a

_high

Curie

_temperature.

The

opposite

would be the case o. These

arguments

hold not

_only

in the case of

_long

_range

order,

but

_they

are

_applicable

also to

_alloys

in which

_only

short _range order has

_developed.

An

_example

of the case when the Curie

tempera-ture of the ordered state is

_higher

than in the disor-dered state is

_Ni3Mn

and

_FeNi3’

In the first

_{[ 12],}

the Curie

_temperature

is raised from around room

temperature

to near 5ooo C.

_Lowering

of the Curie

temperature

on

_ordering

occurs in

_Fe,Al

_(see

_paper

by

Sucksmith in the

_report)

and CoPt

_alloys.

In the first

_alloy

the Curie

_temperature

is lowered

_{[ 17]}

from 55oo C to 5ooO C while in the

latter,

the ordered

phase

is

_non-magnetic

_{[2], although}

the random

phase

has a Curie

_temperature

near 6no° C.

These observations can be

_compared

in a

qualita-tive manner with the conclusions which we can draw

about the various

_{exchange integrals.}

We know

that the Curie

_temperature

in the face-centered iron-nickel

_alloy

reaches a broad maximum around

60 _per cent nickel and _{falls off very}

_rapidly

on the iron side and less

_rapidly

on the nickel side. This behavior is similar

enough

to the existence of a

maximum at 50 : 50 and it indicates that is

larger

than either or in accord with

the

_requirement,

_~~

> o. _{In the}

_{nickel-manganese}

alloy,

one

_{expects the}

to be

_negative

₅₎

and

compensate

almost

_exactly

the so that

and %

> o. In the iron-aluminum

_alloy,

the

_only

positive

interaction is

_undoubtedly

due to the Fe-Fe

pair,

while the others are _zero,

_{thus J}

o,

_again

in

accord with

_experiment.

The CoPt

_alloy

in the

ordered state is

_tetragonal

with alternate

_layers

.of Co and Pt atoms. It is a well-known result of

modern theories of

_{cooperative phenomena}

that

there are no two-dimensional

_{ferro-magnets,}

the

interaction between the cobalt

_layers

across the

platinum layers being negligible.

In the random CoPt

_alloy,

on the other

hand,

there are

_enough

three dimensional cobalt clusters to make the

alloy ferromagnetic.

d.

_{Magnetic anisotropy.}

- A

good

example

of the

change

of

_anisotropy

on

_ordering

is

FeNi,

which

in a random condition is

essentially isotropic [lI],

while in the ordered state the

_[

i i

1 ]

direction becomes

the _{direction of easy}

_{magnetization.}

The

_theory

of

_{magnetic anisotropy}

is,

in

_spite

of much recent

progress, not

_sufficiently

well

_developed

to allow a

speculative

analysis

of the

_ordering

effect

_[18].

4. Structure

_{dependent properties.}

- So

far,

we have been concerned

_only

with the influence of order on structure

_independent

_properties.

There are,

however,

several structure

_{dependent properties}

which

_change

on

_ordering.

_Among

_them,

we shall

consider

_only

coercive force and

_{briefly permeability.}

a. Coercive

_force.

- There

are _{many theories} of

coercive

force;

the one best

_applicable

here is that

given

by

Becker

_f8],

who relates it to internal

stresses v,

magnetostriction X

and saturation

mo-ment M in the formula :

where p is a

proportionality

constant. On

_ordering,

all three

_quantities

may

change,

and so it is difficult

to see which one is the most

_important.

However,

one would

_expect

both ~ and M to reach their extreme

values when the order is

_complete

while o- can reach very

high

values

during ordering

and,

in

fact,

it _may be

_again

lowered when the order is reached.

We shall discuss bere

_only

the influence of order on

coercive force

_through

a

_change

of a- since a

_change

of a, or M is

_really

a

special

instance of a

_change

of a

structure

_{independent quantity. High}

coercive force is

_usually

_produced

in

_{alloys by high}

stresses due to

precipitation, ordering

reaction or both.

_Typical

examples

of the role of

_ordering

are found in

CoPt,

FePt,

FeCo

_powder,

and in numerous commercial

alloys.

The CoPt which was

_recently

investi-gated [21,

shows that

_during

the _{process of}

_ordering

platelets

of the

_tetragonal

ordered

_phase

form and

grow within the random cubic

_phase

in such a _way

that

_they

are

_parallel

to the

_{(i io)}

_planes.

The stresses set up

by

this condition are _very

_high

and lead to a maximum coercive force of around 3ooo Oe

and

_(BH),,,,,

over

_6.4

X 106 for a

partially

ordered

alloy (~ o

h at 6ooo

_C).

A similar situation

presumably

occurs in FePt

_[20].

On the other

_hand,

in the FeCo

_pressed

_powder

_[21],

the situation is more

_complicated

sin cein that case the

coercive force

_depends

on _{many additional factors.}

b.

Permeability.

- Initial

permability

is rather

closely

related to coercive force and

_depends

on

magnetostriction

1B,

strains a- and

_{magnetization}

in the

_following

_way

where c is a constant.

_Usually

a

_high

_permeability

is

_interpreted

as due to low strains. It _appears,

however,

that the other _{factors may}

_{play a}

_very

important

role

and,

as shown

_by

Goldman in

_{FeNi 31}

permalloy,

the

_{high permeability [22]}

can be attri-buted to low

_{magnetostriction.}

The heat treatment

of

_permalloy

is such as to _{suppress the}

_ordering

reaction which occurs at that

_composition

and which is

_{accompanied by}

a

_large

increase of magne-tostriction.

5. Influence of

_{Magnetic properties}

on order.

- In our discussion of the influence of order on

Curie

_temperature,

it was

_tacitly

assumed that the distribution of atoms in an

_alloy

will be

_governed

by

the

_binding

_energies

and the

_magnetic

inter-actions will

_adjust

themselves to the

_existing

condi-tions. This was

justified,

since near the Curie

temperature

the

_magnetism

is weak and would not

exercise much influence on the

_binding.

However,

there are _many

_opposite

instances where the critical

temperature

T,. of the ordered lattice is much below the Curie

_temperature,

as for instance in FeCo

where the

_{corresponding}

values are

_76oo

C and I 100° C

(extrapolated) respectively.

Under these

condi-tions,

one

_might

_expect

an

important

contribution

of the

_magnetic

interactions to the

_preferential

distribution of the two kinds of atoms.

This

_magnetic

_{interaction may appear in} two

ways :

first,

the energy of the bonds etc.

is,

in most theoretical

_{approximations, independent}

of

temperature,

and it is

_only

the thermal

_agitation

which counteracts order. On the other

_hand,

the

magnetic

interaction,

which is now

_part

of the

ordering

energy, may be

_strongly

_temperature

dependent

and so it may alter the

_dependence

of

order on

_temperature

as

_compared

to the behavior in a

_{non-magnetic alloy.}

No such studies have

been

_{reported. Secondly,}

the bond _energy in the

idealized

_theory,

or

_V,

is

_{essentially symmetric}

with

respect

to the 50 : 5o

_composition,

while the Curie

temperature,

and thus the

_exchange

_integrals,

are

known to _{vary very}

_rapidly

across the

_phase

diagrams.

Thus,

since the

_regions

of order

_usually

extend over io, 20 and more _per cent

_composition,

397

with

_respect

to the stoechiometric

_{compositions.}

Some of the

_{binary alloys}

between metals of the iron group should

provide

an excellent check of

these

conclusions,

unfortunately

the _{ranges of}

existence of order are

_mostly

unknown because of

difficulties with the

_X-ray

methods. The neutron

diffraction work _{should be very useful in this field.}

Certain

_predictions

can be made

using

as a

_guide

the

dependence

of Curie

_temperature

on

_{composition :}

one would

_expect

the FeCo ordered

_phase

to be

symmetrical

because the true

_{(extrapolated)}

Curie

temperatures

in that

_system

seem to _be

symme-Fig. 9. - The cobalt-platinum diagram

(after K6ster and Gebhardt, revised by J. B. Newkirk).

trical

_[7]

around the 50: 5o

_{composition. By}

similar

_reasoning

in

FeNi,

the ordered

_region,

if _any,

’

should be broad on the nickel-rich

_side,

while in

_FeNi:;

the iron-rich side would be _{broader. A}

symme-trical behavior may be

expected

in the cobalt-nickel

system.

Other

_alloys

as CoPt and NiPt show a _very

pronounced assymetry

(fig. g).

However,

in these

cases the Curie

_temperatures

are below

_T,.,

and it is

not certain whether the 11

_short-range

order "

of

spins

which exists above the Curie

_temperature

is

an

_important

factor. The effect

is, however,

in

accord with

_{expectation :}

the ordered

_phases

are

non-magnetic

where A stands for Co or

_Ni;

thus the

_stronger

the

_magnetism

at a

given

temperature,

the

_stronger

is the

_tendency

to

randomness in that

_system.

It is

_interesting

to _{compare the} above mentioned

assymetry

of the ordered

_region

in the NiPt

_system

(similar

to

_Co-Pt,

_{f g. g)}

with the

_assymetry

of the

miscibility

gap in the Ni-Au

system.

There the

minimum

_temperature

of

_complete

_miscibility

occurs

at about 3o at. _per cent nickel and the

_miscibility

gap extends more towards nickel than it does towards

_gold.

Both these

_assymetries

can be

considered as due to the same cause,

namely

to the

preference

of nickel atoms for a nickel-rich

neigh-borhood.

_{By forming}

either a solid solution with

gold

or an ordered lattice with

_platinum

the

nickel-nickel distances are increased

_by

o to 15 _per cent.

Thus

_{by splitting}

into a nickel-rich and a

_gold-rich

phase

in the Au-Ni

_system

or

_by

_preferring

a random

solution to an ordered

_phase

on the nickel-rich side

of the 50 : 5o

_composition

in the Pt-Ni

_system

the

number of the more normal distances between

nickel atoms is increased. In the Pt-Ni and Pt-Co

systems

this factor may be more

_important

than the

magnetic

effect

_previously

considered.

Remarque

de M. Goldman. -- Some of the ideas

presented

in this paper

by

Smoluchowski can be

extended to the case of

_alloys

of

_{ferromagnetic}

elements with

_{nonferromagnetic}

elements. For

example,

in Fe-Si

_alloys

Fallot finds discontinuities in the moment vs

_compositon

curve at

_compositions

at which the onset of a

_superlattice

is

_suspected.

For low silicon

content,

the silicon atoms

_simply

replace

Fe-atoms and decrease the moment

_linearly.

However,

at

_higher

silicon

contents,

the moment

decreases more

_rapidly.

We think this can be understood on the basis of N6el’s

_theory.

In

body-centered cubic

structures,

Néel finds that for atomic

spacings

found in iron and some of its

_alloys,

posi-tive

_exchange

results

_only

from next nearest

_neighbor

interactions. In

_{Fe , Si,}

_however,

_{1 f3}

of the Fe-atoms

have

_only

Si atoms as next nearest

_neighbors.

Hence,

they

will not contribute to the

_magnetic

moment.

_According

to this

interpretation,

the

anomalous decrease in moment would commence at

a

_composition

where the

_probability

of

_finding

an

iron atom with silicon atoms as next nearest

_neighbors

becomes

_significant.

Remarque

de 111.

_{0. Berg.}

- It is essential to

distinguish

between

_just

ordered

_phases

and

inter-metallic

compounds.

FeCo is not an intermetallic

compound

whereas

_Fe,Si,

FeSi,

Fe,Al,

FeAl,

and

Ni;Mn

are. In all cases of intermetallic

compounds

ordered

_phase,

i. e. the intermetallic

compound,

is

considerably

low er than that of the disordered

phase.

This is

_particularly

_apparent

in the case

of

_{Fe~W. Quenched specimens}

which do not

contain have a

_higher

_magnetic

moment than

annealed

_specimens

which contain

_Fe2W.

_{The pure}

compound

is

_{non-magnetic,}

at least at room

temperature.

Data

_given

on

_{Ni,Mn by}

various authorities

_spread

to the extent that it is difficult to

draw safe conclusions. It is

_probable

however that

the ordered

_phase

_Ni,Mn

is an intermetallic

com-pound

the formation of which decreases the

_magnetic

moment.

Reponse

de M. Goldman. -- In the

measurements

on Fe-Co the state of order was estimated

_by

means

of neutron diffraction and I _agree with Dr

_Berg

that

this

_system

is not of the intermetallic

_type

of order.

However,

the

_{applicability}

of this

_theory

to such

alloys

as Fe-Co seems to make

_plausible

the notion herein introduced

that,

at least as

_{regards magnetic}

properties, specifically

the

_magnetic

_moment,

a

nea-rest and next nearest

_{neighbor approximation,}

which has certain features similar to an

inter-metallic

_compound

treatment,

is valid.

Réponse

de Guillaud. - J’ai étudié _un

MnNi3,

en

_{surstructure,}

obtenu en

_partant

d’elements très

purs. Le recuit n6cessaire pour obtenir le maximum de moment est très

_{long (trois}

semaines a

_45oo

_C).

Des

_{qu’on d6passe}

_{4800 C,}

1’aimantation

_spontan6e

disparait

et au refroidissement

MnNij

n’est

_plus

ferromagn6tique (6tat d6sordonn6).

Remarque

de M. Went. - Nous

avons trouv6 que

l’ordre

_peut

aussi se manifester dans la forme des

courbes de 1’aimantation

_{spontanée a}

en fonction de

la

_temperature

T. Les courbes

_{(c, T)}

ont été d6ter-min6es pour le nickel

put

et _{pour des solutions}

solides de nickel avec

Al, Si, V, Cr, Cu, Mo, Sn, W,

Mn, Pd,

Fe ou Co. On _{sait que pour le nickel pur}

cette courbe coincide d’une maniere satisfaisante

avec la courbe

_th6orique

bien connue

correspondant

a s =

Pour

tous les

systèmes

binaires mentionn6s la courbe

_{(~, T)}

est moins concave vers 1’axe des

temperatures.

Les d6viations sont

_{proportionnelles}

a la teneur de 1’element

_ajoute;

elles

_dependent

de

la nature de cet element. Dans la s6rie

Co, Fe, Mn,

Cr

la _{deviation par} atome

_ajout6

_augmente

du Co au Cr.

Nous croyons que ces

_ph6nom6nes

sont 6troitement

lies aux fluctuations dans la concentration locale des atomes

_{magn6tog6nes.}

Par

_consequent,

on doit

s’attendre a ce

_qu’une

solution solide

_{complètement}

ordonn6e

_pr6sente

une courbe

(~, T)

normale.

En

_effet,

nous avons

trouv6,

pour

FeNi3,

que dans 1’etat

_{complètement}

ordonn6,

la courbe

_{(a, T)}

coincide

_pratiquement

avec celle du _{nickel pur,}

tandis que pour 1’etat

trempé

des d6viations

consi-d6rables ont été constat6es.

Remarque

de M.

_Meyer.

-- 11 est

remarquable

de

constater que,

d’apr6s

Marian,

l’alliage

Ni;Pt

possède

a 1’etat d6sordonn6 un moment

_sup6rieur

a celui de 1’etat

ordonn6,

tandis _{que le}

_phénomène

inverse se

_produit

_pour

_Ni,Mn

et

_Ni,Fe.

Réponse

de M. Smoluchowski. -The results obtai-ned

_by

Dr Went are most

_interesting

and

_they

should make it

_possible

to check the

_theory

here outlined.

A similar case has been

_recently

considered

_by

T. Muto et al

_(J.

de

_Phys.

Soc.

_{Jap." 1 g!~8, 3, 277-284).}

I cannot _{agree with} Dr

_Berg

that the distinction between ordered solid solutions and intermetallic

compounds

is so clear cut. The

_{quoted examples}

represent,

at

best,

a

_quantitative

rather than a

quali-tative difference in the

_equilibrium

condition. The kinetics of transformation

show,

on the other

hand,

a

_greater

_variety

and there a more close

_analysis

is necessary

[ see

for instance NEWKIRH J.

B.,

SMOLU-CHOWSKI R. et

al.,

Trans. A. I. M.

E’.,

ig5o,

188,

J.

_{A ppl. Phys. (in press)}

and Acia

_Cryst.

_(in

press);

also H.

SAT6,

Sc.

Rep.

Res. Toh6ku

Univ.,

1,

4o5l.

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