On ferromagnetism, paramagnetism and cohesive energy of transition metals and their alloys

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Submitted on 1 Jan 1951

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On ferromagnetism, paramagnetism and cohesive energy

of transition metals and their alloys

T.G. Owe Berg

To cite this version:

ON

_{FERROMAGNETISM,}

PARAMAGNETISM AND COHESIVE ENERGY

OF TRANSITION METALS AND THEIR ALLOYS

By

T. G. OWE BERG,

Avesta

_(Sweden).

Sommaire. - La

représentation usuelle du moment _{magnétique 03BC} des _alliages binaires en fonc-tion du nombre _atomique ou de la densité _{électronique} est étendue aux _{composés intermétalliques.}

A cet effet, le nombre S des électrons d _{« unpaired} » de l’atome _{métallique, qui} est _{égal à 03BC,} est exprimé en une fraction 03BD du nombre S° des électrons célibataires _(unpaired) de l’atome _libre,

donnant 03BC = $$ 03BD

03A3r

S0r (Mr), (M,.) étant la fraction _atomique de l’élément M,. de _l’alliage. La

fonc-tion 03BD = _{f(Z), Z} étant le numéro de groupe fractionnel Z =

_03A3rZr(Mr)

de _l’alliage, est calculée à

r

partir des données _{expérimentales.} La formule est _appliquée à des alliages binaires pour lesquels des observations expérimentales sont _disponibles. Dans le cas des alliages binaires du fer, la formule est

applicable dans un domaine 03B1, qui est _regardé comme contenant des solutions solides _parfaites. Dans un domaine suivant 03B1’, contenant une _phase « metastable », 03BC suit une courbe de _simple dilution. L’extension de ce domaine est

20/N-2

pour 100 _atomique, N étant le nombre des électrons cohésifs de

l’élément _d’alliage. Suivant le domaine _03B1’, un composé intermétallique se forme.

Le nombre d’électrons cohésifs N est égal à la somme des nombres d’électrons s et d’électrons d

cou-plés (paired), N = 2 + (1-03BD)S°. Les _énergies de cohésion L observées des métaux de la quatrième période donnent L = 25 N kcal: mol. Comme LT =

L0-~T0CpdT,

le nombre d’électrons cohésifs

à la température T peut être supposé NT = N0-1 25~T0

_{Cp dT et, par conséquent, ST=S0+1 25~T0}

CpdT.

Cette formule est utilisée pour le calcul de S à des températures élevées _qui est introduit dans la formule de la susceptibilité paramagnétique, donnée sous _{la forme ~} =

$$n03BC2BS(S+2)/k[T-03B8~S(S+2) 2].

Cette formule est _appliquée à la _{susceptibilité} du nickel et les valeurs calculées _{de ~} sont

compa-rées aux valeurs _{expérimentales} de Fallot. La formule _empirique ~ - 03B1 =

C/T-03B8

est déduite comme une _{approximation} de cette formule.

JOURNAL PHYSIQUE f2,

1951,

1. Introduction. -- The usual

representation

of the

_magnetic

_{moments ,u}

of

_{binary alloys}

as a

function of the electron

_{density, represented}

_by,

for

_instance,

the group number

Z,

is very useful in some

_{applications.}

In

_fact,

most of the results of section 3 can be derived from the function

However,

its

_{applicability}

is limited to ideal solid

solutions,

e. _g. it is not

_applicable

to intermetallic

compounds.

Furthermore,

it _says

_nothing

of the

mechanism

_by

which the

_magnetic

moment varies with the electron

_density.

It is

_possible,

however,

as will be shown in the

following,

to _express the function

₍₁₎

in a _{way that}

accounts for intermetallic

_compounds

as well and that

_gives

some information on the role of the d

elec-trons in

_{ferromagnetic}

substances.

The

_magnetic

_{moment u-}

at oo K and infinite field

strength

per atom of the substance

expressed

in

multiples

of the Bohr

_magneton

is

_equal

to the

number of

_unpaired

d electrons S. In an

_alloy

of elements M,. of atomic fractions

_(Mr),

the

_magnetic

moment is

In a metallic substance the number of

_unpaired

electrons S may be

expressed

as a fraction v of the

number of

_unpaired

electrons in the free atom

_So,

The fractian v can be

_expressed

as a function of the electron

_{density, represented by}

_{the group number}

Z,

where

Since,

for a solid

solution,

So is a function of

Z,

we obtain

(1)

after elimination of ) and

_So,

i. e.

For an intermetallic

_compound,

_however,

So is

not

_{given by (6)}

but

_depends

on the mechanism

by

which the

_compound

is formed. It has been

suggested

by

Fo6x

_[1]

that in the intermetallic

compounds

of Si with transition metals the Si

atoms 11

give

off "

their two _{p electrons} to the holes

in the d shell of the transition metal atoms.

Hence,

for

_Fe3Si,

For a solid solution of the same

_composition,

So would be

4

(Fe)

= 3. In section 3 this scheme will be

applied

to the

_{compounds Fe3Si, Fe3Al,}

and

_{Fe2 Ni.}

The

_computed

values of the

_magnetic

moments of these

_compounds

are in

_satisfactory

_agreement

with observation.

The function v

_{= j (Z)}

will be determined from

experimental

data in section 2.

We will now turn to the

_physical

_significance

of the function v.

The free atom of a _{non-transition metal may be}

described in terms of electron shells. The

valence, s,

electrons occupy an

_incomplete

shell exterior to the last

_completed

inert _gas shell. When such atoms

are

_{brought together}

in the

metal,

at distances smaller than the diameter of the s

_shell,

the s

elec-trons of

_neighbour

atoms _{will interact very}

consi-derably.

In

_fact,

it is no more

_possible

to

asso-ciate them with individual

atoms,

but

_they

will form an electron

_community

shared

_by

the

_neighbour

atoms. The cohesion of the metal is due to this electron

_community.

The electrons inside the

inert gas shell are little affected

_by

the

_neighbour

atoms.

The free atom of a transition element

contains,

in addition to the valence electrons and the

_complete

inert gas

shell,

an

_{incomplete d}

shell. The diameter

of the d shell can be

_computed

from formulae

_given

by

Slater

_[2].

The d shell is

_{considerably larger}

than the _{inert gas}

_shell,

and the sum of the radii of d and s shells is

_greater

than the distance between

neighbour

atoms in the metal.

Hence,

in the

_metal,

the d and s electrons of

_neighbour

atoms will

inte-ract as well as do the electrons. This interaction is

_generally

referred to as an

_overlap

between s a nd d

energy bands in the metal.

In the free atom of a transition element the d shell contains electrons of two

_spin

directions. As the d shell is

_gradually

_{built up, the first} 5 electrons

enter with one

_spin

direction and the

_following

5

elec-trons with the

_{opposite spin}

direction. The number of

_{unpaired spins}

_is,

thus S° = n or So = 10 - n

depending

upon whether

n ~

5. The effect of the s-d interaction referred to is to reduce the number of

_{unpaired spins.}

The function v defined

_{by (3)}

is a measure of the

_overlapping

of s-d _energy bands. Since S = 1) So d electrons remain

_unpaired

in the

metal,

(i - v)

S° d electrons have become

_paired

due to the s-d interaction.

In the

_{overlap region, s}

and d electrons can not

be

_{distinguished.}

Since it is

_part

of the s

band,

all its _{electrons may be looked upon} as s electrons.

Since it is also

_part

of the d

_band,

all its electrons may be looked upon as d electrons. _{We may,}

therefore,

consider them as s or d electrons

depen-ding

upon from which

point

of view we

regard

them.

However,

any way we look at

them,

they

belong

to the electron

_community.

_Hence,

due to

the s-d interaction

_{(i - v) S°d}

electrons enter the electron

_community

which

is, thus,

formed

_by

the s electrons

_{plus (I - 1) So}

d electrons. Since the cohesion of the metal is effected

_by

the electron

community,

we

_have,

_therefore,

on the _average and

supposing

the number of s electrons is two _per

atom,

binding

electrons _per

_atom,

i. e. N electrons _per

atom in the electron

_community.

It should be

_emphasized

that from the

_point

of view discussed

_above,

it is

immaterial,

if the

_y

--

1) )50

electrons are «looked upon as d electrons or s

elec-trons.

_Important

is

_only

that

_they

become

_paired

in the

metal,

i. e. that

_they

do not contribute to

the

_magnetic

_moment, and that

_they

act as

_binding

electrons.

It is

_customary

to assume that

_part

of the d electrons become

_{paired by s}

electrons

_entering

the d holes.

Thus,

in metallic

iron,

the number of

unpaired

d electrons

_being

2.2, the number of s

elec-trons should be 0.2. On this

assumption

we would

have o.2 s

electrons,

1.8 s electrons turned into d

electrons and 1.8 d electrons

_{paired by}

the s

elec-trons. This

_gives

a total of 0.2 ₊ 1.8

_{+ 1.8 == 3.8}

electrons involved in the cohesion due to their

being s

electrons or to their

_{being engaged}

in the

pairing.

For our _{purpose this}

_assumption

is of little

signi-ficance,

since we are interested in two electronic

properties only :

the

_magnetic

moment and the

N==2

₊₍₄

-

2.2)

= 3.8

_binding

electrons and 2.2

unpaired d

electrons. If the 3.8

_binding

electrons

are s or d electrons is of little

_importance

from this

_point

of view.

It will be shown in. section that the

_experimental

values of the cohesive energy L of the elements of the fourth

_period

can be

_{expressed by}

the

_simple

formula

N

_being

the number of

_binding

electrons

_given

by

(8)

for the transition elements and the number

of valence electrons for the non-transition elements.

In section 3 a further instance will be

_given

showing

the

_significance

of the formula

_(8).

There is in most

_binary

_systems

of iron a

_region

in which

the added metal acts

_effectively

as a diluent on the

magnetic

moment. The extension of this

_region

is

In many cases the value of N

_suggests

itself,

e. _g. 3 for

Al, 4

for

Si, 5

for

_{V, 7}

for Mn. In order that the

_alloys

of iron with

Ni, Pt,

Rh

_comply

with

₍₁₀₎

_{it is necessary} to take N from

_(8).

So far we have considered conditions at absolute

zero

_only.

At

_higher

_temperatures

the cohesive energy decreases

according

to the formula ’ °

where Lo is the cohesive _energy at oo K and

_Cup

is the

_specific

heat of the metal at constant _pressure.

Assuming

(9)

to be valid at all

_temperatures

we can

_compute

_1V~

from

₍₁₁₎

This

_gives

with

₍₈₎

The value of S

_computed

from

₍₁₃₎

will be used in section 5 for the calculation of the

_paramagnetic

susceptibility

of nickel.

2. Determination of the function ~. - It has

been found

_{experimentally}

that the

_alloys

in the

same column of Table I have the same

_magnetic

moments,

at least within a certain range _of

compo-sition.

TABLE I.

Hence,

the values of Z and So should be the same

for all elements within the same _{group of} the

_periodic

system.

The

_similarity

between the

_alloys

listed in columns 7 and

8,

respectively,

of Table I

_suggests

that Ti and V do not behave as transition elements

in their Ni

_{alloys. Similarly}

Cr, Mo,

W behave

as non-transition elements in their Ni

_alloys

but,

on

_{the . contrary,}

as transition elements in their Fe

alloys.

This dualism must be taken into account

when

_ascribing

values of Z and So to those elements. Table II

_gives

the values of Z and So obtained from these considerations.

The values of Z and So

_given

in Table II inserted in

_{(2), (3)}

and

_{(5) give}

with the

_experimental

data

on

Fe-Cr,

Co-Ni and Ni-Cu

_alloys.

’

For Z 6.36 and Z > ~ 0.606 we have o.

Two values are

_given

for v in

_(14),

one for Co in cubic lattices and one for Co in

_hexagonal

lattices.

Apart

from the

_hexagonal

Co

_alloys,

all

_alloys

on

which measurements are available are

_cubic,

face-centered or

_{body-centered.}

In the

_region

8

Z ~

g the function v

= j (Z)

has been

_extrapolated

from both sides

until

the two branches interesect. The function v =

f (Z)

thus

_computed

is shown in

_figure

I. In

addition,

the values of v

_computed

from a few other

_alloys

have been

_plotted

in

_figure

I. In

principle,

one

system

from each _group of Table I has been chosen for this

_plot.

In order to illustrate the

_agreement

between the

_systems

within one column of Table I

the

_experimental

data for

_Ni-Ti,

Ni-Si and Ni-Sn

alloys

are shown in

_figure

2 with the curve

_computed

from

_{(2), (3), (4),}

₍₅₎

and

_(14).

In _{the paper}

_{originally presented}

to the

_Congress

the

_magnetic

moments of the

_systems

_{Fe-Cr, Fe-W,}

TABLE If.

Fig. r , _ . ,, = f ~Z~,

of the

_{general properties}

of those

_systems

as shown

by

equilibrium diagrams

and other

_pertinent

data. Since it is not

_possible

to include all that information in the Transactions of the

_Congress,

it can

_only

be stated that the calculated values of _1-4 are in

satis-factory

agreement

with the observed values.

3.

_Application

to

_binary

_systems

of iron.

--’rhe

_majority

of the

_binary

_systems

of iron contain intermetallic

_compounds

and

_regions

of "

metas-table "

_phases,

slow transformations and

422

section I for the calculation

_{of u}

are

_applicable.

Fig. 2. -

Magnetic moments of Ni-Sn, Ni-Si, and Ni-Ti alloys.

The

_magnetic

moments of the

_alloys

of Fe with

Al,

Si, Mn, Ni, Pt, Co, Rh, Ru, Os, Ir, V,

Cr and

W,

are shown in

_figures

3-io. ,

Fig. 3. - Magnetic moments of Fe-Al alloys.

In the

_general

case, there are three different

_regions

in the Fe

_{systems :}

1. a

_region

« in which _;~ follows the formulae of

section 1.

2. a

region

«’ in which _1J. decreases

_linearly

with

increasing

content of the added metal

towards,

essentially,

the value of ioo _per 100 of the added

metal,

i. e. the added metal acts,

essentially,

as a mere diluent.

Fig. 4. -

Magnetic moments g, lattice _paremeter a,

and anisotropy constant K, of Fe-Si alloys.

423

3. a

_region

in which an intermetallic

_compound

Fe3Si, Fe2Mn(?), Fe2Ni, Fe2Pt, FeV]

is

formed,

and in which the value

_{of ,u}

decreases

linearly

to that of the intermetallic

_compound.

The extension of the

_region

a’ is

_{given by (10).}

Table III shows the limits of the various

_regions,

the

_composition

of the intermetallic

_compound

and the value of N to be inserted in

(10).

Fig. 6. -

Magnetic moments of Fe-Ni and Fe-Pt alloys.

TABLE III.

The values

_{of rj-}

have been

_computed

for the intermetallic

compounds

Fe3AI, Fe3Si,

FeSi,

Fe2Ni,

Fe2Mn according

to the mechanism

_{suggested by}

Fo6x as discussed in section I. These calculations

are

_given

in Table IV.

TABLE IV.

The extension of the oc’

_region

as

_computed

from

₍₁₀₎

is in

_satisfactory

_agreement

with

Fig. ;. -

Magnetic moments of Fe-Co and Fe-Rh alloys.

moment measurements are not

_reliable,

since

_they

were made on

_alloys

that could not

_possibly

have been in

_equilibrium.

This

_applies

also to Fe-Ni as

studied

_by

Peschard

_[3].

The

_points

observed

_by

Mathieu

_[4]

on Fe-Ni

specimens

which were

_severely

cold worked at low

_temperatures

and which were,

therefore,

closer to

_equilibrium

_{agree better} with

values of Fe-Pt

_{alloys given by}

Fallot

_[5]

and with the

_X-ray

measurements

_by

Owen

_[6].

Fig. i o. -

Magnetic moments of Fe-Cr and Fe-W alloys.

The limits of the « and «’

regions

in the

system

Fe-Mn

may be taken from observations on other

_properties.

According

to

_X-ray

measurements

_by

Troiano and Mc Guire

_[7]

the limit of the «

region

should be

3.4

at

_%

Mn. The curves on most

_properties

show

a break at about

_7.5

at

_%

Mn as shown for

ins-tance in the coefficient of thermal

_expansion

_given

by

Schulze

_{[8] (fig. i i).}

The

_{compound Fe~}

Mn has

not been observed. The

_computed

curve shown in

figure

5

is, therefore,

somewhat

_arbitrary.

The

_computed

values of _~. of the intermetallic

compounds given

in Table _{IV agree within} the

experimental

error with observation. The _~. curve

according

to Fallot

_{[9] (fig. 3)}

on Fe-Al

_alloys

indicate that

_Fe,Al

can be formed in two _ways,

the Al atoms 11

_giving

off "

their one _{p electron} or

all their three s, p electrons to the Fe d holes.

In

_figure

3 the notations are a the a’ curve, b the

curve

_computed

for annealed

_alloys

on the

suppo-sition that Al "

gives

off "

one electron in

_Fe,Al

and three electrons in

_FeAl,

and d the curve

between

_Fe,Al

and FeAl

_assuming

that Al "

_gives

off "

three electrons in

Fe3Al

and FeAl. The curve c

is valid for

_quenched

alloys

in which the formation of

_Fe3AI

is inhibited.

It follows from the considerations on oc"

_regions

and the formation of intermetallic

_compounds

that

there is no intermetallic

_compound

in the

_system

Fe-Co. There is an ordered structure FeCo which is not an intermetallic

_compound,

however. Nor is

the break at 23.5 atoms

_%

Co in the curves _on

f1-and lattice

_parameter

associated with a

_phase

change.

This is

_apparent

from the v curve

(fig.i)

which shows no such break. ’ ‘

Fig. 11. - Coefficient of thermal

expansion of Fe-Mn alloys.

The

_region

oc’ contains in the

_systems

Fe-Ni and Fe-Mn a metastable

_phase

which has been observed with

_X-rays.

Little is known of this

_phase.

It

gives

rise to broad and diffuse

_X-ray

lines

_indicating

a distorted lattice. In the

_alloys

of Fe with

tran-sition metals it is associated with

_{irreversibility}

and

_sluggishness

of the y « oc transformation.

4. The cohesive energy of transition metals

and their

_alloys.

-- The

cohesive energy

Lo

at oo K

is defined as the difference between the _energy of the vapour

(free atom)

and that of the metal.

Hence,

Lo

is

_equal

to the heat of sublimation

at 0° K. Observed values on the cohesive _energy

(at

room

_temperature)

are

given

in Table V for the

metals in the fourth

_period.

The data have been taken from Landolt-B6rnstein

_[10],

Seitz

_[11]

(quoted

from Bichowski and Rossini

_[12),

and

Fr6hlich

_{[ 13~.}

The values

of L

underlined in Table V are

N

within 10

%

of

25 kcal / g

atom. The values of

N within brackets

_(for

_{V, Cr,}

_Mn)

have been obtained

_{by dividing}

the

_experimental

values

of L

_{by a5.}

In those metals a d-d interaction must

be assumed in addition to the s-d interaction.

Although

the

_spread

is

considerable,

the data

TABLE B.

The metals of Z > 11 have a

_considerably

lower cohesive energy than that

_{given by (9).}

This a

probably

due to

_repulsion

between the filled d shells. Table VI

_gives

the values of L and 25

N - L,

N

_being

the number of valence

electrons,

for a

few metals of this

_type.

TABLE ill.

The difference a5 N - L is very

nearly

the same,

23 kcal

_/

_g

_atom,

for all those metals.

For

_alloys

no data are available on cohesive

energies.

In

_figure

12 are shown calculated values

of L and observed values of Brinell hardness

_[14]

of Fe-Cr

_{alloys. According}

to

_figure

1 o the

magnetic

moment, i. e. the number of

_unpaired

d electrons

decreases

_linearly

with

_increasing

Cr content and becomes zero at 82 at

_?’6

Cr.

Hence,

the number of

paired

electrons

and, therefore,

N increases

_linearly

up to 82 at

%

Cr as shown in

_{figure 12.}

From 82 at

%

Cr,

L

_drops

to the value _{of pure Cr.}

6. The

_{paramagnetism}

of nickel. - At

high

temperatures

the

_paramagnetic

_{susceptibility}

is

given by

~ ~.

-where Tc is a " I _critical

temperature

".

A

_being

the

_exchange

_energy, 0 the Curie

tempe-rature. The value of S in

₍₁₅₎

_{may be taken} from

_(13).

Fig. I2. - Coesive

energy L and Brinell hardness H of Fe-Cr _alloys.

The classical formula

_{corresponding}

to

₍₁₅₎

con-tains

_ja§

S~.

_Quantum

_{mechanics, however,}

_gives,

as a

_rule,

the

_{product k}

_{(k + i)}

when classical

mechanics

_gives

_{the square of} a

_quantum

number k2.

Hence,

we _{may consider the} atom of S

_unpaired

d electrons as

_having

a classical

_magnetic

moment

of

~.~ "~ t’~ 3 + ~ ~ ~

. In

analogy,

we may take for

the

_exchange

_{energy between} one electron and an

atom of S

_unpaired

electrons

Hence,

we _{may define the Curie}

_temperature

for

a metal as

and the 11

427

TABLE

(19)

inserted in

_{(15) gives}

This formula contains no

_arbitrary

constant. For 0

we will take the Curie

_temperature

from

_specific

heat _{measurements,} 0 -- 6260 K for nickel.

Table VII

_gives

the observed _{values Xobs} for nickel

given by

Fallot

_[15],

the values of

computed

from

_specific

heat

_{data, S}

as

_computed

from

_(13),

the Curie constant

the " critical

_temperature

"

Fig. r 3. --~ Xcälc

as a function of _temperature for nickel.

Xobs

Fig. r 4. - Curie

temperature 0 computed from (20) and observed values of x.

values _Xcalc, the ratio between calculated and obser-ved values and the value of 0

_{(denoted 0’)}

as

X hs

computed

from

(20)

and

_Figure

13 shows a

plot

of and

_{figure 14}

a

_plot

of 0’ as functions of

;t (,bs

temperature.

The

_agreement

between _Z,,,, and Zobs is within the

_experimental

error above 13000 K.

It has become

_customary

to _express

_empirical

data

_by

the

_empirical

formula.

where a is a constant

_{paramagnetism.}

This formula

Putting ~~

-

QT,

=

C~

(T ---

Tl)

and

_neglecting

the

Fig. 5. -I = _{I (T), -} I = f (T) and Q = f (T) for nickel,

z

’

a _computed from _(22), 0 = _T~ = from (15).

quadratic

term

_{(T -}

_Tl)2

we

obtain-(21)

from

₍₂₀₎

with

Hence,

if we

_compute

_ST1

and

_Ch1

at

_temperature

_T,

from

₍₁₃₎

we can

_compute

C’ and a from

_(22).

With

Tl

=

1073°

K _we obtain

a===2-57

_x from

1

(22)

and,

from the

_slope

of the curve

_{= f (T),}

X - cl

C’ =

3784

_X zo-s. The intersection of the _curve

with the abscissa

_gives

Ta

=

721.5°

K. The value of C’

_given

_{by (22)}

is C’ = 2835 _x io-1. If the

qua-dratic term is not

_neglected,

we obtain I

in better

_agreement

with the

_slope

of the curve.

A number of

_{investigators}

have

_{applied (21)}

to

their data and chosen a so as to

_give

the best fit. The values of a

_{given by}

Fallot

_[15],

_Terry

_[16],

and Gustafsson

_[17]

are 2.12 _{X 10-6,}

_2.~~E

x 10-6 and

3 .3 g .

i _o-G,

respectively.

Above the

_{melting point}

_1728°

K we obtain from our formulae in the range

i~oo-i 600~ ~,~=3.28x10"~

C =

4187

_x 10-6. The _curve

gives

G’ =

4150 x

_io-G,

0 = 3o8O K.

The values of 0

_plotted

in

_figure

15 have been

computed

from

_{(15), (13)}

and observed values

_{of z}

putting

0 =

T,.

Acknowledgement. -

The work

_reported

in this paper was carried out in the Research

Labo-ratory

of Avesta Jernverks

_{AB, Avesta,}

Sweden. I am indebted to the

_Management

of that

_Company

for

_permission

to

_publish

this

_report.

_My

thanks

are due to Mr G.

_Lilljekvist,

Avesta,

for valuable discussions on,

_{particularly, metallurgical}

and

metal-lographic questions.

I am

_particularly

indebted to

Professor E.

_Rudberg,

Director of the

Metallo-graphic

Institute,

Stockholm,

for his constant

encouragement

during

the _progress of the work. Since this work was

_completed

in the

_early

_part

of

₁₉₄₉

the more recent

_{publications pertaining}

to the

_subjects

treated have not been taken into

account,

nor included in the list of references.

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