Short-range order, atomic displacements and effective interatomic ordering energies in TiN0.82

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Short-range order, atomic displacements and effective

interatomic ordering energies in TiN0.82

T. Priem, B. Beuneu, C.H. de Novion, A. Finel, F. Livet

To cite this version:

Short-range

order,

atomic

_{displacements}

and effective

interatomic

_{ordering energies}

in

_{TiN0.82 (*)}

T. Priem

_(1),

B. Beuneu

_(1),

C. H. de Novion

_(1),

A. Finel

₍₂₎

and F.

_{Livet (3)}

(1)

C.E.A./IPDI/DMECN/DTech., Laboratoire des Solides Irradiés, Ecole

_{Polytechnique,}

91128 Palaiseau

_Cedex,

France

(2)

O.M./ONERA, BP 72, 92322 Chatillon Cedex, France

(3)

Laboratoire de

_{Thermodynamique}

et

_{Physico-Chimie}

_{Métallurgiques,}

_{I.N.P.G., BP 75,}

38402 Saint Martin _d’Hères, France

(Reçu

le 5 octobre 1988, révisé le 26 avril 1989,

accepté

le 27 avril

₁₉₈₉₎

Résumé. 2014 La section efficace de diffusion diffuse

_élastique

de neutrons _par un monocristal

TiN0,82

a été mesurée à

_{l’équilibre thermodynamique}

à

_700,

800 et 900 °C. L’intensité diffuse est

maximale aux

_points

de type

(1/2

1/2

_1/2)

du réseau

_réciproque.

Les lacunes d’azote se mettent

préférentiellement

en

_positions

de 3es voisins sur le sous-réseau métalloïde c.f.c. Des

_énergies

effectives de mise en ordre de

_paires

d’atomes ont été calculées, par

l’approximation

de

_champ

moyen, par simulations de Monte-Carlo et _{par la Méthode de Variation} d’Amas ; les deux dernières

_techniques

ont donné des

_énergies

très

_{proches, indépendantes}

de la

_{température :}

V1

~ 82 et

V2

~ 62 meV pour les

premiers

et seconds voisins azote

_{respectivement ;}

_V3

et les

énergies

d’interaction

_plus

lointaines sont très faibles. Les atomes de titane

_{s’éloignent,}

et les

atomes d’azote se

_rapprochent

de leurs lacunes

_premières

_voisines, de

_0,042

et

_{0,024 Å}

respectivement.

Abstract. - The elastic diffuse neutron

scattering

of a

_TiN0.82

_{single crystal}

has been measured in

thermodynamic equilibrium

at _700, 800 and 900 °C. The diffuse

_intensity

is maximum at the

_(1/2

1/2

_1/2)

type

reciprocal

lattice

_{positions. Nitrogen}

vacancies are found to situate

_{preferentially}

as

third

_neighbours

on the metalloid f.c.c. sublattice. Pair interaction

_{ordering energies}

were

calculated

_by

mean-field

_{approximation,}

Monte-Carlo simulations, and Cluster Variation

Method ;

the last two methods

_give

_{very similar temperature}

_{independent pair energies :}

V1

~ 82 and

V2

~ 62 meV for first and second

nitrogen

neighbours

respectively ; V3

and further

interaction

_energies

are _{very small. Titanium} atoms are found to _{relax away, and}

_nitrogen

atoms to relax toward their first

_neighbour

vacancies,

by

respectively

0.042 and 0.024

A.

Classification

Physics

Abstracts 61.60 61.70B -

64.60

(*)

Experiment performed

at Laboratoire Léon Brillouin

_(laboratoire

commun

_CEA-CNRS),

Saclay,

France.

1. Introduction.

Titanium mononitride

_TiNx

is a metallic

_compound

with the f.c.c. NaCI

_type

_crystal

structure.

It shows extreme

_hardness,

a

_{high melting}

_{point (up}

to 2 950 °C for x =

1)

and

superconduc-tivity

(T,

= 5.49 K for stoichiometric

TiN) [1].

TiN has

important applications

_as thin films

and

_coatings

for mechanical wear resistance of

_cutting

tools.

TiNx

displays

large departures

from the stoichiometric

_composition,

and can be found with

a

_nitrogen

concentration x between 0.5 and 1.15.

_{Non-stoichiometry}

is accommodated

_by

nitrogen

vacancies for 0.5 x 1 and

_by

titanium vacancies for x:> 1

[1].

It has been shown

that these vacancies affect

_{thermodynamic,}

mechanical, electrical,

magnetic

and

supercon-ducting properties.

For

_example,

_(i)

the

_{superconducting}

critical

_temperature

decreases with vacancy

concentration,

(ii)

the lattice

parameter

afcc increases with

nitrogen

concentration for

x «-- 1,

attains a maximum value

4.24 Â

for stoichiometric

TiN,

and then decreases for x > 1

_[1].

In transition metal carbides and

nitrides,

vacancies are not

_randomly

_distributed,

but are

organized

in a

_long

or

_short-range

order.

Furthermore,

vacancies induce also small local

distortions of the lattice.

_{Experimentally,}

the

_short-range

order

_(SRO)

and the local

distortions

_give

rise to a diffuse

_intensity

in the

_electron,

_X-ray

or neutron

_scattering

_spectra,

which has been observed

_[2].

In

_{particular, Billingham et}

al.

_[3]

have seen electron diffuse

scattering

surfaces in

_TiNx

for x between 0.5 and

0.75 ;

this

observation,

later extended to

0.65 =s= x 0.88

by Nagakura

and Kusunoki

[4],

has been

tentatively explained by

a

short-range order model

[5].

However,

(i)

electron

scattering

does not allow

_quantitative

measurements of the diffuse

_intensity,

_(ii)

_X-ray

measurements in

_{NbCo.72 [6]}

and

_elementary

considerations on

_{scattering amplitudes}

₍

_{1 fmetal 1 > 1 fN 1 , 1 fc 1 )}

show that

_X-ray

and electron diffuse intensities are dominated

_by

metallic atom

_{displacements.}

Neutrons are therefore

necessary to

_study

the

_short-range

order of metalloid atoms in transition metal carbides and nitrides.

At low

_nitrogen

content

_{(0. 5 -- x -- 0.61 ), TiNx}

annealed below 800 °C shows a

,6’-Ti2N

superstructure

of space group

I41/amd,

where

_nitrogen

vacancies are

_long-range

ordered

_(LRO).

This

_{superstructure}

is characterized

_by

₍₁

1/2

₀₎

diffraction

_spots

in the

reciprocal

lattice

_[2,

4, 7,

8].

&’-Ti2N

is

metastable,

as at

_{750 °C,}

the

_{following phase}

transformation sequence was

_directly

observed

_by

neutron diffraction

_{[9] :}

Quenched

where 6 = disordered rocksalt _structure

TiNx,

and

e-Ti2N

is a stable

_phase

with the

_tetragonal

antirutile structure

_{[10] (1).}

A theoretical work has

_recently

been undertaken to _{understand this vacancy}

_ordering

in transition metal nitrides and carbides from their band structure. Calculations based on the

generalized perturbation

method and in the coherent

_potential

_{approximation}

were

_developed

by

Landesman et al.

_{[13, 14].}

These authors used an

_Ising

model where

_{only pair}

interactions

are taken into account, and assumed

_temperature

_equal

to 0

K,

the lattice

_rigid

and the transfer

_{integrals independent}

of concentration x. _{The calculations gave the effective}

_ordering

pair energies V n

(defined

precisely

in Sect.

₅₎

between n-th

_neighbour

metalloid sites versus

the relative

_{filling Ne}

of metal 3d-metalloid

_2p

bands

_(see

_Fig.

_1).

These

_energies

decrease

(1)

Two other

_TiNx

_phases

in the

_composition

range 0.3 -- x:5 0.5 have been

recently

discovered

_[11,

rapidly

with

_distance,

so that for n -- 5

_they

are

_negligible.

For

_TiNo.82,

0.43,

and one obtains from

_figure

1 :

Fig.

1. - Pair interaction

energies Vi

versus band

_{filling Ne}

for transition metal nitrides

M6N5,

calculated

_by

the

_{generalized perturbation}

method

_(G.

_{Treglia, private}

_{communication).}

The

_short-range

order structure of

_TiNo.g2

can be

_predicted

from the

_{stability diagrams}

of

Clapp

and Moss

_{[15, 16]}

mean field

_{approximation}

_depicted

in

_figure

2 : from the above

Vl, V2

and

_V3

_energies,

it is found in the

_(1/2

1/2

_1/2)

field,

i.e. the diffuse

_intensity

is

predicted

to be maximum at the

_(1/2

1/2

_1/2)

_type

_positions

in the

_reciprocal

lattice. On the other

_hand,

the theoretical calculations mentioned above did not succeed to

_explain

the occurrence of the LRO

_{(1 1/2 0)}

_{6’-Ti2N phase,}

which

_{requires V, -- 2 V2 e 0 (if}

V3

is

_small).

The same

_difficulty

was encountered in a theoretical calculation based on the

Fig.

2.

_{- Clapp}

and Moss

_{stability diagrams}

of

_short-range

order in f.c.c. solid solutions

_{[16] a)}

versus

V

and V2

(V 3

=

0) ; b)

_versus

V 2/V

and

V 3/V l’

In each

_region

is indicated the

_point

of the

reciprocal

lattice where the diffuse

_intensity

is maximum.

_{Representative points}

for

_{TiNo.82 :}

_(e)

theoretical calculation

by

the

_{generalized perturbation}

method

_(from

Fig.

1), (0)

from

_{analysis of}

_present neutron

diffuse

_scattering

data

_by

the

_Clapp

and Moss formula,

(x)

idem,

by

Monte-Carlo and inverse CVM methods.

reasonable

_limits),

the

_{8 ’-Ti2N}

_phase

could

_only

be found stable in a narrow electron concentration range

(Ne c-- 0.5),

with a _{very small}

_stability

_energy.

The

_problem

is now to determine

_{experimentally}

the

_short-range

order of

_TiNo.82

and the values of the

_pair

interaction

_potentials.

_Preliminary

diffuse neutron

_scattering

data on

powder samples

TiNx

[17]

gave rather

unphysical

results

_(negative

first four SRO

_{coefficients,}

see

_{Sect. 6).}

In

_[18],

we have

_found,

_by

neutron

_scattering

at room

_temperature

on a

TiNo.82 single

crystal

(in

a

_quenched

state)

maxima of diffuse

intensity

in

(1/2

1/2

1/2).

In

neutron

_{scattering experiments,}

the

_strong

_high

temperature

phonon scattering

can be

eliminated

_by

_energy

_{analysis ;}

for this reason, the diffuse

_scattering

_experiments

can be

carried in the

_region

of the

_{phase diagram}

where the

_system

is disordered and where

_high

problem

of

_samples).

In the

_present

_work,

the

_short-range

order

_parameters

afmn and the

mean static atomic

_{displacements}

have been determined for the same

_TiNo.82

_single

_crystal,

in

situ,

at

700,

800 and 900 °C. Then the

_pair

interaction

_energies

were calculated

_by

three

different methods

_(Clapp

and Moss mean-field

_{approximation,}

Monte-Carlo simulation and Inverse Cluster Variation

Method).

2.

Experimental.

The

_TiNo.82

_{single crystal}

was

_prepared

_by

a

_{zone-annealing technique}

_[19].

Details on

_sample

characterization are

_given

in reference

_[18].

The diffuse neutron

_scattering

_experiment

was

_performed

at three

_temperatures

( T

=

700 °C,

800 °C,

900 °C)

_on the two axis

_spectrometer

G4-4

[20]

at Laboratoire Léon

Brillouin

_(2),

_{C.E.N.-Saclay,}

France.

_{The {001}}

_{and (lÎ0)}

_planes

of the

_{reciprocal lattice}

were

_explored

with an incident

_{wavelength À}

=

2.56 A

(resolution Ak /,k

4 _x

10- 2).

The

sample

was in a vacuum chamber

( -

10-6

torr).

48

He3

detectors of diameter 50 mm with

time-of-flight analysis

and rotation m of the

_sample,

allowed to obtain the elastic diffuse

cross-section for :

0 . 5 _

_Q

= 4 7T sin 0

/,k --

4.5 Â

with

_steps

2.50

°

in 2(J and 3 °

in co. The furnace was a

niobium foil heater.

The data were calibrated to absolute units

_{by comparison}

with a vanadium standard after

background

correction,

and were corrected for effective

_absorption,

total incoherent

scattering

and

_Debye-Waller

factor,

as described below.

The

_intensity

in the two

_{reciprocal planes}

is calculated in Laüe units

₍₁

Laüe =

x (1 - x )

b 2

= 0.130

barns,

_see

Sect. 3)

by

the formula :

where :

1 TiN,

Iv,

IBN, I vac

are

_respectively

the numbers of detected neutrons when

_sample,

vanadium

_standard,

boron nitride

_{(neutron absorbant)}

and vacuum are in the beam.

. _{a TiN and av} are the effective transmission coefficients for

TiNo.82

and vanadium

(due

to

nuclear

_absorption

+ incoherent + Laüe

_{scattering) :}

CTIN = 0.46 and

av = 0.57 for

À =

2.56 Â

(calculated

from the data of Ref.

[21]).

. C is a normalization coefficient which has the

following expression :

is the incoherent differential cross-section in Laüe units :

u oh, U nc

and

_{u MS}

are the

_coherent,

incoherent and total

_{multiple scattering}

cross-sections

(the

multiple scattering

cross-section was calculated

_by

the method of Blech and Averbach

[22]).

NT;

and

_Nv

are the numbers of titanium and vanadium atoms in the beam.

BT;N

and

_Bv

are the

_Debye-Waller

factors for titanium nitride and vanadium

_{(BTiN =}

BTi BN

= 0.82

Â2

for T =

700 °C,

0.89

Â2

for T =

800 °C,

0.97

Â2

for T = 900 °C

[23],

Bv

= 0.57

Â2

at 300 K

_[24]).

The

_{inelastically}

scattered neutrons are due to the interaction with the

_phonons.

_{The energy} resolution in our

_experimental

conditions was AE == 25 meV for

_phonon

annihilation and

7 meV for

_phonon

creation : for most of the

_{time-of-flight}

spectra,

it is

_relatively

_easy to

separate

the elastic and the non-elastic neutrons

_{(Fig. 3) ;}

in

_particular,

the

_{optical phonons}

(60-80

meV)

and the acoustical

_phonons

near the Brillouin zone border

(30-40 meV)

are eliminated.

Nevertheless,

close to the

_Bragg

reflexions,

as the

_phonon

_energies

are

_low,

the

separation

between the

_phonon

_scattering

and the elastic

_scattering

cannot be done.

_By

comparison

with the

_phonon

spectrum

of TiN

_[25],

one can deduce that the low energy

acoustic

_phonons

are not

_separated

within a zone of dimensions = 0.2-0.3

(in hl,

h2, h3 units)

around the

_{Bragg peaks.}

This is the zone where

_{abnormally high}

intensities are

observed

_{(Fig. 4) ;}

the

_{corresponding}

data had to be eliminated _{in the least squares fit}

_(3).

The normalization factors C

_(for

_Eq.

₍₁₎₎

of both

_experiments

in the

_{001}

and

{IIO}

planes,

are

_adjusted

in order to obtain the same diffuse

_{intensity along}

the common

line of the two

_planes

_(the

_correction,

due to uncertainties on the values of

_Nv,

NTi,

a v aTiN, is about

10 %).

Fig.

3. - Number of counted neutrons versus

_{time-of-flight}

_per meter _{and energy transfer} to the

neutron recorded in a

He3 detector

on _spectrometer

_{G4-4, Orphée}

_reactor,

_Saclay,

for

_TiNo.82

at 900 °C.

3. Results.

Preliminary

thermal

_cycles

without

_{time-of-flight}

were carried out between 700 and 900 °C.

They

have shown that the diffuse

_intensity

is reversible with

_temperature

and that the

nitrogen

sublattice is in

_{thermodynamic equilibrium}

within a few minutes in this

_temperature

range.

(3)

A

high

resolution energy

experiment

at room _temperature,

_performed

on _{spectrometer D7,}

I.L.L.,

Grenoble, confirmed that the

large

diffuse

_intensity

observed in

_TiNo.82

close

tô

the

_Bragg

The elastic diffuse differential cross-section

_{du /dn}

of

_TiNo.82

for two

_temperatures

T = 700 °C and

900 °C,

measured with

_{time-of-flight analysis,}

is shown in

_{figure 4.}

_(For

T = 800

°C,

the

spectrum

has an

intermediary shape

between the 700 and 900 °C

ones.)

. In the

{001}

plane,

the diffuse

intensity

is concentrated

along

circles centered on the

reciprocal

node

_(110).

No maxima are observed at the

_{Ti2N superlattice}

reflexion

_positions

(1 1/2 0).

. In the

{110}

plane,

the diffuse

intensity

streaks are

_{roughly parallel}

to the

(001)

direction. The

_intensity

is maximum at the

_(1/2

1/2

_1/2)

type

positions.

. The

_high

_{temperature spectra}

have the same

_shape

as for the

_{quenched sample}

studied in

[18].

Diffuse

_scattering

is concentrated on the surface observed

_by

electron

diffraction,

but

the

_intensity

is far to be constant

_along

this surface and far to be

_periodic

in the

_reciprocal

lattice.

. The maxima of the diffuse

intensity

decrease with

increasing

_temperature.

The elastic diffuse

_{scattering intensity}

contains two terms. The first one is the

_short-range

order

contribution,

periodic

in the

_reciprocal

lattice. The second one is the non

_periodic

contribution of the static atomic

_{displacements.}

In

_TiNo.BZ,

the two terms have a

_comparable

importance.

Expanded

up to the second order with

_respect

to the static

_{displacements,}

which are much

smaller _{than afcc’ the elastic diffuse} cross-section for

_TiNx

has the

_{following expression}

_{[26] :}

where :

.

f,

m, n are the coordinates of the ideal atomic

_positions

in the

_{crystal :}

(a,

b,

c unit vectors of the f.c.c.

_cell).

9

hl,

h2, h3

are the coordinates of the

_scattering

vector

_Q

in the

_{reciprocal lattice}

_(of

unit

vectors

_{a *, b *,}

_{c *) :}

e _atmn is the

_{Cowley-Warren}

SRO coefficient ’ . atmn =

1 - IFtmn ,

where

Ftmn

is the

Fig.

4. -

Elastic diffuse

_intensity

(in

Laüe

_units)

for the

_{reciprocal planes}

_{001}

_(left)

and

{ll0}

(right)

of

_TiNo.82,

measured at

_high

_temperature

_by

neutron

_scattering.

_a)

T = 700

°C,

b)

T =

900 °C,

c)

reconstructed cross-section from the fit _parameters for T = 700 °C.

Bragg peaks

are

found at

_{hl=h2=h3=1; h1=2, h2=h3=0; h1=h2=2, h3=0;}

etc...

conditional

_probability

to find a _vacancy at the

_(f,

m,

n )

lattice

position,

if a

_nitrogen

is at the

Figure

4

_(continued).

Lém

and

_M#§gf

_represent

the

components

along

a and b of the relative

displacement

uPmn of A and B atoms

_{separated by Rpmn}

(... )

represents

the space average of the

quantity

into brackets.

. N = number of Ti _atoms,

bN

and

bTl

=

scattering

amplitudes

of N and Ti

(respectively

+ 0.94 x

10-12

cm and - 0.34 x

10-12

cm

_[21]).

The a imn’ ’yQmn and

5 imn

parameters

have been determined from the

_experimental

cross-section

_do-/df2

_by

the

_{least-squares}

method.

Due to _{low energy}

_{phonon scattering}

_(see

Sect.

_2),

measurements in small

_spheres

of the

reciprocal lattice

close to the

_Bragg

_peaks

_(radius

0.3 in

_{hi units)}

had to be eliminated from the fit.

In the

_fit,

the

_conditions

_aimn = 0

_{(including a«)}

has been

_{assumed ;}

if not, too _many

Imn

Bragg

peaks

!).

In

_fact,

from statistical

models,

the sum of the

_short-range

order

_parameters

can _{vary between 0 and 1}

₍₀

in the canonical

_system,

1 in the

_{grand-canonical}

_system).

In any case, in the

least-squares

fit,

the calculated

parameters

do not differ

_{significantly}

if we

_modify

the value of K

_{(0 =s=}

K --

_{1 )}

in the

_{condition E}

_{a Imn} = K. Imn

The fit was made with several sets of _{atmn’ ypmn and}

_{8 pmn}

_parameters.

As shown in table

I,

the first (X Qmn and Ytmn values were found

practically

insensitive to the number of

_parameters

chosen for the fit. The second order _{contribution is very}

small,

and as can be seen in table

_I,

the introduction of

_8tmn

_parameters

does not influence much the _{atmn and Yt.n}

_{values ;}

_only

two

_8tmn

have been considered.

_Contrary

to _{Ytmn, the}

_8tmn

increase with

_temperature,

which

means that

_{they incorporate}

some

_{phonon scattering.}

The Btmn

_parameters

do not _appear

because the first ones

_(eool

and

_~100)

are

_equal

to zero

_by

_symmetry.

Table I.

_{Effect o f the}

number

_{o f}

_parameters

_{o f}

the

_fit

on the

_first

_{a 1,,,n} and _’yemn

_(700°C).

Data in the two

_reciprocal

_planes

_{001}

_{and {110}}

have been used. These data lead to a

least-squares

fit matrix of rank

_equal

to the number of unknown

_parameters.

The atmn’ , y $mn and

8 Qmn

with (Ltmn)

and Llmn)

values

(from

the fit with

24 a,

19 y and 2

6)

are

given

in tables II and III.

In

fact,

the fit was made

_using

the data of every two detectors _{and every} two rotations w of the

_sample.

The four sets of

_parameters

_{obtained gave} an estimate of the statistical errors from

_experimental

data : these statistical errors for _atmn are

_given

in table II.

The reconstructed cross-sections from the fitted

_parameters

are shown in

_figure

4. The fit gave a« c= 1.06 to

_1.15,

instead of

1,

the theoretical value. This

_discrepancy

is

mainly

due to the difficulties to determine

_exactly

the real number of atoms of the

_sample

in the neutron beam and the transmission coefficient. If the error on the value of the incoherent

cross-section

_doi.,/df2

is assumed

_negligible,

this

_discrepancy

_{may be}

_interpreted

as a scale

factor for the cross-section unit. For this reason, the atmn, Ytmn and

8emn

parameters

have

Table II. - Non corrected atmn

parameters

and estimate

_{o f}

the statistical error

_{(into brackets)}

for

T =

700,

800 and 900 °C. For T = 700 °C are also

_given

the measured _atmn

_parameters

corrected

_{by formula}

_(2),

and the _atmn calculated

_by

Monte-Carlo

_from

the

_four

M.C.

V;

i

o f

table V.

The correction is about 3 x

10-3

on the first

_neighbour

_a011

_parameter,

and less for the

following

afmn ; the corrected afmn

parameters

are

_given

in table II for T = 700 °C.

4. Atomic

displacements.

As can be seen in

_figure

4,

for

_TiNo.82

the contribution of the atomic

_{displacements}

to the diffuse

_intensity

is not

_negligible

_(the

_intensity

_maps are not

_periodic

in the

_reciprocal

_{lattice) ;}

this contribution is shown in

_figure

5 for T’ = 700 °C. From table

III,

the static atomic

displacements

(Z.)

are seen to be

_independent

of

_temperature.

The average atomic

displacements

around a _vacancy

(Lp,;A

(A

= Ti or

_N)

are determined

Table III. - Values

_{o f}

_-yim,,

_8tmn,

_(Ltmn>

and

_(Llmn)

_for

T =

700,

800 and 900 °C.

The

_displacement

field around a

_nitrogen

_{vacancy deduced from the}

(L)

in

_TiNo.$Z

is shown in

_figure

6 : the nearest titanium atoms

_«al2,

_0,

₀₎

_position

and

_equivalents)

move

away from the vacancy, whereas the nearest

_nitrogen

atoms

_«a/2,

_{a/2, 0)}

_position

and

equivalents)

move _{towards the vacancy. The atomic}

_displacement

field around a _{vacancy in}

TiNo.82

is

_quite

similar to that observed in SRO transition metal carbides

_[26]

and to the one

calculated

_by

a lattice statics method in the

_{isomorphous compound}

UC

_[27].

For this reason, an

_attempt

has been made to calculate static atomic

_{displacements}

around a

nitrogen

vacancy in titanium nitride. The formalism

previously developed

for uranium carbide

_[27]

and based on the Green function method for lattice statics

[28]

has been used.

The elastic constants determined from the

_phonon

_spectrum

of

_TiNo.98

are

_given

in

reference

_[25].

The

_experimental

and calculated first

_neighbour

atomic

_{displacements}

are

compared

in table IV

_(columms

1 and

_2).

The calculated value of the first

_neighbour

titanium

_displacement

is in

_good

_agreement

with

Fig.

5. - First order static

displacements

contribution to the elastic diffuse

_intensity

_(TiNo.82,

T =

700 °C,

Laüe

units).

Fig.

6. - Static atomic

displacements

around a

_nitrogen

vacancy in

_TiNo.82

deduced from the l’ fmn coefficients.

(D) :

vacancy,

(0) :

nitrogen,

(.) :

titanium.

nitrogen displacement

differs in

_sign

from the

_experimental

one. _{This may be due} to the fact

that

_only

the Ti-Ti and the Ti-N first

_neighbour

elastic constants are known

_{(see [25]).}

A more exact evaluation of the atomic

_{displacements}

around a _vacancy

_requires

an

estimation of the elastic constants between N-N first

_neighbours.

The latter are connected to

the relative variation of lattice

_parameter

_{8afcc/ afcc}

induced

_by

metalloid vacancies

_by

a

relationship

which is

_given

in reference

_[27]

for rocksalt structure carbides and nitrides. For

N-N first

_neighbour

elastic constants :

₆₃

= - 0.6 in

_{2 eZ/aicc}

unit

_{(e :}

electron

_charge,

for notation see

_[27]).

The new set of atomic

_{displacements}

calculated with this value of

₆₃

is

_given

in table

IV,

column 3 : it is in better

_agreement

with

_experimental

data

_(Tab.

IV,

column

_1).

Table IV. -

Experimental

and calculated static atomic

_{displacements}

around a

_nitrogen

vacancy in

TiNo.82 (in a/2

units ;

a/2

= 2.114

À).

In

fact,

the

_experimental

data

_give

_{the average atomic}

_{displacements}

around a _{vacancy in}

the

_largely

non-stoichiometric

_compound

_TiNx,

whereas in the lattice statics

method,

one

considers the atomic

_{displacements}

around an isolated _{vacancy in stoichiometric TiN. In the}

simplest

model

_(additivity

of the

_displacement

vectors due to each

_vacancy),

two

configur-ations for the first atomic

_displacement

_{L °ô}

in

_TiN,,

are obtained :

or

The mean atomic

_displacement

L °ô’

is connected to the

_displacement

_LPOOTi

induced

_by

an _{isolated vacancy}

_through

the relation :

For x = 0.82 and a2()o

0.114,

one obtains :

Lioo

= 0.91

L °ô’.

Therefore the

(LPOOTi)

value calculated

_by

lattice statics for

_TiNo.82

is 0.013

_afcc/2

_{(=0.028 À),}

to be

compared

to 0.020

_afcc/2

₍₌

0.042

_Â)

determined

_{experimentally.}

5.

_{Short-range-order}

_parameters

and effective interatomic

_{ordering energies.}

The

_short-range

order contribution

_{a (Q)}

to the diffuse

_intensity

for T=700"C

_{(order 0}

term)

is shown in

_figure

7a for the two

_{reciprocal planes}

_{001}

_{and {110}}

where the

_intensity

(vacancy

concentration x

fixed),

the Hamiltonian which describes the order of vacancies

(Ising Hamiltonian)

is written as follows :

where the

_{sums £ , £ , £ , £}

run

_{respectively through}

the

first, second,

third and fourth 1 2 3 4

neighbour

pairs

in the metalloid sublattice.

_Ci

is the

_occupying

factor :

In this model the

_pair

interactions further than the 4th

_neighbours

are

_neglected

and so are

the

_triplet

and other cluster interactions

_[13].

Three methods are used here to deduce the

_Vi

from the

_(corrected)

measured

CY p"tn : :

_Clapp

and Moss mean field

_{approximation,}

Monte-Carlo

simulations,

and inverse

Cluster-Variation Method

_(CVM).

5.1 CLAPP AND Moss APPROXIMATION. - The

_Clapp

and Moss formula

_{(4) [15, 16]}

is the

simplest

mean-field

_{approximation.}

In this

formula,

the

_{short-range-order}

contribution

a (Q )

to the diffuse

intensity

is a

simple

analytic

function of the Fourier transform of the interaction

_potentials

where

vQ :

first Brillouin zone

_{volume ;}

_kB

is the Boltzmann constant and T the absolute

temperature.

From the measured

_{a (Q),}

a Fletcher and Powell _{least squares fit method}

_[30]

has been used to determine the

_Vi.

The extension of the interaction is assumed to be limited to a few

neighbouring

shells.

The

_{energies Vi}

obtained when

_limiting

the interaction to four

_neighbour

shells are

_given

in table V : the first and second

_neighbour

_{energies Vi}

_{and V2}

are

_positive

and much

_stronger

than the others :

_Vi

₌ 56

_meV,

_{V2/VI =}

0.71,

_V3/Vl

and

V4 /Vi s

0.03. A fit

_performed

with

_{eight energies}

does not

_{modify V}

_{and V2,}

but affects somewhat

_V3

and

_V4

which remain small

_(see

the values

_given

in

_caption

of Tab.

_V).

The

_intensity

calculated with the

_Clapp

and Moss formula

₍₄₎

and the

_Vi

of table V is

_given

in

_figure

7b for T = 700 °C.

According

to the

Clapp

and Moss maps

(Fig. 2),

the

short-range

order for

_V2/Vl:=t

0.71 and

_{V3/V1 -=-}

0.02 should be of the

_(1/2

1/2

_1/2)

_type,

in

_agreement

with the

_experimental

results

_(see

_Fig.

_7a).

5.2 MONTE-CARLO SIMULATIONS. - As the measured fluctuations are not _very

_strong

in this

system,

the Monte-Carlo method

_provides

_precise

_{values of afmn from known}

_Vi.

The Inverse Linearized Monte-Carlo method described in

_{[31, 32]}

has been used to

Fig.

7. -

Short-range

order contribution

_do-SRO/df2

to the elastic diffuse cross-section of

TiNo.g2

at 700 °C in

_{the {001} (left)}

_{and {1 10}}

_(right)

_reciprocal

lattice

_planes

_{(Laüe units).}

a)

Reconstructed from the

_experimental

_{a tmn’}

_b)

Calculated from

_Clapp

and Moss

_{ordering energies}

Figure

7

_{(continued) .}

Table V.

_{- Mean-field (Clapp-Moss),}

Inverse Monte Carlo and Inverse C.V.M.

_effective

nitrogen-nitrogen

ordering

energies

for

the

_four

_first

_neighbour

shells in

_TiNo.82,

calculated

_from

elastic neutron

_diffuse

_scattering

measurements. The uncertainties deduced

_from

« statistical errors » on _atmn

_{(Tab. II)}

are

_given

into brackets in the case

_{of C.V.M.}

calculations. The

_fit

_by

the

_Clapp

and Moss

_formula

₍₄₎

with 8

_potentials

at 700 °C _{gave :}

_V1

_{1 =}

_57,

_V2

=

41,

.Calculation

_from

the

_{four first}

_{atmn. - Table V} shows the results obtained in

_calculating

the four first

_Vi

from the four

_{equations provided by}

the four first measured _atmn’ The short-range order

intensity

can be recalculated from these

_Vi

_{(Fig. 7c) :}

it appears

_hardly

distinct

from the

_experimental

one

(Fig. 7a).

The set of _{Cr pmn}

_parameters

obtained from these

the four first a tmn

only,

the

following

a Imn

parameters

are in

_agreement

with the

experimental

ones

_(in

_particular

the

_sign

alternance is well

_reproduced).

This result

_supports

the

_ability

of a

model hamiltonian with four

_Vi

to describe the measured

_short-range

order.

2022 Calculation

fiom

the 24

first

_{atmn. - The}

_Vi

_{1 were also calculated} at 700 °C from the 24 _atmn

_given

in the first column of table

_II,

_by

a

_{least-squares}

fit method

_{(see [31, 32]).}

The

covariance matrix of this

_{least-squares}

fit leads to an estimate of the range of variation of the

Vi

which can account for the a

_{(Q ).}

The results are :

V

= 78 ± 9

meV,

V2

= 61 ± 6

meV,

V3

= - 2 ± 3

meV,

V4

= 1 ± 2

meV,

in

good

agree-ment with the values from 4 « pmn

given

in table V. A

X 2 test

on this fit

_suggests

that the errors on the _atmn

_parameters

must be about twice the statistical ones

_given

in table II.

It must be noticed _{here that if any}

_Vi

is

_changed

_{in the range of} error

_given,

the other ones

must be also modified. For

instance,

V3

and

_V4

cannot be let

_{simultaneously}

to zero.

5.3 CLUSTER VARIATION METHOD RESULTS. - It has

_long

been shown that the CVM

_[33]

is

a _very

_precise

_technique

for

_studying

statistical

_problems

in

_crystalline

_solids,

_provided

the basic

_clusters,

_{upon which the}

_algorithm

is

_build,

are

_{large enough. Basically,}

a CVM

_study

consists in

_minimizing

a free _{energy functional} where the exact

_entropy

has been

_replaced

_by

a linear combination of

_entropies

of finite clusters included in a

_given

basic cluster.

Until

_recently,

most of the CVM studies were done with interactions limited to lst and 2nd

neighbours,

and,

consequently,

with

_relatively

small clusters

_(up

to six

_{points) (for}

a recent

review,

see

_[34]).

In the

present

case, we want to consider

_pair

_{interactions up} to the 4th

neighbours.

According

to an

_optimized

_procedure

for

_selecting

the basic clusters

_{[35, 36],}

the

smallest clusters that we must use in such a case are the face-centered cube

_itself,

that contains

14

_points,

_and,

_{simultaneously,}

the 13

_point

cluster formed

_by

one site surrounded

_by

its

twelve first

_neighbours

_(see

_Fig.

_{8a) ;}

this

_{approximation}

is referred to as the 13-14

_point

Fig.

8. - Basic clusters for C.V.M. in f.c.c. lattices.

a)

13-14

point

approximation.

b)

TO

approximation.

These basic clusters are rather

large

and

lead,

in the disordered

_phase,

to a

free energy functional which

_depends

on 742 correlation functions.

We have used this 13-14

_point

_{approximation}

in an Inverse CVM

_algorithm

to determine

the

_pair

interactions up to

_V4.

The main

_steps

of that

_algorithm

are as follows

_[37].

From

guessed

values V =

(Vl,

V2, V3,

V4)

for the

pair

interactions,

_a direct minimizatibn of the

CVM free energy leads to the correlations included in the 13- and

_14-point

clusters ;

in

particular,

we calculate the

_short-range

order

_parameter

set a =

( a 1,

_{a 2, a 3,}

a 4 ).

_Suppose

now that a small variation & V is

_applied

to the set

_{V ;}

the induced variation Sa on the set a is

given,

to first

order,

by :

where X _

a a

is the

_{generalized susceptibility}

matrix : av

Hence,

if Aa is the deviation between the calculated a and the

_experimental

ones,

the variation d V needed to _{make up the deviation da is}

_given,

to first

_order,

_by

the solution of the linear

_system

The next trial value for the interaction set is then V + âV. We

stop

the process when the deviation âa is smaller than the

_experimental

error. In

_general,

this

_algorithm

_{converges very}

rapidly

(typically

5 or 6 iterations in

V).

We have used this Inverse CVM in the case of

_TiNo.g2.

The

_input

_parameters

are the first

four measured correlation functions _{atmn- Our results for the three} different

_temperatures

are

presented

in table V where the uncertainties

_correspond

to the statistical errors on

atmn

given

in table II. Note that the calculated interactions do not

_depend

on

T,

which proves,

first,

that the

_short-range

order in this

_system

can be

_precisely

described

by

an

_Ising

model

with

_pair

interactions limited to

_V4

_and,

_secondly,

that the

_experimental

data are _very

_precise.

We note also that the CVM results are similar to those obtained

_by

the Inverse Monte-Carlo method.

In order to have a more

_{complete comparison}

with the

_experimental

data,

we have

computed,

within the CVM

framework,

the

_short-range

order diffuse

_intensity,

which,

in Laue

units,

is

_{given by}

a

(Q ) (the

Fourier transform of the coefficients a

(R )).

_But,

and

that

point

is

_important,

_{a (Q)}

has not been

_{computed by}

a

_direct

Fourier transform of the coefficients

_{a (R)}

calculated

_during

the CVM

minimization, but, instead,

by

using

the

fluctuation-dissipation

theorem which says that

a (Q)

is

_proportional

to the

_staggered

susceptibility

X (Q) (for

more

_details,

see

_{[35, 38]).}

_However,

due to

_practical

reasons

(limitation

of

_computer

_{possibilities,}

even with a

CRAY-XMP),

the

_{susceptibility}

_{X (Q)}

cannot be

_{computed readily}

within the

_13-14-point

_{approximation.}

But we

remark,

at this

point,

that

_V3

and

_V4

are much smaller than

_VI

and

_V2

(V3/Vl =- V4/Vl =-- 0.04)

(see

Tab.

_V).

_Therefore,

_{for the purpose of}

_{computing X (Q)}

and as a first

approximation,

we can

use a lower order

CVM,

namely

the tetrahedron-octahedron one

_{(TO) (see Fig. 8b)}

_[39].

More

_precisely,

within the TO

_{approximation,}

interactions

_Vi

and

_V2

are treated

_correctly

mean-field-like _way

_(the

_{corresponding}

correlations are factorized on the

_point

correlation

functions) .

We have then

_computed

the

_short-range

order diffuse

_intensity,

via

_X(Q)

in the TO

approximation,

with

_Vi

= 83

meV,

V2

= 62

meV,

V3

= 3

meV,

V4

= 3

meV,

for the two

temperatures

T = 700 °C and T = 900 °C and x = 0.82

(Fig. 7d).

We _{can see} that these maps are

_hardly

distinct from the

_experimental

ones

(Fig. 7a).

5.4 DISCUSSION OF THE APPROXIMATIONS. - The three

_{approximations}

used here are « self

consistent » : in each case, a

(Q)

can be recalculated with the same

_{approximation}

and the

results are

_extremely

close to the

_experimental

ones.

The uncertainties on the Monte-Carlo

_Vi

were discussed above. We have seen in

section 5.2 that the total errors on the _atmn are

_probably

about twice the statistical ones

_given

in table II :

therefore,

the true uncertainties on the

_Vi

calculated

_by

C.V.M. should be twice

those

_given

in table V. We also evaluated the effect of normalization errors : at 700

°C,

the

computed

ordering energies

varied from

_VI

= 100

meV,

V2

= 72

meV,

V3

= 7

meV,

V4

= 6 meV from uncorrected

atmn, to

90, 67,

4 and 5 meV

respectively

from atmn corrected

by

formula

_(2).

In

conclusion,

the Monte-Carlo and C.V.M. results are in

_good

_agreement,

_V1

1 and

V2 being

determined with accuracies of

_{approximately ±}

10 %.

On the other

_hand,

the

_Clapp

and Moss

_{approximation gives}

much lower values of

V

and V 2.

This is

_{explained by}

the

_strong

« frustration » of these interactions in a fcc lattice

A, - x B x

(Fig. 9).

As the mean-field

_{approximation}

does not take into account neither the

frustration between first

_neighbour

_pairs,

nor the

_competition

between

_{positive V}

_and

V2,

the

_Clapp

and Moss formula

tends,

for a

_given

set of

_{a parameters,}

to underestimate the

pair

interactions. On the other

hand,

the basic clusters used here in the CVM

_{approximation}

are

_{large enough}

to include these frustration effects.

Fig.

9. - Frustration

phenomena

in a f.c.c. lattice with

_negative

first and second

_neighbour

SRO

parameters a

1 and

a 2.

a)

First

neighbours. b)

Second

neighbours.

5.5 DETERMINATION OF THE ORDER TEMPERATURE AND OF THE FUNDAMENTAL STATE.

Finally,

we consider now the

_problem

of the critical

_temperature

_Tc

of the order-disorder

transformation.

According

to the mean-field

_{approximation}

_(formula

_(4)),

the

_quantity

« -1

1

(Q

= 1/2 1/2

1/2 )

should be _a linear function of

1 / T

and intersect the horizontal axis for

the

_{reciprocal spinodal}

_temperature

_{1 / T$}

_(Ts

is in fact a lower bound of the order-disorder

transition

_Tc*

that could be determined in the mean-field

_{approximation).}

Our

_experimental

data

_{give effectively}

a

_straight

line

(see Fig. 10)

which,

by

extrapolation,

leads to

Fig.

10. - Plot of measured

_1/IsRo

_(1/2

1/2

_1/2)

versus

_{1 / T}

_(x

103)

in

_TiNo.82.

Using

the

_pair

interactions obtained

above,

it is of course attractive to

_compute

a

_phase

diagram

for the Ti-N

_system

around the

_{composition x}

= 0.82. Within the C.V.M.

framework,

this

_requires

the

_knowledge

of the

_ground

state at T = 0. As

V3

and

V4

are _very

_small,

this

_ground

state is

_certainly

_imposed

_by

interactions

_Vi

and

V2,

whose ratio

_V2/V1

is here

_equal

to 0.73.

_According

to the

_{stability diagram}

in

Vi

and V2,

which is known

_exactly

_[40,

41,

42],

the

_ground

state for

_{V 2/ V 1}

= 0.73 and around

x = 0.82 is _an

ASB

_type

_structure

(in

the

_present

_case, A =

nitrogen,

B=

vacancy).

This

structure can be described as follows : it

_consists,

_along

_{a (110)}

_direction,

in a basic sequence

of two _pure

₍₁₁₀₎

A

_planes

followed

_by

a mixed AB

_plane

and this sequence

_repeats

itself

along

direction (110)

(see

Fig.

11).

In

fact,

this

_A5B

structure is

_infinitely

_degenerated,

even

if

_V3

and

_V4

are taken into account

_(for

_example,

three different

_A5B

LRO

structures,

degenerated

with interactions up to

_V4,

have been

_proposed

for

_V6C5

and

_{Nb6C5 [43, 44]).}

But

at finite

_temperature,

from

_entropic

effects

_only

one

_{A5B phase}

should survive

(although

the free

_energies

of all the

_{A5B phases}

are _{very close}

together).

Therefore,

in order to limit the number of correlation

functions,

we have chosen to

consider,

for the C.V.M.

_{phase diagram,}

the

_{simplest A5B}

_phase

_{(Fig. 11).}

In the TO

approximation,

this

_phase

is defined

_by

94 correlation functions

_{(4) (the}

use of the TO

Fig.

11. -

A5

B structure

_{(orthorhombic}

unit

_cell).

_{(0) :}

B atoms

_{(vacancy) ; (e)}

A atoms

_(nitrogen).

ail (112) , bll (112) ,

cil (110)

& . For each orthorhombic unit cell vector a, b, c, are indicated its

components in the disordered f.c.c. cell.

Fig.

12. -

TiN,,

phase

diagram

around

_{composition x ===}

0.82, calculated

_by

C.V.M. from

_V,

= 82 meV

and

_V2

= 60 meV.

approximation

has been

_justified

_above).

The

_{resulting TiNx phase diagram,}

around

x =

0.82,

is

given

in

figure

12.

The Monte-Carlo method has been used to simulate the low

_temperature

_phase

of this

system

at 460 K with the interaction

_{potentials given}

in section 5.2 at the

_A5B

_{(- Ti6N5D)}

composition.

The Monte-Carlo calculated atmn exhibit a

_long

_range structure is not zero

at

_large

_distance)

which is

_compatible

with the

_A5B

structure until the 28th f.c.c.

_neighbour

shell

_(see

Tab.

_VI).

Table VI. - Values

_{o f}

calculated

_by

Monte Carlo simulation

_{for TiNo.82}

in the LRO state at 460

_K,

_compared

to the theoretical values

_for

the

_perfectly

ordered

_A5

B structure.

In any case, this transition should be very difficult to observe

_{experimentally}

for kinetic

reasons, the

nitrogen

mean free

_path

for one hour at 285 °C

_being

about 1

Á

from atomic diffusion data.

6. Conclusion.

In this

_work,

we have for the first time

_{quantitatively}

measured

_short-range

order

_parameters

at the

_equilibrium

_temperature

_by

neutron diffuse

_scattering

in a

refractory

non-stoichiomet-ric

_compound,

and determined

_precise

effective

_ordering

interatomic

_energies.

In

_TiNo.82,

vacancies

_prefer

to be on third

_neighbour

_positions

from a _vacancy

Recently,

room

_temperature

_powder

neutron diffraction studies of

_short-range

order were made on

TiNo.67

and

_TiNo.75

_previously

annealed at 1 110-1 200 °C

_[17].

For

TiNo.75,

the

SRO

_parameters

found were : = -

0.012,

= -

0.032,

all2 = -

0.006,

= -

0.112,

... These results are inconsistent with ours

(especially

for that we found

positive),

which _{may be due} to the fact that it is difficult to estimate

_short-range

order

parameters

from a

_powder

_diagram.

We found maxima of diffuse

_scattering

at

_{(1/2 1/2 1/2 )}

and

_{equivalent positions.}

This is in

agreement

with the theoretical

_{prediction by}

Landesman et al.

_{[13, 14].}

On the other

hand,

the

_long-range

order observed

_{experimentally}

in

_Ti2N

is of the

_type.

The fact that the

type

of order

_changes

between

_Ti2N

_(long-range

order of 1 1/2 0

_type)

and

_TiNo.g2

(short-range order of 1/2 1/2 1/2

type)

is

probably

due to a variation of the

_pair

interactions with

composition ;

the ratio

_{V 2/ V}

l should increase from less than 0.5 for

Ti2N

to = 0.75 for

TiNo.82.

The

_generalized

_{perturbation theory predicts}

the correct

_trend,

but

_disagrees

quantitatively

= 4.2 for

Ti2N

and 6.0 for

TiNo.82,

see

[14]

and

_Fig.

1).

The static atomic

_{displacements}

are similar to those observed in transition metal carbides

[26] ;

the

_displacement

field calculated

_by

the lattice statics method

_{gives good}

results if we

estimate the

_missing

coefficient

_S3

from the value of These static

_{displacements}

are

not taken into account in the

_{ordering Ising}

Hamiltonian

_{(3) ;}

nevertheless,

they play

probably

an

_important

role in the total energy of

_ordering

_(in

_{particular they}

are much

_larger

in

_Ti2N

than in

_TiNo,g2).

We have used three different inverse methods to calculate the

_pair

interactions from the

experimental

data : the

_Clapp

and Moss

formula,

the inverse CVM and the Monte-Carlo simulations.

Our first remark is that the results

_always

confirm the calculation of Landesman et al.

_{[14] :}

V 3, V 4

and further

_potentials

are _{very weak and}

_(although

the numerical values

_disagree

with those deduced from

_Fig.

₁₎

the

_ordering

is dominated

_{by V}

_and

which are

_positive

in the

present

case. As in transition metal

_alloys,

the effective

_{ordering energies}

decrease much

more

_rapidly

with distance than in normal metal

_alloys,

this

_damping

_being

due to electron

mean free

_path

effects

_[47].

Secondly,

we note that the

_Clapp

and Moss formula leads to interaction ratio

which

are

_quite

_good,

even if the interactions themselves are not correct. This is due to

the fact that the

_experimental

data have been collected at a

_{relatively high}

_temperature

700

_{°C ),}

in

_comparison

to the order-disorder transition

temperature

(Tc

= 285

°C,

according

to the CVM

_phase

_diagram,

see

_Fig.

_12).

The

_Clapp

and Moss formula has then

been used in a

_regime

where it is

_expected

to be

_{relatively precise.}

This is confirmed

_by

the fact that the

_plot

of

_(Q

= 1/2 1/2

1/2 )

_versus

1 / T,

obtained from

experimental

data,

is _a

straight

line,

as in the

_Clapp

and Moss formula.

Finally,

we note that the calculated interactions do not

_depend

on

_temperature.

This proves, without any

ambiguity,

that the order-disorder effects in our

_system

are

_precisely

described

_by

an

_Ising

model with

_pair

interactions limited to

_V4.

_Moreover,

the

_good

agreement

between the CVM and the Monte-Carlo results shows that these two inverse methods are _very reliable.

Acknowledgements.