The location of interstitial carbon in austenite

HAL Id: jpa-00246587

https://hal.archives-ouvertes.fr/jpa-00246587

Submitted on 1 Jan 1992

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The location of interstitial carbon in austenite

B. Butler, J. Cohen

To cite this version:

B. Butler, J. Cohen. The location of interstitial carbon in austenite. Journal de Physique I, EDP

Sciences, 1992, 2 (6), pp.1059-1065. �10.1051/jp1:1992195�. �jpa-00246587�

Classification

Physics

Abstracts

61.12E 61.55H 61,708

The location of interstitial carbon in austenite

B. D. Butler

(*)

and J. B. Cohen

Department

^of Materials Science and

Engineering,

Robert R. Mccormick School of

Engineering

and

Applied

Science, Northwestem

University,

Evanston, 1L60208, U.S.A.

(Received18

October 1991,

accepted

in

final form

3 January 1992)

Abstract.

High

resolution neutron

time-of-flight powder

diffraction pattems were obtained for the alloy Fe-13 _wt. pct. ^{Ni-I wt.} pct. C, in the austenitic

phase.

The interstitial site location of the carbon atoms was refined

using

Rietveld

analysis

and it _was found that carbon resides

primarily

on the

octahedrally

coordinated interstices. Uncertainty ^{in the refinement allows} a maximum of 5 pct, ^{of the total number of carbon} atoms to occupy tetrahedral interstitial sites. The other

refined structural and instrument parameters were determined to be

relatively

uncorrelated with these

occupation

fractions. The known total concentration of carbon in this

alloy

_was not needed

as a constraint _on the carbon

occupation,

indicating that this refinement _was

extremely

stable with

regard

to these parameters.

I. Introduction.

Austenite is _a

high

temperature face-centered

(FCC) phase

of steel which _can be stabilized to room

temperature

and below with the addition of

alloying

elements such _as Ni and Mn.

Carbon is soluble in austenite _to

approximately

2 _wt, pct. ^and

occupies

the interstices of the

lattice. There _{are two}

positions

in the FCC lattice where the carbon atoms would

likely

reside:

octahedrally

coordinated sites located at the unit cell _center and

edges,

and tetrahedral sites located

midway

between the cell center and _comers, The octahedral site is

larger

than the tetrahedral site. It _can accommodate a 0,53

I

_radius

_sphere

_{without distortion}

whereas the tetrahedral site _can

only

accommodate _a

sphere

of radius 0.29

I.

_{The covalent}

diameter of C is 0.77

I larger

than either _can accommodate without distortion _so it

might

be

expected

that the octahedral site which

requires

the smallest lattice distortion would be favored.

Electromigration experiments [I

and electronic calculations of C in Fe martensite

[2]

suggest ^that interstitial carbon is

positively charged

and _so

might

have _a radius smaller than this covalent radius. The radius of C+ is 0.29

I

_{and of C+} ~ 0.15

I.

_{The size of the} interstitial

site, therefore, might

not be the _most

important

consideration.

Recently

Rosato

[3], using

molecular

dynamic

simulations of C in

austenite, computed

_a total

configuration

_energy for

(*)

Currently

at the Research School of

Chemistry,

Australian National

University,

Canberra, Australia.

1060 JOURNAL DE

PHYSIQUE

I N° 6

carbon _on octahedral interstitial sites of -7.0eV which compares

favorably

with the

measured value of 6.7 eV

[4].

No

comparison

was

made, however,

with the

configurational

energy of carbon in tetrahedral sites.

As the martensite

phase

inherits the concentration of the parent ^austenite

phase,

carbon atom location is vital information in _our

understanding

of this transformation.

Experimental

evidence of the interstitial site location in austenite _came first from Petch

[5]

who

performed X~ray powder

diffraction measurements

using photographic

film. Based _on trends in the

relative intensities of I I

Bragg

reflections he concluded that C resides _on the octahedral sites.

As

X-rays

scatter

weakly

from carbon their effect is of the order of

2pct,

on low

angle

reflections and is

negligible

at

large scattering

vectors.

Therefore,

this first evidence is _not

convincing.

Other

investigators

have used neutron diffraction

[6, 7]

where the diffraction contrast from C is much

greater

and have observed behavior that is

qualitatively

consistent with octahedral

occupation.

^The most recent neutron diffraction

study [8]

was the most

quantitative

to date.

An Fe-Mn-C austenitic

alloy

_was

employed

in which 6

powder

reflections _were measured and used to refine three structural

parameters including

the relative

proportion

of carbon _on the octahedral and tetrahedral sites. The author

reported

that his data _{was most} consistent with octahedral

occupation

but that

uncertainty

in the

analysis

would allow up to 10

pct.

^{of the} carbon _on the tetrahedral sites. It _was

reported

that texture in this

sample

created _an

approximately spct,

^{variation in} the intensities.

Considering

this fact and that

only

six reflections could be measured the

uncertainty

of lo

pct,

seems to be _an underestimate.

However,

this measurement does

support

^{the idea that} most carbon atoms reside _on the octahedral

sites,

but it still allows for _a

relatively large portion

to exist _on tetrahedral sites. In

fact,

Ino _et al.

[9] interpret

their Mossbauer pattems ^from

virgin

martensite _as

indicating

some tetrahedral

occupation

which should then be true in the

prior

austenite

phase,

_as the transformation to martensite is diffusionless.

This

study

_was undertaken to reduce the current

experimental

uncertainties. Neutron

powder

diffraction measurements on the

alloy

Fe~13 wt, pct. Ni-I _wt, pct. ^C were made

using

a neutron

time-of-flight spectrometer

which allowed the collection of data _{over a} range of

wave-vectors that

spanned

49

Bragg

reflections of austenite. These data _were

analyzed using

Rietveld

techniques

and the

occupation

fractions of carbon atoms on the octahedral and tetrahedral interstitial sites were refined.

2.

Experimental

details.

2.I SAMPLE PREPARATIONS. The material used in this

study

_{was an}

alloy

of

approximate

composition

Fe-13 wt,

pct.

Ni, I wt,

pct.

^{C which} was obtained _{as an} extruded billet of

rapidly

solidified

powder.

This

alloy

is from the _same batch _as that used in _an

investigation by

Hartfield

[10].

A section of the billet 100 _mm

long

^and 16 _mm in diameter _was

separated

from its steel outer

casing

and

homogenized

for 2 hours at 1353 K inside _an evacuated

quartz

tube and then

quenched

into _{water at room}

temperature.

^Preferred orientation _was

present

^{in this} material _{as a} result of the extrusion process and _so it _was unsuitable _{as a}

powder

diffraction

sample.

The billet _was therefore

ground

into fine

chips using

a hand-held electric

grinding

tool

with carbide bit. Care _was taken

during grinding

to prevent ^{the material from}

heating

appreciably

and that _no

sparks

_were

generated

from the carbide bit in contact with the

alloy.

Further,

the

alloy

rod _was rotated

during grinding

to

produce chips

with random orientation relative to the texture axis. These

chips

_were then converted to

powder using

_an alumina ball mill,

Contamination introduced

by

the ball mill and

grinding operation

_was

easily

removed

by

washing

the austenite

powder

in methanol and

collecting

the

powder

with _a strong

permanent

magnet. (This alloy

in the austenitic state is

paramagnetic

but the

powdering

_processes

introduced

enough

stress induced

ferromagnetic

martensite to make this

procedure possible).

Finally,

the cleaned

powder

_was reaustenitized at 1273 K for 20 min inside _an evacuated quartz ^{tube and}

quenched

in _room

temperature

water

immediately

after removal from the

fumace.

X-ray powder

diffraction _scans

using

LiF monochromated

Cr-Kai

radiation

confirmed the

purity

of the

sample.

A small amount of

A1203 (

l vol,

pct.)

from the ball mill remained but _no martensite _or bcc ferrite _was detected in the _scans.

2,2 DATA coLLEcTioN. Measurements _were made _on the General

Purpose

Powder

Diffraction

(GPPD)

at the Intense Pulsed Neutron Source

(TANS), Argonne

National

Laboratory.

This unit is _a

time-of-flight (TOF) spectrometer consisting

of _a series of

~He

detectors which _are mounted in vertical banks around the diffraction circle 1.5 _m from the

sample position.

^The detector banks in the

backscattering position (~

± 150° 2

b) provide

highest

resolution and are

capable

of

sampling

diffracted wave-vectors in the range 0,17

i~

~ sin b

)/A

_~ l.7

i~ corresponding

to

crystallographic d-spacings,

0.3

I

~ d~~i ~ 3.0

I.

_{Data from these}

backscattering

banks _were used in the

analysis.

A thin walled vanadium _can 50 _mm

long

with diameter I I _{mm was}

loosely

filled with the austenitic

alloy powder.

The vanadium container _was then secured _{to a}

sample stage

at the

center of the diffractometer and the entire

sample

chamber

(a cylinder

I _m

deep

and 0.6 _m in

diameter)

_was evacuated to

approximately 10~~

_{torr to minimize the}

scattering

of _neutrons

by

the

surrounding

air. Data _were collected _{at a}

temperature

^{of 300 K} for 2 hours of

operation

of the

spallation

_source, sufficient _to accumulate

greater

than

lo,

000 neutron counts at the

peaks

of

nearly

all available

crystallographic

reflections. The

powder

diffraction

peaks

_were

sharp

limited in resolution

only by

the instrument and well resolved down _to

crystallographic d-spacings

^of 0.33

I.

A _test for

preferred

orientation

along

the _transverse axis of the power

sample

_was made

by comparing

the relative

integrated

intensities of the first 4

powder

reflections obtained from detector banks _at 90° with those in the

backscattering positions.

No detectable differences were found _as would be

expected

for this loose metallic

powder.

No check _was made

along

the

long

axis of the

sample.

The incoherent

scattering

from _a standard vanadium

powder sample,

which is

relatively

insensitive _{to neutron} energy, was used to determine the incident _neutron flux distribution.

This incoherent

scattering

_was then fit _{to an}

empirical

function of the

flight

time which _was in tum used to normalize the diffraction data. The neutron

time-of-flight

was converted to units of

crystallographic d-spacing using

_a standard relation with

parameters

^determined

by fitting

the spectrum ^of a Si

powder sample.

3. Data

analysis.

The data _were

analyzed using

Rietveld

analysis [I1, 12],

_a full diffraction

pattem least-squares fitting procedure.

The computer programs used for the

analysis

_are described in detail in

reference

[13].

The

following

variables _were

employed

in the refinement _{: a} five parameter

background

function used to correct for detector noise and incoherent

scattering

_{processes, a}

scale

factor,

_a six

parameter peak shape

^function derived from the convolution of _a Gaussian with _an

empirical

instrumental

peak shape composed

of

leading

and

trailing exponentials,

the

austenite unit cell

length,

the carbon site

occupation

_on both the octahedral and tetrahedral interstitial

sites,

and

isotropic Debye temperature

^{factors for both the}

Fe/NI

and C atoms. No allowance _was made for thermal diffuse

scattering (TDS)

contributions to the

integrated

intensities _as these _are

likely

small _{at room}

temperature

^and

only

_vary

monotonically

with d-

spacing.

The effect of carbon varies

non-monotonically

with hkl.

The Fe and Ni atoms were assumed _to occupy the FCC lattice sites

randomly

_so that the

refinement treated this

alloy

as a two

component

« Fe »-C system ^{where the} « Fe

» neutron

1062 JOURNAL DE

PHYSIQUE

I N° 6

scattering length

_was

computed

_{as a}

composite

^{of the}

^scattering lengths

of both Fe and Ni

weighted according

to their atomic

percentages.

^{A recent}

single crystal X,ray investiga-

tion

[14]

of _an

alloy

of similar

composition

and heat treatment showed that _no

long,range

or

detectable

short-range ordering

of Fe and Ni _were

present.

A total of 624

independent

data

points

in the range ^0.44 ~ sin

(b )/A

_~ l.47 _were used in the refinement which includes the 49 austenite

powder

reflections from the 311 out to the 666

peak.

The first three low order diffraction

peaks (I I1, 200, 220)

were not included in the refinement _to avoid

complications

from

possible magnetic scattering

and extinction. An

absorption

correction factor _was included in

early

refinements but found to be within _one standard deviation of _{zero as} would be

expected

for this loose

powder

that _was

only 1/sth

the

density

of solid Fe. Since this

parameter

^refined to _a small and

unphysical negative

value it

was removed from

subsequent

refinements.

The

least-squares

refinement _was

performed

in _a number of

steps. First, only

the lattice

parameter

^and

background

function _were allowed to vary ^and then

gradually,

in later

refinements,

each of the other

parameters

was set free _so that reliable convergence was

obtained. This

procedure

was

repeated

_a number of times

using slightly

different initial guesses and

altering

the order of the

pattem

^of

setting parameters

^free to insure that a stable and

unique least-squares

solution resulted.

Typically,

convergence would be obtained

using

three _or four refinement steps ^each

requiring

about four

least-squares cycles.

Indication of the

success of various refinements _was

through

a set of

least,squares

R-factors _:

R-factor

(Rietveld)

=

3

( II

~bs I

~~ic

)/3(1~bs

I

background ,

R-factor

(Weighted Profile)

=

(Ii

_w

_(I~bs Icaic)~)/3(WIjbs ))

^~~

,

j~

R,factor

(Structure Factor)

₌ ³

Fjbs _F(arc( )/~i(Fjbs)

,

R,factor

(Expected)

₌

(degrees

of freedom

)/3(wIjbs)

^~~

,

where F~~~ ^and

F~~~

_are ^{the observed and} calculated structure

factors,

_I~~~ and

I~~~

are the measured

(normalized)

and calculated

intensities,

and

w is _a

weighting

factor obtained from the

counting

statistics of the _raw data. The number of

degrees

^of freedom is calculated

by subtracting

the number of

parameters

^{used in the refinement}

(~17)

from the number of

independent

observations

(1 624).

The Rietveld R-factor is the statistic that is minimized in the Rietveld

refinement,

the

weighted profile

R-factor is the usual

crystallographic

R-factor and describes the

goodness,of-fit

_on a

point by point basis,

and the _structure factor R-factor

measures how well the

integrated peak

intensities fit the refined structural parameters.

4. Results and discussion.

The

expected

R-factor for this data set is 1.5 pct.

Typically,

the refined R-factors _were much

higher

than this around 7 pct. ^{for the} structure factor and

weighted profile

R-factors and _as

high

_as 12

pct,

^{for the Rietveld R,factor. A}

plot

from _one refinement is

presented

in

figure

I.

Along

the bottom of this

figure

the difference between the observed and refined diffraction

pattem

^is

presented, Upon inspection

of the

figure

the _reason for the

discrepancy

between

expected

and actual R-factors becomes obvious. Under each

Bragg

reflection the least- squares fit underestimates the

intensity

of the

peak

_on the

leading edge

and overestimates the

intensity

on the

trailing edge producing

a

sharp

oscillation about _zero in the difference

plot

under each

Bragg

reflection. This

systematic

_error in the fit arises because the

peak shape

function used does not

reproduce precisely

the observed

peak shapes.

This is _a

relatively

recent

problem

in Rietveld

analyses

of GPPD data that appears ^{to be associated with the}

installation of _{a new} booster

target. Fortunately,

this

fitting

_error does _not effect the

w

Fe-13 4%Nl- I.I)t%C AustenJte _al 3(X)K

w

~4~

57

~

~~ ~

TM jm

E 6 Z

lllllllll II Ill _II II

iii? I I %I § I I

U341 DAD? D 473 D 539 D6D5 O671 O?3? D.8a3 D 869 1.DDI ' OS? 1.133

d-Spacing j A )

Fig.

1.

Least-squares

fit of the neutron

powder

spectrum ^{of Fe-13} Ni-1C austenite

using

Rietveld

analysis.

The (+)

symbols

indicate the vanadium normalized data and the

plot

near the bottom of the

figure

is the difference between the calculated and measured diffraction pattem.

integrated peak

intensities _a

great

deal

(and

thus the refined structural

parameters)

because the overestimation _on the

trailing edge

is

mostly compensated

for

by

the underestimation _on the

leading edge.

It does however

degrade

the

computed

R-factors

significantly.

A

second,

but less

important,

_reason the R-factors _are

larger

than

expected

is that there is _a

slight,

but

noticeable,

diffuse undulation in the

background likely

the result of static atomic

displacements

caused

by

the interstitial carbon atoms that cannot be described

using

the

background

function of the Rietveld

analysis

_program. These diffuse oscillations _were removed from the

background using

_a Fourier

filtering technique [15]. Applications

of this

filtering

_process reduced the refinement R-factors somewhat but had _no effect

(within

estimated standard

deviations)

_on the refined structural

parameters.

The

only

structural parameters ⁱⁿ the refinement of this

high

_symmetry lattice _are the unit cell

length,

the

Debye

temperature factors of the Fe and C

sites,

and the

occupancies

of the octahedral and tetrahedral interstitial sites. Since the motivation of this

study

_was to determine the carbon

occupation

fraction _on the two interstitial

sites, particular

attention _was

paid

to the

stability

and

dependence

of these two parameters ^with

respect

to

changes

in the other free parameters ⁱⁿ the refinement. A series of refinements showed that the interstitial

occupation

numbers _{were not} correlated with either the

background function,

the

peak shape function,

the scale

factor,

_or the unit cell

length

and _were

only slightly

correlated with the temperature ^{factors. A refinement of the austenite} diffraction pattem ^{in which both the Fe} and C

isotropic

temperature ^factors were allowed _to vary was

compared

to a refinement

where

they

_were constrained to have the _same value. In the latter _case the temperature ^factor

was refined _as,

Bj~~= 0.559(6),

and in the _case where both _were allowed to vary,

B~~~(C)

=

0.6(2)

and

B~~~(Fe)

₌

0.560(6),

The carbon

occupation

_parameters in both _cases

were

equivalent

within the estimated standard deviations of the refinement. Since the

uncertainty

in the C

temperature

^factor was

large (and overlapping

the value of the Fe

temperature factor)

these two values _were constrained to be

equivalent

in later refinements.

Initial refinements locked in the unit cell

length

of this

alloy

at

3.6057i.

_{The total}

interstitial carbon fraction is related

linearly

to the lattice parameter so this result _was used _to

1064 JOURNAL DE

PHYSIQUE

I N° 6

obtain _an accurate value for the carbon concentration in the

alloy. Using

the

unalloyed

austenite lattice parameter ^of 3.572

A

_{and linear}

expansion

coefficients of 0.0003

A/wt.

_pct.

Ni and 0.033

A/wt.

_{pct. C}

_[16]

_{the carbon} concentration is

computed

to be 0.9 wt.

pct.

^which

corresponds

to a sublattice fraction of 4.3

pct.

^{This result} was used _to fix the total sublattice

fraction of carbon

(I.e.

_{x~~ + x~~~}

= 4.3

pct.)

The results of this refinement _are

presented

in the first column of table I. The octahedral interstitial concentration _was

4.3(2)

_pct. and the tetahedral

concentration, 0.0(2) _pct., indicating

that all of the carbon _atoms in this

alloy

reside _on the octahedral interstitial sublattice. The refined tetrahedral

occupation

was

slightly negative

but the estimated standard deviations _on the

occupation

_parameters would allow 0.2 pct.

occupation

of the tetrahedral

lattice,

⁵ pct. ^{of the total number of carbon} atoms in this

alloy.

Table I. Results

ofRietveld refinement of

carbon _atom sublattice

inaction for

the

alloy

Fe- 13 _wt.

pct.Ni-0.91

_{wt. pct.} C.

(Least-squares

_errors in the last

displayed digit

_are

given

in

parentheses.)

~~~~fl'~t~~ x~~ + x~~ = 4.3 pct.

~°~°~~"i~~~

^°~

ccupation

ao

(A)

_3.60575

₍₄₎

_3.60575

₍₄₎

B;~~ ^0.560

(6)

0.559

(6)

x~~

(pct.)

4.3

(2)

4.2

(2)

Xtet

(PCt.)

_{o_o}

₍₂₎

_0.3

₍₃₎

R-factor

(pct.) (Structure Factor)

5.90 5.86

R-factor

(pct.) (Weighted Profile)

5.93 5.93

R-factor

(pct. ) (Rietveld)

1. 1.2

R-factor

(pct.) (Expected)

1.49 1.49

A _test of the

stability

of these

occupations

_was made

by eliminating

the constraint _on the total sublattice fraction and

allowing

the

occupations

to _assume any real number. The results of this refinement _are

presented

in the second column of table I and _are

nearly

identical to the

refinement where the total carbon concentration was used _{as a} constraint. Here the

octahedral carbon

occupation

refined _to

4.2(2) pct.

^{while the} tetrahedral

occupation

refined to

-0.3(3)pct.

Even

though

the tetrahedral fraction in this refinement is

unphysically negative

it is still within _one standard deviation of _zero. The octahedral fraction is

nearly

identical to the refinement with constrained

occupations.

This remarkable result demonstrates

the extreme

stability

of these

occupation

fractions and demonstrates

convincingly

that

essentially

all of the carbon atoms in this

alloy

indeed reside _on the octahedral interstitial sublattice.

5. Conclusions.

Rietveld

analysis

of TOF neutron

powder

diffraction data demonstrates that carbon atoms in austenite reside

exclusively

_on the octahedral interstitial sites. This determination is _more

accurate than

previous investigations

as 49 reflections and 624 observations _are included in the

analysis

_as

compared

to 6

integrated

reflection intensities in the most

thorough previous

diffraction

study

of austenite

[8].

The refinement R-factors _were

significantly larger

than would be

expected

from the statistical accuracy of the data _as there is _some

problem

with the

current

peak shape

function used in the

analysis.

Determination of structural parameters ⁱⁿ the refinement _{was not}

strongly

effected

by

^this

problem

_as the

integrated peak

intensities _are

relatively unchanged by

this

peak shape fitting

_error.

The carbon

occupation

fractions _were found to be

relatively

insensitive to the values of

other parameters ⁱⁿ the refinement

including

the

Debye

temperature ^{factors. It} was also

discovered that refinement of the

occupations

where the total _amount of carbon in the

alloy

was used _{as a} constraint gave

nearly

identical results _{to a} refinement where the carbon interstitial fractions _were allowed to have any value. This fact demonstrates the

stability

and accuracy of these parameters.

Although

these refinements _are consistent with all interstitial carbon

occupying

the octahedral interstitial

sublattice,

the estimated standard deviations from the Rietveld refinement indicate that 5 pct. ^{of the total number of carbon} atoms could reside

on the tetrahedral lattice.

Acknowledgements.

This research _was funded

by

the U-S- National Science Foundation under grant ^NSF-DMR-

814796l. TANS is

supported by

the U-S-

Department

of

Energy.

Assistance with the

measurements in this

study

was

provided by

Drs R. Hitterman, J.

Richardson,

and J.

Jorgen-

son. Their

help

_was

greatly appreciated.

The authors _are

pleased

to have the

opportunity

to celebrate Prof. Guiniers's 80th

birthday.

One of _us

(JBC)

first met him 35 years ago, as a

young «

post-doc

_»,

just beginning

to understand the power of diffraction

techniques.

^It was a

year that had _a

strong

influence _on his _career.

This research represents a

portion

of _a thesis submitted

by (BDB)

in

partial

fulfillment of the

requirements

for the Ph. D.

degree

at Nordhwestem

University.

References

[I] DAYAL P., DARKEN L. S., Trans. AIME 188 (1950) l156-1158.

[2] MORINAGA M., YUKAWA N., ADACHI H., MURA T., J. Phys. F _: Mel Phys. 17 (1987) 2147-2162.

[3] ROSATO V., Acta Metall. 37 (1989) 2759-2763.

[4] MCLELLAN R. B., KO C., Acta Metall. 35 (1987) 2151-2156.

[5] FETCH N. J., J. Iron Steel Inst. London 145 (1942) II I-117.

[6] KANDARPA V., POWELL G. W.. ERICKSON R. A., SPRETNAK J. W., Trans. AIME 236

(1966)

1021- l023.

[7] ENTIN I. R., SOMENKOV V. A., SHIL'SHTEIN S. Sh., Sov.

Phys. Doklady

17

(1973)

840-842.

[8] MAzzONE G., J. Appl.

C~yst.

20 (1987) 187-190.

[9] INO H., ITO T., NASU S., GONSER U., Acta Metal. 30

(1983)

9.

[10] HARTPiELD S. E., M-S- Thesis, Massachusetts Institute of

Technology, Cambridge,

MA. U-S-A- (l988).

[I II RIETVELD H. M., J.

Appl. C~yst.

2 j1969) 65.

[12] PRINCE E., J. Appl.

C~yst.

14 (1981) 157.

[13] ^ROTELLA F. J., Users Manual for Rietveld

Analysis

of

Time-of-Flight

NeuIron Powder Diffraction Data _at IPNS,

Argonne

National Laboratory (1988).

[14] BUTLER D. B., COHEN J. B., ZSCHACK P., Met. Trans. 22A (1991) 2807-2809.

[15] RICHARDSON J. R., Jr., FABER J., Adv.

X-ray

Anal 29

(1985)

143-152.

[16] RUHL R. C., Ph. D. Thesis, Massachusetts Institute of

Technology,

Cambridge, ^{MA. USA} (1967).

JOURNAL DE PHYSIQUEI T 2, N' 6, JUNE 1992 40