Local order and mean vibrational state of atoms in substitutional metallic solid solutions

HAL Id: jpa-00236680

https://hal.archives-ouvertes.fr/jpa-00236680

Submitted on 1 Jan 1962

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.

Local order and mean vibrational state of atoms in substitutional metallic solid solutions

V. Synecek

To cite this version:

V. Synecek. Local order and mean vibrational state of atoms in substitutional metallic solid solutions.

J. Phys. Radium, 1962, 23 (10), pp.787-790. �10.1051/jphysrad:019620023010078700�. �jpa-00236680�

LOCAL ORDER AND MEAN VIBRATIONAL STATE OF ATOMS IN SUBSTITUTIONAL METALLIC SOLID SOLUTIONS

By ^V. SYNECE K,

Institute of Solid State Physics, Czech. Acad. Sci., Prague 6, Cukrovarnicka 10, Czechoslovakia.

Résumé.

Le carré moyen de l’amplitude effective des vibrations thermiques des atomes dans le facteur habituel de température Debye-Waller ^a été calculé en utilisant le modèle d’interaction entre les plus proches voisins pour des solutions solides métalliques binaires, ^de substitution, cubique ^{face centrée.} 11 est donné en fonction du paramètre

^{d’ordre à courte} distance, ^{de l’entou-}

rage de première coordination et des constantes de force des plus proches ^voisins gAA, gBB et gAB.

Le paramètre de structure

et le paramètre dynamique gAB des solutions solides métalliques peuvent

être déduits des mesures d’intensités intégrées de rayons X pourvu que ^nous connaissions _{gAA et gBB} a partir ^{de mesures} identiques ^sur les métaux purs ^{A et} B.

Abstract.

The effective mean-square amplitude of thermal vibrations of atoms in the usual

Debye-Waller temperature factor has been calculated using ^{the nearest} neighbour interaction model for the binary substitutional f. c. c. metallic solid solutions. It is given in terms of the

short-range ^order parameter 03B1 of the first coordination shell and the nearest neighbour ^force

constants _{gAA, gBB} and _gAB. The structure and dynamical parameters

^and gAB of the metallic solid solutions can be estimated from the X-ray measurement of the integrated intensities pro- vided we know gAA ^and gBB from similar measurements on pure metals A and B.

PHYSIQUE 23; 1962,

Introduction.

The mean vibrational state of atoms in the simple ^{monatomic cubic} crystals ^can

be expressed by ^the mean-square displacement ^U2

of ^an atom from its average position. This quan-

tity ^{can be} obtained from the measurement of the

integrated intensity ^of X-ray reflection hkl. The

resulting diffraction amplitude ^{can be written}

where .K denotes a normalizing ^factor depending

both on the experimental arrangement ^{and the} symmetry ^{of the} lattice, ^{Fhki and} fhki ^{are the}

structure and atomic scattering factors respec’"

tively. M is ^the Debye-Waller température ^factor

which can be expressed ^{in terms of U2} by

where U2 ^{is the} component ^{of U2 in} the directlon of the normal to the (hkl) planes, ^{0 is the} Bragg angle ^{and Àis thé} wavelength ^{of the} radiation used

(James, 1954).

Assuming ^{that the} bounding ^forces between the atoms of the lattice can be approximated by ^cen-

tral elastic forces between nearest neighbours ^then

the corresponding ^{force constant can be} expressed by.

where k is the Boltzman constant and T the abso- lute temperature ; 4Y(z) ^{is the} Debye funtion and

x := 0 , ^{where 0 is} 0, ^the Debye characteristic T

temperature. ^The approximation [3] ^{can be of}

some use for simple ^{f. c.} c. lattices (Leibfried, 1957).

The use of the Debye theory ^{has been often}

extend ed to the study ^{of lattice} vibrations in substi- tutional metallic solid solutions by introducing

further simplifying assumptions, ^{such as the} equa-

lity ^{of thermal} amplitudes of both the A and the B atoms (e.g. Iljina ^and al., 1951) ^or by ascribing ^the

différent U2 ^{to A and B atoms} respectively (e.g.

Murakami, 1953), ^etc.

The aim of the present study ^{is to take into} account ^the aperiodicity ^{in the} distribution of the A and B atoms in the binary metallic solid solutions for deriving the proper Debye-Waller

factor corresponding ^{to various} degree ^{of local}

order and nearest neighbour ^force interaction.

Theoretical part.

^{The atoms of both metallic} components A and B in the substitutional solid solution can be surrounded by ^différent proportion

of like and unlike neighbours ^{due to certain ran-}

domness in the distribution of the A and B atoms among the crystallographically equivalent ^lattice positions. ^{The fraction of A atoms} having say k B atoms and j-k ^{A atoms as nearest} neighbours ( j denotes the coordination number) will be for a

given composition ^{of the} alloy determined by ^the degree ^{of local} order. These A atoms will be held to the surrouding ^lattice through ^{the k A-B and} j-k ^{A-A bonds if we} adopt ^{the nearest} neighbour

interaction model of the simple ^{f. c. c.} solid solu-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010078700

788

tion. The corresponding two-dimensional dyna-

mical model is schematically represented ⁱⁿ figure ^1.

The present discussion will be restricted to alloys

Fie. 1.

Two-dimensional dynamical model of the binary

solid solution with nearest neighbour interactions.

with no atomic size effect. The force interaction with the surrounding ^{lattice and hence the} ’U-2- value, ^is then assumed ^to be the same for all atoms of the given type having ^{the same} proportion ^of

like and unlike neighbours.

The fraction of A ^atoms with k B atoms as nearest neighbours ^{will be} equal ^{to mA kPg where mA}

is the atomic fraction of A atoms in the solid solu- tion and _kPg is the probability ^of finding ^{k B atoms}

among the j ^nearest neighbours ^{of an A atom} (j

^{12 for} simple ^{f. c. c.} lattices). TheU2 -velue

of this fraction ^of A atoms will be deroted bYk U2 Analogous quantities ^{mB jbpB} and kCUi ^{i ef er to B} atoms, the index k denoting ^{the nuii ber of their}

nearest B atoms again. The difh action amplitude corresponding ^to Bi agg reflection h k 1 ^can be then

easily calculaued with the result

where S == 4n sin e lÀ. Equation (4) may be sometimes sirrplified ^{with a} good appioxin,ation

into the form

which can be obtained by expanding the expo-

nentials in (4) ^{into the} potential series, by ^rear- ranging ^{of terms and} by using ^the approximations f A

ZA Î ^and f B

^ZB Î (where Î ^{is the} unitary

atomic scattering ^{factor znd ZA and ZB the atomic}

numbers of A, and B atoms respectively). ^The

effective mean-square displacement U-1 of ^atams (5)

will then be given by

To proceed further the probabilities kpA and _kPB must be expressed ^{in terms} of the local order and the mean-square displacements k‘U.A ^and kflL§ by

means of the force interactions of the central A

or B atom with k B and j-k ^{A atoms} in its first coordination shell. The quantities kPA and _kpB

can easily ^be determined from the well known

expression ^{for the} probability pAB of finding ^{a B}

atom in a certain position ^{on the first} coordination shell of ^an A atom

where ce is the short-range ^order parameter oc,_ of the Cowley’s theory (Cowley, 1950). ^The proba-

bilities pAA, pBA and pBB defined in an analogous

way ^are then given by

so that the probability ^{of a} given configuration

of k B atoms and j-k ^{A atoms in} the first coordi- nation shell of an A atom is simply pAB PAA . By taking ^into account that the k B atoms can be

arranged ^{there into} (1) ^different configurations

one obtains readily (k)

expression ^{for the}

one obtains readily ^the explicit expression ^{for the} probability kpA (and similarly ^for kpB) ^defined

eailier in teins of corr position ^and bhort-range

or der par arr, et er ^{oc as follows :}

The most simple assumption ^as regards ^{the de-} pendence ^of kU-’ ^and kflLi ^{on the number of A}

and B atoms in the first coordination shell of the central atoms is to suppose that the effective ^force constant kgA of ^an A atom surrounded by ^{B atoms}

is equal ^tu the arithn.etic rrean of force constants

cori ei pondirg ^to individual contacts of this A atom

with its neaieht neighbours. ^{It holds then}

By using ^relation (3) ^for expressing ^{force cons-}

tants in (9) by proper values of mean-square dis-

placements, ^{one obtains} readily ^{the desired rela-}

tions for kflL§ ^and kUi ^{as follows :}

The effect of thermal vibrations of atoms on X- ray reflections ^can be thus explicitly given ^{in terms}

of structure (ce) ^and dynamical parameters (gAA,

gBB and gAB). This is obvious by inserting (8)

and (9) (on using (10)) into , (4) ^or (6). Only ^a

and gAB ^are generally a priori unknown, ^because gAA and _gBB can be determined from the same kind of measurement on pure metal components ^{A or B} respectively.

Results and discussion.

The present ^relations

are now being ^{used for the} study of lattice vibra- tions in solid solutions of silver in aluminium

(Simerskâ, 1961), ^{where the} size effect is practi- cally unimportant. ^The X-ray intensity ^measu-

rements on equilibrium ^{solid solution at 525 °C} yield,ed ^the value gAlA,, which is about 60 % ^{of thé}

mean value of gAlAi and gAgAg. The parameter

a was practically equal ^{to zero} up ^to 10 at % Ag ^at

that temperature. ^{The lower} figure of gglAg ^as

compared with pure metallic components, ^{is in} qualitative agreement ^{with the} segregation ^ten- dency ^{of silver atoms even at} temperatures ^above

the solid solubility ^limits (Walker, ^{Blin and} Guinier, 1952). ^{To have a} quantitative test, ^the

increment of vibrational entropy ^{due to the for-}

mation of solid solution ^was expressed ^{in terms}

of kft1§1, k‘U2Ag, kpAi and _kPA.g. The entropy ^incre-

ment determined as a function of silver content from the measurements of U2S gave the quantitative agreement ^{with the excess} entropy ^{curve found} by Hillert, ^{Averbach and} Cohen, ^{1956 from elec-}

trochemical measurements.

The measurements of flLg ^on supersaturated ^solu-

tions of Ag ^{in Al are now in} progrèss. ^{To have}

some estimate of the expected effects, the theore- tical curves of U2s against ^{différent oc-} and gAB- values are given ⁱⁿ figure ² (negative ^{values of oc}

are not included in spite ^of segregation ^{effects in} Al-Ag alloys) corresponding ^{to 12 at.} % Ag ^{in Al.}

The room temperature ^{values of} Us2 ^{of Al} (Bensch

and al., 1955) ^{and of} Ag (Simerska., 1961) ^{were used}

for these calculations. The lower values of a and

a greater difference of gAB, ^as compared with gAA and gBB, are ^{thus more} promising ^for finding ^an

effect on Us2- by segregation processes ⁱⁿ solid solu- tions.

FIG. 2.

The plots ^{of the mean} square amplitude.

a) Against gAB for different degree ^of segregation. (En

abcisses lire 9AB b) Against

for different force constants au ^{lieu de gA1-Ag) -} _gAB given

in 104 dyne ^{cm-l. The} curves are calculated by ^assu- ming ^the validity ^of (9) and fitted to force constants of Al and Ag.

The parameters ^oc and gAB could be in principle

estimated from the measured intensity by using

relation (4). ^The comparison ^{of measured} U2s

values obtained for some compositions of all ys after different thermal treatment with the curves

in figure ^{2 seems more effective in} estimating ^the

desired parameters. There may be some doubt as

to the at least approximate validity ^{of relations} (9)

especially ^{at lower} temperatures. This could be

790

perhaps ^infered e.g. from Gerold’s two-phase ^model

of supersaturated ^{solid solutions} (Gerold, 1961).

Fortunately ^{in these cases the} parameter ^{oc can be} easily calculated with the result

where p denotes the volume fraction of the alloy occupied by ^G.-P. zones, ml and _m2 are the _concen-.

Fié. 3.

Short-range ^order parameter

ⁱⁿ supersaturated

solid solutions of Ag ^{in AI} corresponding ^to annealing temperatures ^of (a) ²⁰ oC, ( b) 100°C, (c) ^{226 OC as a}

function of silver content.

trations of A atoms inside and outside of the A-rich

zones. In deriving (11) ^a random distribution ^of both kinds of atoms is assumed in all regions ^of

the lattice with différent proportions ^{of the A} component. ^{In the Gerold’s} model p

1 and _ml- and m2-values ^{can be read from the} metastable solid

solubility ^{limits. The curves of oc} against ^{mAg are} given ⁱⁿ figure 3 for solid solutions of Ag ^{in Al.}

These were calculated by using ^relation (11) ^and m1-and m2-values given by Baur ^{and Gerold} (1961).

The measured thermal amplitudes ^can be then used for estimating ^the dependence ^{of force constants}

kgg and kgB ^on the number of like and unlike nearest neighbours ^{of an atom.}

The integrated intensities provide ^{much less}

informations than can be drawn for the mentioned

parameters ^{from the diffuse} scattering ^of X-rays.

Although ^the quantitative agreement ^{can be some-}

times achieved (e.g. for thermal entropy ^of Al-Ag alloys), it is desirable to compare ^more relatively

than absolutely ^{the obtained} quantities ^{for ’A-B} alloys ^{of différent} compositions ^and after different thermal treatment. The advantages ^of using ^the integrated intensities for these purposes is in the

possibility ^of obtaining informations in a much shorter time as compared ^{to the} diffuse intensity

measurements. An advantage is also the possi- bility ^of application ^to alloys with metallic com-

ponents ^{of not} much different atomic numbers.

REFERENCES BAUR (R.) and GEROLD (V.), ^Z. Metallkde., 1961, 52, ^671.

’

BENSCH (H.), ^WITTE (H.) ^{and WÖLFEL} (E.), ^Z. Phys. Chem., 1955, 4, 65.

COWLEY (J. M.), ^J. Appl. Physics, 1950, 21, ^24.

GEROLD (V.), Physica status solidi, 1961, 1, ^37.

HILLERT (M.), ^AVERBACH (B. L.) ^{and COHEN} (M.), ^Acta Met., 1956, 4, 31.

ILJINA (V. A.), ^KRICKAJA (V. K.) and KURDJUMOV (G. V,), Problemy metallovedenija ⁱ fiziki metallov II (Moscow, Metallurgizdat), ^{1951, p. 222.}

JAMES (R. W.), ^The optical principles ^{of the} diffraction ^of X-rays (London, Bell), 1954, p.193.

LEIBFRIED (G.), ^Handb. Physik, 1955, ^{Bd. 7} (Berlin, Springer), ^{p. 155.}

MURAKAMI (T.), ^J. Phys. ^Soc., Japan, 1953, 8, 36.

SIMERSKÁ (M.), ^Acta Cryst., 1961, 14, 1259.

SIMERSKÁ (M.), ^{Czech. J.} Phys., 1963 (in press).

WALKER (C. B.), ^BLIN (J.) and GUINIER (A.), C. R. Acad.

Sc., 1952, 235, ^254.