On the spatial distribution of isotropic Guinier-Preston zones

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On the spatial distribution of isotropic Guinier-Preston zones

V. Syneček

To cite this version:

V. Syneček. On the spatial distribution of isotropic Guinier-Preston zones. J. Phys. Radium, 1962,

23 (10), pp.828-829. �10.1051/jphysrad:019620023010082800�. �jpa-00236689�

828.

ON THE SPATIAL DISTRIBUTION OF ISOTROPIC GUINIER-PRESTON ZONES

By ^V. SYNE010CEK,

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

Résumé.

Dans le modèle à deux phases ^{des zones} sphérivues, ^{il est} possible ^{de calculer la}

distance moyenne ^{entre zones} à partir de la concentration de l’alliage ^{et de la taille} expérimen-

tale des _zones, L’effet prévu pour les interférences interzones est en accord avec les courbes observées de diffusion aux petits angles,

Abstract.

Accepting ^{the two} phase ^{model of} Gerold, the average distance between ^zones

can be calculated from the concentration of the solid solution and the experimental ^{size of}

the spherical ^{zones. Then the} predicted ^{effect for the} interparticle interferences explains ^satis- factorily the observed small angle scattering ^curves.

LE JOURNAL DE PHYSIQUE ^{ET LE RADIUM}

^T0$fE 23, ^OCTOBRE 1962,

A preliminary ^{note is} given ^{here on the calcu-}

lations ^and experiménts we are now carrying ^out dealing ^{with the structure and} spatial distribution of isotropic ^{G.-P. zones in} supersaturated ^solid

solutions of Ag ^{or Zn in aluminium.}

As ^it is well known the Guinier’s three-phase

model of these solid solutions ^can explain ^reaso- rably ^the rïng-sha,ped small-angle scattering ^{(S. A.}

S.) ^diffuse intensity by assuming the individual

zones scatter X-rays independently. ^{There were}

nevertheless some attempts ^to explain ^{the observed} intensity distribution as an interparticle ^inter-

ference effect (e.g. ^Webb (1959)). ^New support

for this idea are the absolute intensity ^measu-

rements of integrated ^{S. A. S.} which led Gerold

(1961) ^{to the} suggestion ^of two-phase ^{model of}

G.-P. zones in these alloys. With this model the

explanation ^{of the observed} intensity distribution of S. A. S. is only possible ^{due to the} interparticle interference, ^{i.e. some} spatial correlation in the distribution of the zones must exist in this case.

We have therefore tried to find some evidence for a simple distribntion of zones which could satis-

factorily explain ^{the mentioned} experimental

curves and which would be at the same time in

agreement with concentrations of both metallic

components ^{in the zones as} they ^{are given at}

the limits of the metastable miscibility gap ^corres-

ponding ^{to the} température ^of ageing.

To have some idea of the influence of ordering

of the ^{zones on} S. A. S. intensity, ^on one-dimensional

case was treated at first. The zones are assumed to be identical and disposed ^{in such a} way that the probability ^of finding ^{the nearest zone in a}

distance 2B2±r ^{from a} given ^{zone follows a Gaus-}

sian curve 1/B.exp(- rcIB2.r2) centred at 2R2. ^Then corresponding probabilities ^{for all more distant} neighbours ^{can be then} easily calculated. The

resulting intensity ^{from an} assemblage ^{of N zones}

is then given by

where h

2 sin 6 jA, F(h) denotes the structure

factor of a single ^{zone, and} oc(h) represents ^the interference function which can be expressed by

Even by choosing ^a very great value of B (e.g.

60 % ^of 2R2 ^and N -->c>o), ^{there was found a} very

strong interparticle interference effect with a(h) decreasing ^from high ^values at very small angles

to about zero value, increasing ^{then to a maximum}

at h

1/2R2 ^and tending ^to unity ^{at still} higher angles.

The three-dimensional ^{case is now} in progress, but by analogy ^{we can} expect ^the position ^{of the}

maximum of «(h) ^{to be now at}

Because the interference function a(h) ^{will be} again equal ^to unity by ^{h-values not much} greater

than h..., ^{the radius} R1 ^{of the} Ag - ^{or Zn-rich}

zones can be determined using ^{the Guinier} approxi-

mation to the real I(h) - ^curve.

By adopting ^{the mentioned} close-packed spatial

distribution of G.-P. zones and by using ^{the two} phase Gerold model of solid solution, the quan-

tity R2 ^{can be} readily calculated from Ri ^{and the} atomic ^fractions ml and _m2 of Ag ^{or Zn in the solute} rich and soluté impoverished ^zones corresponding

to the metastable solubility limits with the result

where mA is the _average solute concentration.

Because the steep ^{descent of} a(h) towards the low values of h from the maximum at hmax ^is

expected ^{to be} present also in the three-dimen- sional _case, it _may be expected that the actual

position hmax ^{of the maximum of} I(h) ^{will be not}

much shifted ^{to smaller} diffraction angles through

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010082800

829

the eff ect of F2 (h), ^{so that at first} approximation h:nax

^hm’Bx. On the other hand the knowledge

’

of R, enables the correction to be made for deri-

ving ^{the true hmax.}

Some experimental ^{curves of S. A. S. were}

examined for testing ^the simple relations indicated above. The results are summarized in the following

table.

The first column gives ^the ageing température

and time, ^{column 2} the average silver concentration,

column 3 the radii of silver-rich zones determined

by ^Guinier approximation ^{to the} measured S. A. S.

curves (it ^was found that Webb gives ^half figures

for Ri by reexamination ^{of his} published ^{S. A. S.-} curves), ^{column 4 the} xneasured angular position ^of

the maximum, ^{column 5 the values of} Rz according

to (4) ^with ml and ID2 read from the metastable

solubility ^limits reported by Baur and Gerold (1961)

and column 6 the calculated angular positions ^of

maxima by using (3). ^The agreement ^{between the} figures of columns 4 a 6 is satisfactory especially

as regards ^the measurements of Freise and al.

(1961) ^{and of} ours, where the slit-height ^correc-

tison in the S. A. S. curves was made. This was net the case by ^Webb’s curves, where the shifts of the maxima towards the lower angles ^due

to the slit height ^is clearly visible, especially by ^the

curve corresponding ^to 5,8 ^at. % Ag. Fortunately

Guinier and Walker (1953) ^{carried out measu-}

rement on the same alloy treated in a similar _way.

Their S. A. S.-curve is much ^less influenced by ^the

slit height, 1 hey give F-’O . ⁼⁼ 2,33 ^{which is in} good agreement ^with the value ce;:.’1 ⁼ 2,48,

calculated from ^the present close-packed ^distri-

bution of the G.-P. ^zones.

REFERENCES BAUR (R.) ^{and GEROLD (V.), Z.} Metallkunde, 1961, 52, ^671.

FREISE (E. J.), ^KELLY (A.) and NICHOLSON (R. B.), ^Acta Met., 1961, 9, ^250.

GEROLD (V.), Phys. ^stat. sol., 1961, 1, ^37.

WALKER (C. B.) and GUINIER (A.), ^Acta Met., 1953, 1, ^568.

WEBB (M. B.), ^Acta Met., 1959, 7, ^748.