# Average contrast method for the determination of the α-planar fault displacement vector

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R. M. SOFRONOVA and V. I. CHERVINSKII: a-Planar Fault Displacement Vector 103

phys. stat. sol. (a) 68, 103 (1981)

Subject classification: 1.4; 10

Baikov Institute of dletdlurgy, Academy of Sciencea of the UBSR, Mascou,

Average Contrast Method for the Determination of the a-Planar Fault Displacement Vector BY R. M. SOFRONOVA and V. I. CHERVINSKII

A method for the determination of the displacement vector of a-planar faults in cubic crystals is propoeed. This method is based on the comparative contrast analysis. The evenness of the average intensity functions of the transmitted and scattered beam in dynamical approximation is used for contrast classifying.

Ein Verfahren zur Bestimmung des Verzerrungsvektors von planaren a-Fehlern in kubiechen Kri- stallen wird vorgeschlagen. Dem Verfahren w i d eine Vergleichsanalyse der Kontraste zu Grunde gelegt. Die Geradheit der mittleren Intensitiiten fur eine Transmissions- und Reflexionswelle in dynamischer Niherung wird fur die Klassifikation der Kontraste benutzt.

1. Introduction

The displacement vector of stacking faults and other a-planar faults can be deter- mined by the invisibility criterion a = 2zn, where n is an integer, if the information about the contrast extinction is sufficient [l]. In some cases an unambiguous deter- mination of displacement vector demands a large number of two-beam diffraction conditions using low, high, and very high reflections [2]. However, the operations with high reflections are connected with some difficulties. Besides, the displacement vector with irrational components cannot be determined by the invisibility criterion.

The method of displacement vector determination based on the diffraction spot satellites can be applied only for periodic shear structures which create cooperation diffraction effects [3,4]. This method does not allow to determine the displaoement vector for single faults. In the present paper a method of displacement vector determination for a-planar

faults in cubic crystals is proposed for the cases when the displacement vector cannot be determined by the invisibility criterion as well as in absence of diffraction spot satellites. The method is based on the comparative analysis of average contrasts only for low reflections when o FZ 0. It is desirable to use all operating vectors of the same type for systematical observations.

2. Determination of the Displacement Vector 2.1 Average contlvurt

The average contrast is determined as

(Jb - ( J ) ) / J , 3 (1) where Jb is the background intensity, (J) is the average intensity across the fault. According to [6] the bright-field intensity distribution can be written as

J, = exp (- x) 2nt [JF) + J f ) + Ji3)] ,

104 R. M. SOFRONOVA end V. I. C~EWINSKII

where

(3) 0 2 + cos2 u p

2(1 + W2)2 JLU = [ch (2n AKit + 40) + cos (2n AK,t + 2p)] , sin2 a12

J ( 2 ) = [Oh ( 4 ~ AKit) + cos (4n A K f ) ] , O 2(1 + W2)2

x sh [2n AKi($ - 1) + 20]+sin [2n AK, (+ - t ) + 911 x x sh AKi (+ + t ) + 20]},

0 1 t g p = ____ tga/2 , ah20 = w , AK - f1 + W 2 i - 5 ; f m

(4)

(5)

where 01 = ZngR is the phase angle, g the operating vector, t the thickness of the crystal, t = tl - 11% the distance between the foil centre and the fault, tl the distance between the upper surface of the foil and the fault, w the dimensionless parameter of the deviation from the Bragg position, 5, the extinction distance, 5; and 5; are eb- sorption parameters. The dark-field intensity is written in the same way,

2nt J g = exp ( - x) [J;) + J f ) + J?)] , where

sin2 1x12 J ( 2 ) = [ch (4n AKit + 40) - cos (4n AK,t)] , 2(1 + 0 2 ) 2

x sh [ 2 n A K i ( + - t) - 201 - sin[2nAKr($ - t ) + q] x

x sh p n A K , ($ + t) + 281). The average intensity with respect to t is used for contrast determination

t/2

(9)

Determination of the a-Planar Fault Displacement Vector 105

It can be shown that (J,(a, w ) ) and ( J (a, w ) ) are odd functions relative to a when w =+ 0. If w x 0 the functions (2) an3 (7) are given as follows:

+ sin? (L - t')sh & 2 (+ + t ' ) ] } ,

J, =- 1 exp ( - 7) {cos2 4 [ ch rg) - cos re)] + 2

Fig. 1. The functions (J,,(a)) and (Jg(a)> at w = 0 and the corresponding scale of average intensity subdivided into K zones in accordance with the number of contrasts

106 R. M. SOFRONOVA and V. I. CEERVINSKII

The average values of the functions (12) and (13) can be expressed as

+ 2nt &sin2 '[ 2 sh (2E) +,sin E: . (?)I - + + nt[(E,/Ep + 11 Eg-

(J,(oL)) = - exp ( - 'F;) - [ cos2 ;- [ch (T) - cos re)] + 2 As it follows from expressions (14) and (15) the function (J,(cx)) is odd relative to oc but this oddness can be neglected for t > 45. The function (J,(oc)) is even for all values of thickness. The functions (14) and (15) are schematically presented in Fig. 1. The average contrast (1) is also an even function, i.e. it does not depend on the sign of the operating vector.

2.2 Ttrpe of the displacement vector

The total number K of different contrasts for the fault with displacement vector of the general type (XYZ) is PI2 at maximum due to the evenness of the average contrast function (P is the multiplicity of the operating vector (hkl)) . Reducing the generality of the displacement vector causes the number K to decrease. One can predict the displacement vector type using Table 1 if the number K and some charac- teristics of the contrast are known. The contrast extinction is marked in Table 1 as e. The contrast corresponding to the phase angle OL = n(2n + 1) is similar to the anti- phase boundary contrast marked as a, and this contrast can be also easily distinguished from the other contrasts [5]. The figures before e and a indicate the multiplicity of the extinction and APB contrast. The following figures without letters indicate the multiplicity of other intermediate contrasts. The sum of all figures is PIS. In most cases the displacement vector type can be determined unambiguously by Table 1. However, the ambiguous determination is also useful since it effectively narrows the search as well. If necessary the table can be extended.

2.3 Magnitude of the displacement vector components

The intervals [-1/2; 01, [O; 1/21 on the axis of the normalized phase angle d = a/2x are arbitrarily divided into K regions 'D where k = 1, 2, ..., K (Fig. 1). The regions kD correspond to k zones on t,he average intensity scale. In the case of extinction the region lD is represented by the point lD = 0. The normalized phase angle d, = = 'a', + n, corresponds to a, = 2ng,R, where n5 is an integer, and j is an index corresponding to the operating vectors of the same type, j = 1,2, ... , (1/2) Ps=(h,kltl). The values 'd , must satisfy the following conditions :

' d j c 'D , (16) 0 5 I'djl < I2d,l < ... < I%gl 5 f . (17)

dj = g$ = gzjx + gyjy + gzjz = 'df + nj (18) The system of equations of the kind

Determination of the a-Planar Fault Displacement Vector 107

Table 1 Total number of contrasts for low reflections of cubic crystals (K)

type of R ~~~ ~

type of operating vector

K 110 K 111 K 200 K 220

X Y Z x x z x x x X Y O x x o x o o 112 Y z 112 Y Y 112 Y 0 1/2 112 z 112 112 112 1/2 112 0 1/2 0 0 114 Y 2 114 Y Y 114 Y 0 114 114 Z 114 114 114 114 114 0 114 0 0 1/4 1/2 0 114 114 1/2 114 112 1/2 113 113 113

6 4 2 4 3 2 4 3 3 2 1 2 2 6 4 4 4 2 3 2 2 3 2 2

1,1,1,1,1,1 4 le, 1, 2, 2 3 3e, 3 2 1, 1, 2, 2 2 le, 1, 4 2 2e, 4 1 1, 1, 2, 2 2 1, 1, 4 2 2a, 2, 2 1 2e, 4 1 6e 1 2e, 4a 1 2e, 4a 1 l,l, 1, l,l, 1 4 le, 1, 2, 2 3 1, 1, 2, 2 2 le, la, 2, 2 2 3e, 3a 1 le, la, 4 2 2e, 4 1 2a, 4 1 le, la, 4 2 2e, 4 1 3e, 3 2

1, 1, 1, 1 3 1, 1, 2 2 1, 3 1 2, 2 3 2e, 2 2 4 2 2, 2 3 2a, 2 2 4 2 4 2 4a 1 4e 1 4a 1 1, 1, 1, 1 3 1, 1, 2 2 2, 2 3 2, 2 2 4 1 2e, 2a 2 4 2 4 2 2e, 2a 2 4 2 le, 3 1

1, 1, 1 1, 2 3 le, 1, 1 le, 2 2e, 1 le, 1, 1 le, 2 28, 1 2e, 1 3e 3e 3e la, 1, 1 la, 2 le, la, 1 2a, 1 3a le, 2a 2e, la 2e, l a le, 2a 2e, l a 3

6 4 2 4 3 2 4 3 2 2 1 1 1 4 3 3 2 1 2 2 2 2 2 2

1,1,1,1,1,1 le, 1, 2, 2 3e, 3 1, 1, 2, 2 le, 1, 4 2e, 4 1, 1, 2, 2 le, 1, 4 2e, 4 2c, 4 6e 6e Be 1, 1, 2, 2 le, 1, 4 2 4 2, 2 2e, 4 6e 2e, 4a 2e, 4a 2e, 4a 2e, 4a 2e, 4a 3e, 3

can be supplemented with analogous equations for another reflection type

108 R. M. SOI~RONOVA and V. I. CHEF~VINSKII: a-Planar Fault Displacement Vector

I Fig. 2. The contrast scale on edge fringe in bright field; dark contrast,

brighl contrast

n - 112 < d 46,). The idea of the average contrast was used for the explanation of the asymmetry contrast when w > 0 [6], however, the average contrast in [6] was presented in the kinematical approximation.

The average contrast method described in the present paper has been used for the determination of the nature of planar faults in a dilute Mo-B alloy 7,8].

References [I] P. B. HIRSCH, A. HOWIE, R. B. NICOLSON, D. W. PASHLEY, and M. I. WHELAN, Electron

[2] C. A. FERREII~A LJMA and A. HOWIE, Phil. Mag. 34,1057 (1976). [3] J. VAN LANDUYT, R. DE RIDDER, R. GEVERS, and S. AMELINCKX, Mater. Res. Bull. 5, 353

[a] R. DE RIDDER, J. VAN LANDUYT, and S. AB~ELINCKX, phys. stat. sol. (a) 9, 551 (1972). [5] J. VAN LANDUYT, R. GWERS, and S. AYELINCKX, phys. stat. sol. 7, 519 (1964). [6] H. FOLL, C. B. CABTES, and M. WIL.KENS, phys. stat. sol. (a) 68, 393 (1980). 171 I. I. NOVIKOV, M. M. KANTCJR, R. M. SOFBONOVA, and V. I. CHERVINSKII, Roc. Europ. Congr.

[S] V. I. CHERVINSJKII, M. M. KANTOR, I. I. NOVIKOV, and R. M. SOFBONOVA, to be published.

Microscopy of Thin Crystals, Butterworths, London 1965.

(1970).

Electron Microscopy 1980, Vol. 1 (p. 204).

(Received February 10,1981)

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