THICKNESS DETERMINATION OF SURFACE FILM FOR TEXTURED SPECIMENS BY X-RAY DIFFRACTION

A simple and convenient X-ray diffraction method is proposed to determine the thickness 
of surface film for textured specimens. The analysis result for a synthetic specimen with 
surface film has proved that the method is applicable and reliable.


INTRODUCTION
Improving and strengthening usabilities of materials by surface coating technique plays an important role in modem materials science and engineering, In order to control and estimate the production cost, one has to determine the thickness of the surface film.When the film or the matrix of the coated materials is crystalline and the film thickness is smaller than the depth penetrated by X-rays, the thickness can simply be determined by X-ray diffraction method, i.e., the absorption effect of incident and diffracted beams through the film (Friedman and Birk,  1946; Keating and Kammerer, 1958; Borie and Sparks, 1961).Usually the diffraction phase in these materials exhibits a preferred orientation of crystallites, namely texture, the diffracted intensity will depend on not only the film thickness itself, but also the texture.For this, some Corresponding author.effective methods to avoid the influence of texture on intensity were proposed and developed (Keating and Kammerer, 1958; Tao et al.,  1984).The principle is not more than that one can eliminate the texture factor and finally obtain a unitary functional relation of diffracted intensity to film thickness by measurement of the intensity of a dif- fraction using two incident radiation with different wavelengths or measurement of the intensity from two orders of a diffraction.In order to further simplify the measurement procedure, a new method is in- troduced in the present work, by which neither the double-wavelength diffraction nor the d0uble-order one, as has been stated above, is necessary.In the following we shall describe the fundamentals of the method in detail.

Experimental Arrangement
The first measured position of a specimen is arranged after the back- reflection technique (Schulz, 1949) with the specimen orientation {a fl), as shown in Fig. (a).Here c is the rotation angle of the specimen about the intersection of the specimen surface and the diffraction plane, containing the incident and diffracted beam vectors, and fl is the rota- tion angle ofthe specimen about the normal to its surface.Then keeping c and g/fixed, the specimen will be rotated about the diffraction vector over 90 as the second measured position (see Fig. l(b)).Obviously, there will be the same texture factor and different paths of X-rays in the surface film for an analysed specimen in the two measured positions.

Correction of Diffracted Intensity
In back-reflection technique, using a flat specimen, a defocalization effect occurs due to an a-tilt of the specimen.In the first case of measurement the defocalization may be negligible by the limit of a main slit in front of the specimen holder (Chernock and Beck, 1952).But in the second case a deviation of the irradiated specimen surface from the focussing sphere cannot be limited by the main slit (Liu, 1991).Hence, when the angle c is larger, one has to consider the effect of defocaliza- tion on measured intensities.In addition, the irradiated volume in the specimen will become different when there is a change from the first experimental position to the second one, that may also result in an evident influence on the measured result.
One can quantitatively consider the above-mentioned two effects on the intensity in the following way.Suppose Ic and I' are the diffracted intensities from a correcting specimen in the absence of film, separately in the first and second cases (with certain c and 0), then we take c=g/ic as the correction factor of the intensity in the second experimental case, with respect to the first case, for the corresponding c and 0.

Film Diffraction
Figure 2(a) shows the diffraction geometry in the section of a specimen with surface film along the diffraction plane, corresponding to the first measured position.Suppose the thickness of the film is D, the visible film thickness in the section should be D/cos oz.Consider the diffraction at a particular lattice plane (hkl) in a thin layer of the film with the thickness dx, located at a depth x below the surface (as indicated by shading in Fig. 2(a)), the total length of the path passed by the incident and diffracted beams with Bragg-angle 0 in the film is 2x/sin 0. Given that the irradiated area on the specimen surface for c 0 is A0, and the film has an absorption coefficient/zf, the diffracted intensity from the thin layer may be described in the form dI Io Vd Rhkl Phkl (Oz [) exp(--2#fx/sin 0) Ao dx, (2) where I0 is the incident intensity, Vd the volume fraction of the dif- fraction phase in the film, Ra,t the reflectivity of the lattice plane (hkl), Phkt() the (hkl)-pole density in the specimen direction {cz/3}, exp(-2/rx/sin 0) the absorption factor for film diffraction, A0 dx the volume of the considered thin layer in the film.After Schulz's con- sideration, the elementary volume is independent of cz in this case (Schulz, 1949).The total intensity, diffracted by all the irradiated film, can be obtained by integrating the above intensity dI over x from 0 to D/cos a, i.e., over the depth penetrated by X-rays through the film: I Io Vd Rhkl Phkl(a ) AO sin 0/(2#f) { exp[-2#fD/(cosa sin 0)]}. (3) For the film diffraction in the second measured position (see Fig. 2(b)), the path lengths of the incident and diffracted beams separately become x/sin(0 + a) and x/sin(0 a).The intensity of the (hkl)-diffraction from the thin layer, as shown by shading in Fig. 2(b), is dF Io Vd Rhil Phkl(a ) exp{--#fx[1/sin(0 + a) + 1 / sin(0 a)]}CA0 dx, (4) where exp{ #fx[1/sin( 0 + a) + 1/sin( 0 a)] } is the absorption factor of the film in the new measured position.Due to a change of the irradiated volume and a defocussing effect, the correction factor C should here be introduced in Eq. ( 4).The total intensity of film dif- fraction in this case will be equal to the integration of dI' over x from 0 to D: I' Io Vd Rhkt Phkt (a )CAo[cos(2a) cos(20)]/(4/zf sin 0cos a) x {1 exp[4#fD sin0 cosa/(cos 20-cos 2a)]}.
Substituting the experimentally measured intensities, I and I', into Eqs.

Matrix Diffraction
The optical geometries of matrix diffraction with respect to the two experimental arrangements are seen in Fig. 3(a) and (b).Similar to the discussion about film diffraction, we take the (h'k'l')-diffraction from the thin layer in the matrix, as shown in Fig. 3, into account, then the absorption factors of the specimen corresponding to the two mea- sured positions are, separately, exp{ 2/sin 0' [/zfD/cos a +/ZmX]} and exp{-(#fD+p,mX)[1/sin(O'+a)+ 1/sin(O'-a)]}, where /m is the absorption coefficient of the matrix.By integrating the diffracted in- tensity of the considered matrix element over the whole irradiated volume in the matrix (in other words, over x from 0 to o), the total Y.oS.LIU AND Y. ZHAO intensities of matrix diffraction in the two measuring cases are ex- pressed separately as I Io V Rh'k'l' eh'k'l'( ) Ao sin 0'/(2/Zm) x exp[-2#fD/(cosa sin 0')] ( 6) and I' Io V Rh,k,l, eh,k,l,( Ot )CA0[cos(2c) cos(20')]/ (4/Zrn sin O' cos c) exp[4#fD sin 0' cos c/(cos 20' cos 2c)]. (7)  The film thickness D may also be determined from the two experimen- tally measured intensitites I and I' by the simultaneous Eqs. ( 6) and ( 7).

EXPERIMENT AND RESULT
In order to check the introduced method, a synthetic specimen with surface film was prepared by combining a rolled nickel film and a rolled copper sheet with strong texture, which are on hand, with each other to simulate coated materials.The thickness of the nickel film is 5.521 tm, determined from its weight, density and surface sizes.
The experiment was carried out with Cu Kc-radiation in the dif- fractometer D/MAX-3A.For measurement the tilting angle a of the specimen was 5 , and a fl-rotation with higher speed was executed to increase the experimental statistical accuracy.The texture factor, thus, is independent of ft.The diffracted intensity from the lattice plane (111)   of the 'matrix' Cu (0-21.38)was measured.This peak is well re- solved from neighbouring peaks.Due to a broadening of the diffrac- tion peak caused by various reasons, a step-scanning method was applied to obtain the complete peak profile.Then the Kc-peak was separated from Ka2, and the integral intensity of the former was sub- stituted for I in Eq. ( 6) or I' in Eq. ( 7), depending on the corresponding measured position.The correction factor C was obtained by the in- tensities of the same diffraction from a copper sheet without film in the two measured positions.The thickness of the nickel film was finally determined by Eqs. ( 6) and ( 7).The experimental and calculated results are shown in Table I: Taking the possible errors, such as the inhomogeneity of the film thickness or the roughness of the specimen surface, into account, we may think the calculated result is satisfactory, as compared with the above-mentioned weight method.

CONCLUSIONS
The thickness of a surface film on textured materials can easily be determined by the X-ray diffraction method, recommended in the paper, when the film thickness is smaller than the penetrated depth of X-rays.
The advantage of the method over others is to need neither the double-wavelength nor the double-order diffraction, that may further simplify the experimental procedure.
The analysis for a synthetic specimen with surface film indicates that the result is accurate enough and the method is applicable.