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Observation of pendellösung fringes in reflection-section topographs of bent silicon crystals
Volume
4, number
MATERIALS
2
LETTERS
February
1986
OBSERVATION OF PENDELLOSUNG FRINGES IN REFLECTION-SECTION TOPOGRAPHS OF BENT SILICON CRYSTALS Haydn
CHEN
Department oj Metalhug), and Mining Engineering and Materials Research Laboratory, Unirwsity of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Received
26 November
1985
Pendellbsung fringes have been observed in reflection-section topographs of bent silicon single crystals. The number of fringes increases while the separation distances between fringes decrease with decreasing radius of curvature. Excessive bending of the lattice planes adjacent to a dislocation produces finer fringes. These fringes may be employed as a means to detect minute lattice strains.
1. Introduction The usefulness of X-ray diffraction topography for assessing the degree of perfection of single crystals is well known [ 11. The high sensitivity of X-rays to lattice distortion is exploited to produce images with the diffracted rays. Contrast is visible when lattice perturbing defects are present. There are many different experimental geometries for the production of X-ray diffraction topographs. The common features of the various techniques are: a collimated X-ray source, a specimen, and an imaging recording medium. Variables which distinguish the different techniques are : the characteristics of the X-ray source (divergence, spatial extent, range of wavelengths present), the specimen setting (stationary, translated, rocked) and the detector position with respect to the sample surface (Laue versus Bragg geometry). Transmission (Laue) topographs are produced from the entire thickness of the specimen whereas reflection (Bragg) topographs are produced from the near surface regions of the sample, with the depth imaged depending upon the diffraction vector, absorption properties of the specimen and the degree of lattice perfection. A projection topograph, produced by translating the specimen and the recording medium in tandem across incident beam, reveals the defect distribution throughout the specimen, while a 0 167-577x/86/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
section topograph, produced with a stationary specimen and narrow incident beam, reveals the depths of defects within a narrow slice. Reflection techniques are useful when the regions of crystal near one surface are of interest (as in the cases when specimen’s other surface is severely damaged or physically inaccessible, or when the specimen is too thick to permit transmission of X-rays). Reflection-section topography is particularly useful because it allows an exact localization of kinematical lattice distortions as well as high-contrast imaging by spatial separation of the images from perfect and distorted parts of the crystal [2]. In the present study, we have’ focused on the Pendellasung fringes produced by the wave fields propagating along the same directions inside of a crystal. Although PendeKsung fringes are readily observable in transmission-section topographs, they are seldom detected in reflection-section topographs [2-51. Besides, the physical origins of these fringes and the experimental conditions under which they were observed in reflection-section topographs are not identical. Batterman and Hildebrandt [3] observed oscillations (fringes) in the tails of the rocking curve from thin specimens of flat silicon for planewave incident conditions. They provided a theory which involves the interference of wave fields on the same branch of the dispersion surface as portion of 65
Volume 4, number 2
MATERIALS LETTERS
the incident beam can be scattered backward from the specimen back surface and interfere with the regularly diffracted beam. Sample thicknesses of the order of a few tens of microns or less are required for visibility of this type of fringe pattern. In a more general treatment by Uragami [4] using Takagi’s theory 161, the prediction of crystal thickness independent intensity oscillations in the Bragg diffracted beam resulting from a spatially narrow, but divergent beam incident upon a perfect crystal, was made. Fringes were also observed by Zaumseil [2] in elastically bent quartz crystals using spherical beam geometry. Theoretical calculation was not offered, but application of Takagi theory [6] to weakly bent crystals was suggested. Recently, a theoretical treatment of fringe formation in spherical (divergent) beam Bragg diffraction from bent crystals has been carried out by Chukhovskii et al. [7] using a Green’s function approach to the Takagi theory [6]. Their results indicate that, in the presence of uniform strain gradient, some of the wave fields will propagate in “wave guides” back to the surface in an oscillating manner. Calculations of an expected intensity profile for a simulated Bragg section experiment were performed by Gabrielyan and Chukhovskii [8]. Secondary maxima at one side of the diffracted peak in reflectionsection topographs are predicted. Interpretation of the Green’s function solution reveals that as the radius of curvature of the crystal decreases, the period of the intensity oscillations decreases and they approach the main peak [8]. The present study provides experimental evidence for the predictions of Chukhovskii et al. [7,8] that Pendellosung fringes (intensity oscillations) are observed adjacent to the main diffraction peak in reflection-section topographs of a bent crystal using a spherical incident beam. The variation of the positions and the spacings of these fringes were examined as a function of the bending radius of the crystal. Fringes from the strain field adjacent to a dislocation were also observed.
2. Experimental
procedures
Two types of samples were used in the present study. The first specimen was a perfect Si(ll1) single66
February 1986 Dtffracting /
Planes
53z&*--.
Recorded dlffrocted Image
SM
incident beam defining slit
Sample
Film
Fig. 1. Simplified schematic of the production of a reflection topograph using a Lang camera. Collimation of the incident beam to a vertical line is shown.
crystal wafer one inch in diameter and 0.2 mm thick. Mechanical bending [9 ] was applied to achieve an approximately cylindrical curvature. The other specimen was a similar Si(ll1) wafer which had a 1060 A thick film of epitaxial Pd,Si on one side. The epitaxial silitide film was grown by solid-state reaction of a 750 A thick Pd film evaporated on Si(l11) substrate at 250°C in a purified He furnace. The latter specimen had a radius of curvature of 38 m [9]. All reflection-section topographs were taken on a Rigaku Lang camera, illuminated by a Picker microfocus generator (focal spot size 1.4 X 0.10 mm). MO Kar radiation was used with an incident beam collimating slit of dimensions 15 /.ur~X 2 cm located 4 cm from the sample. Topographs were recorded on 50 p Ilford L.4 nuclear emulsion plates placed perpendicular to the diffracted beam. The microfocus generator was operated at 46 kV and 3 mA emission current. The Si(l11) symmetrical reflection was used. Fig. 1 is a schematic diagram of the experimental arrangement.
3. Results and discussion The reflection-section topographs are shown in figs. 2-4. The dominant feature in the flat, film-free sample (fig. 2a) is the strong line of Kol, diffracted intensity. A weak, narrower line appears parallel to the main beam on the higher 6’ angle side. This is not a KCY~beam, but apparently an artifact with origin at the incident slit. This was shown in 3 ways: (1) Reflection-section topographs taken with the Ka2 component of the incident beam also contained the high
MATERIALS LETTERS
Volume 4, number 2
:uary 1986
100 fim
a
b
Fig. 2. Si( 111) reflection-section topographs of: (a) flat Si crystal, (b) same Si crystal as in (a) but bent to a radius of curvature of 35 m, (c) Si substrate bent to a radius of curvature of 38 m by a PdaSi film.
0 line, (2) ray tracing, by varying the recording film distance from the sample indicated the beam was diverging from the slit, and (3) the direct beam profile also showed the line. Fig. 2b is a topograph from the same sample but which now has an imposed curvature corresponding to a radius of 35 m. Parallel fringes are visible on the low 13angle side of the main peak. Additional curvature (R = 16 m) has the effect of reducing the spacings between fringes as illustrated in fig. 4a. The reflection-section topograph obtained from the sample bent to R = 38 m by the PdZSi overlayer [9] is shown in fig. 2c. Comparison with fig. 2b indicates that not only are the fringes visible from deformed Si as a result of the deformation, but also that their separations are determined by the magnitude of the curvature. The fringes in fig. 2c are perturbed in regions where lattice defects exist. Fig. 3 shows one such region at higher magnification. The disruption of the fringes is evident and, upon closer inspection, a second more closely spaced set of lines appears adjacent to defect
Fig. 3. Higher-magnification view of a region in the Si(ll1) reflection-section topograph of the R = 38 m sample, fig. 2c. Closely spaced fringes are visible on the low 0 side of defect A.
A on the low f3 angle side. Defect A has been identified as ( 1 lo)-type dislocation by the g-b = 0 criterion using Lang projection topographs, fig. 5, produced from the three sets of { 111) planes inclined to the substrate surface. The expected 60’ angular differences between the { 111) traces are indicated in the figure. Fig. 4b shows a reflection-section topograph of the same sample as in fig. 2c after the Pd,Si fti had been chemically removed. Bending to R = 82 m yields fringes with spacings following the trend that as R increases, spacing increases. Table 1 lists the fringe positions measured from the photographic plates in a direction perpendicular to the main beam. With increasing distance from the main beam, both the spacings and the intensities are seen to decrease, in agreement with the predictions of Chukhovskii et al. [8]. 61
Auxiliary maxima were not observed on the low 8 side of the surface diffracted beam from the flat Si(l11) crystals. This observation is not inconsistent with the theory of Uragami [4] for perfect crystals. The positions of the auxiliary maxima depend on the ma~itude of the extinction distance which predicts that the first fringe should appear at a distance less than 5 m from the main peak on the photographic plate. Since the width of the main peak is a few times larger than this, the oscillations that would exist in a flat crystal are expected to be smoothed out in the present case. On the other hand, the work of Lang et al. [5] suggests that fringes would be visible from the flat crystal with carefully prepared incident beam. It is useful to estimate the sensitivity of these fringes to the strain produced by the bending of the
4 Fig. 4. Si( 111) reflection-section topographs of: (a) Si crystal bent to a radius of curvature of 16 m, (b) Si crystal bent to a radius of curvature of 82 m.
Fig. 5. Lang reflection-projection topographs, using three different diffraction vectors, of a Si substrate containing (1 I1 )< 110) dislocations. The majority of line directions are parallel to the traces of the { 111) slip planes. The traces make angles of 60” with one another. The lower contrast of the 111 and 1Trtopographs is due to high fluorescent background intensity during these exposures. (MO Kcvr radiation, incident slit width 100 Mm,)
February 1986
MATERIALS LETTERS
Volume 4, number 2
Table 1 Distance of subsidiary maximum from the Bragg position (pm). Conditions: (a) Film is normal to the diffracted beam. (b) Estimated error in film measurement is * 2 pm. Note: Distances parallel to the crystal surface may be obtained from those in the table by multiplying by 8.8 3 Sample
Xl
(1) Si(ll1) flat (2)Si(lll) with R = 35 m (3) Si(ll1) with R = 16 m (4) Si(111)/1070 A PdzSi with R = 38 m (5) Si( 11 l)/etched to remove silicide and back with R = 82 m (6) Defect in Si(111)/1010 A PdzSi with R = 35 m
19.4 62.1
45.5
61.6
19.4
49.6
82.5
95.1
inter-maximum
e=t/2R.
(1)
In the current study, t = 0.2 mm and fringes are seen for bending radii of the order of 100 m. This result suggests that strain on the order of 1O-6 can be readily detected by the appearance and positions of these fringes. The strain versus the distance between the first fringe and the main peak is shown in fig. 6 with the dashed line serving as a guide to the eye.
I
( 10
0 Pd$i/Si b Si
x4
X5
90.4
91.3
--
(not observed) 41.6 61.6 50.4 35.9
crystal lattice. For a thin plate of thickness t bent cylindrically to a radius R, the amount of shear strain, E, may be calculated using elementary beam theory [ll]:
20 )
X3
x2
spacing = 3.4
Based upon the theoretical work of Chukhovskii et al. [7], and the data presented here, we believe we have experimental evidence for the propagation of wave fields, generated in the bulk by a spherical incident beam, back to the entrance surface. The qualitative behavior of our fringe separations agrees well with the theoretical predictions. In addition, the fringes adjacent to the kinematical defect image in fig. 3 may arise due to the strain field of the defect, superimposed on the macroscopic bending strain field. The smaller fringe separations would be consistent with the expected higher local strain surrounding a dislocation. In view of the sensitivity of these fringes to the bending curvature of the crystal, it appears possible to utilize these fringes for the detection of minute strain fields and/or bending of a single crystal in the reflection geometry.
Acknowledgement
XI (ad Fig. 6. Strain versus X1 showing an inverse relationship between the two quantities.
The author wishes to thank Drs. M. Kuriyama and S.R. Stock for valuable discussions. This work was supported by the Materials Science Division of the Department of Energy under contract numbers: DE-FG02-84ER45098 and DE-AC02-76ERO1198. The topography experiments were performed at the University of Illinois Materials Research Laboratory’s Center for Microanalysis of Materials which is supported by the Materials Science Division of the Department of Energy. 69
Volume 4, number 2
MATERIALS LETTERS
References [l]B.K. Tanner, X-ray diffraction topography (Pergamon Press, London, 1976), and references therein. [2] P. Zaumseil, Kristall Technik 13 (1978) 983. [3 ] B.W. Batterman and G. Hildebrandt, Acta Cryst. A24 (1968) 150. [4] ST. Uragami, J. Phys. Sot. Japan 27 (1969) 147. [5] A.R. Lang and Z.-H. Mai, Proc. Roy. Sot. A368 (1979) 313. [6] S. Takagi, Acta Cryst. 15 (1962) 1311; J. Phys. Sot. Japan 26 (1969) 1239.
70
February 1986
[71F.N. Chukhovskii, K.T. Gabrielyan and P.V. Petrashen’, ActaCryst. A34 (1978) 610. I81 K.T. Gabrielyan and F.N. Chukhovskii, Soviet Phys. Cryst. 25 (1980) 607. [91 B. Coulman, Ph.D. Thesis, University of Illinois at UrbanaChampaign (1984). [lo1 H. Chen, G.E. White, S.R. Stock and P.S. Ho, Thin Solid Films 93 (1982) 161. [ill S. Timoshenko and J.N. Goodier, Theory of applied elasticity (McGraw Hi& New York, 1951).
4, number
MATERIALS
2
LETTERS
February
1986
OBSERVATION OF PENDELLOSUNG FRINGES IN REFLECTION-SECTION TOPOGRAPHS OF BENT SILICON CRYSTALS Haydn
CHEN
Department oj Metalhug), and Mining Engineering and Materials Research Laboratory, Unirwsity of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Received
26 November
1985
Pendellbsung fringes have been observed in reflection-section topographs of bent silicon single crystals. The number of fringes increases while the separation distances between fringes decrease with decreasing radius of curvature. Excessive bending of the lattice planes adjacent to a dislocation produces finer fringes. These fringes may be employed as a means to detect minute lattice strains.
1. Introduction The usefulness of X-ray diffraction topography for assessing the degree of perfection of single crystals is well known [ 11. The high sensitivity of X-rays to lattice distortion is exploited to produce images with the diffracted rays. Contrast is visible when lattice perturbing defects are present. There are many different experimental geometries for the production of X-ray diffraction topographs. The common features of the various techniques are: a collimated X-ray source, a specimen, and an imaging recording medium. Variables which distinguish the different techniques are : the characteristics of the X-ray source (divergence, spatial extent, range of wavelengths present), the specimen setting (stationary, translated, rocked) and the detector position with respect to the sample surface (Laue versus Bragg geometry). Transmission (Laue) topographs are produced from the entire thickness of the specimen whereas reflection (Bragg) topographs are produced from the near surface regions of the sample, with the depth imaged depending upon the diffraction vector, absorption properties of the specimen and the degree of lattice perfection. A projection topograph, produced by translating the specimen and the recording medium in tandem across incident beam, reveals the defect distribution throughout the specimen, while a 0 167-577x/86/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
section topograph, produced with a stationary specimen and narrow incident beam, reveals the depths of defects within a narrow slice. Reflection techniques are useful when the regions of crystal near one surface are of interest (as in the cases when specimen’s other surface is severely damaged or physically inaccessible, or when the specimen is too thick to permit transmission of X-rays). Reflection-section topography is particularly useful because it allows an exact localization of kinematical lattice distortions as well as high-contrast imaging by spatial separation of the images from perfect and distorted parts of the crystal [2]. In the present study, we have’ focused on the Pendellasung fringes produced by the wave fields propagating along the same directions inside of a crystal. Although PendeKsung fringes are readily observable in transmission-section topographs, they are seldom detected in reflection-section topographs [2-51. Besides, the physical origins of these fringes and the experimental conditions under which they were observed in reflection-section topographs are not identical. Batterman and Hildebrandt [3] observed oscillations (fringes) in the tails of the rocking curve from thin specimens of flat silicon for planewave incident conditions. They provided a theory which involves the interference of wave fields on the same branch of the dispersion surface as portion of 65
Volume 4, number 2
MATERIALS LETTERS
the incident beam can be scattered backward from the specimen back surface and interfere with the regularly diffracted beam. Sample thicknesses of the order of a few tens of microns or less are required for visibility of this type of fringe pattern. In a more general treatment by Uragami [4] using Takagi’s theory 161, the prediction of crystal thickness independent intensity oscillations in the Bragg diffracted beam resulting from a spatially narrow, but divergent beam incident upon a perfect crystal, was made. Fringes were also observed by Zaumseil [2] in elastically bent quartz crystals using spherical beam geometry. Theoretical calculation was not offered, but application of Takagi theory [6] to weakly bent crystals was suggested. Recently, a theoretical treatment of fringe formation in spherical (divergent) beam Bragg diffraction from bent crystals has been carried out by Chukhovskii et al. [7] using a Green’s function approach to the Takagi theory [6]. Their results indicate that, in the presence of uniform strain gradient, some of the wave fields will propagate in “wave guides” back to the surface in an oscillating manner. Calculations of an expected intensity profile for a simulated Bragg section experiment were performed by Gabrielyan and Chukhovskii [8]. Secondary maxima at one side of the diffracted peak in reflectionsection topographs are predicted. Interpretation of the Green’s function solution reveals that as the radius of curvature of the crystal decreases, the period of the intensity oscillations decreases and they approach the main peak [8]. The present study provides experimental evidence for the predictions of Chukhovskii et al. [7,8] that Pendellosung fringes (intensity oscillations) are observed adjacent to the main diffraction peak in reflection-section topographs of a bent crystal using a spherical incident beam. The variation of the positions and the spacings of these fringes were examined as a function of the bending radius of the crystal. Fringes from the strain field adjacent to a dislocation were also observed.
2. Experimental
procedures
Two types of samples were used in the present study. The first specimen was a perfect Si(ll1) single66
February 1986 Dtffracting /
Planes
53z&*--.
Recorded dlffrocted Image
SM
incident beam defining slit
Sample
Film
Fig. 1. Simplified schematic of the production of a reflection topograph using a Lang camera. Collimation of the incident beam to a vertical line is shown.
crystal wafer one inch in diameter and 0.2 mm thick. Mechanical bending [9 ] was applied to achieve an approximately cylindrical curvature. The other specimen was a similar Si(ll1) wafer which had a 1060 A thick film of epitaxial Pd,Si on one side. The epitaxial silitide film was grown by solid-state reaction of a 750 A thick Pd film evaporated on Si(l11) substrate at 250°C in a purified He furnace. The latter specimen had a radius of curvature of 38 m [9]. All reflection-section topographs were taken on a Rigaku Lang camera, illuminated by a Picker microfocus generator (focal spot size 1.4 X 0.10 mm). MO Kar radiation was used with an incident beam collimating slit of dimensions 15 /.ur~X 2 cm located 4 cm from the sample. Topographs were recorded on 50 p Ilford L.4 nuclear emulsion plates placed perpendicular to the diffracted beam. The microfocus generator was operated at 46 kV and 3 mA emission current. The Si(l11) symmetrical reflection was used. Fig. 1 is a schematic diagram of the experimental arrangement.
3. Results and discussion The reflection-section topographs are shown in figs. 2-4. The dominant feature in the flat, film-free sample (fig. 2a) is the strong line of Kol, diffracted intensity. A weak, narrower line appears parallel to the main beam on the higher 6’ angle side. This is not a KCY~beam, but apparently an artifact with origin at the incident slit. This was shown in 3 ways: (1) Reflection-section topographs taken with the Ka2 component of the incident beam also contained the high
MATERIALS LETTERS
Volume 4, number 2
:uary 1986
100 fim
a
b
Fig. 2. Si( 111) reflection-section topographs of: (a) flat Si crystal, (b) same Si crystal as in (a) but bent to a radius of curvature of 35 m, (c) Si substrate bent to a radius of curvature of 38 m by a PdaSi film.
0 line, (2) ray tracing, by varying the recording film distance from the sample indicated the beam was diverging from the slit, and (3) the direct beam profile also showed the line. Fig. 2b is a topograph from the same sample but which now has an imposed curvature corresponding to a radius of 35 m. Parallel fringes are visible on the low 13angle side of the main peak. Additional curvature (R = 16 m) has the effect of reducing the spacings between fringes as illustrated in fig. 4a. The reflection-section topograph obtained from the sample bent to R = 38 m by the PdZSi overlayer [9] is shown in fig. 2c. Comparison with fig. 2b indicates that not only are the fringes visible from deformed Si as a result of the deformation, but also that their separations are determined by the magnitude of the curvature. The fringes in fig. 2c are perturbed in regions where lattice defects exist. Fig. 3 shows one such region at higher magnification. The disruption of the fringes is evident and, upon closer inspection, a second more closely spaced set of lines appears adjacent to defect
Fig. 3. Higher-magnification view of a region in the Si(ll1) reflection-section topograph of the R = 38 m sample, fig. 2c. Closely spaced fringes are visible on the low 0 side of defect A.
A on the low f3 angle side. Defect A has been identified as ( 1 lo)-type dislocation by the g-b = 0 criterion using Lang projection topographs, fig. 5, produced from the three sets of { 111) planes inclined to the substrate surface. The expected 60’ angular differences between the { 111) traces are indicated in the figure. Fig. 4b shows a reflection-section topograph of the same sample as in fig. 2c after the Pd,Si fti had been chemically removed. Bending to R = 82 m yields fringes with spacings following the trend that as R increases, spacing increases. Table 1 lists the fringe positions measured from the photographic plates in a direction perpendicular to the main beam. With increasing distance from the main beam, both the spacings and the intensities are seen to decrease, in agreement with the predictions of Chukhovskii et al. [8]. 61
Auxiliary maxima were not observed on the low 8 side of the surface diffracted beam from the flat Si(l11) crystals. This observation is not inconsistent with the theory of Uragami [4] for perfect crystals. The positions of the auxiliary maxima depend on the ma~itude of the extinction distance which predicts that the first fringe should appear at a distance less than 5 m from the main peak on the photographic plate. Since the width of the main peak is a few times larger than this, the oscillations that would exist in a flat crystal are expected to be smoothed out in the present case. On the other hand, the work of Lang et al. [5] suggests that fringes would be visible from the flat crystal with carefully prepared incident beam. It is useful to estimate the sensitivity of these fringes to the strain produced by the bending of the
4 Fig. 4. Si( 111) reflection-section topographs of: (a) Si crystal bent to a radius of curvature of 16 m, (b) Si crystal bent to a radius of curvature of 82 m.
Fig. 5. Lang reflection-projection topographs, using three different diffraction vectors, of a Si substrate containing (1 I1 )< 110) dislocations. The majority of line directions are parallel to the traces of the { 111) slip planes. The traces make angles of 60” with one another. The lower contrast of the 111 and 1Trtopographs is due to high fluorescent background intensity during these exposures. (MO Kcvr radiation, incident slit width 100 Mm,)
February 1986
MATERIALS LETTERS
Volume 4, number 2
Table 1 Distance of subsidiary maximum from the Bragg position (pm). Conditions: (a) Film is normal to the diffracted beam. (b) Estimated error in film measurement is * 2 pm. Note: Distances parallel to the crystal surface may be obtained from those in the table by multiplying by 8.8 3 Sample
Xl
(1) Si(ll1) flat (2)Si(lll) with R = 35 m (3) Si(ll1) with R = 16 m (4) Si(111)/1070 A PdzSi with R = 38 m (5) Si( 11 l)/etched to remove silicide and back with R = 82 m (6) Defect in Si(111)/1010 A PdzSi with R = 35 m
19.4 62.1
45.5
61.6
19.4
49.6
82.5
95.1
inter-maximum
e=t/2R.
(1)
In the current study, t = 0.2 mm and fringes are seen for bending radii of the order of 100 m. This result suggests that strain on the order of 1O-6 can be readily detected by the appearance and positions of these fringes. The strain versus the distance between the first fringe and the main peak is shown in fig. 6 with the dashed line serving as a guide to the eye.
I
( 10
0 Pd$i/Si b Si
x4
X5
90.4
91.3
--
(not observed) 41.6 61.6 50.4 35.9
crystal lattice. For a thin plate of thickness t bent cylindrically to a radius R, the amount of shear strain, E, may be calculated using elementary beam theory [ll]:
20 )
X3
x2
spacing = 3.4
Based upon the theoretical work of Chukhovskii et al. [7], and the data presented here, we believe we have experimental evidence for the propagation of wave fields, generated in the bulk by a spherical incident beam, back to the entrance surface. The qualitative behavior of our fringe separations agrees well with the theoretical predictions. In addition, the fringes adjacent to the kinematical defect image in fig. 3 may arise due to the strain field of the defect, superimposed on the macroscopic bending strain field. The smaller fringe separations would be consistent with the expected higher local strain surrounding a dislocation. In view of the sensitivity of these fringes to the bending curvature of the crystal, it appears possible to utilize these fringes for the detection of minute strain fields and/or bending of a single crystal in the reflection geometry.
Acknowledgement
XI (ad Fig. 6. Strain versus X1 showing an inverse relationship between the two quantities.
The author wishes to thank Drs. M. Kuriyama and S.R. Stock for valuable discussions. This work was supported by the Materials Science Division of the Department of Energy under contract numbers: DE-FG02-84ER45098 and DE-AC02-76ERO1198. The topography experiments were performed at the University of Illinois Materials Research Laboratory’s Center for Microanalysis of Materials which is supported by the Materials Science Division of the Department of Energy. 69
Volume 4, number 2
MATERIALS LETTERS
References [l]B.K. Tanner, X-ray diffraction topography (Pergamon Press, London, 1976), and references therein. [2] P. Zaumseil, Kristall Technik 13 (1978) 983. [3 ] B.W. Batterman and G. Hildebrandt, Acta Cryst. A24 (1968) 150. [4] ST. Uragami, J. Phys. Sot. Japan 27 (1969) 147. [5] A.R. Lang and Z.-H. Mai, Proc. Roy. Sot. A368 (1979) 313. [6] S. Takagi, Acta Cryst. 15 (1962) 1311; J. Phys. Sot. Japan 26 (1969) 1239.
70
February 1986
[71F.N. Chukhovskii, K.T. Gabrielyan and P.V. Petrashen’, ActaCryst. A34 (1978) 610. I81 K.T. Gabrielyan and F.N. Chukhovskii, Soviet Phys. Cryst. 25 (1980) 607. [91 B. Coulman, Ph.D. Thesis, University of Illinois at UrbanaChampaign (1984). [lo1 H. Chen, G.E. White, S.R. Stock and P.S. Ho, Thin Solid Films 93 (1982) 161. [ill S. Timoshenko and J.N. Goodier, Theory of applied elasticity (McGraw Hi& New York, 1951).