David R. BlackGabrielle G. Long

960-10Special Publication 960-10

0.5 mm

i

NIST Recommended Practice Guide

Special Publication 960-10

X-Ray

Topography

David R. Black and

Gabrielle G. Long

Materials Science and

Engineering Laboratory

April 2004

U.S. Department of Commerce

Donald L. Evans, Secretary

Technology Administration

Phillip J. Bond, Under Secretary for Technology

National Institute of Standards and Technology

Arden L. Bement, Jr., Director

DEPARTMENT OF COMMERCE

UNITED STATES OF AMERICA
ii

◆ X-Ray Topography

Certain commercial entities, equipment, or materials may be identified in

this document in order to describe an experimental procedure or concept

adequately. Such identification is not intended to imply recommendation or

endorsement by the National Institute of Standards and Technology, nor is it

intended to imply that the entities, materials, or equipment are necessarily the

best available for the purpose.

National Institute of Standards and Technology

Special Publication 960-10

Natl. Inst. Stand. Technol.

Spec. Publ. 960-10

53 pages (April 2004)

CODEN: NSPUE2

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iii

FOREWORD

The study of the interrelationships between processing, structure and

properties of materials is fundamental to the field of materials science and

engineering. The need to understand the microstructure of materials has

driven the development of a wide variety of x-ray based characterization

techniques. While X-ray topography is a very powerful tool for the evaluation

of crystals for technological applications and for many areas of crystal and

thin-film growth and processing, it is perhaps less well known and, therefore,

has historically been underutilized in the United States. The aim of this guide

is to raise the awareness of X-ray topography and, therefore, make modern

x-ray topography techniques more accessible to materials scientists who

would benefit by obtaining the rich variety of microstructural information

it offers. The basic principles and practical aspects of topography are

discussed and demonstrated through examples to help illustrate the diverse

applications of this technique.

Foreword ◆
iv

◆ X-Ray Topography
v

ACKNOWLEDGMENTS

The authors are indebted to Harold Burdette who collaborated on most of

the examples shown in this Practice Guide. X-ray topography would not have

been possible at NBS/NIST were it not for the vision of Masao Kuriyama,

who built the original NBS/NIST x-ray topography laboratory, and the many

efforts of our early collaborators: William Boettinger, Ronald Dobbyn,

Richard Spal and Bruce Steiner.

Acknowledgments ◆
vi

◆ X-Ray Topography
vii

Table of Contents ◆

TABLE OF CONTENTS

List of Figures ...................................................................................viii

1. Introduction ....................................................................................1

2. Principles of the Method ...............................................................3

3. X-Ray Topography Techniques ....................................................7

4. Image Contrast in X-Ray Topography........................................ 11

5. Practical Aspects of the Method ................................................21

6. Sample Geometry for Diffraction Imaging................................23

7. Sample Preparation .....................................................................27

8. Data Analysis and Interpretation ................................................33

9. Problems.......................................................................................41

References ..........................................................................................43
viii

◆ X-Ray Topography

List of Figures

Figure 1. A (111) transmission topograph of an isotopically-pure diamond

crystal, approximately 0.75 mm thick, showing four of the

defects commonly seen in X-ray topography, e.g., individual

dislocations (D), stacking faults (SF), inclusions (I), and surface

damage (SD)..................................................................................... 1

Figure 2. A back-reflection Laue camera used to orient single crystals

is shown in (A) and a typical Laue pattern is shown in (B).

A divergent white X-ray beam from the laboratory source

impinges the sample and the diffracted beams from a variety

of lattice planes form Laue spots on the film. Because the

source is divergent, the structure in the spots may be smeared ....... 3

Figure 3. The geometry for white radiation topography is shown in (A)

and typical images in (B). A collimated white X-ray beam from

a synchrotron radiation source impinges on the sample and the

diffracted beams form images on the film. It is convenient to

define the principle plane of diffraction to be vertical. In general,

the Laue images will not lie in the vertical plane and so will be

skewed. The shape of each spot, amount of foreshortening,

will depend on the angle of incidence to the film. The sample

imaged in (B) was smaller than the incident beam cross

section, and, thus, images of the entire crystal are produced ........... 5

Figure 4. The geometry for section topography. A narrow X-ray beam

is incident on the sample crystal, and the transmitted diffraction

image from a restricted region is recorded. The same geometry

is used for projection topography where the sample and the

film are scanned synchronously as indicated by the double

arrows ............................................................................................... 7

Figure 5. The Berg–Barrett geometry is shown in (A). Asymmetric

diffraction in the expansion or magnification geometry is

illustrated in (B) and in the reduction or demagnification

geometry in (C) ................................................................................. 8
ix

Table of Contents ◆

Figure 6. Double crystal geometry for X-ray topography. The incident

X-ray beam is “collimated” by diffraction from a high-quality

reference crystal. The film is positioned perpendicular to the

diffracted beam to maximize resolution or is placed parallel

to the sample to provide a 1:1 image with no distortion

or foreshortening ............................................................................... 9

Figure 7. A perfect crystal matrix is shown in (a) containing a region, A,

misoriented by more than the rocking curve width of the matrix.

A monochromatic X-ray topograph from this crystal is shown

schematically in (b), where region A does not satisfy the

diffraction condition. If another region, B, is imperfect, but has

the same orientation as the matrix as shown in (c), a reflection

X-ray topograph will show more intensity from region B than

from the perfect matrix as indicated in (d) ..................................... 11

Figure 8. Symmetric reflection (0 0 0 12) topograph from a sapphire

crystal showing dark dislocation images (D). The incident

beam was larger than the sample in the horizontal direction but

smaller than the sample in the vertical direction. Therefore the

kinematic intensity at the edges of the sample is seen on the

sides. The beam edge defines the image at the top and bottom.

The reduced intensity in the lower right hand corner is due to

a slight tilt of the sample diffracting planes with respect to the

monochromator diffracting planes. The lower panel is a

magnified view of the lower right portion of the upper panel.

The large dark feature (FD) is damage in the film emulsion.

The light grey images (RI) are radiographic images from defects

on the beryllium window separating the experimental hutch

from the monochromator chamber ................................................. 13

Figure 9. Symmetric transmission image from the same sapphire

crystal shown in Figure 8. This crystal is 2 mm thick, and the

photoelectric absorption μ0t = 25 at 8 keV, so the image is formed

by anomalous transmission. The dislocation images are white on

a dark background. The sample was mounted on a plastic ring

with a circular aperture, and the reduced intensity, due to

absorption by the plastic, is seen as the lighter regions on the

left and right sides of the image. The lower panel is a magnified

view of a region shown in the upper panel. Dislocations near the

exit surface (D1) are sharper than those closer to the entrance

surface (D2) ................................................................................... 15
x

◆ X-Ray Topography

Figure 10. The effective absorption coefficient for symmetric and

asymmetric Laue diffraction as a function of deviation from the

exact Bragg angle at X = 0. The dashed lines are for Branch 1,

and the solid lines are for Branch 2. Far from the Bragg

angle, the effective absorption coefficient approaches the

photoelectric absorption coefficient ................................................ 16

Figure 11. The symmetric Bragg rocking curve from a crystal with finite

absorption. The solid line is the contribution from Branch 1,

and the dashed line is the contribution from Branch 2. The

labeled positions on the rocking curve are the points where the

topographs shown in Figure 11 were recorded ............................... 17

Figure 12. Symmetric (0 0 6) surface reflection topographs from a bismuth

silicon oxide crystal recorded at the positions indicated on the

rocking curve of Figure 11. The contrast reverses between the

two branches. These images were recorded on a video camera,

rather than X-ray film, and, therefore, they have the opposite

contrast from all the other images shown above. Here,

increased diffracted intensity appears light, and decreased

intensity appears dark ..................................................................... 19

Figure 13. The depth of penetration as a function of incidence angle just

below the critical angle, θc, for total external reflection ................. 21

Figure 14. A (110) stereographic projection for a cubic crystal. Planes

near the center, as indicated, are candidates for asymmetric

reflection while those near the circumference are candidates

for asymmetric transmission. Planes on the circumference

provide symmetric transmission images .......................................... 23

Figure 15. An example of the depth sensitivity of X-ray images above

and below the critical angle for total reflection. This is a

(1129) asymmetric diffraction from a sapphire crystal. The

microstructure that can be seen above the critical angle, the

left image, indicates that the sample is strained. Below the critical

angle, in the image on the right where the near surface region

is imaged, the sample exhibits much more strain and residual

polishing damage. The incident beam was slightly larger than

the crystal and impinged on the aluminum mounting ring.

This produced diffraction images (Al) from individual grains

in the polycrystalline aluminum ....................................................... 25
xi

Figure 16. Surface preparation can have a profound effect on topographic

images. The ZnSe crystal shown in (A) was prepared using a

standard abrasive-polishing procedure. This image is dominated

by surface scratches (S), and the underlying bulk microstructure

cannot be seen. After the crystal was etched to remove residual

damage, the image shown in (B) allows the observation of the

bulk dislocation (D) structure .......................................................... 27

Figure 17. A silicon crystal with surface damage. The damage on the

entrance surface (B) appears blurry whereas damage on the

exit surface (S) appears sharp ........................................................ 28

Figure 18. Diffraction contour mapping of a CdZnTe crystal. The sample

Bragg angle was rotated in 0.015° steps. At each angular

position, an image was recorded to produce a multiple-exposure

composite image of the strain in the sample ................................... 30

Figure 19. Surface reflection topographs from 12 sapphire crystals.

The microstructure of the six Czochralski-grown crystals

consists of subgrains (SG), strained grains showing strain profiles

(SP), and dislocation bands (D). The surface also contains

residual scratches from fabrication and polishing. The six

heat-exchanger-grown crystals contain no subgrains, they are

almost strain-free, and they have a very low dislocation density.

These crystals show damage to their surfaces from inadvertent

mishandling (HD)............................................................................ 33

Figure 20. A transmission topograph from a diamond substrate is shown in

(A). This crystal has a distinctive microstructure consisting of

stacking faults (SF), dislocations (D), and surface damage (SD).

After chemical vapor deposition of a homoepitaxial film, the

composite crystal has become extremely strained, as shown

in (B). The high defect density of the strained film masks the

underlying structure of the substrate. This image is a multiple

exposure, using the same procedure as that used to obtain the

data in Figure 18, where here the angular steps were 0.04° .......... 34

Figure 21. A transmission topograph through an ice crystal about 1 mm

thick. The dislocation mobility can be determined by following

the motion of individual dislocation as a function of applied load

in real time ...................................................................................... 36

Table of Contents ◆
xii

◆ X-Ray Topography

Figure 22. A series of topographs from an aluminum crystal strained to

(A) 0.85%, (B) 1.2%, (C) 1.5%, and (D) 1.91%. The formation

of dislocation structures was correlated with ultra-small-angle

X-ray scattering data ...................................................................... 37

Figure 23. Anomalous transmission topographs of ZnSe crystals grown by

physical vapor transport (PVT) and by grain refinement (GR).

The image in (A) from the PVT crystal shows a large variation

in dislocation density, to the extent that anomalous transmission

does not occur in the central region. A much lower dislocation

density is observed at the edge. The sample shown in (B) from

the GR crystal has a more uniform distribution of defects

and a lower overall dislocation density............................................ 38

Figure 24. A (00018) reflection topograph from a sapphire dome.

Imperfections on the monochromator crystals appear as the

fuzzy light features indicated. Subgrain structure in this large

crystal is seen on the left and growth related dislocation

structures are evident in the upper right ..........................................40
1

1. INTRODUCTION

X-ray topography (XRT) is a nondestructive characterization technique for

imaging, by means of X-ray diffraction, the micrometer-sized to centimetersized defect microstructure of crystals. Topography got its name from the

fact that the diffraction image can resemble a geographical topographic map

with the appearance of different elevations and topographical contours.

However, since diffracted X-rays form the image, its interpretation is not

always straightforward. XRT is a very powerful tool for the evaluation of

crystals for technological applications and for characterizing crystal and

thin-film growth and processing. The aim of this guide is to make modern

X-ray topography more accessible to materials scientists and others who

would benefit from the rich variety of microstructural information that it offers.

In X-ray topography, the crystal sample is illuminated by an X-ray beam and

images of the diffracted beams are recorded. These images are generally

formed from X-ray wave fields interfering with one another inside the crystal.

The image from a perfect crystal is usually completely homogeneous. Changes

in the image contrast are seen if there are imperfections in the crystal which

cause deviations, u(r), from perfect long-range atomic order. In most cases, the

defects themselves are not visible in the image, but rather the lattice deformations

surrounding the defects are seen. An X-ray topograph of a diamond crystal,

shown in Figure 1, displays four of the defects most commonly observed in

X-ray topographs: dislocations, stacking faults, inclusions, and surface damage.

Later, examples are given to illustrate the wide range of applications for

which topography is used to characterize crystals.

Introduction ◆

Figure 1. A (111) transmission topograph of an isotopically-pure diamond crystal,

approximately 0.75 mm thick, showing four of the defects commonly seen in X-ray

topography, e.g., individual dislocations (D), stacking faults (SF), inclusions (I),

and surface damage (SD).
◆ X-Ray Topography

2

At modern synchrotron X-ray sources, microstructural evolution can be

observed on X-ray-sensitive video cameras in real time with subsecond

resolution, e.g., to follow the kinetics of defect motion, phase transitions, and

crystal growth. Depending on the optical parameters of the experiment, the

strain sensitivity can approach ∆d / d ~ 10–8 (where d is the lattice spacing),

the sensitivity to crystal orientation can be a few μrad, the spatial resolution

can approach 1 μm, and sample areas as large as 100 cm2 can be imaged in

a single exposure.

In the reflection (Bragg) geometry, an X-ray topograph emphasizes the

microstructure of the surface region (typically within 1 μm–10 μm) of the

crystal surface. In the transmission (Laue) geometry, an X-ray topograph

displays the bulk microstructure of the crystal. For reasonably good crystals

(what this means will be discussed later), a transmitted diffraction image can

be recorded through several millimeters of material. Information that can be

extracted from topographs includes: the density and distribution of dislocations

(including Burgers vector analysis), the distribution of inhomogeneous strain

(caused, for example, by thermal processing or a variation in impurity

concentration), subgrain structures including misorientation angles, the depth

of defects below the surface, and the presence of voids, secondary phases and

twins. In reflection and transmission topography, defect visibility depends upon

the scalar product, g • u, where g is the diffraction vector. This dependence

can be exploited to determine the magnitude and direction of the local average

atomic displacements, u. [Barrett, 1945; Guinier and Tennevin, 1949; Schultz,

1954; Lang, 1958; Tanner, 1976; Boettinger, et al., 1977; and Miltat, 1980].

X-ray topography, as it is practiced today, has its roots in the classical

laboratory X-ray work [Berg, 1931; Barrett, 1945; Guinier and Tennevin, 1949;

Shultz, 1954; Newkirk, 1958; Bonse, 1962; Lang, 1958 and 1959] of the 1930s

through the 1960s. The technique was developed using characteristic radiation

from laboratory X-ray sources, and much significant topography research is

still done this way. [See J. Phys. D: Applied Physics, 28 (1995) No. 4A].

The development of synchrotron radiation sources has enabled the extension

of the field by making white beam (i.e., polychromatic) topography possible

[Hart, 1975] and by enabling significant improvements to monochromatic XRT

techniques [Tanner, 1977; Sauvage and Petroff, 1980; Kuriyama, et al., 1982].

This guide will discuss primarily monochromatic X-ray topography as practiced

at modern synchrotron X-ray sources [See J. Phys. D: Applied Physics, 32

(1999) No. 10A], after a brief synopsis of the classical techniques from which

it arises. Several good reviews of synchrotron radiation topography [Tanner

and Bowen, 1980; Weissman, et al., 1984; Lang, 1992] are available.
3

2. PRINCIPLES OF THE METHOD

A simple way to understand the creation of X-ray topographic images is

to consider a Laue photograph [Preuss et al.1973]. Laue cameras are

commonly used in the X-ray laboratory to orient single-crystals. (See Figure 2.)

Principles of the Method ◆

Figure 2. A back-reflection Laue camera used to orient single crystals is shown in (A)

and a typical Laue pattern is shown in (B). A divergent white X-ray beam from the

laboratory source impinges the sample and the diffracted beams from a variety of

lattice planes form Laue spots on the film. Because the source is divergent, the

structure in the spots may be smeared.
◆ X-Ray Topography

4

A polychromatic (“white”) X-ray beam, containing X-ray energies from about

6 keV to 50 keV (X-ray wavelengths from approximately 2 Å to 0.25 Å),

impinges on a crystal. The beam is diffracted in many directions, creating

Laue spots. The positions of the diffraction spots appear according to the

Bragg equation:

(1)

where E is the incident X-ray energy (and λ is the incident wavelength)

selected by crystal planes with spacing d, h is Planck’s constant, c is the

speed of light, and θB is the Bragg angle. Each spot contains uniform intensity

if the crystal is perfect. If, however, the crystal is strained, streaks appear

instead of spots due to variations in lattice spacing, ∆d. In fact, each Laue

spot contains a spatial distribution of diffracted intensity attributable to the

presence of defects in the crystal. This distributed intensity is difficult to see

because Laue spots are typically the same size as the X-ray beam pinhole,

and the incident X-ray beam is divergent, but each tiny Laue spot is actually

an X-ray topograph. At synchrotron radiation facilities, a collimated white

X-ray beam can be used to illuminate a sample crystal, and spots with the

much larger cross section (typically 0.9 cm x 2 cm) of the synchrotron X-ray

beam are recorded, Figure 3(A). The resulting data are an array of Laue spots,

as shown in Figure 3(B), each of which is an X-ray topograph arising from a

different set of atomic planes.

If the white synchrotron beam passes through a monochromator, an X-ray

topograph is created when the sample crystal is set to the Bragg angle for

a specific set of lattice planes for the selected X-ray energy. Images from

different atomic planes are acquired by orienting the sample to satisfy the

Bragg condition for those planes and orienting the detector to the new

scattering angle (2θB) to record the image. With monochromatic radiation

only one topograph is recorded at a time, but the experimenter controls the

energy or wavelength of the X-ray beam, the X-ray collimation, the energy

or wavelength spread of the X-ray beam, and the size of the incident beam

on the sample crystal. If the sample is large, the experimenter can enlarge

the cross section of the incident beam so as to illuminate samples up to

about 100 cm2 and record the microstructure of the entire sample in a

single exposure.

hc

E = , or λ = 2d sin θB , 2d sin θB
5

Figure 3. The geometry for white radiation topography is shown in (A) and

typical images in (B). A collimated white X-ray beam from a synchrotron radiation

source impinges on the sample and the diffracted beams form images on the film.

It is convenient to define the principle plane of diffraction to be vertical. In general,

the Laue images will not lie in the vertical plane and so will be skewed. The shape of

each spot, amount of foreshortening, will depend on the angle of incidence to the film.

The sample imaged in (B) was smaller than the incident beam cross section, and, thus,

images of the entire crystal are produced.

Principles of the Method ◆
◆ X-Ray Topography

6
7

3. X-RAY TOPOGRAPHY TECHNIQUES

In practice, there are many ways to create an X-ray topograph. Since each

method has distinct advantages, a brief summary is given here of the classical

XRT experiments. Among XRT techniques, section topography is the most

fundamental. In section topography, diffraction from a restricted volume, or

“section” is taken. The restricted volume is defined by X-ray entrance and

exit slits. The section geometry is shown in Figure 4. It is performed using

a monochromatic and spatially narrow (≈ 10 μm) incident X-ray beam.

Two diffracted beams are shown in Figure 4: the forward-diffracted O-beam,

and the diffracted H-beam. The image from the crystal volume in which these

beams interact (shown in Figure 4 as a triangular section) is recorded. In section

topography, even the topograph of a perfect crystal displays fringes. Image

interpretation usually involves modeling by means of the Takagi and Taupin

equations [Takagi, 1962 and 1969; and Taupin, 1967]. These equations have

the advantage that they can be used to describe general defects. The benefits

of using section topography are that very small lattice strains can be observed

because they result in a distortion of the observed fringes, defects can be

located as a function of depth in the crystal [Lang, 1980], and one can

distinguish between stacking faults and tilt boundaries [Capelle, et al., 1982].

Finally, the analysis is quantitative because of the availability of computer

simulation algorithms [Epelboin, 1999].

Figure 4. The geometry for section topography. A narrow X-ray beam is incident

on the sample crystal, and the transmitted diffraction image from a restricted region is

recorded. The same geometry is used for projection topography where the sample

and the film are scanned synchronously as indicated by the double arrows.

X-Ray Topography Techniques ◆
◆ X-Ray Topography

8

In many applications, images from a restricted volume are not sufficient.

The defect microstructure of the entire crystal specimen must be imaged.

Starting from the section geometry, an integrated image over the entire

specimen can be created if the specimen and the detector are translated

synchronously in the pencil beam. (See Figure 4.) This method, invented by

Lang, has been called “traverse” and also “projection” topography [Lang,

1959]. The traverse image is a superposition of the section images. This

technique has been used to great advantage in X-ray laboratories, and many

X-ray instrumentation manufacturers include Lang cameras as standard

catalog items.

Figure 5. The Berg–Barrett geometry is shown in (A). Asymmetric diffraction in the

expansion or magnification geometry is illustrated in (B) and in the reduction or

demagnification geometry in (C).
9

Analogous to the transmission-diffraction geometry of Lang is the

reflection-diffraction geometry of Berg–Barrett [Berg, 1931 and Barrett, 1945]

topography, shown in Figure 5(A). In this geometry, the incident pencil beam

reaches the crystal at grazing incidence, where it is diffracted by planes

oblique to the sample surface (Figure 5(B)) under asymmetric diffraction

conditions. Although asymmetric diffraction is mentioned here in the context

of Berg–Barrett topography, diffraction can be asymmetric in transmission as

well as in reflection. For either geometry, the asymmetry is specified by the

parameter b ( b = 1/m, where m is the magnification factor) where

(2)

and the angle α is measured from the crystal plane to the crystal surface

(for reflection) and to the crystal surface normal (for transmission). The

advantage of asymmetric diffraction is that no rastering motions are needed

to acquire a Berg–Barrett topograph of the entire crystal. The diffracted X-ray

beam, however, is divergent due to defects in the crystal, so the imaging film or

camera is placed as close as possible to the sample to retain spatial resolution.

The Berg–Barrett method is another XRT technique that has been applied very

successfully in X-ray laboratories.

Figure 6. Double crystal geometry for X-ray topography. The incident X-ray beam

is “collimated” by diffraction from a high-quality reference crystal. The film is

positioned perpendicular to the diffracted beam to maximize resolution or is placed

parallel to the sample to provide a 1:1 image with no distortion or foreshortening.

X-Ray Topography Techniques ◆

sin (θB + α ) –b = sin (θB – α )
◆ X-Ray Topography

10

Improvements to the spatial resolution of Berg–Barrett topography, and a

relaxation of its restrictive geometry, are possible by reflecting the incident

X-ray beam from a high-quality reference crystal before using it to image

defects in the sample crystal. The reference crystal serves to narrow the

angular spread of the X-ray beam incident on the sample crystal. This is

called the “double-crystal” geometry [Kohra, et al., 1962], shown in Figure 6,

where the first crystal acts as a “collimator,” and the second crystal is the

sample. This technique has been used successfully in the laboratory, but the

exposure times to produce a topograph may be measured in hours or even

days. The double-crystal geometry became much more useful with the advent

of intense synchrotron X-ray sources, where the exposure times are reduced

by many orders of magnitude.
11

4. IMAGE CONTRAST IN X-RAY TOPOGRAPHY

Topographic contrast, i.e., the point-to-point variation in diffracted intensity,

is generally understood in terms of two mechanisms: orientation contrast

and extinction contrast [Petroff, 1984]. Orientation contrast can be explained

by the simple application of Bragg’s law. Extinction contrast, however, is

described by means of kinematical [Zachariasen, 1945] and the dynamical

[Batterman and Cole, 1964; Zachariasen, 1945] theories of X-ray diffraction.

Although dynamical theory accounts for most of the image contrast in XRT,

a detailed description of dynamical theory is outside the scope of this guide.

Instead, the important concepts derived from dynamical theory are discussed

as they apply to an analysis of XRT data. The reader may wish to consult

some of the many available resources [Authier, 1970; Kato, 1996; Lang, 1973;

Hart, 1980; Authier, et al., 1996; Bowen and Tanner, 1998] for a more

complete description of dynamical theory with reference to XRT.

First, however, orientation contrast is described. As the name implies,

orientation contrast arises when parts of the sample are oriented such that

they do not satisfy the Bragg condition. To understand orientation contrast,

one can imagine a monochromatic X-ray beam incident on a sample composed

of areas that are crystallographically misoriented (e.g., there exists subgrain

structure or strain) compared to the rest of the crystal. If these areas are

misoriented sufficiently such that they are outside the reflection range (rocking

curve) of the crystal, they cannot satisfy the Bragg condition and therefore

they will not diffract. For areas that are less misoriented, the diffracted

intensity can take on any value from negligible to nearly the intensity diffracted

by the surrounding crystal. To illustrate orientation contrast, an early example

from Lang [1970] is offered here. Consider a perfect crystal containing a

region (A) that is misoriented with respect to the rest of the crystal but has

the identical lattice spacing, Figure 7(a). For monochromatic incident X-rays,

and the sample oriented in the Bragg reflection geometry to diffract from the

crystal matrix, the misoriented region will not satisfy the diffraction condition,

and, therefore, the image arising from this region will appear white against the

matrix image, Figure 7(b).

A qualitative, general understanding of extinction contrast in the Bragg

reflection geometry is available from a continuation of the above example.

Another region (B), Figure 7(c), is imperfect compared to the matrix but has

the same orientation, e.g., the region in the vicinity of a dislocation. In this

case, the diffracted intensity is determined by the local scattering power.

The scattering power from diffraction by perfect and imperfect crystals is

different. The FWHM (rocking curve angular width) of a perfect crystal is

very narrow, the peak reflectivity is close to 100%, and its scattering power

(its integrated reflectivity) is proportional to the structure amplitude of the

Image Contrast in X-Ray Topography ◆
◆ X-Ray Topography

12

Bragg reflection,  F [Cullity, 1967]. By comparison, the FWHM of an

imperfect crystal is broad, the peak reflectivity is much less than 100%, but

the integrated reflectivity is proportional to  F2. Thus, the diffracted X-ray

intensity from imperfect regions of a crystal is much greater than that from

the surrounding perfect crystal matrix (Figure 7(d)). An example of increased

intensity from deformed regions surrounding dislocations is shown in a

reflection topograph from a sapphire crystal in Figure 8. The upper panel in

this figure is a (0 0 0 12) symmetric reflection topograph, and the lower panel

is an enlargement of the lower right portion of the image. Dislocations are

seen as dark lines. Other visible features are described in the figure caption.

Returning to X-ray contrast in Bragg reflection topography, the fundamental

difference between the kinematical and the dynamical theories of X-ray

diffraction is that the scattering at each atomic site is independent when

kinematical diffraction takes place, and the incident and the diffracted waves

Figure 7. A perfect crystal matrix is shown in (a) containing a region, A,

misoriented by more than the rocking curve width of the matrix. A monochromatic

X-ray topograph from this crystal is shown schematically in (b), where region A

does not satisfy the diffraction condition. If another region, B, is imperfect, but has

the same orientation as the matrix as shown in (c), a reflection X-ray topograph will

show more intensity from region B than from the perfect matrix as indicated in (d).
13

Figure 8. Symmetric reflection (0 0 0 12) topograph from a sapphire crystal showing

dark dislocation images (D). The incident beam was larger than the sample in the

horizontal direction but smaller than the sample in the vertical direction. Therefore

the kinematic intensity at the edges of the sample is seen on the sides. The beam

edge defines the image at the top and bottom. The reduced intensity in the lower

right hand corner is due to a slight tilt of the sample diffracting planes with respect

to the monochromator diffracting planes. The lower panel is a magnified view of

the lower right portion of the upper panel. The large dark feature (FD) is damage

in the film emulsion. The light grey images (RI) are radiographic images from

defects on the beryllium window separating the experimental hutch from the

monochromator chamber.

Image Contrast in X-Ray Topography ◆
◆ X-Ray Topography

14

within the crystal interact when dynamical diffraction takes place. Dynamical

diffraction is generally seen in thick crystals of good quality, i.e., with rather

low defect density. In dynamical Bragg reflection, the wave fields diffract

back toward the surface (rather than continue into the sample) where they

add to the reflected beam at the surface. As a result, the x-rays penetrate

much less than would be the case if there were only true (photoelectric)

absorption. This effect is called “primary extinction.” Thus, for primary

extinction, the scattering amplitudes are added before the total intensity is

calculated. On the other hand, kinematical diffraction is generally seen when

the crystal is imperfect. Here the X-rays penetrate further than they would if

the crystal were perfect, but still less than for true absorption. This effect is

called “secondary extinction.” When secondary extinction takes place, the total

intensity is calculated by adding the scattering intensities directly. Since there

is increased diffracted intensity from imperfect regions in a crystal arising as

a result of a reduction in primary extinction, the term “extinction contrast”

[Newkirk, 1958] came into use.

In transmission (Laue) diffraction by thick crystals of good quality, dynamical

theory shows that X-ray energy is transferred between the forward-diffracted

O-beam and the transmitted-diffracted H-beam. As the waves traverse

through the crystal, they can decouple at defect sites and create new wave

fields, thereby changing the appearance of the diffraction image. Thus,

defects are observed as “disruption images” and are seen in topographs as

a reduction in intensity. An example of this is shown in Figure 9, which is

a transmission topograph of the same sapphire crystal shown in a reflection

topograph in Figure 8. The lower image in Figure 9 is a magnified view of

a portion of the upper image. Dislocations in this transmission topograph

are seen as white features. Dislocations near the exit surface (such as D1)

are sharply defined, while those further from the exit surface (such as D2)

appear fuzzy. Interestingly enough, some secondary extinction contrast

(black contrast) can be seen for crystal defects very close to the exit surface

of the X-ray beams, where dynamical diffraction reverts to kinematical.

As indicated above, kinematical theory typically accounts for diffraction

contrast in images from thin crystals and very distorted crystals, whereas

dynamical theory accounts for diffraction contrast in images from thick, good

quality crystals. Here, the linear absorption coefficient, μ0, associated with

the photoelectric effect is used to define “thick” and “thin.” If, μ0 t < 1, where

t is the crystal thickness, the crystal is considered “thin” and the kinematical

theory is usually (but not always) operative; if, 1 < μ0 t < 10, the thickness

is intermediate, and the dynamical theory may or may not be dominant;

if μ0 t > 10, the crystal is “thick” and the dynamical theory is operative.
15

Figure 9. Symmetric transmission image from the same sapphire crystal shown in

Figure 8. This crystal is 2 mm thick, and the photoelectric absorption μ0 t = 25 at

8 keV, so the image is formed by anomalous transmission. The dislocation images

are white on a dark background. The sample was mounted on a plastic ring with

a circular aperture, and the reduced intensity, due to absorption by the plastic, is

seen as the lighter regions on the left and right sides of the image. The lower panel

is a magnified view of a region shown in the upper panel. Dislocations near the

exit surface (D1) are sharper than those closer to the entrance surface (D2).

Image Contrast in X-Ray Topography ◆
◆ X-Ray Topography

16

The dynamical theory requires the solution of Maxwell’s equations inside

the crystal consistent with the periodicity of the crystalline lattice and with

continuity of the electric field across the crystal/vacuum boundary. The

interaction of the diffracted wave fields within the crystal produces a number

of important effects that are observed in topographs. First, as mentioned

above, when a crystal is oriented so that it diffracts X-rays, the absorption

is very different from photoelectric absorption. In this case, one calculates

absorption from the imaginary part of the wave fields, which leads to two

important results: the diffraction “extinction length” and the existence of

“anomalous transmission.” The diffraction extinction length is analogous to

the optical 1/e rule in photoelectric absorption, and is given by [Authier, 1996]:

, (3)

where re is the classical radius of the electron and V is the unit cell volume.

The extinction length is a measure of the depth of X-ray penetration when a

Bragg condition is satisfied, and it sets the minimum thickness for extinction

contrast to be observed in images. The size of defect images also scales

with the extinction length, as shown below.

Figure 10. The effective absorption coefficient for symmetric and asymmetric Laue

diffraction as a function of deviation from the exact Bragg angle at X = 0. The dashed

lines are for Branch 1, and the solid lines are for Branch 2. Far from the Bragg angle,

the effective absorption coefficient approaches the photoelectric absorption coefficient.

ξ = πV /λr

e

F e
17

Another result of dynamical diffraction is anomalous transmission or the

“Borrmann effect” [Borrmann, 1941 and 1959; Borrmann, et al., 1958].

The solutions to Maxwell’s equations inside the crystal are expressed as a

hyperbloid of revolution around the Laue point called the Dispersion Surface.

The dispersion surface has two branches, Branch 1 and Branch 2, which

describe wavefields that are π out of phase with respect to each other. X-ray

transmission topographs show a strong reduction in the effective absorption

of X-rays for Branch 2. The reduction is sufficiently large that a diffracted

beam can be transmitted through a high quality crystal with a photoelectric

absorption as high as μ0 t ≈ 30 – 50. In an illustrative example, Batterman

and Cole [1964] calculated the absorption of wave fields at the center of the

Bragg diffraction range. For the 220 reflection in a 1-mm-thick Ge crystal,

the ordinary absorption μ0 t = 38 for 8 keV photons. The effective absorption

is 74 for diffracted X-rays in Branch 1 and 1.9 for diffracted X-rays in Branch 2.

Thus, the crystal is rather transparent to diffracted X-rays of Branch 2.

This weak absorption is observed for the wave field that has the nodes of its

stationary waves located between the atomic planes. The effective absorption

coefficients for Branch 2 and Branch 1, as a function of diffraction angle,

are shown in Figure 10 for symmetric diffraction (m = 1) and for two cases

of asymmetric diffraction (see Equation 2). Far from the Bragg condition,

Figure 11. The symmetric Bragg rocking curve from a crystal with finite

absorption. The solid line is the contribution from Branch 1, and the dashed

line is the contribution from Branch 2. The labeled positions on the rocking

curve are the points where the topographs shown in Figure 12 were recorded.

Image Contrast in X-Ray Topography ◆
◆ X-Ray Topography

18

the effective absorption for the symmetric case is the linear photoelectric

absorption coefficient. Near Bragg, i.e., near X = 0, the absorption dips

for Branch 2 and peaks for Branch 1. Results for asymmetric diffraction

are more complicated, but similar [Kuriyama and Cohen, 1982].

When the scattering amplitude is calculated for dynamical Bragg reflection,

both branches contribute to the X-ray intensity in the crystal rocking curve.

Branch 1 is shown by a solid line and Branch 2 by a dashed line in the rocking

curve shown in Figure 11. Crossing over from one branch to the other, white and

black contrast is interchanged [Kuriyama and Long, 1984]. At each angular

position along the rocking curve, a distinct topographic image can be recorded.

In Figure 12, (00 6) symmetric reflection topographs of a bismuth silicon oxide

crystal [Steiner, et al., 1988], taken at the positions indicated in Figure 11, are

shown. The best contrast is seen in images taken on the rapidly changing

wings of the rocking curve, where striations created during crystal growth are

seen. The contrast in images (b) and (c) is inverted from that of images (f)

and (g). Images (d) and (e), taken near the peak of the rocking curve, show

mostly surface scratches because here the beams are confined to the near

surface region of the crystal. Images taken far out on the wings of the curve

are referred to as “weak beam” images; there the diffracted beam penetrates

into the sample crystal, and sensitivity to bulk crystal structure is enhanced.

An additional source of contrast, Fresnel imaging due to phase contrast, is

often observed at synchrotron X-ray sources because of spatial coherence in

the incident beam. An image may be formed by a low-absorbing sample or

indeed by any object in the X-ray path. A phase image is created by local

variations in the optical path of the X-rays due to phase variations across the

beam related to microstructures within the object. These images indicate, for

example, the presence of pores, internal boundaries, thickness variations, etc.,

which are made visible depending on the object-to-detector distance. Now

recognized as an imaging technique in its own right, phase contrast radiography

[Cloetens, et al., 1999] is usually a source of unwanted image contrast for XRT.

One final topic regarding image formation in X-ray topography is related to the

frequently asked question: how do atomic-scale defects such as dislocations

produce micrometer-scale images in X-ray topographs? The answer can be

found by comparing the distorted regions around the dislocations to the rocking

curve range for the defect-free regions of a crystal. The following discussion

follows along the lines of examples in Lang [1973], Tanner [1996], and Bowen

and Tanner [1998]. The rocking curve width for symmetrical Bragg diffraction

in the absence of absorption is

∆θ = 2 /gξ. (4)
19

Figure 12. Symmetric (0 0 6) surface reflection topographs from a bismuth silicon

oxide crystal recorded at the positions indicated on the rocking curve of Figure 11.

The contrast reverses between the two branches. These images were recorded on a

video camera, rather than X-ray film, and, therefore, they have the opposite contrast

from all the other images shown above. Here, increased diffracted intensity appears

light, and decreased intensity appears dark.

Image Contrast in X-Ray Topography ◆
◆ X-Ray Topography

20

In a simple model of a dislocation, there is a distorted region surrounding the

dislocation which falls off as the inverse of the distance from the dislocation

core. For a screw dislocation perpendicular to the diffracting planes, the

distortion can be described as

(5)

where B is the Burgers vector and R is the distance from the dislocation core.

The distorted region defines the part of the crystal around the defect that

behaves as an imperfect crystal, with the consequent increased diffracted

intensity in its image. The size of the region depends on the angular range of

the perfect crystal rocking curve. When the distortion exceeds the diffraction

rocking curve width, the intensity distribution is described by imperfect crystal

(kinematical) theory. Therefore, the effective size, W, of the dislocation is

twice the maximum radius as determined from B/2πR ≈2/gξor:

(6)

The general result for any arbitrary dislocation line direction is [Tanner, 1996]:

As an example, consider a (200) diffraction from an aluminum single crystal,

where primary slip has occurred along {111} planes. With an 8 keV incident

X-ray beam, dislocation images are expected to be about 4.7 μm wide. The

large widths of these images, compared to those from transmission electron

microscopy for example, are a direct consequence of the high strain sensitivity

of XRT. The width of XRT images also limit the defect density for which

individual dislocations can be resolved to approximately 104/cm2. Hence, a

“good quality” crystal from the perspective of x-ray topography has a defect

density ≤ 104/cm2. Another result of Equation 6 is that the size of the defect

image is a function of the structure factor as well as the incident X-ray energy.

Since the extinction length depends on both the X-ray wavelength (energy) and

the structure factor (Equation 3), it becomes evident that the image width increases

as the X-ray energy is increased or as the structure factor is decreased.

∆ω = B /2πR . 

W = 2R ≈ Bgξ/2π.

W ≈ B • g ξ/2π for screw dislocations, and

W ≈ 0.88 B • g ξ/2π for edge dislocations. (7)
21

5. PRACTICAL ASPECTS OF THE METHOD

Monochromatic X-ray topography at a synchrotron facility requires the

introduction of a monochromator into the X-ray optical path. The monochromator

serves to select a particular energy or wavelength from the incident beam.

Once past the monochromator, the power in the X-ray beam is significantly

reduced and the probability of radiation damage to the sample crystal is

lessened. In addition to wavelength, λ, selection, the monochromator can be

used to improve the angular collimation so that it is very narrow and the plane

wave approximation is applicable. Asymmetrical diffraction in the monochromator

can be used to increase (Figure 5(B)) or reduce (Figure 5(C)) the physical size

of the X-ray beam on the sample so that, in the first case, large crystals can be

imaged in a single exposure, or, in the second case, the defect microstructure

of small, weakly diffracting crystals can be imaged.

There are a variety of monochromators currently in use at synchrotron

radiation facilities —single crystal and multiple crystal designs. The important

point is that the X-ray optics can be used to prepare the X-ray beam so that

the optimal spatial and angular resolutions are achieved and the information

content of the topographic image is maximized. Since only one topograph is

created at a time, that image can be recorded with the detector very close to

the sample, thereby maximizing resolution.

Figure 13. The depth of penetration as a function of incidence angle just below the

critical angle, θc, for total external reflection.

Practical Aspects of the Method ◆
◆ X-Ray Topography

22

Surfaces and thin films, as well as bulk microstructures, can be probed with

monochromatic X-ray topography. For studies of surfaces, two methods

are available. With symmetric surface reflection, the depth of penetration

decreases the closer the diffraction image is taken to the top of the crystal

rocking curve. As noted earlier, the images in Figure 12(d) and 12(e) near

the top of the rocking curve display the sample’s surface scratches, while the

images in Figure 12(a) and 12(h) display more of the bulk growth-striation

microstructure. The depth of X-ray penetration varies by almost an order

of magnitude as a function of the Bragg angle. The other approach is a

grazing-angle asymmetric Bragg geometry [Imamov, et al., 1989; Novikov,

et al., 1995]. In this case, we are combining asymmetric diffraction with total

external reflection in the same experiment. One prepares a grazing incidence

condition by selecting an asymmetric surface diffraction. Then the incident

photon energy is selected to establish a diffraction condition such that the

incident angle is just above or just below the critical angle for total external

reflection of the X-ray beam. Below the critical angle, the penetration into

the sample can be less than 3 nm, see Figure 13, and the depth of penetration

changes significantly with small adjustments of the incident energy around

the critical angle.
23

6. SAMPLE GEOMETRY FOR DIFFRACTION IMAGING

The geometry of prospective XRT sample crystals generally conforms to

their intended application in technology, rather than to optimal XRT standards.

Nevertheless, it is very useful to understand what the parameters of an ideal

XRT sample are. There are two classes of experiments, depending on the

information that is required: surface microstructure or bulk microstructure.

The ideal sample for imaging the surface microstructure is described first.

An XRT sample crystal is usually in the form of a flat plate with a

well-defined surface of known crystallographic orientation. A surface

reflection topograph can be recorded under three different geometric

Figure 14. A (110) stereographic projection for a cubic crystal. Planes near

the center, as indicated, are candidates for asymmetric reflection while those

near the circumference are candidates for asymmetric transmission. Planes on

the circumference provide symmetric transmission images.

Sample Geometry for Diffraction Imaging ◆
◆ X-Ray Topography

24

conditions: symmetric diffraction, asymmetric diffraction, and grazing-incidence

asymmetric diffraction. With a monochromatic incident beam, only one of

these conditions can be established at a time. The availability of diffraction

planes for imaging is determined by the crystallographic orientation of the

sample relative to its surface, the diffractions which are allowed (i.e., the

structure factor), and the photon energy available from the monochromator.

Most monochromatic XRT facilities can provide X-rays in the range from

5 keV to 25 keV.

The easiest image to obtain is from symmetric surface diffraction. In this

case, the diffracting planes lie parallel to the sample surface, and the diffraction

vector is along the surface normal. The symmetric reflection image yields

information on defect microstructure that has a projection perpendicular to

the crystal surface. The image is not sensitive, however, to defects parallel

to the surface since, in that case, g • u = 0.

Asymmetric diffraction is used to image microstructure parallel to the

surface. The asymmetric diffractions that are available depend on the

surface orientation. These can be found from a stereographic projection

as in Figure 14. Crystallographic planes with normals close to the surface

normal are candidates for asymmetric surface diffraction images. On the

stereographic projection, these candidates are close to the center. Of course,

the inclination angle of the selected diffraction planes with respect to the

crystal surface must be smaller than the Bragg angle for the incident X-ray

energy. For a silicon (110) surface, there are {211} planes inclined 30° from

the surface and the Bragg angle for a (422) diffraction is about 44° for 8 keV

incident photons. Therefore, the (422) asymmetric diffraction is possible with

θin = 14°. However, for a silicon (111) surface, there are {311} planes ≈ 29.5°

away from the surface. For an 8 keV incident beam, the Bragg angle for the

{311} planes is about 28°, so asymmetric (311) diffractions are not possible.

If a (311) diffraction image is required, lower energy incident photons will

bring the Bragg angle for (311) diffraction into the accessible range.

The third reflection geometry is the grazing-incidence asymmetric diffraction

geometry, which is an extreme form of the asymmetric geometry just

described. For monochromatic radiation, the diffracting plane and the incident

photon energy are chosen so that the X-ray angle of incidence is extremely

small. Then the depth of X-ray penetration is also very small. (See Figure 13.)

Also, the microstructure parallel to the surface will be imaged because it

has the largest projection in the direction of the diffraction vector for that

diffraction. Further, the specific depth of penetration can be selected by

means of the incidence angle. An example of the depth sensitivity of grazingincidence asymmetric diffraction imaging is shown in Figure 15. The depth of

penetration in the left panel (above the critical angle) is much greater than that
25

Figure 15. An example of the depth sensitivity of X-ray images above and below

the critical angle for total reflection. This is a (1129) asymmetric diffraction from a

sapphire crystal. The microstructure that can be seen above the critical angle, the

left image, indicates that the sample is strained. Below the critical angle, in the image

on the right where the near surface region is imaged, the sample exhibits much more

strain and residual polishing damage. The incident beam was slightly larger than

the crystal and impinged on the aluminum mounting ring. This produced diffraction

images (Al) from individual grains in the polycrystalline aluminum.

in the right panel (below the critical angle). While both show strain patterns,

the right panel shows that the strain closer to the surface is greater, and there

is residual surface damage from surface processing.

Next, samples for experiments to image the bulk microstructure are described.

The transmission geometry is used in this case, where it also is divided into two

cases: symmetric diffraction imaging and asymmetric diffraction imaging. The

candidate planes for transmission diffraction imaging lie in an annulus near the

circumference of the stereographic projection such as in Figure 14. Diffraction

Sample Geometry for Diffraction Imaging ◆
◆ X-Ray Topography

26

from planes indicated on the circumference is symmetric, and diffraction from

all others is asymmetric. For a cubic crystal, a (110) surface is particularly

useful in that symmetric transmission diffraction from {001}, {110} and {111}

planes are available. Returning to the example of dislocation structures, it

is often necessary to describe the dislocation images in X-ray topographs in

terms of the dislocation type (edge or screw) and their interactions. Both

types of dislocations are lines, but screw dislocations have no X-ray contrast

if g • B = 0 and edge dislocations have no contrast if g • (Bxl) = 0, where B is

the Burgers vector and l is the dislocation line direction. Thus, the dislocations

in XRT images can be sorted according to type by finding where they are

invisible. With a (110) sample surface, symmetric transmission images can

be taken in each of the principle directions.

If the sample thickness is such that μt ≥ 10, image interpretation is

straightforward. For a silicon crystal and 8 keV incident photons, μ/ρ, the

mass absorption coefficient, is 61.8 cm2/g and the density is 2.33 g/cm3.

To achieve μt ≥ 10, the sample should be at least 0.7 mm thick. If the

thickness cannot be changed, the incident photon energy can be changed to

bring the experimental conditions closer to μt ≥ 10. For a silicon crystal, for

example, the mass absorption coefficient changes by a factor of 33 between

6 keV and 20 keV. However, the defect image width also increases with

increasing X-ray photon energy.
27

7. SAMPLE PREPARATION

Sample preparation is a key element of successful XRT. To achieve

topographic images representative of the defect microstructure to be studied,

great care must be taken to avoid introducing damage into the sample during

sample preparation and mounting. This section is divided into two parts: the

preparation of sample surfaces, and the handling and mounting of samples.

A resource on this subject is Chapter 13 in Tanner and Bowen [1980].

Figure 16. Surface preparation can have a profound effect on topographic images.

The ZnSe crystal shown in (A) was prepared using a standard abrasive-polishing

procedure. This image is dominated by surface scratches (S), and the underlying

bulk microstructure cannot be seen. After the crystal was etched to remove residual

damage, the image shown in (B) allows the observation of the bulk dislocation

(D) structure.

Sample Preparation ◆
◆ X-Ray Topography

28

The goal of sample preparation is to obtain a flat, smooth, damage-free surface

that will not contain features that mask the bulk or surface microstructure of

interest. The degree to which surface damage can interfere with the required

data depends on the geometry of the measurement. Asymmetric surface

diffraction topographs are extremely sensitive to surface structure, and thus

an improperly prepared surface, if present, will dominate the data. The

transmission geometry, by comparison, can be less sensitive to surface damage

depending on the severity of the damage and whether the damage is on the

X-ray beam entrance or exit surface. Minor damage on the entrance surface

in transmission geometry may cause little obscuration of data. The main effect

is that the overall beam intensity is lowered. The quality of the exit surface,

however, is critical for transmission imaging. Even minor damage on the exit

surface will dominate the transmission image, and, as with reflection images,

may obscure the microstructure of interest.

In general, a crystal sample should be polished and etched to prepare as clean

a surface as practical. A note of caution: an optically-smooth polished surface

is not always optimal. In the example shown in Figure 16, a ZnSe crystal was

prepared for XRT by polishing to a mirror finish. This process left considerable

residual scratches, as seen in Figure 16(A). After this crystal was etched to

remove the polishing damage, the bulk dislocation structure became visible in

the transmission topograph, Figure 16(B).

Figure 17. A silicon crystal with surface damage. The damage on the entrance

surface (B) appears blurry whereas damage on the exit surface (S) appears sharp.
29

The difference between the appearance of entrance surface damage and

exit surface damage in the transmission topography geometry is shown in

Figure 17. This silicon wafer under investigation has damage on both surfaces.

The entrance surface damage appears diffused or blurred, whereas the exit

surface damage appears crisp.

The guiding principle when preparing and handling a crystal for XRT is to

perturb the crystal as little as possible. While it may be straightforward to

mount a crystal rigidly to a support, it is not particularly straightforward to do

this without introducing strain. XRT is sensitive to orientational variations as

small as 5 μrad, which means for a 1 cm diameter sample, a radius of curvature

as slight as 1000 meters can reduce the diffracted image area. Therefore, only

the most gentle (but secure) mounting methods can be used. Furthermore, for

the highest quality images, the sample must also be stationary in the X-ray

beam since small vibrations can cause the sample to flutter in the beam and

cause the image to become blurred. For some orientations, the sample is likely

to be rotated almost upside down, so the mount must support the full weight of

the crystal. For transmission images, both surfaces should be free of mounting

materials over as large an area as possible.

The requirement of a strain-free yet rigid mounting is a non-trivial problem.

No single method works for all samples. For many applications, a trial and

error process is needed to find the right mounting. This is an opportunity to

be creative. Some of the successful mounting products that have served over

the years include: a glue made of styrene packing material (peanuts) dissolved

in zylene, a rubber-based double-stick adhesive tape, a variety of putties and

clays, crystal mount, “blue” tack, surface tension mounting with viscous fluids,

and diffusion bonding of a metal crystal to a high-purity backing plate of the

same material [Black and Burdette, 1986]. To avoid obscuring or shadowing

the incident or the diffracted beam, the sample mount should also have a low

profile. In general, samples will be mounted to a goniometer head of some

type. Typical goniometer heads are those from Huber* and Rigaku.*

Given a sample or samples with the appropriate orientation and thickness,

one is ready to record images. A prime consideration is to get the recording

medium as close as possible to the sample to maximize spatial resolution.

The image can be captured digitally from a real time video camera, acquired

digitally from an integrating CCD camera, or recorded on high-resolution X-ray

film. To find and optimize a particular diffraction condition, it is extremely

useful to have a real-time video camera to observe the diffracted image as

the Bragg condition is changed. Cameras are available with a pixel size

from 9 μm with an image area of 58 mm2 up to an image area of 6348 mm2

with a pixel size just under 150 μm. While electronic imaging detectors are

extremely useful, the best resolution is still obtained by photographic means.

Sample Preparation ◆
◆ X-Ray Topography

30

Figure 18. Diffraction contour mapping of a CdZnTe crystal. The sample Bragg

angle was rotated in 0.015° steps. At each angular position, an image was recorded

to produce a multiple-exposure composite image of the strain in the sample.
31

The standard photographic medium is Ilford* L4 nuclear emulsion, with a

nominal thickness of 25 μm, on glass substrates which can be cut to the

required size. The grain size is 0.17 μm, which allows subsequent photographic

image magnification. The image can be recorded with the film perpendicular to

the diffracted beam, yielding the highest spatial resolution at the cost of image

foreshortening, or with the film parallel to the sample, providing a one-to-one

image with a consequent broadening of the features, as shown in Figure 6.

Two other useful techniques for extracting information from topographs include

recording stereo pairs [Armstrong, 1984] and performing diffraction contour

mapping [Larson and Black, 1995]. A stereo image of the defect structure in

a sample crystal is obtained by recording two images from the same diffraction

differing only by a slight rotation, ≈ 10°, about the diffraction vector [Haruta,

1964]. Contour mapping is used for samples with inhomogeneous strain,

where a single diffraction image will cover only a small part of the sample.

Multiple exposures, taken at successive angular positions, display the contours

of the strain field [Bowen and Tanner, 1998]. An example of such a multiple

exposure topograph, called a zebra-stripe image, is shown in Figure 18.

This is a multiple-exposure strain-contour topograph from a CdZnTe crystal.

The sample was rotated in 0.015° steps, where an image was recorded at

each angular position. Broad stripes indicate lower strain and narrow stripes,

such as those seen at the edges, indicate severe strain.

Sample Preparation ◆
◆ X-Ray Topography

32
33

8. DATA ANALYSIS AND INTERPRETATION

Data analysis and interpretation of X-ray topographs takes many forms.

The product of an XRT experiment is the diffraction image, which is captured

on film or on a digital camera. In many practical applications, the images

record changes that have occurred in the sample as a result of the process or

processes of interest. Such processes can include modifying crystal-growth

parameters, changing surface treatments or annealing conditions, varying

dopant concentration, examining device growth and operation or thin-film

growth, and the application of electromagnetic fields or mechanical testing.

Consider the X-ray topographs of 12 sapphire crystals shown in Figure 19.

The dislocation structure is markedly different for the two growth methods.

The Czochralski-grown crystals contain subgrains, strained grains and

dislocation bands. The heat-exchanger-grown crystals contain no subgrains,

almost no strain and very low dislocation density. The appearance of

additional surface damage can be traced to inadequate sample-handling

protocols. Finally, the effectiveness of polishing procedures and the

appearance of specific surface defects can be related to the mechanical

strength by correlating the topographic evidence for the existence of defects

with strength data.

Figure 19. Surface reflection topographs from 12 sapphire crystals. The microstructure

of the six Czochralski-grown crystals consists of subgrains (SG), strained grains showing

strain profiles (SP), and dislocation bands (D). The surface also contains residual

scratches from fabrication and polishing. The six heat-exchanger-grown crystals contain

no subgrains, they are almost strain-free, and they have a very low dislocation density.

These crystals show damage to their surfaces from inadvertent mishandling (HD).

Data Analysis and Interpretation ◆
◆ X-Ray Topography

34

Figure 20. A transmission topograph from a diamond substrate is shown in (A).

This crystal has a distinctive microstructure consisting of stacking faults (SF),

dislocations (D), and surface damage (SD). After chemical vapor deposition of a

homoepitaxial film, the composite crystal has become extremely strained, as shown

in (B). The high defect density of the strained film masks the underlying structure of

the substrate. This image is a multiple exposure, using the same procedure as that

used to obtain the data in Figure 18, where here the angular steps were 0.04°.
35

Rather than attempt an exhaustive description of the data analysis for the wide

variety of topographic information that is available, a few selected examples

will be shown. A good resource for contemporary XRT experiments and their

interpretation can be found in special issues of Journal of Physics D: Applied

Physics that appeared in 1995 and 1999, in which collections of original papers

presented at the biennial conference on High Resolution X-Ray Diffraction and

Topography are published.

XRT studies of microstructure development as a function of processing have

had significant impact on processing technology. Here an example is given

of CVD growth of homoepitaxial diamond. A transmission X-ray topograph

of a substrate diamond is shown in Figure 20(A). The data reveal that the

substrate crystal contains stacking faults and single dislocations, and there

is also a small amount of surface damage. The overall quality, however,

is quite good since the microstructure of the entire crystal can be captured

in a single monochromatic transmission topograph. Figure 20(B) is a

monochromatic transmission topograph taken from the identical crystal

and orientation after deposition of a homoepitaxial film equal to the

substrate thickness. The composite crystal was so severely strained

that the microstructure could not be captured in a single monochromatic

transmission topograph. Instead, the image was produced in five exposures,

taken 0.04° apart, in the same procedure used to create the contour map

shown in Figure 18. The post-deposition data indicate the development of

copious dislocation walls and a significant amount of curvature. The defect

density and the strain in the film, which is facing the X-ray detector, mask

the underlying structure of the substrate. The amount of curvature in the

crystal was calculated from the range of Bragg angle over which it diffracts.

The sample diffracts over 0.2° along its length of 8 mm, indicating a

curvature of about 2.3 m.

The next example is one of dynamic in situ topography, in which the

dislocation mobility in ice is followed [Xiahong, et al., 1995]. The experiments

were conducted in real time with the ice crystal under tension. The distance

that individual dislocations moved under a known load was used to derive the

mobility. A monochromatic transmission topograph of an ice crystal is shown

in Figure 21 where the real time data were recorded directly by an X-ray

video camera.

Data Analysis and Interpretation ◆
◆ X-Ray Topography

36

Figure 21. A transmission topograph through an ice crystal about 1 mm thick.

The dislocation mobility can be determined by following the motion of individual

dislocation as a function of applied load in real time.

In another example of in situ topography, the formation of dislocation

structures in single-crystal aluminum was followed. Figure 22 shows a

sequence of (220) transmission diffraction images of a 1-mm-thick aluminum

crystal, deformed in situ to a maximum strain of 1.9%. The sequence

shows the evolution of dislocation walls from diffuse intermittent structures

at 0.85% strain to sharply defined walls at 1.9% strain. Since these are

disruption images, the walls are white against a darker background.

The white lines extending from the lower left to the upper right are 111

dislocation walls, parallel, and the less prominent features on the other diagonal

are the 111 dislocation walls. The 111 planes are clearly the dominant slip

planes for this region of the sample. The bright horizontal lines are actually a

series of bright spots. These represent a set of grown-in dislocations that lie on
37

Figure 22. A series of topographs from an aluminum crystal strained to (A) 0.85%,

(B) 1.2%, (C) 1.5%, and (D) 1.91%. The formation of dislocation structures was

correlated with ultra-small-angle X-ray scattering data.

one of the remaining two slip planes. Information from these images was also

used to derive parameters from a complementary experiment, ultra-small-angle

X-ray scattering [Levine, et al., 2000]. The scattering experiment was carried

out on samples crystallographically identical to the one imaged in Figure 22.

Scattering from the interface between the wall and the surrounding material

was identified, but it was necessary to know the orientation of the walls with

respect to the diffraction vector before the true thickness of the interface

could be derived. This information was obtained from the diffraction images

in Figure 22.

Data Analysis and Interpretation ◆
◆ X-Ray Topography

38

Figure 23. Anomalous transmission topographs of ZnSe crystals grown by physical

vapor transport (PVT) and by grain refinement (GR). The image in (A) from the PVT

crystal shows a large variation in dislocation density, to the extent that anomalous

transmission does not occur in the central region. A much lower dislocation density

is observed at the edge. The sample shown in (B) from the GR crystal has a more

uniform distribution of defects and a lower overall dislocation density.
39

X-ray topography has contributed significantly to crystal-growth studies.

The ability to correlate crystalline imperfections such as growth striations,

precipitates and dislocations with growth conditions is essential to the effort

to improve crystal quality. In the example given here, ZnSe crystals grown by

a grain refinement (GR) process and by physical vapor transport (PVT) are

shown in Figure 23. The monochromatic transmission image in Figure 23(A)

of the PVT crystal shows a very inhomogeneous microstructure. The central

region is so severely defected that the X-ray beam does not make it though the

crystal. Outside the strained regions, single dislocations and precipitates are

clearly evident. In Figure 23(B), the GR crystal shows both a more uniform

distribution and lower density of dislocations.

Data Analysis and Interpretation ◆
◆ X-Ray Topography

40
41

9. PROBLEMS

The best X-ray topographs display the defect structure of the sample and are

free from spurious features that detract from the microstructure of interest.

Unfortunately, there are many possible sources of spurious contrast. Of major

concern for MXRT, for example, is the quality of the monochromator crystals.

If these crystals contain imperfections, the image of those defects may be

visible along with the recorded image from the sample, Figure 24. Most often,

the images from such defects will be very diffused, compared to the images

from the sample defects, because the monochromator is typically much further

from the detector than the sample. An obvious method to distinguish between

monochromator and sample images is to translate the monochromator crystals

or the sample and observe when the features move.

Figure 24. A (00018) reflection topograph from a sapphire dome. Imperfections on

the monochromator crystals appear as the fuzzy light features indicated. Subgrain

structure in this large crystal is seen on the left and growth related dislocation

structures are evident in the upper right.

Problems ◆
◆ X-Ray Topography

42

Any object in the X-ray beam may produce an image. One object under the

control of the experimenter is the material enclosing the photographic film.

Often, paper or plastic is used. These sometimes have a texture that produces

a fine contrast in the recorded image due to variable absorption.

Another possible source of contrast, this one beyond the experimenters’

control, is the vacuum window separating the synchrotron optics from the

topography station. These windows may have nonuniform thickness and

produce phase images, and/or they can become oxidized or otherwise

contaminated over time and cause a diffuse contrast to be seen in all images.

An example is shown in Figure 8, where features attributable to a beryllium

window are seen alongside dislocation images from the sample.

Another consideration is the size of the incident beam, which should be

kept as small as possible and still provide the desired image area. When

the beam is larger than the sample, the increased general scatter will reduce

the effective image contrast. If the beam impinges on the sample mounting

material, there is the possibility of recording diffraction images from individual

grains in a polycrystalline mount. An example of this is seen in Figure 15.

Here a sapphire sample was as large as the mount, and some of the beam

impinged on the aluminum holder. This caused the formation of an aluminum

Debye–Sherrer ring along with the microstructure of the sapphire sample.

Since the beam was collimated and monochromatic, only a small number of

grains diffract producing well-defined images of the grains.
43

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David R. BlackGabrielle G. Long

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