research papers

192 doi:10.1107/S0021889809004300 J. Appl. Cryst. (2009). 42, 192–197

Journal of

Applied

Crystallography

ISSN 0021-8898

Received 15 September 2008

Accepted 5 February 2009

# 2009 International Union of Crystallography

Printed in Singapore – all rights reserved

Formalism for the determination of intermediate

stress gradients using X-ray diffraction

Thomas Gna ̈upel-Herold

University of Maryland, Department of Materials Science and Engineering, College Park, MD

20742, USA, and NIST Center for Neutron Research, 100 Bureau Drive Stop 6102, Gaithersburg,

MD 20899-6102, USA. Correspondence e-mail: tg-h@nist.gov

A method is outlined that allows the determination of one-dimensional stress

gradients at length scales greater than 0.2 mm. By using standard four-circle

X-ray diffractometer equipment and simple aperture components, length

resolutions down to 0.05 mm in one direction can be achieved through constant

orientation of a narrow, line-shaped beam spot. Angle calculations are given for

the adjustment of goniometer angles, and for the effective azimuth and tilt of the

scattering vector for general angle settings in a four-circle goniometer. The latter

is necessary for the computation of stresses from lattice strain measurements.

1. Introduction

Stress determination using X-ray diffraction is based on

measuring lattice spacings in different directions within a

chosen specimen reference frame. This is performed experimentally by tilting (angle ) and rotating (angle ’) the

specimen with respect to the incident and the diffracted X-ray

beam. X-ray diffraction is predominantly surface limited

owing to the strong absorption of X-rays in most materials,

and the resulting penetration depths are of the order of

micrometres. Stress gradients with significant changes within

the penetration depth can be determined (Genzel, 1994, 1999);

however, the evaluation of strain/stress gradients for length

scales much larger than the penetration depth is mostly limited

to neutron diffraction, synchrotron X-ray diffraction and

laboratory X-ray diffraction in connection with layer removal.

Neutron diffraction is commonly used with spatial resolutions 1 mm but measurements with resolutions of 0.1 mm

have been reported (Foecke & Gnaupel-Herold, 2006;

Gnaeupel-Herold et al., 2004; Oliveira et al., 2006). Unless new

focusing optics become available (Gibson et al., 2002) further

improvements are unlikely owing to limitations of the primary

beam intensity. Synchrotron X-ray diffraction offers primary

beam intensities superior both to neutron diffraction and to

laboratory X-ray diffraction. It has been used successfully to

determine strain and stress gradients at all length scales

ranging from 103 to 101 mm (Larson et al., 2002; GnaeupelHerold et al., 2005; Jakobsen et al., 2006; Levine et al., 2006;

Martinschitz et al., 2008; Rahman et al., 2008). However,

availability and cost limit the widespread use of both techniques. Thus, the most common diffractive method for the

determination of intermediate stress gradients is laboratory

X-ray diffraction in connection with electrochemical layer

removal from the surface (Suominen & Carr, 2000). Although

layer removal is a destructive method which causes perturbations of the stress field, there is often no alternative.

In this work a different approach is outlined, based on

measuring the lattice spacings on the cross-sectional surface

using collimated beams in a diffraction geometry that minimizes or keeps constant the size and location of the spot of the

incident beam. This is implemented in two different modes,

one in which the width of the beam spot stays constant and

one in which the spot width varies. The angle calculations

governing the diffraction geometries are derived.

2. Diffraction geometry

For the following considerations, biaxial stress states are

assumed with no contributions for all normal components i3.

For the sake of simplicity, it is also implied that the sample is

elastically isotropic. First and foremost, high spatial resolution

(in one dimension) requires (1) excellent beam definition, (2)

close proximity of the primary beam aperture to the specimen

Figure 1

Diffraction geometry in a four-circle configuration. The primary beam

should be pictured as rectangular, and it is within the YZ plane (typical

beam dimensions Z = 3 mm, X = 0.04 mm). On the right, the magnified

specimen is shown. The zs axis is parallel to the ’ axis. The azimuth  is

the angle between xs and the normal projection of the scattering vector q

onto the specimen surface. The tilt is the angle between zs and q. The

angle introduced here is the rotation of the beam spot at ’ = constant

with respect to (! + ) = 90 where = 0.
and (3) low beam divergence. Requirements (1) and (2) can be

easily fulfilled through simple collimator tubes with attached

slits; for (3) it is sufficient to use fine-focus X-ray tubes with

focal spot sizes of 0.4 mm. For example, in a routinely used

experimental setup the aperture to specimen distance was

30 mm, the focal spot to specimen distance was 350 mm and

the aperture size was 0.04  3 mm. The spatial resolution was

evaluated through moving a specimen into the beam and

measuring the background-corrected integral intensity,

resulting in a measured resolution of 0.05 mm. In order to

translate the beam definition (dimensions) into constant

spatial resolution at all tilt angles, the proposed diffraction

geometry requires combinations of tilt, azimuth and ! rotation

angles that can be produced only in a four-circle goniometer

(Fig. 1).

The naming of angles follows that given in the classic

treatment of four-circle geometry by Busing & Levy (1967). In

order to avoid conflicts between the naming of Euler rotations

(!, , ’) and the commonly used names for the azimuth and

tilt angle of the scattering vector in the specimen reference

frame, the capitalized letters  (azimuth) and (tilt) are

used, as shown in Fig. 2. With all diffractometer angles set to

zero, the specimen frame from Fig. 2 coincides with the

laboratory frame from Fig. 1. The angle  is the Bragg angle

from the equation

 1⁄4 2dhkl sin  ð1Þ

in which  is the wavelength, hkl are the Miller indices and dhkl

is the lattice spacing. Two strategies/geometries exist in

conventional diffraction-based lattice strain measurements. In

the omega mode, the specimen tilt is performed such that the

scattering vector is within the scattering plane (i.e. the axis of

rotation is perpendicular to the scattering plane), and

rectangular beams are often

employed. The use of measurements with defined spatial resolution is limited because the

orientation of the spot is constant

but not its width, which increases

with 1/cos(Wilson, 1963). This

broadening of the beam spot

creates complex correlations

between the tilt angle and the

spatial distribution of lattice

strains.

In the psi mode, the scattering

vector tilts perpendicularly to the scattering plane, i.e. the tilt

axis is within the scattering plane. The beam is usually circular

or quadratic in cross section, and both the size and the

orientation of the spot change. In both the omega and the psi

modes the stress tensor components can be separated by

choosing specimen orientations with  = 0 or  = 90 as

follows from equation (2) (see Noyan & Cohen, 1987; Hauk,

1997):

"ðhklÞ1⁄4ð1=2Þs2ðhklÞ 1⁄2ð11 cos2  þ 22 sin2  þ 12 sin 2Þ

 sin2 þ 33 cos2 þ ð13 cos  þ 23 sin Þ

 sin 2 þ s1ðhklÞð11 þ 22 þ 33Þ: ð2Þ

The use of the geometry shown in Fig. 1 for residual stress

determination has been described in the literature (Reimers,

1992; Hauk, 1997; Welzel et al., 2005), and it is similar to the psi

method with omega tilt described by Hauk (1997), where

equations are derived for a transformation equivalent to

equation (2). The equations given by Hauk (1997) can be

omitted through the expressions for  and derived in the

following. Similar expressions for  and were also given by

Welzel et al. (2005). It should be noted that the rotation of

the beam spot occurs at all angles  61⁄4 90 and (! + ) 61⁄4 90 in

addition to any sample rotation ’.

For the following considerations, it is assumed that (1) the

stresses involving the surface normal component are zero

within the X-ray penetration depth and (2) within the

measurement field the stresses are constant in one direction.

The reference frame used here is defined in Fig. 3.

An example of such a stress field is found in the homogeneous (constant radius of curvature) bending of a sheet.

Conditions (1) and (2) are also fulfilled for many types of twodimensional deformations and the resulting plane stresses

such as deformed membranes (deep-drawn sheet metal) or

deep-reaching (> 1 mm depth) peening stresses. For example,

the changes of the stress component xx in the y direction can

be measured on surface (1) and the y dependence of

component zz can be measured on surface (2). Stresses

caused by shallow (comparable to the X-ray penetration

depth) two-dimensional surface deformations do not fall into

this category because the stress in the in-plane direction that

becomes the surface normal on a side surface is subject to

relaxations [zz on surface (1)]. For measurements on the

surfaces (1) and (2) it is assumed that the measurement field is

research papers

J. Appl. Cryst. (2009). 42, 192–197 Gna ̈upel-Herold  Determination of intermediate stress gradients 193

Figure 2

Left: schematic example of a through-thickness stress field that can be determined with the technique

described in this work. Right: magnified measurement field with rotation of the beam spot as the

specimen (left) is tilted with  at (! + ) 61⁄4 90. The beam-spot size is minimal at (! + ) = 90.

Figure 3

Accessibility of gradients of in-plane stresses from side surfaces. Note

that the same reference frame is used for all three surfaces, which causes

each normal component to have a different index x, y, z.
sufficiently small and there is sufficient distance from the

edges for the stresses to be unaffected by the left and right

edges. If the stresses change in both directions within the

beam-spot area [for example, if xx changes in both the x and

the y directions on surface (1)], then the length of the beam

spot on the specimen surface has to be decreased such that

xx ’ constant within the length of the beam spot in this

direction. This can be done either through a smaller X-ray

beam or by means of thin (< 10 mm) lead foils attached to the

specimen surface such that the illuminated area is constant as

suggested by the rectangular measurement field in Fig. 2 (left).

Thus, in order to evaluate the change of the stresses xx in the

y direction, the primary X-ray beam has to be shaped such that

it forms a beam spot long in the x direction and narrow in the y

direction. The xy dimensions of the beam spot are derived

from the requirements of spatial resolution for a particular

stress field. The orientation of the beam spot has to be kept

constant in order to maintain the spatial resolution at each tilt

angle. It should be noted that edge effects need to be avoided,

as depicted by using only the middle region of the specimen in

Fig. 2. For example, xx becomes zero at both the upper and

the lower edges.

A constant orientation of the beam spot on the specimen

surface requires that the ’ angle is adjusted by as outlined in

Fig. 2. The adjustment can be calculated from the intersection

of the YZ plane in the laboratory reference frame and the

specimen surface, given by the two vectors xs and ys (see

Fig. 1). Using the transposed R matrix (see Busing & Levy,

1967), all four vectors can be expressed in the specimen frame

[the phi frame of Busing & Levy (1967)] in terms of instrument angles:

Y 1⁄4 R~ 01

0

0

@

1

A 1⁄4

sinð! þ Þ cos  cos ’ þ cosð! þ Þsin ’

sinð! þ Þ cos  sin ’ cosð! þ Þ cos ’

sinð! þ Þsin 

2

4

3

5;

ð3Þ

Z 1⁄4 R~ 00

1

0

@

1

A 1⁄4

sin  cos ’

sin  sin ’

cos 

0

@

1

A: ð4Þ

At ’ = 0, xs and ys become

xs 1⁄4 100  T

; ð5Þ

ys 1⁄4 010  T

; ð6Þ

from which the following expression for the angle is derived:

1⁄4  arccos1⁄2cos  sinð! þ Þ þ sin  tan  sinð! þ Þ

=



cos2

ð! þ Þþ1⁄2cos  sinð! þ Þ

þ sin  tan  sinð! þ Þ21=2

: ð7Þ

Note that ( + !) is the total specimen rotation, i.e. one has

symmetric diffraction at ! = 0. The following quadrant

convention applies:

< 0



0 < ð þ !Þ < 90; 0 << 90;

90 < ð þ !Þ < 180; 90 << 180; ð8aÞ

< 0



0 < ð þ !Þ < 90; 90 << 180;

90 < ð þ !Þ < 180; 0 << 90: ð8bÞ

The angle is independent on ’ which is why ’ can be set to

zero for the purpose of calculating . Equation (2) is not

defined for  = 90 but from Fig. 2 it is obvious that = 0 for  =

90. The angle is an additional rotation of ’, i.e. it is additive,

and is independent of the actual orientation of the primary

beam. For example, equation (2) is also correct if the plane of

the primary beam is chosen such that it is parallel to the XY

plane.

If left uncorrected, the changing orientation of the beam

spot on the specimen would be equivalent to measuring the

lattice spacings over a larger effective area given by the beam

dimensions. In effect, a smoothing of the stress gradient takes

place. The effective increase b of the beam spot with  and ! is

shown in Fig. 4.

By adjusting ’ at every tilt, the size of the beam spot

remains unchanged as well (at large angles, the part of the

beam spot further away from the apertures may become

broader as a result of divergence). Thus, the constant orientation and size of the beam spot make measurements with high

resolution actually feasible. The resolution is only limited by

the closeness of the beam apertures and the beam divergence,

as well as the willingness to accept the necessary data acquisition times.

A constant orientation of the beam spot has the consequence that the tilt plane is inclined against the specimen

surface (Fig. 5). Thus, the azimuth  and tilt are generally

not identical to ’ and (90), respectively. This tilt mode

and the equations for  and tilt have been described in the

literature (Hauk, 1997; Welzel et al., 2005); however, new

expressions for  and are developed to allow for the

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194 Gna ̈upel-Herold  Determination of intermediate stress gradients J. Appl. Cryst. (2009). 42, 192–197

Figure 4

The rotation of the rectangular beam spot due to changing orientation of

the specimen with . The result is an effective broadening b. It is assumed

that the beam spot is 0.10 mm wide in the X direction of the laboratory

system and 1.0 mm high in the Z direction. The broadening is

proportional to the height of the beam, e.g. a 2.0 mm beam will produce

twice the broadening. The lengthening of the beam spot with is

disregarded because it can be easily limited through a lead foil window

attached to the specimen surface. The graphs were calculated using

equation (2).
different reference frames used, and to include the correction

. Both  and can be calculated using the transformations

described by Busing & Levy (1967) by expressing the vectors

q, xs, ys and zs, in terms of instrument angles. is the angle

between the surface normal (assumed to be parallel to the ’

axis) and the scattering vector q,

q 1⁄4 R~ 10

0

0

@

1

A 1⁄4

cos ! cos  cos ’ sin ! sin ’

cos ! cos  sin ’ þ sin ! cos ’

cos ! sin 

0

@

1

A: ð9Þ

The matrix R is given explicitly by Busing & Levy (1967). One

obtains for 

cos 1⁄4 cos ! sin : ð10Þ

With the correction applied,  is the angle between the

projection of q on to the xsys plane and the vector xs:

cos  1⁄4 1⁄2  cos ! cos  cosð’ þ Þ sin ! sinð’ þ Þ

ðcos2 ! cos2  þ sin2 !Þ

1=2 : ð11Þ

Equations (10) and (11) allow the calculation of azimuth and

tilt for any specimen orientation. If the orientation of the

beam spot is not of interest then one can set = 0. The

symmetric case is where ! = 0 corresponds to the psi mode and

one has ’ + =  and cos 1⁄4 sin . In the omega mode, one

has  = 90 and cos = sin(’ + ). The latter is the result of

the choice of axes here and it represents an offset of /2, i.e.

 = /2 + ’ + .

3. Strategies for high spatial resolution

3.1. General considerations

If the intensity of the primary beam was abundant, one

would simply choose aperture settings that give the primary

beam the necessary dimensions to provide a defined spatial/

area resolution on the specimen. This is not the case for

conventional X-ray sources, and one has to compromise by

having good spatial resolution only in one dimension while the

aperture in the perpendicular direction remains as far open as

possible to provide both sufficient diffracted intensity and

good grain statistics. The latter is a result of the beam-spot

area being less than 1 mm2 which, for large-grained materials,

reduces significantly the number of grains contributing to a

particular diffraction peak at a given orientation. According to

the simple scheme used by Gnaeupel-Herold et al. (2004) for a

grain number calculation, one can estimate that for typical

experimental settings the total number of grains in a reflecting

orientation is of the order of 1 (linear position-sensitive

detector with 2 vertical opening, 10 mm penetration depth,

grain size 20 mm, 0.10  5 mm beam). Such low grain numbers

almost always require the use of oscillations in ! and  in

order to improve the grain statistics. The grain statistics are

essential for stress determination because they affect not only

the accuracy of the measurement of the lattice spacing but also

the validity of the underlying assumptions about polycrystal

elastic behavior (Luzin et al., 2002). In the above example the

average number of grains contributing to one peak can be

increased to more than 20 by using  = 2 oscillation in 

and ! = 5 oscillation in ! (1 interval, determined from a

single grain rocking curve). The correction is performed at

each oscillation  and !. If necessary, and if the specimen

geometry allows, the statistics can be improved further

through shifting of the measurement field (e.g. along the x

direction in Fig. 2).

Although the spatial resolution can be calculated from the

aperture settings and then beam divergence, the effective

spatial resolution should be verified because of the possible

misalignment of the beam spot with a specimen reference

direction (i.e. the edge of the specimen should be parallel to

the line-shaped beam spot). This is done experimentally

though an edge scan in which the specimen is moved incrementally into the beam. The integrated intensity of a reflection

increases as more of the specimen surface is illuminated, and

the slope of this function is used to evaluate the effective

spatial resolution [see Wang et al. (1998) for the treatment of

neutron beams].

3.2. Constant orientation/minimum size beam spot

Two modes of operation exist in which the small beam-spot

dimension stays both constant in orientation and at its smallest

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J. Appl. Cryst. (2009). 42, 192–197 Gna ̈upel-Herold  Determination of intermediate stress gradients 195

Figure 5

The meaning of  and for an arbitrary tilt scan in the specimen

reference frame. The psi mode is a tilt within the  plane at ! = 0; the

omega mode (not shown) is within the yszs plane with  = 90. Each

orientation of the scattering vector produces combinations (1,1;

2,2; 3,3 ...) in which  changes together with .

Figure 6

Tilt-scan options for a constant orientation and a minimum beam spot.

For both tilts the plane of the primary beam given by its height/width

directions is always perpendicular to the specimen surface. (a)  tilt, (b) !

tilt.
possible size in a tilt scan: one by setting (! + ) = 90 and one

by keeping  = 90. This is outlined in Fig. 6. The specimen

surface and the primary beam plane are perpendicular to each

other. The – motion is produced either through the  circle

(Fig. 6a) or through the ! rotation (Fig. 6b). The main

difference between Figs. 6(a) and 6(b) is that in Fig. 6(a) the

scattering vector is not parallel to the tilt plane because it

would require  = 90 and ! = 0.  = 90 implies perfect

backscattering and it is experimentally impossible.

The  tilt yields tupels (, ) in which both depend on (!,

, ’) through equations (10) and (11). From equation (2) and

Fig. 5, it follows that stress tensor components cannot be

separated in this mode. However, as implied earlier, the

formalism requires that, within the length of the beam spot,

the stress field is constant in the direction of the long

dimension of the beam spot. Thus, this component can be

determined from an omega-mode measurement.

The ! tilt in Fig. 6(b) produces scans in which (,  =

constant) which allows the selective measurement of stress

tensor components (in Fig. 6: 11). It should be noted that

often the  tilt (Fig. 6b) is preferable to the ! tilt (Fig. 6b)

because of superior resolution /tan due to the small size of

the beam spot within the scattering plane. Therefore, the ! tilt

in Fig. 6(b) should only be used if a  rotation is not available

or if the selective measurement of stress tensor components is

required with  = 0 or  = 90.

3.3. Constant orientation/minimum azimuth variation

The measurement geometries in Fig. 6 are preferable if the

smallest possible beam spot is a priority. An alternative can be

found in the symmetric mode similar to Fig. 6(a) with the

exception that ! = 0. The symmetric mode differs from the psi

mode only by the correction applied to ’ at every angle . As

shown in Fig. 7 the variation of the effective azimuth 

(including the correction ) is much smaller for ! = 0 than for

(! + ) = 90. For large values of 2 the angle  is sufficiently

small to allow the selective evaluation of stress tensor

components, i.e. by setting  = 0 or  = 90.

4. Discussion

The non-destructive evaluation of gradients of in-plane stress

fields up to 1 mm depth/thickness has been frequently

challenging because of the scarcity of suitable experimental

resources (synchrotron, neutron sources). The approach

presented here circumvents many of the difficulties associated

with the evaluation of these stress fields by measuring the

depth dependence of the lattice strains, not from the surface

by means of penetration radiation, but from the side of a

specimen using a well collimated beam of weakly penetrating

X-rays. The need for a fixed orientation of a line-shaped beam

spot creates complex relationships between specimen orientation and the directions of the measured strains. This, in turn,

mostly prohibits a simplistic data analysis such as linear

regression of lattice strain versus sin2

 . As a result, explicit

assumptions have to be made about stress tensor components

or additional measurements have to be performed in different

directions . The first approach can be used for many

problems in which boundary conditions permit only negligibly

small stresses in one direction. Also, symmetric scans with ! =

0 and intermediate tilts (see Fig. 7a) permit only small

contributions by 22 and 12.

The capabilities of a four-circle goniometer are another

requirement for the implementation of this technique. The

limits of applicability, i.e. a stress gradient can be measured in

only one direction and resolutions of 0.05 mm, still include a

wide variety of materials problems such as sheet metal

formation, laminated structures and layers, certain surface

treatments such as laser peening or shot peening, and the

study of edge effects. A notable advantage of the method is

that X-ray goniometers with the necessary capabilities are

widely available, thus providing a simpler way of studying the

aforementioned class of residual stress problems.

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Figure 7

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