1) The document describes modifications made to an X-ray diffractometer to enable local strain measurements on small, curved samples like steel wires. 2) A custom sample holder was designed with adjustable angles to decouple the theta-theta scan and allow scanning at different psi angles relative to the stress directions. 3) Techniques like ray screening and X-ray capillary optics were used to confine the X-rays to small areas on the curved wire samples and enhance intensity. Measurements of residual strain on steel valve springs were then carried out with the modified diffractometer.

1. 52
Excerpt about micro-focus X-ray diffraction strain
measurements and capillary optics from report on
microsystem reliability analysis
[Paper X] refer to the list of publications by J. Janting
2.2.4 X-ray strain measurements
The next paragraphs of this report are a description of hitherto unpublished work on
modifications of an X-ray diffractometer for local strain measurements which was
conducted during the authors’ employment at IFA, University of Aarhus, 1991-1993.
2.2.4.1 Introduction
X-rays are probably the most versatile and therefore also most widespread materials
analysis tool existing. It is far more widespread than SAM. A general introduction to X-
ray science and technology is given in [10].
X-rays are used to analyse e.g: structure (crystallographic, texture by diffraction),
mechanical properties (X-ray Elastic Constants (XEC), by diffraction), elemental
composition (by X-ray absorption, X-ray Photoelectron Spectroscopy (XPS)), chemical
composition (by X-ray Absorption Fine Structure, XAFS), grain/particle size (by
diffraction), strain/stress (by diffraction), interior structure (by X-ray absorption). Most
X-ray analysis is based on diffraction [11-13]. Some selected descriptions of strain, grain
size and surface analysis on crystalline and amorphous materials are given in [Paper 6],
[14-17].
The monitoring of stress has previously in this report been highlighted as very important
to control the reliability of microsystems [Papers 9, 21, 27], [1-4]. General analytical
expressions for strains and stress can be found in [18]. It can be measured by different
means [19, 20] e.g. XRD [21, 22] and curvature methods [Paper 9], [3]. An overview of
experimental and theoretical aspects of X-ray strain measurement and analysis methods is
given in [23-25] and [26-36] respectively. In [37] different XRD methods have been
compared. Measurement of strain in thin layers is of particular interest for microsystems
[Papers 9, 21, 27], [1-4]. Several papers have been written on the origin of thin film strain

2. 53
[38, 39] and measurement methods by XRD [37, 40]. Examples on technological
interesting Mo, Cr and TiN thin films for other uses than microsystems are given in [41-
44] and [45, 46] respectively.
This work was motivated by the need to evaluate the effect of different surface treatments
of 3.8 mm diameter steel wire for car engine valve springs. The starting point for the
measurements was a common θ-2θ Philips PW1050 X-ray diffractometer. This
equipment is suited for ordinary identification of materials by powder diffraction on flat
samples. Measurement of strain on small samples with high curvature requires
modifications and/or procedures like those described in the following paragraphs. Errors
generally introduced in θ-2θ XRD strain measurement on curved samples have been
analysed in [21, 47, 48], and practical considerations and procedures for measurement on
drawn steel wires are given in [47, 49-52]. X-ray strain measurement on steel in general
has been reviewed in [53].
2.2.4.2 Theory of X-ray residual strain measurement
In fig. 5 the basic Philips PW1050 X-ray diffractometer is shown. X-rays are generated in
the X-ray tube tower with the radiation sign in the middle of the picture, by accelerating
electrons from a tungsten filament cathode into a water cooled anode which is typically
copper where K shell electrons are exited. When the electrons fall back to the K-shell
energy level X-rays are emitted. Some of the X-rays pass through the low absorbance Be
tube widow in the direction of the sample. In the tower/tubeshield the X-rays pass a Ni
window which absorbs the slightly higher energy CuKβ radiation. To control the
divergence of the rays they pass a slit of adjustable width on their way to the sample. At
the sample they are diffracted on crystallite planes parallel to the surface of the sample
when the Bragg condition (1) below is satisfied:
)1(sin2 λθ nd =
Fig. 5: Philips PW1050 X-ray diffratometer.

3. 54
where d is the plane spacing, θ is the angle between the sample surface and the incoming
rays, n is an integer, λ is the X-ray wavelength, λ(CuKα) = 1.54 Å. The diffracted rays are
first received by a Ge monochromator crystal on which they are further diffracted to a
detector (upper right in fig. 5). By simultaneously changing the angle θ and the angle 2θ
between the incoming rays impinging the surface and the diffraction direction towards
the detector, detection of diffracted rays from crystallite planes with different d spacing is
ensured. Each time the Bragg condition is satisfied it gives rise to a peak in the intensity-
2θ XRD spectrum.
In a stressed sample the d spacings are altered i.e. measuring the strain ε and knowing
Youngs modulus E the stress σ can be determined basically from:
)2(Eεσ =
Measuring strain implies the “same” d spacings to be measured at different inclinations to
the stress directions. Further, this means that determining stress with the PW1050
equipment requires θ-2θ decoupling at different angles ψ to the surface normal. To
determine all elements of the full stress tensor σij where i, j =1, 2, 3 it is also necessary to
turn the sample at different angles φ in the sample surface plane. For an isotropic sample
the full strain equation is [21]:
( )
( )
( ) )3(2sinsincos
1
1
sinsin2sincos
1
2313
33221133
2
33
2
2212
2
11
0
0,
,
ψϕσϕσ
ν
σσσ
ν
σ
ν
ψσϕσϕσϕσ
ν
ε ϕψ
ϕψ
+
+
+
++−
+
+
−++
+
=
−
=
E
EE
Ed
dd
where d0 is the unstrained d spacing. From this, it is seen that if the stress tensor is biaxial
in the surface this can be written:
( ) )4(sin
1
21
2
0
0,
, σσ
ν
ψσ
ν
ε ϕ
ϕψ
ϕψ ++
+
=
−
=
EEd
dd
where σϕ is
)5(sincos 2
22
2
11 ϕσϕσσϕ +=
or
)6(sin2sincos 2
2212
2
11 ϕσϕσϕσσϕ ++=
That is, the biaxial in-plane stress σϕ can be determined from the tilt of linear plots of
dψ,ϕ vs. sin2
ψ. This equation is the most frequently used in X-ray stress determination.

4. 55
“ψ-splitting” i.e. unlinear dψ,ϕ vs. sin2
ψ plots symmetrically distributed around a centre
line for negative and positive ψ indicate the presence of the shear stresses σ13, σ23, in the
stress tensor. Further analysis of (3) reveals the separate σij for specific φ.
2.2.4.3 Results
The strategy followed to make reliable strain measurements on 3.8 mm wire diameter
steel valve springs surface treated in different ways on the PW1050 Philips X-ray
diffractometer has been to:
1) Make a sample holder with accurate θ-2θ decoupling.
2) Make accurate sample holder and alignment accessories.
3) Make a sample holder accessory for φ movement.
4) Develop ray screening methods to confine the X-rays to small areas.
5) Enhance the X-ray intensity on small areas with X-ray capillary optics.
The following chapters describe the developed tools, procedures and theoretical
considerations together with some measurements made with the new sample holder.
2.2.4.3.1 PW1050 θ−2θ decoupling
Figs. 6-8 show the assembled and disassembled sample holder which enables θ-2θ
decoupling. After mounting the sample holder, θ = 0° was found by mounting a double
knife i.e. two sharp metal edges at equal distance to the Focusing Circle (FC) centre, and
observing maximum intensity equal to half the unrestricted intensity when turning the
holder. Note that for a given ψ tilt a scan over a peak corresponding to dψ,φ is still a θ-2θ
scan. Figs 9, 10 show the alignment and confinement procedures respectively. For the
Fig. 6: Assembled sample holder with optional θ-2θ decoupling. Ψ can be varied ±65°
in steps of 5° with a precision of ±0.05°. With the same precision φ can be varied
between ±20°, ±45°, ±65°, and ±90° with respect to the plane defined by the
incoming and received diffracted rays. In this picture the fixture is mounted which
enables measurements on the circumference of springs (outer rim of cut out spring
pieces).

5. 56
alignment (fig. 9) the sample can be
moved up and down in the sample
holder with an Allen key. Proper
placement of the sample surface in
the FC centre is detected by
establishing electrical contact to
another surface temporarily placed in
the FC centre. This method is of
cause limited to materials that is at
least semi conducting. At the bottom
of the samples electrical contact to
the equipment can e.g. be achieved
by using placement in carbon
powder. The precision of this
procedure is limited by the surfaces
roughness. To confine the radiation
to the very top (1 mm width) of the
lying down cut out spring segments
the sides (3 pieces in fig. 10) were
covered with a heavily absorbing
substance, here Ta2O5. By
determining σφ and σφ±α, σ11 and σ22
can be determined. To do that with
smallest error φ = 65.9° should be
used [26]. This was also the idea
behind the optional φ = 65° in this
sample holder.
Fig. 7: Basic sample holder axle prepared for θ-
2θ decoupling. The dents correspond to the
different ψ positions. A spring actuated steel ball
falls into these when the sample holder is turned
and holds the position.
Fig. 8: Accessories for θ-2θ decoupling and φ movement. Dimensions are compared
with that of a coin.

6. 57
Fig. 9: Positioning of sample surface in the FC centre.
Fig. 10: 3 cut out segments of a spring ready for X-ray diffraction stress measurement.
The sample is aligned with its surface in the FC centre and the sides of the spring
segments are covered with heavily absorbing Ta2O5 to ensure diffracted X-rays from
the top of the segments only.

7. 58
2.2.4.3.2 X-ray capillary optics
The ideal shape of a sample in a diffractometer like the Philips PW1050 with Bragg-
Brentano geometry is concave with a radius of curvature equal to that of the FC since this
ensures diffraction focus on the FC. The geometry of flat/convex and concave samples
with radius of curvature r greater than the FC radius rFC results in a 2θ downward peak
shift component in the XRD spectra corresponding to virtual compression residual stress.
Concave samples with r < rFC results in a 2θ upward peak shift component in the XRD
spectra corresponding to virtual tension residual stress [48]. One way of coping with
errors due to high sample curvature in X-ray strain measurements is by only irradiating a
small area [47, 48, 54]. However, this most often requires restricting the beam with small
divergence slits near the source or screening of the sample as with Ta2O5 mentioned
above. In [47] this has also been used to evaluate the stresses in car suspension spring
wire with a radius of curvature of 6 mm. The curvature is a source to virtual ψ tilts as
illustrated in fig. 11. For the spring segment where only 1 mm is exposed horizontally to
the X-rays we have:
)7(3.1590mm5.0cosmm9.1 oo
=−=⇒=⋅ ωψω
That is, the total ψ uncertainty is Δψ = ±15°. Usually, no matter how and where the beam
is restricted the consequence is low beam divergence at the cost of intensity on the
sample. X-rays can not be collected and guided inside an ordinary fibre like light in the
Fig 11: Cross section sketch of wire segment prepared for X-ray stress
measurement by coverage of the sides with absorbing Ta2O5 and exposure
of a region of only 1 mm width. Nevertheless this result in a ψ tilt error of
±15° corresponding to the tilt with respect to the vertical line of crystallite
planes parallel to the surface tangent at the left and right end of the
exposed area, see the text.

8. 59
optical frequency regime because the index of refraction n is less than 1 in optically
dense media like glass due polarisation of the electrons by the electric field. The index of
refraction n for X-rays is in its simplest form determined from:
)8(1 δ−=n
where δ is [17, 55, 56, 57]:
)9(
2
2
2
2
λ
π
δ
mc
ZNe
=
where Z is the atomic number, N the number of atoms per cm3
, e is the unit electronic
charge, m is the electronic mass, and c is the speed of light. This can be rewritten to:
)10(
2
2
2
2
λ
π
ρ
δ
Mmc
eNZ A
=
where ρ is the density, NA is Avogadros’ number, M is the molar mass. Since n < 1 total
external reflection without energy loss is according to Snells law observed above a
critical incident angle Ic or below a grazing angle θc, see fig. 12.
That is we have (assuming n = 1 for X-rays in air) according to Snells law:
)12(2
)11(5.01cos)90cos(1sin 2
δθ
θθδ
≈
−≈=−=−==
c
cccc InI
c
o
Fig. 12: Definition of critical grazing angle θc for total external reflection of X-rays.

9. 60
Further, since n < 1 there is no π/2 phase change of the reflected waves eliminating
undesirable interferences [17, 58]. θc in degrees can be calculated from:
)13(0917.0
2
180
2
180 ½
½
2
2
2
λρλ
π
ρ
π
δ
π
θ ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=≈≈
g
cm
Mmc
eNA A
c
where A is the atomic mass number. As can be seen from equation (13) θc depends
inversely on the X-ray energy. θc calculated from (13) is compared for different materials
and X-ray energies in table 1:
Material θc (CuKα) / ° θc (CrKα) / °
SiO2 0.23 0.35
Al 0.23 0.35
Ge 0.31 0.46
Ag 0.44 0.65
Au 0.57 0.85
Pt 0.60 0.89
Ir 0.62 0.91
Table 1: θc for different materials and energies.
This can be used to guide X-rays. Confining and guiding the X-rays to the sample by
collecting them near the source inside a small hollow fibre i.e. a capillary by total
external reflection the loss of intensity can be circumvented but now at the cost of higher
beam divergence on the sample, see fig. 13. The critical angle for total external reflection
Fig. 13: Principle of X-ray intensity gain by use of capillary capture near the source.

10. 61
or the acceptance angle of the capillary, which is e.g. made of glass, is quite low which
means that although the divergence increase within the same small area as irradiated
without use of a capillary, it is still so low that it is not contributing significantly to
diffraction line broadening. Referring to fig. 13 the beam divergence increase from θ1 to
θc or radius of irradiated area increase from r2 to r1 is determined by the factor:
)14(1
sin
sin
2
2
11 d
R
Rd
r
d
r
cc
+=
+
=≈
θ
θ
θ
θ
Likewise, the intensity gain G by use of the capillary is determined from the ratio of solid
angles subtended by the capillary at each end:
)15(1
sin
sin
2
2
1
2
1
2
2
⎟
⎠
⎞
⎜
⎝
⎛
+=≈=
Ω
Ω
=
d
R
G cceff
θ
θ
θπ
θπ
So, the capillary effectively brings the X-ray source and sample closer to each other, like
in the alternative Seeman Bohlin diffraction geometry [40]. Other ways of achieving high
intensity and low divergence X-rays on small areas are by use of more expensive
equipment like that with rotating anode for better heat dissipation or that of synchrotron
radiation where the radiation is emitted from accelerated electrons.
Here 138 mm long glass capillaries with r2 = 0.1 mm starting at d = 30 mm from the
source close to the Be X-ray tube window were used. Based on equations (13), (14), and
(15) the theoretical intensity gain G performances for hypothetical samples placed right at
the exit end of these capillaries and others are compared in table 2 below:
Table 2: Intensity gain with capillaries with r2 = 0.1, d = 30 mm at different lengths
using CuKα, CrKα radiation and different capillary surface materials.
Special holders for the capillaries were constructed, see figs. 14, 15. First, the capillary
was placed on an aluminium bar in a V-groove which was fitted into the fine movement
holder, fig. 14. This assembly was then fitted into the gross tilt holder in fig. 15 to be
mounted on the X-ray tower/tubeshield top. Unfortunately the test measurements using
the capillary set-up are no longer available at the time of writing this report.
G(CuKα) G(CrKα)
Capillary surface material
R/mm SiO2 Ge SiO2 Ge
50 10 19 24 41
100 27 50 63 109
138 46 83 105 182
150 52 95 121 209
200 85 154 197 341
250 126 230 293 505
300 176 319 406 702

11. 62
Fig. 15: Gross tilt holder for the holder in fig. 14.
Fig. 14: Capillary holder with x-y
translation movement and tilt along
two orthogonal directions in the x-y
plane.

12. 63
Capillary positioning (irradiated position, capillary tilt) were performed by repeated
confirmation of FC centre irradiance and ray direction using a flat sample coated with a
fluorescent paint and a double knife in the sample holder respectively.
2.2.4.3.3 Strain measurements
The functionality of the sample holder/goniometer described in chapter 2.2.4.3.1 was
satisfactorily verified on standards. It has been used to analyse the Ni coatings in
[Paper 5] and the graphite coatings in [Papers 6, 8]. However, most strain measurements
were made on the engine valve springs; see table 3 and figs. 16-18.
A variation in compressive stress over the mid-position between the inside and outside
surfaces of untreated springs was generally detected. The variation indicates the expected
tensile residual stresses on the inside surface and the compressive residual stresses on the
outside surface of springs.
Within the first 8 µm depth the compressive stresses are a factor of 3.5 higher in duplex
peened springs than in single peened. However, this needs to be verified with more
measurements.
It was attempted to transfer compressive stresses by coating the springs and flat polished
spring wire pieces with PVD Cr films. Unfortunately no XRD lines from the films and no
stress transfer could be detected.
Virgin straight wires have a small tensile stress around 9 MPa in the surface. Deeper
inside the value is higher, up to 154 MPa was found, table 3. Peening results in high
compressive stress deep inside the material. Values of e.g. -382 MPa (straight piece, table
3) and -520 MPa (spring segment, fig.16) have been found. Fig. 16 is a typical stress plot
showing the compressive stress achieved in the surface of the αFe springs as a
consequence of duplex shot peening. Both CuKα and CrKα radiation was used in these
measurements. Using only electropolishing has no significant effect on the stress level;
see table 3 and fig. 17. Fig. 17 is some typical measurements on straight spring wire
showing the differences between surface treatments. Electropolishing of virgin pieces
expose layers of higher tensile stress. Electropolish after peening do not change the
compressive stress level achieved by the peening significantly, table 3. Electropolish and
subsequent Cr implantations do also not have a significant effect on the stress as
evidenced by backside investigations of the irradiated samples, table 3. N+
implantations
introduce significant compressive stresses, however not to as high a level as the peening
process. This is also evidenced by backside sample investigations; see table 3 and fig. 18.
Fig. 18 shows the effect of N+
implantation on the residual stress level in spring wires.
The stress determined from the linear tilt is changed from slightly tensile to slightly
compressive.

13. 64
Treatment σφ / MPa
Virgin
From all points in plot:
50
From first three points from left in plot:
154
From last three points from left in plot:
9
Peened -382
Electropolished 103
Peened + electropolished -405
Electropolished + Cr implanted (2·1017
ions / cm2
)
89
Electropolished, Cr implanted (2·1017
ions
/ cm2
), backside investigation
103
Electropolished, N implanted (1018
ions /
cm2
)
From all points in plot:
-110
From first three points from left in plot:
57
From last four points from left in plot:
-166
Electropolished, N implanted (2·1018
ions /
cm2
)
From all points in plot:
-55
From first three points from left in plot:
116
From last three points from left in plot:
-114
Electropolished, N implanted (1018
ions /
cm2
), backside investigation
103
Table 3: Overview of typical results on stress introduction in straight 3.8 mm diameter
spring wire pieces treated in different ways and measured with the sample holder
described in chapter 2.2.4.3.1. Some of the plots corresponding to these measurements
are shown in figs. 16-18. The results are not corrected for errors due to curvature,
alignment precision etc. which might be significant, however without changing the
general trends.

14. 65
Fig. 17: Stress plots for 3.8 mm diameter BECROVA straight wire treated in different
ways, see values in table 3. The high compressive stress values are comparable to the
value found for a spring segment in fig. 16.
Fig. 16: Stress plot using the sample holder described in chapter 2.2.4.3.1 and equation (4).
The residual stress in a duplex peened and electropolished spring segment was determined
to be σφ = -520 MPa ± 40 MPa.

15. 66
2.2.4.4 Discussion
Compressive surface stress is attractive because it tends to close openings, cracks etc.,
which might otherwise begin to propagate eventually leading to failure. Shot peening is a
common way of introducing compressive stress [59-61] extending hundredths of µm into
samples depending on the shot size, energy etc., and the results presented above are
generally in accordance with the expectations. However, the compressive stress obtained
in spring wires with N+
ion bombardment and the lack of the same with Cr implantation
is a new and yet unexplained observation.
Electropolishing generally removes surface defects like small cracks and inclusions prone
to crack propagation and therefore prolong the lifetime of components. In the process
layers are removed which can be used to reveal the stress depth profile of samples. In
such a profiling the stress in the removed layers will have to be corrected for. In fig. 17
electropolish of straight virgin pieces expose a surface where the slightly tensile stress
seems virtually unchanged. However, generally d<211> is smaller indicating how a virgin
sample with tensile surface stress have more tensile surface stress isotropically within ψ
(all over smaller d<211> values) after electropolish. For a peened and subsequently
electropolished sample it is the opposite way around in accordance with previous
observations [59, 60]. Further, the bend curve form seen for this virgin plot indicate
Fig. 18: Stress plot for 3.8 mm diameter BECROVA straight wire treated in
different ways.

16. 67
presence of shear stress which could be confirmed by negative ψ tilts (cf. ψ splitting).
Note also, that as indicated in the figure, the probed depth (intensity reduction to ½ that
of the incoming beam) varies with θ and ψ which often explains unlinear stress plots.
The lack of diffraction lines from Cr thin films on spring and flat polished wire pieces is
believed to be explained by strong texture in these films, which needs to be verified in
future measurements.
In fig. 18 the d<211> spacing is within ψ almost isotropically changed to a larger value on
irradiation, which in itself indicates introduction of pronounced compressive stress. The
curve tilts indicate less compressive stress introduction. Again, the bend form of the
implantation plots can be explained by shear stress (ψ splitting) due to the limited depth
distribution of the N+
ions. This also corresponds well to the common oscillatory
behaviour of more or less compressive stress - tensile stress - compressive stress…as a
function of depth, which is also seen in the untreated and peened samples [59, 60].
However, as indicated, at the same time the implantation change the stress in the
direction of more compressive at all probed depths. The wire pieces were not implanted
on the one side, which as indicated in the figure was used to verify the effect.
It should be emphasised that straight capillaries merely constitutes a means of
transporting/guiding the X-rays in a manner without energy loss corresponding to
bringing source and sample closer i.e. the radiation at the two ends are the same
concerning beam divergence and cross section radius or area. For other geometries this is
not the case. The propagation and physics of X-rays guided by different capillary
geometries has been described in [57, 62-68]. Use of straight capillaries for XRD
purposes has also been described in [69], but they have mainly been used for Energy
Dispersive X-Ray Fluorescence (EDXRF) [56, 70-73].
According to Liovilles theorem from classical mechanics the volume in 6 dimensional x,
y, z, px, py, pz phase space of a Hamiltonian ensemble of points/states is preserved in time
[74]. This is expressed by equation (16):
)16()(...)()(...)( 2211∫∫∫∫∫∫ ∫∫∫∫∫∫ ∧∧=∧∧ tdptdxtdptdx zz
where t1 and t2 indicate two different times. The points/states behave as an
incompressible fluid. Coordinates x, y, z are the ordinary space coordinates, px, py, pz are
impulses i.e. direction coordinates. Within optics Liouvilles theorem is conveniently
expressed in 4 dimensional phase space [75] in terms of the “geometrical vector flux”
[76-78]. Then for an optical system with n = 1, symmetry around the z axis, and uniform
illumination up to the angle θz with respect to the z axis the conserved quantity can be
calculated from [76] to be:
∫∫ ∫∫∫∫∫∫∫∫ Ω==Ω== )17(sincos 2
AAdxdyddxdydpdpdAJ zzyxz θπθ
where
)18(∫∫= yxz dpdpJ

17. 68
is the z scalar component of the flux vector J, dΩ is an element of solid angle implied by
dpxdpy, Ω is for small angles the solid angle subtended by all the rays, and A is an area
parallel to the x-y plane that the rays passes. This conserved quantity is also called the
“throughput” or “étendue” for the optical system. For optical beams the theorem can be
expressed by two dimensional area projections of the 6 dimensional volume i.e. by saying
that the product of beam divergence and size is constant [74]. This is often illustrated by
two dimensional (θ, x) projection of 6-dimensional phase space as in fig. 19, where then
the area of the parallelepiped representing the beam is preserved on transformation.
The transformation illustrated in fig. 19 can be written:
)19(
10
0
1
1
1
1
1
12
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
+
+=⎥
⎦
⎤
⎢
⎣
⎡
=⎥
⎦
⎤
⎢
⎣
⎡
θθθ
r
d
R
d
Rr
M
r
c
where M is the transfer matrix for which det (M) = 1 in accordance with Liovilles’
theorem. As mentioned other geometries than straight capillaries e.g. conical, bent have
also been analysed and the above mentioned principles have been used several times in
concentrator optics [79-86], also in considerations including beam focus by total internal
reflection [79-81]. Another example is the one in fig. 19. In all cases due to Liovilles’
theorem (throughput preservation) intensity gain will always be at the expense of higher
Fig. 19: The straight capillary in fig. 13 corresponds to collecting e.g. with a conical
capillary X-rays with divergence θ1 from a circular area with radius r1 and focusing
them to a divergence θc on a circular area with radius r2. The throughput or area of
phase space projected on the (θ, r) plane is conserved: θcr2 = θ1r1.

18. 69
beam divergence. However this is as underlined earlier not critical for X-ray strain
measurements since high divergence rays are filtered out by the low θc [17, 79].
At the time of these experiments and considerations the use of capillaries to guide X-rays
was quite new. Today capillaries of different geometries are specifically produced for
that purpose [87, 88].
2.2.4.5 Conclusion
The functionality of a new sample holder/goniometer and accessories for strain
measurement in a PW1050 Philips diffractometer has been verified. With the equipment
it is possible to determine the full surface stress tensor for 3.8 mm thick car engine steel
valve springs which have been subject to different surface treatments. However, until
now only σφ values have been determined. It has been found that shot peening introduces
large compressive stresses, N+
ion implantation less, and electropolishing virtually none.
Procedures to avoid significant measurement errors due to sample curvature have been
identified, analysed and tested. The most promising is the feasibility of using capillaries
of different geometries to gain high intensity on small areas. The equipment for mounting
and positioning capillaries on a PW1050 diffractometer has been built but only
preliminarily tested. The tests were encouraging.
2.2.4.6 Acknowledgement
This work has been part of the EU BRITE/EURAM 1 project BREU0435 entitled
“Advanced surface engineering processes to enhance the dynamic performance of
springs”.

19. 70
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