1. science
2. mathematics
3. geometry

1
Modelling pseudo-strain effects induced in strain measurement using time-of-flight
neutron diffraction
S.R. Creek, J.R. Santisteban, L. Edwards
Department of Materials Engineering, Faculty of Technology, The Open University, Milton Keynes MK7 6AA, UK
Abstract
The penetrating nature of thermal neutrons makes neutron strain scanning a unique tool for the sub-surface measurement of elastic
strain. However, it is also important to be able to measure strain in near surface regions. The gauge volume represents the region of
diffracting material over which a measurement is made and is described by the intersection of incident and diffracted beams. In a nearsurface measurement the gauge volume is incompletely filled with material, resulting in a pseudo-strain. Pseudo-strains due to
incomplete filling of the gauge volume can be large (1x10-3 (1000µε)) in comparison to the resolution of instruments used to measure
such strains (50µε), thus requiring correction. This paper develops a model of a system using a radially collimated diffracted beam and
evaluates the pseudo-strain that would be recorded. A Fortran 90 program has been developed allowing real-time correction of such
pseudo-strains during the measurement process.
1. Introduction
Thermal energy neutrons have wavelengths similar
to atomic spacings, allowing diffraction experiments to
be performed. Neutron strain scanning determines the
vector strain field in a material by measuring the latticeplane spacings, dhkl, within a small volume of the
material (gauge volume) using neutron diffraction. These
values are then compared to the spacings of the same
planes measured in an unstressed sample, do hkl to
calculate the strain, ε, as in equation (1):
d − d o hkl ∆d hkl
ε = hkl
=
d hkl
d hkl
Thermal energy neutrons with a velocity v have a
momentum p [1], which is classically given by:
mL
(2)
resulting in a de Broglie wavelength, λ:
λ=
h
hτ
=
mv mL
λ = 2d hkl sin θ
(3)
(4)
to be rewritten:
hτ
2mL sin θ
(1)
1.1 Strain measurement using time-of-flight neutron
diffraction
τ
This wavelength allows Bragg’s law of diffraction:
d hkl =
As a probe of atomic structure, neutrons have a
distinct advantage over X-rays and electrons. In contrast
to electrons and X-rays, which interact with the electron
cloud, it is the interaction of neutrons with atomic nuclei
that results in diffraction. As a consequence, neutrons are
only weakly attenuated by materials allowing strain
measurements to be performed deep within samples
where other techniques would permit only surface
measurements.
p = mv =
where m is neutron mass, v is neutron velocity, L is total
flight path length (incident plus scattered), τ is the
measured time-of-flight (TOF) and h is Planck’s
constant.
(5)
where 2θ is the Bragg angle – the angle between the
incident and scattered beams. It is clear that for constant
L and θ, the spacing of a lattice plane, dhkl, is
proportional to the TOF, τ. This allows the lattice strain
to be evaluated as,
ε=
∆τ hkl
d hkl − d o hkl
τ − τ o hkl
= hkl
=
d hkl
τ hkl
τ hkl
(6)
The energy of a neutron is inversely proportional to
the square of its wavelength. A spallation source
provides a pulsed ‘white’ beam of neutrons that the TOF
technique uses for diffraction. This beam consists of a
wide range of wavelengths, and hence energies. The high
energy (low wavelength) neutrons arrive at the sample
first so, for a given angle, the diffraction peaks
corresponding to small d-spacings arrive at a detector
first, with the larger d-spacings arriving later.
Equivalently, considering a constant wavelength, it can
be seen that the angle at which Bragg’s law is satisfied
sweeps out in the direction of increasing 2θ as time
proceeds. Each pulse therefore produces a complete
diffraction profile of low intensity in a detector mounted
at constant 2θ. To recover a significant profile many
detectors are mounted to cover a range of 2θ and then a
2
technique known as electronic time-focussing [2] is
employed to merge the profile from each detector into a
single d-spacing spectrum.
maintain resolution. An xyzω positioner table accurately
located the observation point within the sample to within
± 0.1mm.
From equation (5), for the ith detector located at 2θi
with a neutron flight path length of Li,
The ENGINX system currently being installed at ISIS
follows the same overall design, but has 4
interchangeable collimators and a redesigned detector
bank including an order of magnitude increase in the
number of detectors.
τi =
2md hkl
Li sin θ i
h
(7)
Processing of this measured time can yield a
pseudo-time τ*, which is the TOF that would have been
measured by a detector located at a reference position Lo,
θo, (usually 2θo=90o), according to equation (8):
τ* = τi
Lo sin θ o
Li sin θ i
(8)
The intensities of these pseudo-time profiles are then
summed and a Rietveld refinement [3] is conducted to
calculate the lattice cell size.
Reference was made above to identifying the
observation “point” within the sample at which a strain
measurement is made. This concept needs clarification,
as it is apparent that the incident beam is of finite width
in 2 dimensions and the collimator blades limit the
volume of space at the collimator front end from which
diffraction events may be observed. The intersection of
the incident beam with this volume of space at the
collimator front end defines a diffracting volume known
as the “gauge volume”. This gauge volume can be
considered to consist of many such diffracting points.
Consider a coordinate system set up as in Fig. 1.2.
1.2 The ENGIN and ENGINX systems
An example of a system that operated successfully
to measure strain using TOF neutron diffraction is the
ENGIN instrument [4,5] at ISIS, the pulsed neutron
facility located at the Rutherford Appleton Laboratory. A
schematic of the ENGIN instrument can be found in Fig.
1.1.
Fig. 1.2. Coordinate system in the ENGIN instrument.
The widths of the collimating slits in the incident
beam define the dimensions of the gauge volume in the x
and z directions, while the y dimension is determined by
the field of view of the radial collimator.
Fig. 1.1. Schematic of the ENGIN instrument.
The ENGIN pulsed neutron strain scanner consisted
of an incident neutron beam with small angular
divergence, defined by slits placed in the beam, and two
large radial collimators centred about a Bragg angle of
2θ = ±90o. The function of the radial collimators was to
provide a large surface area over which to detect
neutrons while restricting the angular field of view of
individual detectors in the large detector banks to
To define the gauge volume mathematically the
spatial resolution function (SRF) [6] is introduced. The
SRF defines the relative detected intensity of scattered
neutrons as a function of position within the sample.
SRF(r ) = Pi (r ) Ps (r ) Pd (r )
(9)
where Pi(r)δV, the incident beam resolution function, is
the probability of incidence of a neutron on an
infinitesimal volume δV centred at r, Ps(r)δV is the
probability that a neutron will be scattered within δV and
Pd(r)δV, the detector resolution function, is the
probability of detection given that a scattering event has
3
occurred within δV. The dimensions of the gauge
volume then follow naturally from the volume in space
containing 68% of the probability density of the SRF
(equivalent to ± one standard deviation for a onedimensional probability distribution) when the scattering
volume is completely immersed in the sample.
To determine the performance of the instrument it
is useful to set Ps(r) equal to 1 so that the SRF is only
affected by instrumental parameters, the attenuation of
the beam by the sample material and the location of the
sample. It is important to note that the SRF is a strong
function of the attenuation coefficient and specimen
position as this will cause it to be biased to areas of
shorter path length and will return a value of zero for
regions of the gauge volume not immersed in the sample.
This is a particularly important consideration when
making measurements of near surface strains.
computation time required to produce such a result
would be substantial and prohibitive to real-time
measurement correction. The problem can be greatly
simplified by noticing that the angular term in the
pseudo-strain is independent of the azimuthal angle, Φ,
and the change in path length resulting from
incorporation of the third dimension will be small. Also
it is unnecessary to model each detector element
individually due to the time-focusing routine ultimately
combining the signals from all detectors. Modelling the
pseudo-strains for individual collimator channels will
suffice, reducing the number of calculations required by
an order of magnitude.
It is the purpose of this paper to develop a 2dimensional model to predict and correct for pseudostrains introduced by attenuation and surface scanning
with the ENGIN/ENGINX systems.
1.3 Pseudo-strain
2. The Spatial Resolution Function
For the purpose of assigning a lattice strain to a
point in the sample material, the most representative
location of the scattering volume is the neutron-weighted
centre of gravity (ncog), which takes into account
variations in intensity due to attenuation or absence of
sample material in the gauge volume. The ncog is
evaluated as the centroid of the SRF. Variation in the
position of the ncog due to attenuation effects or the
incomplete filling of the gauge volume will cause
variations in the values of L and θ for scattering from the
ncog to each detector. It is clear from equation (5) that
this will cause variation in the measured lattice-spacings,
independent of residual stress, and hence will introduce
error in the ultimate evaluation of the lattice strain
according to:
As outlined above the spatial resolution function is
obtained as the product of the incident beam and detector
resolution functions, including the effect of attenuation
where necessary. From this it is possible to identify the
gauge volume and calculate the location of the ncog. A
Fortran 90 subroutine has been written to evaluate the
SRF and locate the ncog for the 2-dimensional region of
space over which the gauge volume is defined.
∆τ
τ
=
∆d
d
+
b
∆ L sin θ
L sin θ
g
Peak shift = lattice strain + pseudo - strain
(10)
where the pseudo-strain, , can be decomposed into
spatial and angular components:
b
∆ L sin θ
L sin θ
g = ∆L + cot θ∆θ = ~ε
L
(11)
Where attenuation effects are very strong or there is
partial filling of the gauge volume it is necessary to
correct for this pseudo-strain. To do this accurately a 3dimensional model of the system should be constructed
in order to evaluate the pseudo-strain recorded by each
detector element from each point in a 3-dimensional
gauge volume. This would be a very complex task,
requiring the evaluation of the incident beam resolution
function, the detector resolution function and the
attenuation effect in a 3-dimensional sample. The
2.1 The Incident Beam Resolution Function
The shape of the gauge volume in the x direction is
determined by the divergence of the incident beam. This
is assumed to have a gaussian angular profile. According
to Withers et al. (2000), upon passing through the final
collimating slit this shape is described by the convolution
of a top hat function (for the slit) and a gaussian spatial
function (to account for the beam divergence). For a slit
of width s located at (xs,ys), with a beam of angular
divergence σθ propagating in the positive y direction, the
incident beam resolution function is given by:
Pi ( x , Σ ) =
b
1
sΣ 2π
z
FG
H
x−x + s2
s
exp −
x − xs − s 2
IJ
K
t2
dt
2Σ 2
g b g
Σ( y) = y − y s tan σ θ
(12)
(13)
For the purpose of numerical evaluation it is
convenient to note that this is equivalent to:
b g
Pi x , Σ =
F F
GH GH
IJ
K
FG
H
x − xs + s 2
x − xs − s 2
1
erf
− erf
2s
Σ 2
Σ 2
IJ I
K JK
(14)
where Σ is defined as in equation (13).
4
The incident beam resolution function is illustrated
in Fig. 2.1 for a slit of width 1mm, located at (0,-60) with
an angular divergence of 1o. Note the difference in scales
of the x and y axes used for illustration.
Fig. 2.1. The incident beam resolution function for a slit with
s=1mm, σθ = 1o, (xs,ys) = (0,-60) and a beam propagating in the
positive y direction.
Considering first a single channel, a neutron
scattered from a point at the front end will only be
detected if it can pass through the channel without
incidence on a blade (assumed to be a perfect neutron
absorber). For convenience the channel is oriented such
that its axis is co-linear to that of the coordinate axis.
Assuming isotropic scattering from all points in front of
the channel, the probability of detecting a neutron will be
proportional to the angular range through which the
neutron may be scattered and detected (acceptance angle,
Ψ). The channel geometry is such that from a given point
source of scattered neutrons the acceptance angle is
always limited by two of four defining points – the two
front, and two rear ends of the blades. The acceptance
angle is calculated using ray tracing from the scattering
point to the two appropriate limiting points.
The collimator function, Ψ(x,y), describes
analytically the acceptance angle of a channel as a
function of scattering position and the four collimator
parameters which locate the limiting points – R, l, α and
n. The function is continuous, but divided into 4 regions
depending on which limiting points are imposing
constraint. The situation is illustrated in Fig. 2.3 for a
channel with zero blade thickness.
2.2 The Detector Resolution Function
A radial collimator consists of radially aligned
neutron absorbing blades. These blades define channels
of constant angular thickness along which neutrons may
pass. Detectors are mounted at the far end of the
channels to record the arrival of diffracted neutrons at
fixed values of 2θ. A radial collimator can be described
by 5 parameters: the radius from the geometrical centre
to the end of blades, R; the length of the blades, l; the
total acceptance angle, α; the number of channels, n; and
the blade thickness, t, as shown in Fig. 2.2, illustrated
schematically for a collimator with only 5 channels.
Fig. 2.3. Schematic of a collimator channel with zero blade
thickness showing the 4 regions of space at the collimator front
end that are defined by the 4 limiting points.
Consideration of the effect of finite thickness
blades on the acceptance angle of a channel reveals
several interesting features, illustrated in Fig. 2.4.
Fig. 2.2. Schematic of a radial collimator demonstrating the 5
parameters with which it can be completely described.
5
for each channel. The detector resolution function for the
point (x,y) is computed according to equation (17),
∑ Ψb x , y g
n
Pd ( x , y ) =
k
k
(17)
k =1
where Ψ(xk,yk) is the value of the collimator function at
the rotated scattering point (xk,yk), representing the
contribution from the kth channel due to scattering from
the point (x,y).
Fig. 2.4. Schematic of a collimator channel with finite blade
thickness showing how the 4 regions are altered by the
reparametrisation of the limiting points in terms of the fifth
parameter t.
The overall effect of introducing blade thickness is
to redefine the location of the limiting points in terms of
a fifth parameter, t. The first feature is that the overall
acceptance angle is reduced, as the thickness of the blade
narrows the channel. Secondly the focal point of a
channel (the intersection of all 4 regions) is displaced
towards the channel by a distance d:
t
d=
2 sin
FG α IJ
H 2n K
(15)
The final effect is that while the channel has the
same angular width, α/n, and blade length, l, the radius
of the channel is reduced to give an effective radius, R′:
R' = R −
t
2 tan
FG α IJ
H 2n K
Fig. 2.5. Equivalent constructions of the collimator.
(16)
Explicit evaluation of the collimator function is
outlined in the appendix §A.1.
The detector resolution function describes the
probability of detecting a neutron at the far end of the
collimator assuming uniform irradiation and isotropic
scattering from a non-attenuating sample. It is calculated
as the sum of angles through which a neutron scattered
from a point in front of the collimator can be detected at
the end of the channels, divided by the total angle
through which a neutron can be scattered (assumed 2π
for isotropic scattering). From the model of a single
channel the detector resolution function can be
calculated for a point in space either by summing the
contributions from n channels rotated about the origin by
the angular thickness of a channel, α/n, or by summing
the contributions from n rotated observation points in
front of the single channel, as in Fig. 2.5. For ease of
computation the second method was adopted, requiring a
simple rotation of the observation point about the origin
One benefit of detecting neutrons over a large
surface area with the radial collimator system, as
opposed to slit based systems commonly found on
reactor neutron sources, is that it substantially reduces
the time required to detect statistically significant
intensity. It may then seem counter-productive to place
neutron-absorbing blades in the collimator that absorb a
large proportion of the scattered intensity. However,
evaluation of the detector resolution function using the
method outlined above for the ENGIN collimator
parameters in table 2.1, both with and without blades,
clearly demonstrates the blades’ purpose.
Radius
R
(mm)
Blade
length
l (mm)
750
600
Total
acceptance
angle
α (o)
17.5
No.
channels
n
Blade
thickness
t (mm)
41
0.16
Table 2.1. Parameters for ENGIN collimator
6
µ=
ρσ T
A
(19)
where ρ is the material density, σT is the collision cross
section and A is the atomic weight. In reality I0 and σT
are both wavelength dependent [7], but this fact will
initially be ignored.
Strongly attenuating materials severely reduce the
overall detected intensity and cause the ncog to be
shifted towards regions of shorter path length. The shift
of the ncog results in an attenuation related pseudostrain. This pseudo-strain is in addition to that which
occurs when the gauge volume is partially filled during
surface scanning.
Fig. 2.6. Detector resolution function for the ENGIN collimator
without blades – note the 4 regions defined above visible in the
description of a single channel.
Fig. 2.7. Detector resolution function for the ENGIN collimator
with blades.
The blades provide spatial resolution, reducing the
full width at half maximum in the y direction at the
geometrical centre from 57mm to 1.4mm. This is
essential to provide strain resolution as the relative dspacing spread in a Bragg peak for a given reflection has
a contribution from the relative uncertainty in the total
flight path length of a neutron which is largely
determined by the collimator resolution in the y
direction.
A subroutine has been incorporated into the
program that returns both the incident beam and
diffracted beam path lengths in the material in order that
the attenuation effect can be applied to the detected
intensity. This subroutine is quite versatile as it can
accommodate both convex and concave samples of any
size, shape and orientation to the system. At present the
sample is described by the coordinates of the vertices of
a single simply-connected polygon in the frame of the
system. The polygon represents the intersection of the 3dimensional sample with the 2-dimensional plane normal
to the z-axis, passing through the centre of the
collimator. This defines the extent and position of the
sample relative to the system. The order of labelling
sample vertices must be in the clockwise direction for
calculation to be correct. The sample surface can then be
scanned through the gauge volume by adjusting the
sample coordinates in accordance to the movement of the
sample positioner. A full discussion of this subroutine is
presented in the appendix §A.3.
3. Calculation of Pseudo-strains
Two methods of calculating pseudo-strain, , have
been investigated. The first approximates all scattering to
the ncog and calculates pseudo-strain relative to this
point, while the second is an integration of the pseudostrains over the entire gauge volume. Both models
attempt to imitate the measurement process by
calculating ∆L and ∆θ with reference to a gauge volume
immersed in the centre of an 11mm diameter silicon
cylinder (µ=0.083mm-1), which is the reference material
used in the creation of time-focussing routines.
2.3 Attenuation
Considering a path through a neutron-attenuating
material, the neutron intensity as a function of path
length, x, through the material is given by:
bg
b g
I x = I 0 exp − µx
(18)
3.1 The centroid method
As the ncog is the point most representative of the
scattering volume the first model evaluates pseudostrains by calculating the ncog of the silicon run and the
appropriate values of L and θ for each channel for the
paths from the incident beam slit to the centre of the
illuminated area at the end of a channel, via the centroid
7
position, as in Fig. 3.1. The procedure is then repeated
for the sample material in the orientation of interest and
the pseudo-strain is returned for each channel as in
equation (11). The mean pseudo-strain < > is then the
channel intensity-weighted sum of the individual channel
pseudo-strains, normalised by the total detected intensity,
as in equation (20),
~
ε
centroid
=
c
c
k
c
i
c
c
k
c
c
i
c
k =1
x,y
F
∑ GGH
n
k =1
zz
k
i
x, y
(21)
4.1 The Collimator
n
k =1
=
I
b g b g b g JJ
K
I
Ψ b x , y g P b x , y gdxdyJ
JK
~
ε k x , y Ψk x , y Pi x , y dxdy
c
k =1
∑ Ψ bx , y g P bx , y g
integration
zz
4. Results
∑ ~ε b x , y gΨ b x , y g P b x , y g
n
k
~
ε
F
∑ GGH
n
c
(20)
where (xc,yc) is the position of the ncog, k(xc,yc) is the
pseudo-strain of the kth channel evaluated using the
position of the ncog and Ψk(xc,yc)Pi(xc,yc) is the
probability of detecting a neutron in the kth channel due
to scattering from the ncog, which is proportional to the
detected intensity.
The collimator model has allowed evaluation and
direct comparison of the detector resolution functions for
the ENGIN and 4 ENGINX collimators. This is
particularly useful in assessing the performance of the
new instrument. Figs. 4.1–4.5 show the detector
resolution functions for the 5 collimators and Fig. 4.6
shows how the full width at half maximum (FWHM) of
each resolution function varies as a function of
displacement from the focal point (defocus). A
comparison of ENGIN and ENGINX collimator
parameters is given in table 4.1.
ENGIN
ENGINX
R (mm)
l (mm)
α (o)
n
t (mm)
750
450/510
/660/840
600
350
17.5
32
41
160
0.16
0.05
Table 4.1. Comparison of the ENGIN/ENGINX collimator
parameters.
Fig. 3.1 Schematic of system illustrating evaluation of L and θ
for each channel.
3.2 The integration method
The second method, described in equation (21)
calculates the mean intensity-weighted pseudo-strain for
each channel by integrating the product of the pseudostrain and the channel intensity, both as a function of
position, over the gauge volume. The overall mean
pseudo-strain is then evaluated as the sum of these
values over the number of channels, normalised by the
total detected intensity. In practise, this integration must
be performed numerically, introducing slightly increased
computation time. As before, ∆L and ∆θ are calculated
for each channel and at each scattering point with
reference to the ncog of a silicon run.
Fig. 4.1. Detector resolution function for the ENGIN
collimator.
8
Fig. 4.2. Detector resolution function for the ENGINX
collimator R=840mm.
Fig. 4.5. Detector resolution function for the ENGINX
collimator R=450mm.
Fig. 4.6. Graph of FWHM against Defocus for the ENGIN and
4 ENGINX collimators.
Fig. 4.3. Detector resolution function for the ENGINX
collimator R=660mm.
Fig. 4.4. Detector resolution function for the ENGINX
collimator R=510mm.
It is important to consider the general symmetry
properties of the detector resolution function and the
effect this has on the centroid position regardless of
attenuation. The entire collimator has symmetry in the y
direction resulting in a zero y-centroid coordinate.
Although on initial examination it appears that there is
also symmetry in the x-direction, this is not the case. The
model of a single collimator channel does not possess
this symmetry and consequently on construction of the
collimator from a single channel there is slight but
definite asymmetry present. This causes a small positive
displacement of the x-centroid coordinate of order 104
mm.
It is also interesting to note the effect of blade
thickness on the transmitted intensity. Fig. 4.7 shows the
transmitted intensity of neutrons scattered from the
geometrical centre of the collimator as a function of
blade thickness, while Fig. 4.8 represents the total
transmitted intensity over the region defining the gauge
volume using a 2mm wide incident beam slit.
9
suitable approximation for the overall transmitted
intensity when considering collimator design.
Fig. 4.7. Effect of increasing blade thickness on the detected
intensity from neutrons scattered at the geometrical centre of
the collimator.
A secondary consequence of blade thickness is that
the focal point (point of greatest intensity) is displaced
towards the collimator. This is an interesting effect as it
is in direct contradiction to the Monte Carlo analysis
previously mentioned. However there would seem
adequate physical justification for the phenomenon
according to the analysis already presented in §2.2. The
Monte Carlo analysis predicted an almost complete
translation of the resolution function away from the
collimator and hence a corresponding shift in the
centroid position of order 0.5mm in that direction. The
analytic model presented here predicts a significant shift
of the focal point and its immediate surroundings
towards the collimator of order 0.5mm for the ENGIN
and 0.1mm for the ENGINX collimators, but without
translation of the majority of the detector resolution
function. This results in a much smaller shift of the
centroid towards the collimator of order10-4mm.
4.2 The Gauge volume and shift of the centroid position
Fig. 4.8. Effect of increasing blade thickness on the detected
intensity from neutrons scattered over the entire gauge volume
using a 2mm wide incident beam slit.
It is apparent that increasing blade thickness
decreases the width of channels, so reduces detected
intensity from all points. What is particularly notable is
the strong linearity between the blade thickness and the
transmitted intensity from the geometrical centre. This
has been previously observed by performing a Monte
Carlo simulation of the collimator [8]. Attention was
drawn at that time to the similarity between the
transmitted intensity from the geometrical centre and the
ratio between the front end width of a channel to the
blade thickness. Analysis of the collimator presented
here has shown this to be no coincidence and a proof that
the relative transmission from the geometrical centre as a
function of blade thickness obeys the equation,
I (t ) = 1 −
Fig. 4.9. Gauge volume for the ENGINX collimator R=450mm
with a 2mm wide incident slit. The highlighted area is the
common rectangular approximation.
t
b R − lg αn
The SRF is described by the intersection of the
incident beam and detector resolution functions and is
illustrated in Fig. 4.9 for the R=450mm ENGINX
collimator with a 2mm wide incident beam slit. From
this the shape of the gauge volume can be obtained as the
area containing 68% of the probability density of the
SRF. Although the shape of the SRF appears quite
complex in Fig. 4.9, the gauge volume, as defined, is
excellently approximated to the rectangular area
superimposed on the contour plot as most of the
probability density is located within this region. It should
be noted that the dimensions of the gauge volume are
highly dependent on the collimator used. For the
R=840mm collimator the width in the y direction at x=0
is approximately 8 times greater than that of the
R=450mm collimator.
(22)
where t is the blade thickness and (R-l)α/n is the width of
a channel at the front end, is presented in the appendix
§A.2. Fig. 4.8 demonstrates that this linearity is only true
for the geometrical centre, although it may suffice as a
When near surface measurements are made so that
the gauge volume is incompletely filled, or attenuation
effects are strong, the zero position of the sample
positioner table does not correspond to the location of
the ncog. It is necessary to be able to relate the positioner
10
table reading to the location of the ncog in order to
determine the point at which the strain field is sampled.
Fig. 4.10 shows the effect attenuation has on the
relationship between the distance of the ncog from the
surface of the sample and the positioner table reading for
the ENGIN collimator (0 is taken as the sample surface
passing through the centre of the gauge volume). Fig.
4.11 shows how this phenomenon varies between
ENGINX collimators.
4.3 Pseudo-strain calculation
The pseudo-strain as a function of the distance of
the ncog from the surface of the sample for both the
centroid and integration methods is plotted in Fig. 4.12
for the ENGIN instrument. The graph shows 3 levels of
attenuation, µ=0mm-1 (a good approximation to Al:
µ=0.087mm-1), µ=0.118mm-1 (Fe), and µ=0.5mm-1. This
plot is a constructive test of the model as it can be
compared against the experimental data obtained while
the ENGIN instrument was in operation. This test is
particularly important as no data is yet available for the
ENGINX instrument. Fig. 4.13 shows a comparison
between ENGINX collimators assuming an attenuation
level equivalent to Steel (µ=0.118mm-1) for both
methods of calculation. Both graphs are plotted using a
2mm wide incident beam with negligible angular
divergence.
Fig. 4.10. The relationship between the distance of the ncog
from the sample surface to the positioner table reading,
illustrated for the ENGIN collimator with a 2mm wide
collimating slit, showing 3 levels of attenuation.
Fig. 4.12. Pseudo-strain as a function of distance of ncog from
sample surface for the ENGIN collimator.
Fig. 4.11. Comparison between the distance of the ncog from
the sample surface to the positioner table reading for the
ENGINX collimators. Values shown are for Steel
(µ=0.118mm-1).
It can be seen that even where attenuation is zero, if
the gauge volume is only partially filled then there is
significant deviation of the location of the ncog from the
positioner reading. Attenuation reinforces this fact,
introducing a systematic error of 0.06mm for steel using
the R=450mm ENGINX collimator and 0.5mm using the
R=840mm collimator when the gauge volume is
completely filled.
Fig. 4.13. Comparison of the pseudo-strain as a function of
distance of ncog from sample surface for the ENGINX
collimators.
It is clear that the magnitude of the pseudo-strain is
strongly dependent on the dimensions of the gauge
volume and hence collimator used. For the ENGIN
instrument strain resolution was ±50x10-6 (±50µε) and
needed correction when the gauge volume was
incompletely filled. Considering the anticipated
11
improvement in resolution of ENGINX then from Fig.
4.13 it is obvious that pseudo-strain correction is
essential, particularly for the collimators with larger front
end radii.
5. Discussion
zz
SRF( x , y )dxdy
Gauge
Volume
Fortran 90 subroutines have been written allowing
evaluation of the incident beam and detector resolution
functions for slits and radial collimators respectively.
This then permits calculation of the spatial resolution
function as the product of the above functions. The
spatial resolution function is of great importance when
undertaking strain measurement using neutron
diffraction for three reasons:
1. When designing collimators it allows a direct
comparison of performance both in terms of spatial
resolution and total transmitted intensity. Spatial
resolution contributes to ultimate d-spacing
resolution and hence the strain resolution of the
instrument, while total transmitted intensity dictates
experimental duration. The detector resolution
function program permits the study and optimisation
of collimator design.
2.
neutrons. The 2-dimensional equivalent, relevant to
the model presented above, is defined by equation
(25).
(25)
Fig. 5.1 shows the relative total detected intensity as
a function of rear end radius, R, with data points
corresponding to the 4 ENGINX collimators. It is
interesting to note that this relationship is almost
perfectly quadratic in nature, intersecting the x axis at
the value corresponding to the blade length, 350mm,
such that the front end radius would be zero. The
relationship would suggest that the time required to
perform an experiment, as a function of R, is
inversely proportional to this quadratic expression.
When planning an experiment, the SRF enables
prediction of the d-spacing resolution and minimum
experiment duration.
The resolution of a diffractometer, ∆d/d, can be
calculated from equation (5), assuming all variables
act independently [9],
∆d
=
d
LMFG ∆τ IJ + bcot θ∆θ g + FG ∆L IJ OP
H L K PQ
MNH τ K
2
2
2
Fig. 5.1. The relationship between rear end radius, R, and the
total detected intensity from the gauge volume defined using a
2mm wide slit and each of the ENGINX collimators.
1
2
(23)
where the contribution to ∆τ is from the moderation
time of the neutron and the contribution to ∆θ is from
the angular width of a detector as observed from the
scattering point, but ∆L is determined by the
dimensions of the gauge volume for a given detector.
As SRF(r)δV represents the probability of an incident
neutron being diffracted and detected from the
volume δV located at r (as in equation (9)) then
equation (24) represents the total probability that an
incident neutron is scattered from the gauge volume
and detected at the far end of the radial collimator.
zzz
SRF(r )dV
Gauge
Volume
(24)
This is clearly proportional to the total detected
intensity, which will be inversely related to the time
required to detect a statistically significant number of
The study of total transmitted intensity brings to
attention the need for further consideration of the
collimator parameters. It is apparent that there are
three geometrical variables that directly influence the
acceptance angle of a channel – the front and rear
radii, Rf & R respectively, and the blade length, l. It
is also apparent that only two of these variables are
independent due to the relationship,
Rf = R − l
(26)
It is then necessary to determine which two of the
three variables best characterise the response of a
channel. It would seem most appropriate to use Rf
and R as it is these values which directly locate the
limiting points of a channel, defining the four
collimator function domains. Although a full analysis
of the effect of varying collimator parameters on the
spatial resolution function is not presented here, were
it to be performed it would seem most appropriate to
proceed in terms of R, Rf, n, α and t.
12
3.
On completion of a measurement the SRF allows the
evaluation and correction of pseudo-strain effects due
to attenuation or partial filling of the gauge volume.
Two methods of evaluating the pseudo-strain have
been investigated and tested against experimental
data obtained for the ENGIN instrument. While not
perfectly consistent, both methods have been shown
to produce results that are accurate to within the
instrumental resolution and the difference in
calculation time has been found to be slight, both
allowing real-time correction. No data is yet available
for the ENGINX instrument, but it is essential that
the model is tested against new experimental data
once the instrument is operational. The improved
resolution of ENGINX may determine which method
of evaluating pseudo-strain most accurately
represents the phenomenon.
would substantially increase computation time, but
comparison of 3-dimensional theoretical data to
experiment would prove conclusively whether or not
a 2-dimensional model is really sufficient for realtime correction.
6. Conclusions
In this paper a model has been developed of the
ENGIN/ENGINX strain scanning systems. This model
permits evaluation of the spatial resolution function
using a Fortran 90 program. The spatial resolution
function has three direct applications to neutron strain
scanning:
•
There are several logical extensions to this project
that should be considered:
•
•
•
•
The first is to develop the computer program to allow
for other than simply-connected polygons so that
hollow specimens may be accommodated. This
should be straightforwardly achieved (see appendix
§A.3).
The second development is to account for pseudostrain introduced by using a white beam of neutrons.
Both the incident intensity and the attenuation
coefficients are a function of wavelength. As the
pseudo-strain contribution from a channel is
weighted by the received intensity, variations in
detected intensity due to wavelength will affect the
pseudo-strain calculated. A study of this for the
ENGIN instrument revealed that any pseudo-strains
introduced were smaller than the experimental error,
but the improved resolution of ENGINX may make
this correction necessary.
A third logical development would be to conduct the
study of the effect of collimator parameters on the
SRF that has been made possible by the creation of
the Fortran subroutine.
The final development would be to extend the model
presented above to accurately represent the system in
3 dimensions. Incorporation of the z coordinate
•
•
Equipment design – the direct comparison of the
performance of radial collimator designs, both in
terms of resolution and transmitted intensity, and the
study of the effect of incident beam properties in
terms of beam width and angular divergence.
Experimental planning – the prediction of
measurement resolution and minimum experimental
duration.
Correction of pseudo-strains – the evaluation and
correction of pseudo-strain effects due to attenuation
and incomplete filling of the gauge volume.
Two methods of pseudo-strain calculation have
been presented and compared to available experimental
data, both proving to be accurate to within experimental
error. The need to compare both methods to data obtained
from the ENGINX instrument, once operational, has been
emphasised. Whichever method is determined to most
accurately represent the phenomenon of pseudo-strain, the
speed of calculation is such that the Fortran 90
subroutines will be fully integrated into ENGINX analysis
software allowing real-time pseudo-strain correction.
Acknowledgements
The authors wish to thank M.R. Daymond for his
time and helpful suggestions in the development of the
models and description of pseudo-strain effects.
13
Appendix
A.1 Collimator function
N .B .
R E G IO N 1:
d =
t
F α IJ
2 s in G
H 2n K
&
t
R'= R −
2 ta n
FG α IJ
H 2n K
D o m a in :
g FGH 2αn IJK < y < b x − d g ta n FGH 2αn IJK
F α IJ + d
d < x ≤ b R '− l g c o sG
H 2n K
F R ' s in FG α IJ − y I
F R ' s in FG α IJ + y I
G
J
G
JJ
H
H 2n K
2n K
Ψ = a ta n G
+ a ta n G
J
α
α
GG R ' c o s FGH IJK − x + d JJ
GG R ' c o s GFH IJK − x + d JJ
2n
2n
H
K
H
K
b
− x − d ta n
(A 1 )
R E G IO N 2:
D o m a in :
F α IJ
ta n G
α I
F
b x − d g ta n GH 2 n JK ≤ y ≤ Hl 2 n K FGH − x b 2 R ' − l g + 2 R ' b R ' − l g c o s FGH 2αn IJK + d b 2 R ' − l gIJK
F α IJ + d
d ≤ x ≤ b R '− l g c o sG
H 2n K
A ls o :
F α IJ ≤
− b x − d g ta n G
H 2n K
ta n
y ≤
FG α IJ
H 2 n K FG − x b 2 R ' − l g + 2 R ' b R ' − l g c o s FG α IJ + d b 2 R ' − l g IJ
H 2n K
l
H
K
x < d
Ψ
F b R ' − l g s in FG α IJ − y I
F R ' s in FG α IJ + y I
G
G
J
JJ
H
H 2n K
2n K
+ a ta n G
= a ta n G
J
GG b R ' − l g c o s FGH α IJK − x + d JJ
GG R ' c o s FGH α IJK − x + d JJ
2n
2n
H
H
K
K
(A 2 )
R E G IO N 3:
D o m a in :
FG α IJ
H 2 n K FG x b 2 R ' − l g + 2 R ' b R ' − l g c o s FG α IJ − d b 2 R ' − l gIJ ≤
H 2n K
l
H
K
F α IJ + d
d ≤ x ≤ b R '− l g c o sG
H 2n K
ta n
b
g FGH 2αn IJK
y ≤ − x − d ta n
A ls o :
ta n
FG α IJ
H 2 n K FG x b 2 R ' − l g + 2 R ' b R ' − l g c o s FG α IJ − d b 2 R ' − l gIJ ≤
H 2n K
l
H
K
y ≤
b x − d g ta n FGH 2αn IJK
x < d
Ψ
F R ' s in FG α IJ − y I
F b R ' − l g s in FG α IJ + y I
G
J
G
JJ
H
H 2n K
2n K
+ a ta n G
= a ta n G
J
α
α
F
I
F
I
GG R ' c o s GH JK − x + d JJ
GG b R ' − l g c o s GH JK − x + d JJ
2n
2n
H
K
H
K
R E G IO N 4:
D o m a in :
b x − d g ta n FGH 2αn IJK <
b
(A 3 )
g FGH 2αn IJK
y < − x − d ta n
x < d
Ψ
F b R ' − l g s in FG α IJ − y I
F b R ' − l g s in FG α IJ + y I
G
G
J
JJ
H
H 2n K
2n K
+ a ta n G
= a ta n G
J
GG b R ' − l g c o s FGH α IJK − x + d JJ
GG b R ' − l g c o s FGH α IJK − x + d JJ
2n
2n
H
H
K
K
(A 4 )
14
A.2 The effect of blade thickness on the transmitted
intensity from the geometrical centre.
The geometrical centre is unique as it is the point of
rotation about which the collimator is constructed. It is
therefore the only point invariant under rotation making
it identical for all channels. This allows the detector
resolution function evaluated at (0,0) to be expressed,
b g
Pd 0,0 = nΨ(0,0)
(A5)
It is also apparent from Fig. 2.4 in §2.2 that the effect of
increasing blade thickness is to translate region 4
towards the collimator. As a consequence Ψ(0,0) is
always described by the collimator function for region 4,
F b R'−lg sinFG α IJ I
G
H 2n K JJ
P b0,0g = 2n atanG
GG b R'−lg cosFGH α IJK + d JJ
2n
H
K
d
(A6)
Substituting,
d=
t
FαI
2 sinG J
H 2n K
& R' = R −
t
2 tan
FG α IJ
H 2n K
F F
I
I
GG G R − t − lJ sinF α I JJ
GG GGG 2 tanFG α IJ JJJ GH 2nJK JJ
H 2nK K
H
P b0,0g = 2n atanG
JJ
I
GG FG
GG GG R − tF α I − lJJJ cosFGH 2αnIJK + Ft α I JJJ
2 sinG J J
GH GH 2 tanGH 2nJK JK
H 2nK K
d
(A7)
The typical angular width of a channel is (for ENGINX)
32/160= 0.2o, so using the small angle approximation,
F F
I
I
GG G R − t − lJ F α I JJ
GG GGG 2FG α IJ JJJ GH 2n JK JJ
H H 2n K K
P b0,0g = 2nG
JJ
I
GG FG
GG GG R − F tα I − lJJJ + F tα I JJJ
GH GH 2GH 2n JK JK 2GH 2n JK JK
d
which on simplifying becomes,
(A8)
b g FGH b R −ntlgα IJK
Pd 0,0 = α 1 −
(A9)
After normalisation by the acceptance angle of the empty
collimator viewed from the origin (α) and slight
rearrangement, this is expressed by,
b g
Pd 0,0 = 1 −
t
b R − lg αn
(A10)
where the second term is the ratio of the blade thickness
to the channel width at the front end.
A.3 The Path Length Problem
In applying the effect of attenuation of neutrons on
the detected intensity it is essential to know the path
length of a neutron through the material both in the
source to scattering point and scattering point to
detection point path segments. The source to scattering
point path length, ps, will obviously vary from point to
point in the gauge volume and the scattering point to
detector path length, pd, will vary, not only with location
in the gauge volume, but also with location of the
detection point. This will be dependent on the channel
through which the neutron passes. Most importantly the
path lengths depend on the location of the sample with
respect to the gauge volume. This would initially seem
quite a complex problem to solve, but separate
consideration of the two path segments for a given
detector reveals that each path segment can effectively
be regarded as the same problem. The problem is defined
by the following:
given the coordinates of the vertices of a polygon
representing a sample Ri = (xi,yi), {i=1,…,nvertices},
a source point outside, Rs, and an observation point
inside the polygon, Ro, to calculate p, the length of
the segment of the straight line joining Rs and Ro
located inside the polygon.
In order to achieve this the order in which the
vertices of the polygon are connected must be known so
that the sample surfaces are formed correctly. It was
decided to label the vertices with increasing subscript in
the clockwise direction.
An edge of the polygon will only contribute to the
path length if the line joining Rs and Ro intersects it.
Criteria must be developed in order to determine whether
intersection occurs for each edge. Defining ΦR, Φn and
Φn-1 in the region [–π, π] relative to the line joining Rs to
the origin, as in Fig. A.1, it can be seen that intersection
only occurs between the line and the nth edge if:
15
φ n ≤ φ R < φ n −1
(A11)
&
y − y n −1
y x − y n x n −1
y0 > n
x 0 + n −1 n
x n − x n −1
x n − x n −1
FG
H
IJ FG
K H
= mn x0 + cn
IJ
K
(A12)
where yn =mnx + cn is the equation of the nth edge and
the “equal to” part of the ≤ sign in the angular criterion is
to accommodate intersection through a vertex.
Else if y s > cn then intersection occurs providing
y0 < mn x0 + cn
(A15)
If intersection occurs then the intersection point can
be obtained simply by solution of simultaneous
equations and the path length calculated using
Pythagoras’ theorem. At this point it is important to
notice a limitation to the model imposed by introducing
the angular criterion. The angles Φi and ΦR must be
defined over some interval relative to an axis. For
convenience the axis has been chosen to be the y axis in
the positive direction from Rs to ∞ (after rotation), and
the interval to be [–π, π]. This introduces a 2π
discontinuity in Φ along the y axis in the negative
direction from ys to -∞. If a sample edge traverses this
discontinuity then the angular criterion as defined above
will not correctly apply. The physical consequence of
this limitation is that the path length will not be correctly
calculated for any line joining Rs and R0 for which there
is no unobstructed line of sight from a point at infinity.
This effect is illustrated in Fig. A.2. In application of this
model to the problem of calculating ps and pd it can be
seen that this limitation will not affect the desired results.
The position of Rs represents a point on the collimating
slit and a point on a detector in the calculation of ps and
pd respectively, both of which are distant points in
comparison to the dimensions of the sample, ensuring
that there is always an unobstructed line of sight from
infinity.
Fig. A.1. Demonstrating the criteria for intersection of an edge
by the path.
However this is only the case in Fig. A.1 as ys < yn(xs)
for all n which may be intersected. It is obvious that if Rs
had been located above the specimen, i.e. ys > yn(xs) for
all n which may be intersected, then the second condition
for intersection would have been,
y0 < mn x0 + cn
(A13)
Thus the second intersection criterion is dependent
on the relative positions of Rs and the two vertices
forming an edge. The second criterion (and evaluation of
the first) can be somewhat simplified by initially rotating
the entire system about the origin by the amount required
to bring the line joining Rs to the origin collinear with the
negative y axis. From this point it is assumed that Ri, Rs
and R0 refer to these rotated coordinates. The second
criterion becomes:
If y s < cn then intersection occurs providing
y0 > mn x0 + cn
(A14)
Fig. A.2. Illustration of the limitations of the solution to the
path length problem.
This limitation would not be an issue if all samples
could be represented by simply connected convex
polygons, as in that case no edges would be able to
traverse the discontinuity. However, this is not the case
as samples often contain concave regions and may be
doubly or triply connected to accommodate holes. At
present the model outlined here accommodates only
simply connected convex and concave polygons.
16
When a polygon is entirely convex then the line
joining Rs and Ro may intersect only one edge or vertex
and the nature of that intersection must be in the positive
sense (i.e. into the sample). If concave regions are
present then it is possible that the line may intersect
multiple edges in both the positive (entering sample) and
negative senses (exiting sample), depending on the
orientation of the sample and the location of Rs and Ro. It
is necessary to adjust the criteria that determine whether
intersection occurs to also determine the sense of the
intersection. This is simply achieved by reconsidering
the angular criterion. It can be seen that for the case in
Fig. A.3. the criteria:
φ n −1 ≤ φ R < φ n
(A16)
&
y0 > mn x0 + cn
(A17)
achieved by repeating the path length algorithm for
additional polygons, representing the holes, that have
vertices labelled with increasing subscript in the
anticlockwise direction. The total path will then be the
sum of the outputs from the iterations due to each
polygon.
determine a negative contribution to the path length.
Fig. A.4. Flow diagram for determining the contribution of a
sample edge to the path length.
References
[1]
[2]
[3]
[4]
Fig. A.3. Demonstrating the criteria for negative intersection of
an edge by the path.
A complete set of outcomes of the two criteria for
different locations of Rs and R0 in relation to the nth edge
described by the straight line, yn =mnx + cn , can be
summarised by the flow diagram in Fig. A.4 (although
slight mathematical adjustment is required to
accommodate vertical edges), and it is this flow diagram,
along with evaluation of the intersection points and then
path segments contributed by an intersected edge, which
provides a solution to the path length problem.
At present the Fortran program that evaluates path
lengths can only accommodate simply connected
polygons. However, it would not be dificult to extend
this to accommodate holes in the sample. This could be
[5]
[6]
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