J Nondestruct Eval (2009) 28: 149–155

DOI 10.1007/s10921-009-0057-5

Techniques for Neutron Stress Determination with High Spatial

Resolution

Thomas Gnäupel-Herold

Published online: 13 November 2009

© Springer Science+Business Media, LLC 2009

Abstract A number of sub-surface strain measurement

problems require spatial resolutions of the order of 10−1 mm

to be meaningful. The use of neutron diffraction can be

both time effective and otherwise advantageous compared to

synchrotron diffraction if measurement directions and beam

shapes are chosen such that scattered intensity is maximized.

The first point refers to those directions and specimen orientations for which the gage volume orientation can be kept

constant. They can be chosen such that preferred orientation boosts the diffracted intensity. The second point refers

to the shaping of the incident beam for maximizing the illuminated volume in the material even for curved specimens.

Guidelines and limitations of such high spatial resolution

measurements are discussed.

Keywords Neutron diffraction · Sub-millimeter spatial

resolution · Shaped aperture

1 Introduction

Many engineering problems require the non-destructive

measurement of sub-surface strain fields that change significantly within millimeter length scales. For example,

through-thickness residual stresses pose a big problem in

sheet metal forming. With sheet thicknesses of ≈1 mm a

depth resolution of ≈0.1 mm is required in order to reflect

T. Gnäupel-Herold

University of Maryland, College Park, MD 20742, USA

T. Gnäupel-Herold ()

NIST Center for Neuron Research, Gaithersburg,

MD 20899-6102, USA

e-mail: tg-h@nist.gov

the strain gradient with sufficient detail. Because of primary beam flux and/or sufficiently low attenuation, neutron

diffraction and synchrotron based high-energy X-ray diffraction (HEXRD) are essentially the only non-destructive

methods capable of providing that resolution. Due to speediness of measurements and spatial resolution unmatched by

neutron diffraction HEXRD represents the predominantly

used experimental tool for the evaluation of strain gradients

up to the scale of 1 mm [1–4]. The main problem associated with HEXRD is that Bragg angles of usable reflections

become small—of the order of 10◦—which leads to a highaspect ratio intersection of incident beam and reflected beam

(gage volume) with lengths up to 1 mm and a corresponding loss in spatial resolution in this direction. This problem

is exacerbated for high-Z materials that require photon energies up to 100 keV and more to achieve sufficient depth

penetration. Another problem is the size of the gage volume

which is of the order of 10−2 mm3 [2]. The actual volume

of grains that fulfill the diffraction condition and scatter into

the detector is much smaller (of the order of 10−6 mm3), and

this has implications both for continuum-based calculations

of quantities such as elastic constants, and for the measured

d-spacing. For example, due to insufficient elastic constant

averaging stress fluctuations of ±5% from the average can

be expected for ferritic steel if the number of contributing

grains is <100 [5]. Regarding the d-spacing one has to expect deviations not related to stress from one measurement

location to the next because the center of gravity of the contributing grains within the gage volume changes.

These considerations together with source availability—

only 3rd generation synchrotron sources offer the necessary flux and photon energy range—make neutron diffraction an attractive technique for depth-resolved strain scanning. Here, the limitations are of a different nature. Primarily stemming from the low primary beam flux, gage volumes
150 J Nondestruct Eval (2009) 28: 149–155

are typically >1 mm3 which implies that the spatial resolution is available only in one or two dimensions [6], thus

limiting the range of materials problems suitable for study

with neutron diffraction. It should be mentioned that some

efforts were made to increase the neutron flux at the sample position through the use of neutron lenses based on total

reflection in bent, hollow fibers [7, 8]. However, due to several factors such as insufficient gains in the focal point, large

beam divergence (affecting the peak width), and high manufacturing cost such a device has not yet been used for depth

resolved strain measurements with neutrons.

As outlined in the following, for a problem suitable for

study with neutron diffraction the experimental conditions

can be chosen such that, using simple aperture systems,

depth resolutions are ≈0.1 mm at data collection times of

the order of minutes per orientation and location.

2 Experimental Technique

The practice of using flat gage volumes in sub-millimeter

spatial resolution strain measurements is quite common at

neutron sources. The incident and reflected beam apertures

define a gage volume of the shape of a wide but thin parallelepiped, and it is used to scan the strains caused by a plane

stress field that varies with depth. The extension of this technique to more complex specimen shapes and stress fields is

outlined in Fig. 1.

The overriding goal of the method is to produce a high

spatial resolution in one specimen direction while providing a gage volume of several mm3 for sufficient scattering intensity. The orientation of the gage volume with respect to the individual specimen orientations is constant.

It is worth noting that the constant gage volume orientation has an equivalent in (surface) X-ray diffraction where

the beam spot is kept constant in orientation and size [9].

The measurement procedure consists of a set of individual

through-thickness strain scans where each scattering vector

orientation (φ,ψ) is produced by means of a combination

of specimen orientation, aperture orientation (rotation about

the beam axis), and a different Bragg angle θ (produced by a

different wavelength). The ability to reproduce a wavelength

with sufficient accuracy is lacking on many neutron instruments; hence it is necessary to measure a reference samFig. 1 The combination of rotations of the specimen, rotation of the

aperture (primary) and different 2θ -values (through different wavelengths) result in different orientations of the scattering vector Q. The

picture can be understood as the incident beam rotating together with

the sample to produce different orientations while maintaining both the

spatial resolution (in the depth direction Z) and the depth coordinate

(Z) of the gage volume
J Nondestruct Eval (2009) 28: 149–155 151

Fig. 2 Upstream (towards monochromator) and downstream (near

sample) parts of primary aperture systems with cadmium used as a

neutron absorber. The front end part produces a specific beam shape

(see the curved aperture on the left) while the upstream part (towards

the monochromator) limits the divergence. The combination of aperture openings, their distance to each other, and the specimen distance

determine the effective spatial resolution

Fig. 3 Stepwise translation of a curved specimen into the gage volume (entering curve). The uncertainties are of the order of the size of the

symbols

ple or unstressed d0-sample at each aperture orientation and

each wavelength. The requirements for measurement uncertainties are significantly higher for reference measurements;

hence, this will add significantly to the total data acquisition time. The incident beam is subject to attenuation; thus,

the location of the gage volume should be as close as possible to the face through which the beam enters the sample.

The guiding principle should be to avoid the disturbances of

the stress field near the edge. For example, for sheets with

1 mm thickness it is deemed sufficient for the outer edge of

the gage volume to have a distance of 3× sheet-thickness

(3 mm) from the edge of the specimen. It is also to be expected that attenuation causes the center of scattering to be

shifted from the center of the gage volume; this requires that

the reference sample reproduces the attenuation conditions

of the specimen (same material, same aperture distance and

gage volume location).

Another feature shown in Fig. 1 is that the primary beam

enters the specimen in a direction of symmetry for the specimen. This way it is possible to shape the primary beam such

that it follows the outline of the specimen surface. In practice this is done by cutting thin cadmium-sheet along the

specimen shape using razor blades. This procedure allows

it to produce almost arbitrary beam shapes. A selection of

apertures is shown in Fig. 2.

The alignment of the slit with respect to the specimen

surface (curved or flat) is done visually. The main purpose

of the shaped slits is the increase of the gage volume without

the loss of spatial resolution that would otherwise occur for

curved specimens. With good alignment of the beam tilt and

beam axis to the specimen tilt and specimen axis it is possible to achieve a spatial resolution <0.20 mm as demonstrated by the entering curve in Fig. 3. A data acquisition

time of 1000 s was necessary to achieve a strain accuracy of

<10−4 in the configuration shown in Fig. 3. The intensity

of the beam diffracted from the gage volume as the specimen enters the gage volume defines a function that is fitted

to the integrated intensities of the Bragg peak at each translation step [10]. The parameters of this function are the surface position, the Bragg angle (fixed), the attenuation (for a
152 J Nondestruct Eval (2009) 28: 149–155

Fig. 4 Measurement configuration and orientation of the scattering

vectors in the (211) pole figure of a AISI 1010 ferritic steel (left). The

axial (X) distance of the gage volume from the sample surface should

be as small as possible; here, the closest edge of the gage volume had

3 mm (3× thickness) distance to the face of the sample

geometry in which the scattering vector is not parallel to the

sample surface) and the effective slit width. The effective

slit width defines the effective size of the gage volume and

it gives a measure of the spatial resolution.

In order to achieve the resolution shown in Fig. 3 it is

necessary both to limit the beam divergence (here: 0.1◦) by

means of additional upstream slits and a minimal distance

between the primary slit and the gage volume location in the

specimen (here: 10 mm from edge). Thus, the primary slit

is usually as close as possible to the specimen (here: 1 mm).

There are no such strict requirements for the diffracted beam

slit. The opening is typically chosen to be equal to the larger

opening of the primary slit and the slit distance should be as

close to the specimen as possible.

Further increases of the scattered intensity can be achieved

by taking advantage of preferred crystallite orientation

through measuring in directions of the intensity poles. This

is outlined schematically in Fig. 4 for the (211) reflection

on a ferritic steel sheet metal. The specimen had undergone

rolling and stretch-bending resulting in a roughly cylindrical shape. The corresponding pole figure is shown on the

left side.

Different orientations of the scattering vector Q are

achieved by combinations of different specimen orientation

and Bragg angles through different wavelengths. The incident beam is aligned such that it is parallel to the surface

in the direction of the cylinder axis and, defined by the slits

shown in Fig. 2, the beam has the same curvature as the

specimen. For most neutron diffraction stress diffractometers the angle 2θ is varied (detector rotation) in the horizontal plane which is defined in Fig. 4 by the incident beam and

the directions (1) and (2). The directions (1) and (2) of the

diffracted beams can be measured in the same specimen orientation shown in Fig. 4. The direction (3) requires that both

the specimen and the primary aperture are rotated together

(see Fig. 1, right).

This approach requires the ability to change the wavelength of the neutron beam such that the neutron beam is

diffracted at different Bragg angles θ with the goal of matching the direction of the scattering vector Q to the direction of increased intensity in the pole figure. Changing the

Q-direction through the wavelength comes with one drawback: on some neutron diffractometers the instrument parameters do not allow the determination of the wavelength

with sufficient accuracy, thus requiring a calibration for each

change of the wavelength or the measurement of the specimen reference (unstressed) d-spacing. Both require comparable amounts of measurement time. Additional time is

needed to establish the position of the surface for each orientation. The initial position accuracy from optical alignment

is <0.5 mm; however, in order to achieve an accuracy better
J Nondestruct Eval (2009) 28: 149–155 153

than the spatial resolution it is typically necessary to perform

a wall scan (entering curve) [10].

The range in the space of Q-orientations is reduced by

the constraints posed by usable Bragg angles (typically 30◦

to 70◦), by the directions of intensity poles, and by the requirements of the specimen symmetry and shape. For example, for optimal spatial resolution in the thickness direction a cylindrically curved specimen with a curved neutron

beam as shown in Fig. 4 must be oriented with the cylinder axis parallel to the direction of the incident beam. If the

specimen is a flat sheet then there is greater flexibility in selecting measurement directions; for example, both the principal in-plane directions X and Y are accessible. However,

regardless of the specimen shape the perpendicular direction Z cannot be measured since this would require a very

small Bragg angle outside of the range that can be produced

by wavelength changes. It should be noted also that since

the azimuth φ and tilt ψ (see Fig. 1 for definition) are generally non-zero the data analysis excludes simple sin2ψ linear

regression [13] for stress determination and it is required to

fit all components of the stress tensor [14] or to make explicit assumptions about their magnitude. For example, thin

sheets with ≈1 mm thickness usually justify setting the normal stress component to zero.

3 Examples

3.1 Side Panel of a U-Channel

The experimental technique outlined above was successfully used to determine the trough-thickness stress distribution in deep drawn U-channel sections shown in Fig. 5

with the geometry and sample shown in Fig. 3, left. The

process and background of the forming process is described

in [11, 12].

The stress gradients are resolved in good detail which,

together with the surface stresses determined by X-ray diffraction, indicates a spatial resolution that is sufficient to indicate the character of the gradients present in the sheets.

Figure 5 demonstrates that neutron diffraction can be successfully used for the non-destructive investigation of subsurface strains for a class of materials that previously were

the domain of destructive methods (X-ray diffraction with

layer removal) or HEXRD. These materials include especially steels, both ferritic and stainless/austenitic and nickel

based alloys. Compared to these materials aluminum alloys

have a small coherent neutron scattering length resulting in

reduced scattered neutron intensity while exhibiting a much

reduced X-ray absorption, thus making depth gradients of

strains in aluminum and its alloys more suitable for investigation with HEXRD [3].

3.2 Laser-Peened Steel Plate

Laser peening generates very high stresses in less steep gradients than peening through mechanical impact; thus it is

well suited for a comparative study of the method. The specimen was a 300M steel, with dimensions 42 mm × 47 mm

× 13 mm. For this sample, two different measurements were

taken: one measurement (1) followed the procedure outlined

here with the flat (6 mm × 6 mm × 0.2 mm, 8 mm from both

edges) gage volume having a constant orientation with respect to the specimen surface. The second (2) measurement

followed the established procedure of a neutron diffraction

stress measurement in which a matchstick-like gage volume

(0.5 mm × 0.5 mm × 20 mm) was used. All measurements

Fig. 5 Through-thickness stress (b) in the side panel of a deep drawn U-channel in the tangential direction (a). The values directly at the surface

(0 mm, 1.0 mm) were determined by X-ray diffraction
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Fig. 6 D-spacings in two in-plane directions for method (1) and (2). Specimen and gage volume orientations are shown on the right

were referenced to the d-spacing measured on iron powder.

No unstressed sample or coupon was available, therefore, a

d0 was used from a calculated value for the location closest to the surface where σzz = 0. Specimen orientations and

d-spacings measured in comparable orientations are shown

in Fig. 6. It was confirmed using surface X-ray diffraction

that σxx ∼= σyy for the entire surface including the measurement locations used here. Also, at locations with distances

≤3 mm from an edge the surface stresses start to decrease.

The location for (1) was safely outside that zone (8 mm distance).

The use of the geometry in 1b) (Fig. 6, right) is certainly

not new but using the same primary beam, only rotated, as in

geometry 1a) has not been reported to the best knowledge of

the author. Geometries 1a) and 1b) together give a constant

orientation of the gage volume within the sample, and with

constant spatial resolution.

The near surface differences in d-spacings in Fig. 6 can

be interpreted as the result of a smoothing effect of a gage

volume too large (0.5 mm × 0.5 mm × 20 mm) to capture the detail of the strain gradient. That is reflected in the

stresses as well (Fig. 7). The measurement with the large

gage volume underestimates the peak stresses by almost

300 MPa!

It should be noted that the gage volumes were nearly the

same ((1): 7.2 mm3, (2): 5 mm3) but the neutron paths differed. Thus, counting times varied for (1) from 20 min at

minimum attenuation to 90 min at maximum attenuation.

For (2), the spread was 40 min and 80 min for minimum and

maximum attenuation.

Fig. 7 In-plane stresses for the laser peened sample

4 Conclusions

It was demonstrated that, through adaptation of the shape

of the primary beam, through measurement in directions of

raised intensity (if preferred orientation exists), and through

maintaining the orientation of the gage volume within the

sample for all specimen orientations the effective spatial

resolution of neutron diffraction strain measurements can

be improved down to the order of 0.1 mm. This can be

done at data acquisition times <1 h even for medium power

(20 MW) neutron sources. The method is particularly well

suited for stresses that are homogenous in a plane (sheet

metal), and in cases where not large distances to edges and

surfaces are required. The demonstrated gain in spatial resolution is to some extent paid for with the need for additional
J Nondestruct Eval (2009) 28: 149–155 155

(and time consuming) reference d-spacing measurements. It

is also often necessary to re-measure the surface position after each change in specimen orientation.

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