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An energy-dispersive technique to measure x-ray coherent scattering form factors of

amorphous materials

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2010 Phys. Med. Biol. 55 855

(http://iopscience.iop.org/0031-9155/55/3/020)

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IOP PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 55 (2010) 855–871 doi:10.1088/0031-9155/55/3/020

An energy-dispersive technique to measure x-raycoherent scattering form factors of amorphousmaterials

B W King1 and P C Johns1,2

1 Ottawa Medical Physics Institute and Department of Physics, Carleton University,1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6, Canada2 Department of Radiology, University of Ottawa, Canada

E-mail: Brian.King@newcastle.edu.au

Received 1 September 2009, in final form 22 November 2009Published 14 January 2010Online at stacks.iop.org/PMB/55/855

AbstractThe material-dependent x-ray scattering properties of amorphous substancessuch as tissues and phantom materials used in imaging are determined bytheir scattering form factors, measured as a function of the momentum transferargument, x. Incoherent scattering form factors, Finc, are calculable for allvalues of x while coherent scattering form factors, Fcoh, cannot be calculatedexcept at large x because of their dependence on long-range order. As aresult, measuring Fcoh is very important to the developing field of x-ray scatterimaging. Previous measurements of Fcoh, based on crystallographic techniques,have shown significant variability, as these techniques are not optimal foramorphous materials. We have developed an energy-dispersive technique thatuses a polychromatic x-ray beam and an energy-sensitive detector. We showthat Fcoh can be measured directly, with no scaling parameters, by computingthe ratio of two spectra: the first, measured at a given scattering angle andthe second, the direct transmission spectrum with no scattering. Experimentshave been constructed on this principle and used to measure Fcoh for waterand polyethylene to explore the reliability of the technique. A 121 kVp x-rayspectrum and seven different scattering angles between 1.67 and 15.09◦ wereused, resulting in a measurable range of x between 0.5 and 9.5 nm−1. Theseare the first measurements of Fcoh made without the need for a scaling factor.Resolution in x varies between 10% for small scattering angles and 2% forlarge scattering angles. Accuracy in Fcoh is shown to be strongly dependent onthe precision of the experimental geometry and varies between 5% and 15%.Comparison with previous published measurements for water shows values ofthe average absolute relative difference between 8% and 14%.

0031-9155/10/030855+17$30.00 © 2010 Institute of Physics and Engineering in Medicine Printed in the UK 855

856 B W King and P C Johns

1. Introduction

X-ray scatter imaging is a new approach to x-ray instrumentation for medicine. Several groupshave been investigating ways to make use of scattered x-radiation to extend the usefulnessof x-ray imaging beyond conventional primary beam imaging (Brateman et al 1984, Hardinget al 1987, Arendtsz and Hussein 1995, Leclair and Johns 1998, 1999, Poletti et al 2002,Davidson et al 2005, Leclair et al 2006, Van Uytven et al 2007, Griffiths et al 2007, Elshemeyand Elsharkawy 2009).

There are two types of x-ray scattering in diagnostic radiology: coherent scattering,which is the basis of x-ray diffraction, and incoherent scattering, which is the Comptoneffect with correction for electron binding. Attempts have been made to use both coherent(Harding et al 1987, Davidson et al 2005, Leclair et al 2006) and incoherent (Bratemanet al 1984, Arendtsz and Hussein 1995, Van Uytven et al 2007) scattering for differentimaging tasks, but coherent scattering has several advantages over incoherent. First, it is verystrongly forward directed, making it relatively easy to collect the scattered photons. Second,because coherently-scattered radiation interferes to give x-ray diffraction patterns, it is stronglydependent on the structural properties of the material. Since there is significant variationbetween the structural properties of different tissues, greater contrast is available using coherentscattering.

Previous work in our group has analysed the improvement in image quality possiblethrough using scattered x rays (Leclair and Johns 1998). As an example, it was shown that todistinguish between a 1 mm target of gray matter versus water, the signal-to-noise ratio can bemultiplied by about 14 by using the coherently-scattered photons as compared to the primary(Leclair and Johns 1999). This analysis was based on the available data for the scatteringproperties of the tissues in question. Specifically, the property of interest is the differentialscattering cross section per electron, deσ/d�, which can be written in terms of the coherent(Fcoh) and incoherent (Finc) scattering form factors:

deσ

d�= deσ0

d�

[F 2

coh(x) + FKN(E, θ)Finc(x)]. (1)

Here, FKN is the Klein–Nishina factor and deσ0/d� is the classical Thomson cross section forscattering from a free electron. Both of these are readily calculable and independent of targetmaterial. The incoherent scattering cross section is material dependent but can be calculatedfor any given material based on its composition using the independent atom model (IAM). Weuse the tabulated atomic values from Hubbell and Øverbø (1979).

The coherent scattering form factor is conveniently tabulated as a function of themomentum transfer argument x which combines the photon energy E and the scatteringangle θ :

x = 1

λsin

(θ

2

)= E

hcsin

(θ

2

), (2)

where λ is the photon wavelength, h is Planck’s constant and c is the speed of light.The standard method for measuring scattering form factors is to use a crystallographic

diffractometer. Previous work in our lab has shown that crystallographic techniques havedeficiencies when measuring scattering from non-crystalline materials (Johns and Wismayer2004). We are investigating alternative methods of measuring coherent scattering form factors.Previously, we have reported on a method using an image plate and polychromatic x-ray beamthat measures the form factor with low resolution in x (King and Johns 2008). Here, we detailan energy-dispersive method that gives x resolution still slightly worse than diffractometermeasurements but with improved accuracy when measuring amorphous materials (King and

An energy-dispersive technique to measure x-ray coherent scattering form factors 857

A JIGFEB D

Transmission

Scatter

C

Lst Ltd

Lt

Y

Y

Z

Figure 1. Experiment schematic showing apertures for form factor measurements in transmission(upper) and scatter (lower) configurations. All equipment except the x-ray tube (A), ion chamber(B) and spectrometer (J) with its aperture (I) are mounted on an optical bench for precise positioningand alignment. The location and dimension of each item can be found in table 1.

Johns 2009). Instead of measuring the scattering as a function of θ using monochromaticx rays, we measure the scattering at a fixed θ using polychromatic x rays and an energysensitive detector. Via equation (2), the form factor is determined as a function of x.

2. Apparatus

Figure 1 is a schematic of our experiment. The measurement of Fcoh requires knowledge ofthe transmission, as well as of the scatter spectrum. The geometry shown allows both to bemeasured without a change in the source or the detector location, by translating the targetlateral position, Y. This configuration is based on earlier work by our group (Hasan 2003,Hasan and Johns 2004). An x-ray tube is used as the source and a high purity germanium(HPGe) spectrometer measures the energy of each photon. The distance from the source tothe target plane is Lst = 69.3 cm, and from the target plane to the detector is Ltd = 75.6 cm.As shown, the rays make angles α and β with the source and detector, respectively; their sumgives the scattering angle, θ . The target is placed in a custom-made station with Pb aperturesat either end to allow measurement at seven different θ . This extends the range of x accessible.

2.1. Apertures

All apertures in the system, except aperture G in figure 1, were punched out of Pb sheet witha Di-Acro mechanical punch (Acrotech, Lake City, MN). Five punches were available. Thedimensions were measured with a travelling microscope and are given in table 2.

Aperture plane D was fabricated with a circular aperture for the transmission configurationand seven different rectangular apertures, allowing seven different scatter configurations.Apertures C and F can be translated laterally for alignment in any one of these eight geometries.This allows for independent measurements of the form factor in different x regions. Thepunches used in each configuration are shown in table 3. Narrower apertures were used for

858 B W King and P C Johns

Table 1. Location of objects in transmission and scatter configurations as shown in figure 1. TheZ position refers to the centre of each object. For more information on the transmission apertureG see section 2.1.

Object Description Z position (cm) Pb thickness (mm)

A Source 0 N/AB Ion chamber 12.8 ± 0.6 N/AC Aperture 13.8 ± 0.6 2.0 ± 0.1D Aperture 64.5 ± 0.3 3.9 ± 0.1E Target 69.3 ± 0.4 40.0 ± 0.5F Aperture 76.9 ± 0.3 3.9 ± 0.1G Aperture 129.3 ± 0.6 See textI Aperture 145.1 ± 0.3 3.9 ± 0.1J Detector 145.8 ± 0.6 N/A

Table 2. Available aperture dimensions.

Punch Shape W , width (mm) H, height (mm)

C1 Circular 1.66 ± 0.03 −→C2 Circular 3.32 ± 0.05 −→R1 Rectangular 2.08 ± 0.04 7.66 ± 0.03R2 Rectangular 3.12 ± 0.07 7.62 ± 0.06R3 Rectangular 3.09 ± 0.04 10.11 ± 0.09

Table 3. Aperture configurations used for form factor measurements. The detector aperture in thetransmission configuration is defined by a separate transmission aperture (G) as shown in figure 1;aperture I is not the limiting aperture when G is present. The dimensions of the apertures can befound in table 2.

Configuration Aperture C Aperture D Aperture F Aperture I

Transmission C2 C1 R1 See textScatter 1 C2 R1 R1 R1Scatter 2 C2 R1 R1 R1Scatter 3 C2 R1 R1 R1Scatter 4 C2 R1 R1 R1Scatter 5 C2 R2 R2 R3Scatter 6 C2 R2 R2 R3Scatter 7 C2 R2 R2 R3

the first four scatter configurations to improve the resolution at small x where more structureis expected for amorphous materials. Wider apertures were used for the last three scatterconfigurations to improve the counting statistics.

Optimal alignment of aperture C was accomplished by using a PRM model D-15 pancakestyle ion chamber (Precision Radiation Measurement Inc., Nashville, TN) in place of the targetat plane E and measuring the exposure for each configuration as the aperture was translatedlaterally.

Optimal alignment of aperture F was accomplished using a short (6.6 mm thick) watertarget and measuring the total number of counts observed by the HPGe detector for each

An energy-dispersive technique to measure x-ray coherent scattering form factors 859

30 40 50 60 70 80 90 100 1100.7

0.8

0.9

1.0

1.1

1.2

1.3

Energy (keV)

Apert

ure

Are

a (

cm

2)

Measured

Fit

Figure 2. Typical energy dependence of effective area of detector transmission aperture, Adt,caused by tungsten spindles. The fitting function is computed from separate straight lines in theenergy ranges 30–69 keV and 69–110 keV. Note that the tungsten K-edge is at 69 keV.

configuration as the aperture was translated laterally. It is important that a short target be usedto ensure that the optimal value is found for the centre of the target. If a long target is used,multiple scattering can cause a false optimum to be found for larger scattering angles.

For measurements in the transmission configuration, where the fluence from the primarybeam is very high, a coarse 5 mm diameter aperture is used at plane I and an additionalaperture (labelled G in figure 1) is used to limit the countrate. This aperture comprises a5 mm thick Pb sheet with a small pinhole (made with a nominal 0.5 mm drill) followed by anopposed pair of 6 mm diameter tungsten carbide tipped micrometer spindles mounted directlybehind the pinhole. The effective area of the pinhole referenced to aperture plane I wasdetermined by computing the ratio of the total counts measured by the spectrometer throughthe pinhole to those measured through a larger aperture with known dimensions, incorporatingthe inverse square law. A 65 kVp spectrum with 3 cm Al filtration and a current of 0.2 mA wasused. This gave a countrate of ∼4000 counts per second and 1% deadtime through the largeraperture.

The micrometer spindles allow precise control of the aperture size for a very smallaperture. The effective area of the detector aperture in transmission mode, referenced to planeI and denoted by Adt, was measured by computing a further ratio of counts in each energybin, measured at the lowest possible current (0.2 mA) at 121 kVp with the spindles in theopen position and in their final position. Countrates and deadtimes were ∼20 000 counts persecond and ∼5% in the open position and much smaller in the closed position. Ideally, theratio would be energy independent, giving a constant value for Adt. In practice, the tungstenspindles add a slight energy dependence which is compensated for by fitting to the measuredarea as a function of energy. This ensures that the tungsten spindles do not affect the measuredform factors. The effective area corresponds roughly to a rectangular aperture of width0.02 mm and height 0.5 mm but this can vary slightly each time the aperture is emplaced, soAdt must be characterized before each transmission spectrum is measured. A typical result of

860 B W King and P C Johns

the measurement of Adt is shown in figure 2. For measurements in the scatter configuration,transmission aperture G is removed from the bench completely.

2.2. Source, detector and target

The x-ray tube is a Machlett–Dynamax rotating tungsten anode tube, powered by a PickerGX550 single phase generator. It was operated at 121 kV and 2 mA using the nominal0.6 mm focal spot for a total exposure of approximately 600 mAs. The tube is mountedvertically such that the heel effect does not affect the tube spectrum as seen at differentscattering angles. The tube has a target angle of 12◦ but has been tilted towards thedetector by 9.83◦ to reduce the apparent height of the focal spot through foreshortening.The output of the x-ray tube is monitored using a PTW N30001 ion chamber (PTW Freiburg,Germany) permanently placed directly in front of the x-ray tube, out of the beam path forthe measurements. The ion chamber position is unaltered across the eight experimentalgeometries. Its readings are used to normalize all measured spectra to the same tube output.

All spectra were measured using an Ortec planar HPGe detector (Ortec, Oak Ridge, TN),with a detector crystal 16 mm in diameter and 10 mm thick. The full width at half maximumof the detector was measured to be 0.41 keV at 59.5 keV and 0.53 keV at 122 keV (King andJohns 2007). The detector is located in a Pb casing to reduce the effect of stray radiation. A Pbaperture, labelled I in figure 1, is placed in the casing to limit the area of the detector exposedto radiation and to ensure that the radiation is absorbed in the central region of the crystal. Thedimensions of the apertures are given in table 3. The signal chain consists of a Canberra 2024spectroscopy amplifier with time constant 0.5 μs, 8077 ADC and an Aptec 2000 series MCA,all from Canberra (Canberra Industries, Meriden CT). The MCA is integrated into a personalcomputer. Deadtimes in the transmission configuration were between 1.5 and 4% while in thescatter configurations were ∼0.2%.

The target is placed on a plastic holder which can be translated laterally using a precisionslide assembly (Velmex, Bloomfield, NY). Liquid targets are placed in a 40 mm longrectangular container made from polycarbonate with 0.5 mm thick entrance and exit windows.The container fits securely in place on the plastic holder to ensure a consistent target location.Solid targets are placed directly on the plastic holder.

2.3. Spectral measurements

Spectra are acquired with bin sizes ∼0.1 keV and then rebinned so that the spectra for allconfigurations are measured on a consistent x mesh. The form factors were measured on a200 point logarithmic mesh ranging from 0.25 nm−1 to 14 nm−1. This allows finer resolutionat small x where more structure is expected in Fcoh.

Some of the x-ray photons that reach the detector will not come from scattering within thetarget. In order to remove the effect of these photons, background spectra must be measuredfor each θ . There are two distinct sources of unwanted photons. First, the number of strayphotons that are present throughout the room can be determined by closing all of the aperturesat plane D and measuring the spectrum. Second, the number of background photons thatscatter along the path of the beam, in the walls of the target container, or from the edges of theapertures, can be determined by measuring the spectrum with an empty target container. Inthis case, the spectrum measured must be corrected for the attenuation that these backgroundphotons would experience when travelling through the target in the actual experiment. Thetransmission factor is determined by measuring the spectra along the transmission path withthe target present and without.

An energy-dispersive technique to measure x-ray coherent scattering form factors 861

Taken all together, this means that five spectra are required in order to compute thecoherent scattering form factor with a given configuration: two spectra along the transmissionpath, one with the target present and one without, two spectra along the scatter path, onewith the target present and one without, and one spectrum with closed apertures to measurethe stray radiation. All measured spectra are normalized to the tube monitoring ion chambersignal and corrected for MCA deadtime.

Three spectral effects could cause artefacts in the measured form factors. Some ofthe photon energy can escape from the spectrometer through interaction processes suchas Compton scattering or fluorescence escape. When this happens, the measured photonenergy is lower than the true energy. It is possible to correct for this effect by measuring anumber of monochromatic spectra to determine a detector response function. However, for Gespectrometers, this effect is mostly insignificant except in the very low-energy region of thespectrum. We investigated two options for dealing with this effect. Option number 1 entailedimplementing a detector response correction based on measured monochromatic spectra whileoption number 2 was simply to exclude the low-energy region, below 30 keV, where the effectis strongest. Both options gave virtually identical results for the form factor measurements.As a result, we chose to simply exclude the low-energy region from all calculations. Wealso excluded energies above 110 keV where a small amount of pulse pileup was present inthe transmission spectra due to their much higher countrates. Furthermore, we will show insection 3 that the form factor can be calculated by taking the ratio of spectra measured in thescatter configuration and the transmission configuration. In strongly peaked regions of thespectrum, such as in the vicinity of the tungsten characteristic peaks, the ratio will be quitesensitive to minute differences in the locations of the peaks. To avoid artefacts caused bythese slight misalignments, energies between 57–60 keV and 67–70 keV are excluded fromthe analysis.

3. Theory

3.1. Model development

Here, we derive expressions for the number of photons measured by the detector in eachenergy range E → E + dE in both the transmission and scatter configurations.

In the transmission configuration, if the differential incident fluence per energy intervalat distance Lst is d�t0/dE, then in the energy range E → E + dE, the detector will measure

dNt(E) = d�t0(E)L2

st

(Lst + Ltd)2 Adt exp[−μt(E)Lt], (3)

where μt is the attenuation coefficient of the target and Adt is the effective area of the detectorexposed to the source in transmission mode as determined, including energy dependence, insection 2.1.

For the scattering configurations, Y > 0. Let the differential incident fluence per energyinterval at distance Lst be d�s0/dE. The differential x-ray fluence at a particular depth l ofthe target before any scattering has occurred is

d�s

dE= d�s0

dE

L2st(

L2st + Y 2

) exp

[−μt(E)l

cos α

]. (4)

The probability of these photons scattering at depth l into a given solid angle � and travellingthrough the rest of the target is

ps(l) = deσ

d�

ρedV (l)

At(l) cos αexp

[−μt(E)(Lt − l)

cos β

]�, (5)

862 B W King and P C Johns

where At(l) is the area of the target at depth l from which scattering can be observed at thedetector, ρe is the electron density of the target material, dV (l) = At(l) dl is an element oftarget volume at depth l, and the solid angle of the detector is given by

� = Ads cos β

L2td + Y 2

, (6)

where Ads is the physical area of the detector aperture I corrected for its finite thickness forphotons incident at angle β. The number of photons observed by the detector in the energyrange E → E + dE from the layer dl at depth l is d�sAt(l) cos α ps(l):

d2Ns = d�s0(E)L2stρeAds cos β(

L2st + Y 2

)(L2

td + Y 2) exp

[−μt(E)Lt

cos β

]deσ

d�

× exp

[−μt(E)l

(1

cos α− 1

cos β

)]At(l) dl. (7)

The total number of photons observed by the detector is given by integrating this expressionover all depths:

dNs(E) = d�s0(E)L2stρeAds cos β(

L2st + Y 2

)(L2

td + Y 2) exp

[−μt(E)Lt

cos β

]deσ

d�

×∫ Lt

0exp

[−μt(E)l

(1

cos α− 1

cos β

)]At(l) dl. (8)

The exponential factor in the integral is very nearly 1 for the small angles present in our caseand can be neglected. The integral then resolves to simply the scattering volume within thetarget, Vt = ∫

At(l) dl.The ratio of dNs(E) to dNt(E) in each energy range E → E + dE is then

dNs(E)

dNt(E)=

[(Lst + Ltd)

2ρeVt cos β(L2

st + Y 2)(

L2td + Y 2

)] [

Ads

Adt

] [d�s0

d�t0

]exp

[−μt(E)Lt

(1

cos β− 1

)]deσ

d�.

(9)

For small β, the exponential factor can be neglected. We checked the validity of neglecting theexponential factors in equations (8) and (9) by numerically evaluating the full integral in theworst case scenario for water: low-energy photons (30 keV) with the largest scattering angleused in our experiment. We found that the two exponential factors in equation (8) constitutedless than a 1% effect on the volume calculation, justifying our decision to ignore them. Usingthe definition of the scattering cross section from equation (1), an expression for the coherentscattering form factor is arrived at based entirely on measurable and tabulated parameters:

Fcoh(x) ={[ (

L2st + Y 2

) (L2

td + Y 2)

(Lst + Ltd)2 deσ0

d�ρeVt cos β

] [Adt

Ads

d�t0

d�s0

dNs(E)

dNt(E)

]− FKN(E, θ)Finc(x)

}1/2

.

(10)

3.2. Parameter calculation

The scattering volume Vt and the scattering angles cannot be measured directly in theexperiment. Instead, we have developed a Monte Carlo ray tracing simulation of ourexperiment in Matlab (The Mathworks, Natick, MA). We perform a numerical integrationof the volume by uniformly sampling N points within a rectangular parallelepiped targetvolume Vtot. For each, a point is chosen within the acceptance area of the detector Ads (this

An energy-dispersive technique to measure x-ray coherent scattering form factors 863

Table 4. Calculated parameters for scattering apertures from the Monte Carlo calculation. Thecalculated uncertainties include variations in all experimental parameters.

Configuration α (deg) β (deg) θ (deg) Y (cm) Vt (cm3)

Scatter 1 0.91 ± 0.11 0.77 ± 0.09 1.67 ± 0.17 1.07 ± 0.11 0.61 ± 0.13Scatter 2 1.70 ± 0.11 1.48 ± 0.09 3.17 ± 0.15 2.04 ± 0.11 0.51 ± 0.10Scatter 3 2.67 ± 0.12 2.36 ± 0.08 5.03 ± 0.15 3.22 ± 0.12 0.39 ± 0.05Scatter 4 3.33 ± 0.12 2.97 ± 0.09 6.30 ± 0.15 4.03 ± 0.11 0.31 ± 0.03Scatter 5 5.31 ± 0.13 4.74 ± 0.11 10.05 ± 0.17 6.45 ± 0.11 0.42 ± 0.03Scatter 6 6.63 ± 0.13 5.95 ± 0.11 12.58 ± 0.17 8.07 ± 0.12 0.31 ± 0.02Scatter 7 7.94 ± 0.13 7.15 ± 0.11 15.09 ± 0.17 9.69 ± 0.11 0.241 ± 0.017

acceptance area depends on β because of transmission effects through the thick aperture Idirectly in front of the detector), and a point is chosen from the spatial distribution of the x-raytube source strength � over the focal spot area Af . We modelled � with a two-dimensionalnormal distribution with widths determined from pinhole images of the focal spot (Johns andSchulze 2009). The number of locations Nt for which the path defined by the source, target anddetector points passes through all of the apertures in the system is then calculated. The targetvolume Vt is then obtained as Vt = Nt/N ·Vtot in the limit of large N. Strictly speaking, becausewe have included the finite sizes of the source and detector in the Monte Carlo calculation,the integral actually calculated is the product of Vt, � and Ads, but since it is this product thatappears in equation (10), the distinction is unimportant. This volume calculation is similar tothat of Balogun et al (2000) with the exception that we have extended the calculation to modelthe effect of rectangular apertures with finite thickness. The effect of thick apertures can bequite significant for the larger angles and must be accounted for properly.

The calculation treats all apertures as ideal, ignoring any penetration through the collimatoredges. In fact, there would be a slight effect from penetration through the edges of eachaperture, which would become more significant for larger θ . This would introduce anadditional energy-dependent effect to our measurements as more x rays will penetrate theaperture edges at higher energies. If this were a significant effect, the Pb K-edge would beapparent in our measured spectra where the penetration would be most significant. As ourmeasured spectra did not show any significant effect around the Pb K-edge, we concluded thatpenetration through the apertures was insignificant.

The volume calculation has a precision better than 0.5%. The same Monte Carlocalculation is also used to determine α, β, θ and the offset Y as well as the statisticaluncertainties and correlations of all parameters. Uncertainties in the location and dimensionsof every element in the experiment are factored into the uncertainty estimates. The calculatedvalues are given in table 4.

3.3. Overall fluence ratio

The Monte Carlo calculation of section 3.2 models the finite size of the x-ray tube focal spotbut it does not include any variations in the size or strength of the x-ray beam at different α.We account for this by measuring the overall fluence ratio d�t0/d�s0 separately.

We measure the exposure from the x-ray tube with a PRM model D-15 pancake styleion chamber (PRM, Nashville, TN) in place of the target at plane E for each configuration intable 3. The signal, S, in each case is proportional to the fluence at the ion chamber and to

864 B W King and P C Johns

Table 5. Measured fluence ratios (transmission/scatter) for scattering apertures. Measurementsfor the ratios were performed three times and averaged. The overall fluence ratio is the ratio of theexpected to the measured values.

Plane D aperture Expected ratio Measured ratio d�t0/d�s0

Scatter 1 7.2 ± 0.3 7.50 ± 0.12 0.96 ± 0.04Scatter 2 7.0 ± 0.3 7.83 ± 0.07 0.90 ± 0.04Scatter 3 6.7 ± 0.3 7.26 ± 0.06 0.93 ± 0.04Scatter 4 6.6 ± 0.3 7.78 ± 0.10 0.84 ± 0.04Scatter 5 9.7 ± 0.4 11.55 ± 0.07 0.84 ± 0.04Scatter 6 9.3 ± 0.4 11.03 ± 0.11 0.85 ± 0.04Scatter 7 8.9 ± 0.4 10.9 ± 0.2 0.82 ± 0.04

the effective volume, v, of the ion chamber visible through the apertures. In the transmissionconfiguration,

St ∝ d�t0

L2st

vt (11)

while in the scatter configuration,

Ss ∝ d�s0

L2st + Y 2

vs. (12)

Taking the ratio of equations (11) and (12) allows an expression to be derived for the ratiod�t0/d�s0:

d�t0

d�s0=

( L2st

L2st+Y 2

) vsvt

SsSt

= re

rm. (13)

The ratio re is the expected scatter to transmission ratio under the assumption that the x-raytube fluence is independent of α. The ratio rm = Ss/St is the actual scatter to transmissionratio measured with the ion chamber. The expected ratio was calculated using a MonteCarlo calculation similar to that in section 3.2. The focal spot was modelled using the samedimensions as in section 3.2 for consistency.

The overall fluence ratio measured using this procedure is shown in table 5. The ratiodecreases with larger angles, meaning that the x-ray tube output increases for larger α. Anassumption with this procedure is that any variation in x-ray tube output is not energy dependentas the ion chamber is energy insensitive.

4. Results

4.1. Individual scattering angles

We have measured the coherent scattering form factors of water (H2O, ρe = 3.3348 ×1023 cm−3 based on mass density at 22.5◦C (Lide 2009)) and polyethylene ([C2H4]n, ρe =3.255 × 1023 cm−3, based on measurement of mass density of our sample) to verify that theenergy-dispersive method is configured correctly. The individual form factor measurementsfrom each scattering configuration are shown in figure 3. Although not shown on the graphs,uncertainties were also calculated for each point, taking into account uncertainties in allexperimental parameters. To confirm the consistency of the individual measurements, χ2

testing was performed, based on the estimated uncertainty in each measurement, for each

An energy-dispersive technique to measure x-ray coherent scattering form factors 865

0 1 2 3 4 5 6 7 8 90

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 50

1

2

3

4

5

6

x (nm−1

)

form

facto

r (f

ree e

− / b

ound e

−)0

.5

Scatter #1

Scatter #2

Scatter #3

Scatter #4

Scatter #5

Scatter #6

Scatter #7

(a)

(b)

Figure 3. Form factor measurements for (a) water and (b) polyethylene from each scatteringconfiguration. Error bars are not shown for clarity. Note the different range of x displayed for eachmaterial.

value of x. The vast majority of x values were found to be consistent, within a confidenceinterval of 95%. For water, only 2 points out of 145 were found to be inconsistent for x between0.5 nm−1 and 9.5 nm−1. Since there are no adjustable parameters in our measurement, thissuggests that our calculations have been implemented correctly. The measured value for eachconfiguration drops off towards zero at large x as the upper energy limit of the x-ray spectrumis approached. We truncate Fcoh for x > 9.5 nm−1 where the uncertainty is too large to providemeaningful results.

The measurements for the more structured material, polyethylene, were made to examinethe limits of the x resolution of our equipment. We again found that the vast majority of pointswere consistent, based on the same χ2 criteria as for water. Excluding the sharply peakedregion around 1.2 nm−1, only 3 points were inconsistent. The differences around 1.2 nm−1

are a result of the limited x resolution for the smaller angles. As can be seen in table 4, theangular resolution of scatter configuration 1 is around 10%, meaning that the sharp peaksin the polyethylene form factor are poorly resolved for this configuration. As the resolutionimproves at higher θ , the sharper peaks become apparent.

866 B W King and P C Johns

12

34

56

7

0

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0.25

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tter #

Fra

ctional err

or

ρe

ZA

ZE

ZI

HI

WI

EYθβ

Adt

dNs

dNt

Vt

Figure 4. Uncertainties in experimental parameters for each configuration from the measurementof the form factor of water at 50 keV. The subscripts on the W, H, and Z parameters refer to theaperture plane in figure 1. The uncertainty in dNs/dNt contains the effect of the fluence ratio aswell. All parameters, with the exception of the ratio of the two measured spectra dNs/dNt, Adtand E itself are energy independent.

The uncertainties were calculated by repeating the Monte Carlo calculation ofsection 3.2 as the geometric parameters were varied and constructing a covariance matrixcontaining all 29 of the geometric uncertainties. Variances of the ratio dNs/dNt and theeffective transmission aperture area Adt were determined using counting statistics. Theuncertainties in each experimental parameter at a specific energy are shown in figure 4.It can be seen from this that the most significant source of error in these measurements comesfrom variation in Vt. The resulting error in the coherent scattering form factor for water isshown in figure 5.

The resolution in x is dependent on both the energy and angular resolution of themeasurements via equation (2). The angular resolution is most significant for small θ whilethe energy resolution becomes more significant for large θ .

4.2. Combining results from individual scattering angles

The individual measurements made with each configuration are combined to yield one datasetover a larger x range. The energy bins of our individual measurements were chosen suchthat the form factor measurements are on a consistent scale. In the overlap regions, we applya χ2 test to check our individual measurements at each x value for consistency with a 95%confidence interval. Consistent values were combined using a weighted average based on theuncertainty of each measurement.

For points where the χ2 test shows that the results are inconsistent, we do one of twothings: if an isolated point is inconsistent, we compute an unweighted average of the individualresults. If a region of points (defined as three or more adjacent points) is inconsistent, we use thesingle measurement with the highest x resolution under the assumption that the inconsistencyis caused by a difference in resolution, such as in the 1.2 nm−1 region of polyethylene.

The combined results are shown in figure 6 compared to previously published values.Water has been studied by several groups, most extensively by Narten (1970). Although some

An energy-dispersive technique to measure x-ray coherent scattering form factors 867

0 1 2 3 4 5 6 7 8 90

0.05

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0.40

0.45

0.50

x (nm−1

)

Fra

ctio

na

l e

rro

r in

Fcoh

Scatter #1

Scatter #2

Scatter #3

Scatter #4

Scatter #5

Scatter #6

Scatter #7

Figure 5. Error estimates for individual coherent scattering form factor measurements of water.

Table 6. Average absolute relative difference between the energy-dispersive technique andmeasurements from the literature.

Material Comparison x range (nm−1) Dav

Water Narten (1970) 0.5–9.5 0.101Kosanetzky et al (1987) 0.5–4.25 0.086Peplow and Verghese (1998) 0.5–9.5 0.140Hura et al (2000) 0.5–9.5 0.137

Polyethylene Kosanetzky et al (1987) 0.5–4.25 0.198

differences exist between the measurements by different groups, the general behaviour of thescattering form factor is known. Polyethylene measurements are compared to those madeusing a diffractometer by Harding’s group (Kosanetzky et al 1987).

Quantitative comparisons are made to various published results (Fref) in table 6 bycomputing the average absolute relative difference

Dav = 1

N

N∑i=1

∣∣∣∣Fcoh(xi)

Fref(xi)− 1

∣∣∣∣ (14)

where N is the number of points that were measured.

5. Discussion

The results of the energy-dispersive measurements are quite encouraging. The good agreementof the individual measurements with each other, as well as the fact that the measurementsapproach the IAM value at large x is very strong evidence of the accuracy of the energy-dispersive measurement. The good correspondence with previously published diffractometermeasurements is also promising.

868 B W King and P C Johns

0 1 2 3 4 5 6 7 8 90

0.5

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ree

e− /

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−)0

.5

This work

Narten

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Kosanetzky et al + IAM

(a)

(b)

Figure 6. Combined form factor measurements of (a) water and (b) polyethylene compared topublished values (Narten 1970, Kosanetzky et al 1987). Error bars are shown for only every 25thpoint for clarity.

Setup and characterization of the experiment is quite time consuming. After the systemhas been characterized, however, measurements are relatively quick. Each set of measurementsrequires a 5-min low-current transmission spectrum to characterize the detector transmissionaperture (aperture G in figure 1), two 1-min transmission spectra, and seven 5-min scatterspectra. Including the time for adjustment for each configuration, a complete measurementcan be made in approximately 1 h.

Our technique measures the form factor absolutely, with no scaling parameters.Diffractometer measurements are placed on an absolute scale by fitting the measured valuesto the IAM results at large x. It is not readily apparent, however, exactly where the IAMregion begins. If the range of x is not large enough then the scaling can introduce a systematicshift in the measurement. This was certainly a problem in the results of Johns and Wismayer(2004) and also possibly with some of the results of Kosanetzky et al (1987). Our energy-dispersive measurement avoids this problem entirely and allows an independent check that theexperiment is configured correctly.

It is certainly possible to scale the energy-dispersive results to the IAM value at largex once the region of validity of the IAM has been established. This could be achieved byscaling the results for the largest scattering angle to the IAM value at large x, then scalingthe next largest results to the refined value in the overlap region, and so on until all of thescattering angles have been refined. Doing this may provide a more precise value of the

An energy-dispersive technique to measure x-ray coherent scattering form factors 869

geometrical parameters (primarily Vt) than can be achieved directly from the measured values,thus giving a more precise measurement of Fcoh. The series of cascaded scaling parameters,however, would mean that the uncertainties would increase as the calculation progresses tosmaller x values. Additional study would be required to determine whether this would be animprovement over the current method.

Even if it is advantageous to scale the measured values to the IAM, it should not be donelightly. The full calculation should be performed first and the self-consistency of the individualscatter results verified, as well as the region of validity of the IAM. Introducing an arbitraryscaling parameter would allow potential experimental configuration problems to be maskedif it is used indiscriminately. The scaling should be used as a refinement of the measuredparameters, not a replacement.

Currently, there is very little benefit to be gained from longer exposures to improve thecounting statistics as the uncertainty is dominated by the volume calculation. If scaling to theIAM value at large x is found to be beneficial, the counting statistics in the scatter spectra maybecome the dominant source of uncertainty and longer exposures could provide more accurateresults.

The fairly long target used in these measurements was chosen to provide the bestbalance between scattering volume and target attenuation. As a first approximation, usingequation (8), the scatter countrate is proportional to the product of the target length and thetransmission factor. The maximum countrate is then for a target length of μ−1 which for watergives a value between 3 and 6 cm. For larger θ this approximation breaks down, however. TheMonte Carlo calculation shows that for the largest scattering angle used in our experiment,the scattering volume is contained entirely within about half of the target length. A shortertarget would therefore provide a better countrate at larger angles, at the expense of a reducedcountrate for the smaller angles and larger uncertainties in the volume calculation.

The long target used here may not be applicable to measurements of some tissues as itmight not be possible to find large enough homogeneous samples of a given type of tissue.In these cases, a shorter target could be used at the expense of larger uncertainties due to thereduced countrate and larger volume uncertainties. For materials that are available in largerquantities, such as fat or muscle, the longer target would give better results.

The combination of the individual measurements into a single dataset has advantages anddisadvantages. Since each measured x value (except at very high and low x) was measured atleast twice, in most cases three or more times, any variations that might be present tend to bereduced in the combined measurement. On the other hand, the difference in the x resolutionbetween different configurations means that, in some cases, the weighted average will not beappropriate. Our method of determining when to combine the results works fairly well butcan sometimes cause small jumps in the combined measurements as adjacent points in x spaceswitch from a single measurement to a combined value.

6. Conclusion and future work

We have developed and characterized a general purpose system to measure coherent scatteringform factors of amorphous materials such as tissues and phantom materials used in imaging.We have compared our measured results to previous work and found good agreement. Ourenergy-dispersive measurements are more reliable than diffractometer based systems forseveral reasons: the measurements are made without the need for any scaling parameters,a wider range of x values is accessible, well into the IAM region, and the transmissionconfiguration allows a more well-defined and quantifiable background to be measured. In

870 B W King and P C Johns

addition, the thorough characterization of the system has provided quantitative uncertaintyestimates.

We are currently using this system to make accurate measurements of several types oftissue to develop an accurate form factor library that will aid in the development of the fieldof x-ray scatter imaging.

Acknowledgments

This work was supported financially by the Natural Sciences and Engineering ResearchCouncil of Canada. Measurements of spectra were performed by Karl Landheer. The authorsthank Philippe Gravelle in the Physics Department machine shop for general fabricationassistance and Dr David Rogers for loan of the dual channel electrometer used for measuringthe overall fluence ratio.

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