This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 141.213.236.110

This content was downloaded on 20/01/2014 at 17:58

Please note that terms and conditions apply.

X-ray coherent scattering form factors of tissues, water and plastics using energy dispersion

View the table of contents for this issue, or go to the journal homepage for more

2011 Phys. Med. Biol. 56 4377

(http://iopscience.iop.org/0031-9155/56/14/010)

Home Search Collections Journals About Contact us My IOPscience

IOP PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 56 (2011) 4377–4397 doi:10.1088/0031-9155/56/14/010

X-ray coherent scattering form factors of tissues,water and plastics using energy dispersion

B W King1, K A Landheer1 and P C Johns1,2

1 Ottawa Medical Physics Institute, 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 18 October 2010, in final form 18 April 2011Published 27 June 2011Online at stacks.iop.org/PMB/56/4377

AbstractA key requirement for the development of the field of medical x-ray scatterimaging is accurate characterization of the differential scattering cross sectionsof tissues and phantom materials. The coherent x-ray scattering formfactors of five tissues (fat, muscle, liver, kidney, and bone) obtained frombutcher shops, four plastics (polyethylene, polystyrene, lexan (polycarbonate),nylon), and water have been measured using an energy-dispersive technique.The energy-dispersive technique has several improvements over traditionaldiffractometer measurements. Most notably, the form factor is measuredon an absolute scale with no need for scaling factors. Form factorsare reported in terms of the quantity x = λ−1 sin(θ/2) over the range0.363–9.25 nm−1. The coherent form factors of muscle, liver, and kidneyresemble those of water, while fat has a narrower peak at lower x, and bone ismore structured. The linear attenuation coefficients of the ten materials havealso been measured over the range 30–110 keV and parameterized using thedual-material approach with the basis functions being the linear attenuationcoefficients of polymethylmethacrylate and aluminum.

1. Introduction

A library of the scattering properties of tissues and phantom materials will be an important toolfor the developing field of x-ray scatter imaging (Leclair and Johns 2001). The differentialscattering cross section per electron, deσ/d�, determines the probability of photons withenergy E scattering through an angle θ . The cross section is determined by a material’scoherent (Fcoh) and incoherent (Finc) scattering form factors (Johns and Cunningham 1983):

deσ

d�= deσ0

d�

[F 2

coh(x) + FKN(E, θ)Finc(x)]. (1)

0031-9155/11/144377+21$33.00 © 2011 Institute of Physics and Engineering in Medicine Printed in the UK 4377

4378 B W King et al

In this equation, deσ0/d� is the classical Thomson cross section for scattering from a single,free electron and FKN is the Klein–Nishina factor for incoherent scattering. Both form factorsare functions of the momentum transfer argument:

x = 1

λsin

(θ

2

)= E

hcsin

(θ

2

), (2)

where h is Planck’s constant and c is the speed of light. For a given material, incoherentscattering can be accurately characterized for all values of x by computing Finc as a combinationof individual free atom form factors, such as those tabulated in Hubbell and Øverbø (1979).This is the independent atom model (IAM) approach. In the case of coherent scattering,however, interference effects dominate for small x, meaning that Fcoh must be measuredexperimentally. Taking the practical limits of scatter imaging in diagnostic radiology in termsof angle and photon energy as 0.5◦ � θ � 179◦, 16 � E � 140 keV, a library of form factorsfor tissues and phantom materials is required from x ∼ 0.1 nm−1 through to the IAM region,x ∼ 10 nm−1.

For pure water, the gold standard for Fcoh is the dataset of Narten (1970) who publisheddata for 0 � x � 12.7 nm−1. For tissues, the seminal paper was that of Kosanetzky et al(1987), who published angle-dispersive diffractometer curves for 0.25 � x � 4.3 nm−1 forpork fat, muscle, tendon, bone, blood, liver, brain white matter, and brain grey matter, andthe synchrotron work of Peplow and Verghese (1998) who studied several normal animaltissue types out to x = 10 nm−1 with the minimum x between 0.42 and 1.08 nm−1, sampledependent. Others looked at a restricted range of materials and/or of x values (Evans et al 1991,Royle and Speller 1995, Westmore et al 1996, Tartari et al 1997, Kidane et al 1999, Lewiset al 2000, Desouky et al 2001, Poletti et al 2002, Fernandez et al 2002, Castro et al 2004,Griffiths et al 2007, Elshemey et al 2010). Most of these measurements used angle-dispersivediffractometers, which were either conventional crystallographic machines or synchrotroninstruments. Angle-dispersive diffractometers have inherent problems in measuring Fcoh foramorphous materials such as tissues and the published results vary significantly (Johns andWismayer 2004). We have reported on alternative methods of measuring form factors that canbe done simply using a photostimulable phosphor plate but are low in x-resolution (King andJohns 2008).

Most recently, we have thoroughly characterized an energy-dispersive method thatprovides reliable measurements for tissue-like materials (King and Johns 2010). In thisapproach, a polychromatic x-ray tube source and energy-dispersive detector are aligned andthe specimen is moved laterally so that photons must scatter at a small θ to reach the detector.This configuration is based on earlier work in our lab (Leclair and Johns 2002, Hasan 2003,Hasan and Johns 2004, King 2009) and a similar approach was taken by Kidane et al (1999)and Leclair et al (2006). Here we report coherent scatter form factors of tissues, water, andplastics based on this energy-dispersive technique.

2. Theory

The basic geometry of the experiment is shown in figure 1. More detailed information on theexperimental configuration and the derivation of the following equations can be found in ourprevious work (King and Johns 2010). An x-ray tube acts as a polychromatic source of x raysand the detector is an x-ray spectrometer. By translating the target laterally, the number ofphotons can be measured in both a transmission and a scatter configuration. The x-ray sourceand spectrometer remain stationary throughout the experiment.

Energy-dispersive measurements of x-ray coherent scattering form factors 4379

Source Detector

Target

Lst Ltd

Y

Lt

Figure 1. Schematic geometry of the energy-dispersive experiment.

In the transmission configuration, we have shown previously (King and Johns 2010) thatif the differential fluence per energy interval from the x-ray source at a distance Lst is d�t0/dE,the number of photons in the range E → E+dE measured by the detector with cross-sectionalarea Adt through a target with linear attenuation coefficient μt(E) will be

dNt(E) = d�t0L2

st

(Lst + Ltd)2Adt exp[−μt(E)Lt] . (3)

In the scattering configuration, if the x-ray source differential fluence at a distance Lst isd�s0/dE, the number of photons measured is

dNs(E) = d�s0(E)L2stρeVtAds cos β(

L2st + Y 2

)(L2

td + Y 2) exp[−μt(E)Lt]

deσ

d�(4)

where Ads is the area of the detector, ρe is the electron density of the target material and Vt isthe scattering volume of the target. This expression is only valid for cos α ≈ cos β ≈ 1. Inour experiment, α is at most 7.94◦ and β is at most 7.15◦ (King and Johns 2010). Then, bycomputing the ratio of the scatter and transmitted spectra and using the definition of deσ/d�

from equation (1), an expression for the coherent scattering form factor can be found:

Fcoh(x)={[ (

L2st + Y 2

)(L2

td + Y 2)

(Lst + Ltd)2 deσ0d�

ρeVt cos β

] [Adt

Ads

d�t0

d�s0

dNs(E)

dNt(E)

]− FKN(E, θ)Finc(x)

}1/2

. (5)

In order to compute Fcoh from this expression, the composition of the target material must beknown so that ρe and Finc can be generated from tables.

Although measurement of Fcoh is the main goal of this work, the data provide additionaluseful information. The differential linear scattering coefficient (Kosanetzky et al 1987,Leclair et al 2006) is analogous to the linear attenuation coefficient μt but is the probabilityper unit distance travelled of scattering into a given solid angle:

μs = ρedeσ/d� . (6)

Both coherent and incoherent scattering information for the material are contained in thisdefinition. From equations (3) and (4),

μs(E, θ) =[(

L2st + Y 2

)(L2

td + Y 2)

(Lst + Ltd)2Vt cos β

] [Adt

Ads

] [d�t0

d�s0

] [dNs(E)

dNt(E)

]. (7)

4380 B W King et al

The advantage of measuring μs is that the composition of the material does not need to beknown. Thus, μs can be extracted more easily from scatter measurements. The quantity μs

is not simply a function of the single variable x, however, but varies with both E and θ due tothe presence of the Klein–Nishina factor FKN in the scattering cross section. For low energies,FKN approaches 1 so this difference is small but at higher energies it is more important.

In order to remove the effect of any background photons (i.e. photons scattered in thecontainer walls or elsewhere), for both the scatter and the transmission configurations wemeasured spectra with none of the target material present.

From our data, it was straightforward to also determine the linear attenuation coefficientof the material μt as a function of energy, using the two transmission measurements, i.e. withand without the target material. In the absence of K-edges, the attenuation coefficient canbe parameterized in a dual-material decomposition (Lehmann et al 1981) using two basismaterials α and β:

μt(E) = aαμα(E) + aβμβ(E) (8)

where μα and μβ are the linear attenuation coefficients of the basis materials and aα , aβ arefitting parameters determined from the measured data. The parameters can also be expressedin polar coordinates r and �.

3. Experimental details

3.1. Apparatus

Our experimental setup was described in detail previously and fully characterized (King andJohns 2010). Here, we summarize some of the important points. We used a Machlett-Dynamaxrotating anode tungsten x-ray tube as a source of polychromatic x rays. A potential of 121 kVwas used with a nominal current of 2 mA. A PTW (PTW, Freiburg, Germany) Farmer styleion chamber was used to measure the beam output. All measurements were normalized to theoutput of this chamber. Apertures before and after the target defined seven different θ as wellas the transmission configuration. Values for θ ranged from 1.7◦ to 15.1◦ (see table 1). AnOrtec HPGe spectrometer (Ortec, Oak Ridge, TN, USA) was used to measure all spectra. Thex-ray tube and spectrometer were aligned for the transmission configuration and were thenfixed in place. A very small aperture was constructed for the transmission configuration bymounting a pair of micrometer spindles with tungsten carbide tips behind a 0.5 mm pinholein a Pb sheet to give an aperture of roughly 20 μm × 500 μm. This aperture was removedduring the measurement of the scatter spectra. The target was translated laterally to align inturn with each scatter aperture. In each configuration, both transmission and scatter spectrawere measured with the target present in the path of the beam and without the target material.For liquid or tissue materials, this background was measured with an empty target container.For solid materials, the background was measured with nothing in the path of the beam. Thescattering volume was calculated using a Monte Carlo ray-tracing simulation of the experimentdeveloped in Matlab (The Mathworks, Natick, MA, USA).

The individual results from each configuration were combined together on a common gridin x space. Where different configurations overlapped, the results were checked for consistencywith a χ2 test. If the data were consistent, a weighted average was computed. Otherwise,for isolated inconsistencies, an unweighted average was used, and for regions with multipleinconsistencies the highest x resolution value was used. The rationale for this procedure isthat a string of inconsistent values most likely corresponds to a sharp peak in the form factorwhere the varying resolutions from different configurations blur the peak to different extents.

Energy-dispersive measurements of x-ray coherent scattering form factors 4381

Table 1. Details of the seven different scatter configurations used in the experiment. The rangeof x accessible is based on a usable spectrum between 30 and 110 keV. The uncertainty in x givenhere is the mean uncertainty over all energies.

Configuration Scattering angle (degrees) x accessible (nm−1) x uncertainty (%)

Scatter 1 1.67 ± 0.17 0.35−1.29 5.5Scatter 2 3.17 ± 0.15 0.67−2.45 3.3Scatter 3 5.03 ± 0.15 1.06−3.89 2.6Scatter 4 6.30 ± 0.15 1.33−4.87 2.4Scatter 5 10.05 ± 0.17 2.12−7.77 2.3Scatter 6 12.58 ± 0.17 2.65−9.72 2.2Scatter 7 15.09 ± 0.17 3.18−11.65 2.2

Table 2. Material parameters, compositions and number of repeated measurements for the materialsstudied. The density of water is given at 22.5 ◦C. The number of repeated measurements for eachmaterial is given as N.

Material Composition ρ (g cm−3) ρe (cm−3) N

Water H2O 0.9982 3.3348 × 1023 5Polyethylene (C2H4)n 0.948 3.255 × 1023 5Polystyrene (C8H8)n 1.042 3.373 × 1023 1Polycarbonate (C16H14O3)n 1.17 3.71 × 1023 1Nylon (C6H11NO)n 1.138 3.753 × 1023 1Fat From ICRP 23 (1975) 0.92 3.09 × 1023 2Muscle From ICRP 23 (1975) 1.04 3.44 × 1023 3Liver From Kosanetzky et al (1987) 1.045 3.48 × 1023 3Kidney From ICRP 23 (1975) 1.05 3.48 × 1023 1Bone From Woodard (1962) 1.85 5.73 × 1023 1

3.2. Sample preparation

The tissues studied were fat, muscle, liver, kidney and bone. Water and four plastics were alsostudied. The composition and physical parameters of these materials can be found in table 2.

All samples were butcher-shop beef except for the fat samples which were pork, and wererefrigerated, but not frozen, before use. All tissue samples except for the bone were placed ina 4 cm long custom built polycarbonate container. The entrance and exit windows were each0.5 mm thick. The length of the container was designed to provide the largest scattering volumeand hence, scattering signal possible. The tissue samples were cut to fit the container as tightlyas possible. To prevent dehydration, the container was filled with phosphate-buffered salineand then the tissue was inserted, displacing the solution without air entrapment. Whereverpossible, single pieces of tissue were used to fill the container.

The bone sample was placed in a shorter container (0.66 cm long) because of the largeramount of attenuation and multiple scattering present in this sample. The entrance and exitwindows were again 0.5 mm thick. The shorter container, however, gave a reduced signal forthe smaller scattering angles and increased the uncertainty of the volume calculation.

4382 B W King et al

0 2 4 6 80

0.5

1.0

1.5

2.0

2.5

x (nm−1)

form

fact

or

(fre

e e

− / b

ound e

−)0

.5

0 2 4 6 80

1

2

3

4

5

6

x (nm−1)

form

fact

or

(fre

e e

− / b

ound e

−)0

.5

Meas. 1Meas. 2Meas. 3Meas. 4Meas. 5

(a) (b)

Figure 2. Results of repeated form factor measurements of (a) water and (b) polyethylene. Errorbars are shown only for every 20th point for clarity.

4. Results

Water and polyethylene were used as control samples with measurements being repeated fivetimes over a period of roughly six weeks to ensure the reliability of the measurements. Theresults are shown in figure 2.

The other plastic materials were each measured a single time. Measured form factors forpolystyrene, polycarbonate (lexan) and nylon are shown in figure 3. Tissue measurementswere repeated multiple times, using different samples, to assess how much variation in the formfactors can be expected. The results can be found in figures 4 and 5. Where measurementswere repeated, the mean value was used and the uncertainties shown in the graphs representthe standard deviation of the results. All of the measured values of Fcoh, μs and μt and theiruncertainties are tabulated in the appendix.

The measured attenuation coefficient of water (table A1) was compared to two standardreferences. When compared to Plechaty et al (1975), μt was larger by 3.3% at 40 keV, 5.6% at60 keV, 5.2% at 80 keV and 2.9% at 100 keV. In comparison to the more recent NIST XCOMdatabase (Berger et al 2010), the differences were 1.2% at 40 keV, 4.9% at 60 keV, 4.9%at 80 keV and 2.8% at 100 keV. These differences are within our measurement uncertaintyat 100 keV and somewhat outside at lower energies depending on which dataset is used forcomparison.

The dual-material decompositions of μt(E), as outlined in section 2, are shown in table 3.The dual-material fitting parameters were computed by performing a χ2 minimization of themeasured attenuation coefficients to those predicted by equation (8) for energies between 30and 110 keV. The basis materials used were Al and polymethylmethacrylate (PMMA). Theattenuation coefficients of the basis materials were taken from Plechaty et al (1975).

Energy-dispersive measurements of x-ray coherent scattering form factors 4383

0 1 2 3 4 5 6 7 8 90

1

2

3

4

x (nm−1)

form

fa

cto

r (f

ree

e− /

bo

un

d e

−)0

.5

0 1 2 3 4 5 6 7 8 9

x (nm−1)

0 1 2 3 4 5 6 7 8 9

x (nm−1)

This workKosanetzky et al

(b)(a) (c)

Figure 3. Measured form factors of (a) polystyrene, (b) polycarbonate, and (c) nylon as comparedto diffractometer-based measurements by Kosanetzky et al (1987). Error bars are shown only forevery 20th point for clarity. Data were taken from the paper of Kosanetzky et al to 4.3 nm−1. Athigh x we used Independent Atom Model results, and at intermediate x a smooth transition betweenthe two.

Table 3. Dual-material decompositions of μt for the ten materials studied. Values of r and � givethe polar coordinates of the decomposition, where � is the angle from the aPMMA axis. There were155 degrees of freedom for each set of fitting parameters.

Material aPMMA aAl r � (degrees) χ 2

Water 0.879 ± 0.013 0.0197 ± 0.0012 0.880 ± 0.012 1.28 ± 0.08 168Polyethylene 0.910 ± 0.012 −0.0200 ± 0.0010 0.910 ± 0.012 −1.26 ± 0.06 148Polystyrene 0.952 ± 0.015 −0.0198 ± 0.0018 0.952 ± 0.015 −1.19 ± 0.10 129Polycarbonate 1.005 ± 0.015 −0.0054 ± 0.0015 1.005 ± 0.015 −0.31 ± 0.09 130Nylon 1.036 ± 0.015 −0.0110 ± 0.0015 1.036 ± 0.015 −0.61 ± 0.08 125Fat 0.823 ± 0.011 −0.0028 ± 0.0010 0.823 ± 0.011 −0.20 ± 0.07 131Muscle 1.059 ± 0.015 0.0113 ± 0.0011 1.059 ± 0.015 0.61 ± 0.06 231Liver 0.952 ± 0.014 0.0265 ± 0.0010 0.952 ± 0.014 1.59 ± 0.07 184Kidney 0.860 ± 0.014 0.0309 ± 0.0019 0.861 ± 0.014 2.06 ± 0.14 136Bone −0.23 ± 0.04 0.84 ± 0.03 0.87 ± 0.03 105.2 ± 2.6 115

4384 B W King et al

0

1.0

2.0

3.0

4.0

x (nm−1)

form

fa

cto

r (f

ree

e− /

bo

un

d e

−)0

.5

0

1.0

2.0

3.0

x (nm−1)

form

fa

cto

r (f

ree

e− /

bo

un

d e

−)0

.5

0 2 4 6 80

0.5

1.0

1.5

2.0

x (nm−1)

form

fa

cto

r (f

ree

e− /

bo

un

d e

−)0

.5

0 2 4 6 80

0.5

1.0

1.5

2.0

x (nm−1)

form

fa

cto

r (f

ree

e− /

bo

un

d e

−)0

.5

This workKosanetzky et alPeplow and Verghese

(a)(b)

(d)(c)

Figure 4. Measured form factors of (a) fat, (b) muscle, (c) liver and (d) kidney. Error bars areshown only for every 20th point for clarity. The results are compared to measurements from theliterature where available (Kosanetzky et al 1987, Peplow and Verghese 1998). Data were takenfrom the paper of Kosanetzky et al to 4.3 nm−1. At high x we used Independent Atom Modelresults, and at intermediate x a smooth transition between the two.

5. Discussion and conclusion

The measured form factors of the tissues are very similar to water, with the exception of thefat sample. This is consistent with the fact that the soft tissues are composed primarily ofwater. The tissues all show a strong increase in the form factor for x � 0.5 nm−1 that wasnot visible in diffractometer results. As this increase is not present in the water and plasticmeasurements, we conclude that this is a real feature of the tissue scattering. This may be aresult of longer range structure present in tissues as compared to water. However, the resultsfor small x values are quite uncertain and should be verified through measurements at lowerenergies.

The size of the sample is an important factor in the design of the experiment. A largerscattering volume will produce more scattered photons and hence a stronger signal. A largersample also leads to smaller geometric uncertainties. As the geometric uncertainties constitutethe dominant source of uncertainties in the results, this is an important factor. The effect of

Energy-dispersive measurements of x-ray coherent scattering form factors 4385

0 1 2 3 4 5 6 7 8 90

1

2

3

4

x (nm−1)

form

fa

cto

r (f

ree

e− /

bo

un

d e

−)0

.5

This workKosanetzky et al

Figure 5. Measured form factor of bone. The results are compared to diffractometer measurementsfrom the literature (Kosanetzky et al 1987). Error bars are shown only for every 20th point forclarity. Data were taken from the paper of Kosanetzky et al to 4.3 nm−1 while at high x we usedIndependent Atom Model results.

0.5

1.0

1.5

2.0

2.5

Co

he

ren

t fo

rm f

act

or

ratio

1 2 30.5

1.0

1.5

2.0

2.5

3.0

3.5

x (nm−1)

Co

he

ren

t fo

rm f

act

or

ratio Bone / Water

Fat / WaterMuscle / WaterLiver / WaterKidney / Water

(a)

(b)

Figure 6. Ratio of (a) soft tissue and (b) bone coherent scattering form factors compared to waterfor 0.363 � x � 3.5 nm−1.

4386 B W King et al

the smaller sample size can be seen by comparing the uncertainties of the tissue measurementsto those of the bone sample. However, a larger sample also makes it more difficult to preparepure samples for measurement. Some types of tissues were unable to be studied because puresamples of the required size could not be obtained. As an example, in our work, grey and whitebrain tissue have not yet been separated clearly enough from each other to be distinguished.

The measured values of Fcoh at large x values are quite noisy. This is a result of the limitednumber of coherently scattered photons in these regions. Extracting the small coherentlyscattered signal from the much larger incoherent contribution is quite difficult. On the otherhand, since the coherently scattered photons represent such a small fraction of those scatteredin this region, an accurate knowledge of Fcoh is less critical than for smaller values of x wherecoherent scattering is the dominant mechanism.

The feasibility of using coherent scattering to distinguish between different types ofmaterials was investigated by computing the ratio of the tissue form factors to that of waterin figure 6. Fat shows the strongest difference from water, owing to the structural differencespresent between the two. The other soft tissues studied are much more similar. All of thesoft tissues exhibit a peak around 1 nm−1 which we ascribe to the fat content. In this region,it appears that imaging will yield the most contrast. Further study of these and other tissuesmay identify other cases where coherently scattered photons can play a diagnostic role.

Acknowledgments

This work was supported by the Natural Sciences and Engineering Research Council ofCanada. Thanks are extended to Philippe Gravelle in the Carleton University PhysicsDepartment machine shop for general fabrication assistance and Dr David Rogers for loan ofa dual channel electrometer.

Appendix A. Measured data

Table A1 gives the measured differential linear scattering coefficients at different scatteringangles for the materials studied in this paper as well as the measured linear attenuationcoefficients. Tables A2 and A3 give the measured form factors. The form factors can be usedto construct scattering cross sections for any combination of scattering angle and energy. Allresults in these tables represent the mean values of repeated measurements.

Table A1. Measured scattering coefficients μs (from equation (7)) and linear attenuationcoefficients μt for the materials studied. The subscripts give an upper bound on the percentageuncertainty for each measurement.

Energy Differential linear scattering coefficient μs (cm−1 sr−1) μt

(keV) 1.7◦ 3.2◦ 5.0◦ 6.3◦ 10.1◦ 12.6◦ 15.1◦ (cm−1)

Water30 0.016610 0.02615 0.05594 0.12213 0.11905 0.07195 0.05349 0.3652

35 0.01686 0.02596 0.08534 0.17232 0.09375 0.05464 0.05197 0.3052

40 0.01694 0.03234 0.14603 0.15372 0.05614 0.05603 0.04586 0.2712

45 0.01595 0.03925 0.16722 0.12443 0.05186 0.04562 0.03856 0.2433

50 0.01657 0.05174 0.14862 0.11792 0.04966 0.03833 0.03566 0.2303

55 0.01624 0.07823 0.12101 0.09712 0.04484 0.03422 0.03414 0.2233

60 0.019513 0.10825 0.11653 0.06264 0.03714 0.03327 0.03099 0.2163

Energy-dispersive measurements of x-ray coherent scattering form factors 4387

Table A1. (Continued.)

Energy Differential linear scattering coefficient μs (cm−1 sr−1) μt

(keV) 1.7◦ 3.2◦ 5.0◦ 6.3◦ 10.1◦ 12.6◦ 15.1◦ (cm−1)

65 0.01858 0.13955 0.11284 0.05022 0.03296 0.03184 0.03008 0.2063

70 0.02086 0.14254 0.08946 0.04965 0.03065 0.03244 0.03168 0.2093

75 0.023612 0.14114 0.06383 0.05085 0.03165 0.03345 0.03167 0.1954

80 0.02858 0.12682 0.04904 0.04795 0.03123 0.03082 0.02873 0.1924

85 0.03488 0.11675 0.04872 0.04468 0.03022 0.03485 0.03098 0.1904

90 0.03218 0.10623 0.04663 0.03867 0.02814 0.02645 0.02618 0.1924

95 0.04066 0.10343 0.04886 0.03415 0.02694 0.02515 0.02857 0.1944

100 0.04719 0.08853 0.04475 0.03067 0.02483 0.024710 0.02137 0.1755

105 0.056812 0.08729 0.04587 0.031914 0.02406 0.02649 0.022112 0.1886

110 0.064511 0.05904 0.03009 0.022515 0.021514 0.019513 0.015316 0.1748

Polyethylene30 0.012011 0.01962 0.10073 0.33817 0.04835 0.04054 0.03656 0.2613

35 0.00998 0.02526 0.52123 0.02617 0.03874 0.03883 0.03332 0.2283

40 0.01088 0.03863 0.04094 0.02302 0.04576 0.03204 0.03603 0.2163

45 0.01136 0.07344 0.02483 0.05264 0.03175 0.04143 0.03292 0.2103

50 0.01285 0.14896 0.02204 0.06446 0.03233 0.03402 0.03153 0.2033

55 0.01426 0.37883 0.02844 0.04174 0.03994 0.03044 0.03073 0.1973

60 0.01614 0.16936 0.03556 0.03186 0.03022 0.02806 0.03013 0.1913

65 0.01756 0.02446 0.05863 0.04693 0.02845 0.02716 0.03014 0.1893

70 0.02426 0.02307 0.03263 0.03203 0.03034 0.02844 0.03144 0.1904

75 0.03537 0.02685 0.03314 0.03287 0.02855 0.03025 0.02968 0.1884

80 0.04028 0.01786 0.03856 0.02793 0.02806 0.02644 0.02806 0.1734

85 0.06325 0.02486 0.03565 0.033418 0.02904 0.02954 0.03187 0.1754

90 0.08565 0.03437 0.02864 0.03495 0.02894 0.02716 0.02606 0.1804

95 0.16248 0.04297 0.02638 0.02739 0.02676 0.02456 0.02575 0.1705

100 0.23616 0.05146 0.026010 0.02976 0.02231 0.02317 0.02384 0.1755

105 0.24965 0.04277 0.02438 0.026310 0.02294 0.020910 0.02653 0.1527

110 0.19786 0.02836 0.026511 0.02307 0.01937 0.018611 0.020411 0.1649

Polystyrene30 0.049823 0.115520 0.285715 0.114515 0.059511 0.048512 0.042913 0.2684

35 0.068323 0.122520 0.188715 0.070115 0.060710 0.044211 0.039512 0.2543

40 0.090223 0.158020 0.096115 0.063115 0.045110 0.039511 0.042512 0.2224

45 0.111023 0.237920 0.067116 0.053115 0.038711 0.042611 0.040012 0.2244

50 0.115623 0.241320 0.058016 0.058614 0.036411 0.044111 0.033912 0.2144

55 0.112523 0.156720 0.055916 0.061214 0.041110 0.036311 0.034012 0.2054

60 0.105623 0.106120 0.053716 0.048316 0.043611 0.030312 0.030014 0.2124

65 0.108123 0.069820 0.059316 0.040415 0.040811 0.031112 0.033512 0.1974

70 0.117723 0.058821 0.056317 0.042117 0.036112 0.035813 0.027715 0.2005

75 0.148523 0.049721 0.051517 0.038717 0.039412 0.043213 0.035214 0.2015

80 0.160823 0.046121 0.037918 0.031818 0.028312 0.035513 0.033914 0.1876

85 0.201023 0.045921 0.033819 0.038518 0.031012 0.036713 0.032415 0.1846

90 0.202923 0.048621 0.035019 0.037718 0.029013 0.036713 0.030115 0.1756

95 0.188823 0.048222 0.027720 0.034819 0.029713 0.032014 0.025217 0.1767

100 0.179123 0.051222 0.029522 0.045120 0.034314 0.027717 0.033017 0.1997

4388 B W King et al

Table A1. (Continued.)

Energy Differential linear scattering coefficient μs (cm−1 sr−1) μt

(keV) 1.7◦ 3.2◦ 5.0◦ 6.3◦ 10.1◦ 12.6◦ 15.1◦ (cm−1)

105 0.136624 0.047124 0.036923 0.039123 0.029617 0.030918 0.027121 0.2068

110 0.093926 0.049327 0.014541 0.034331 0.023324 0.019129 0.036325 0.19312

Polycarbonate30 0.057823 0.118320 0.275215 0.160315 0.074011 0.068812 0.052713 0.3303

35 0.058823 0.160620 0.155015 0.104115 0.079810 0.048612 0.051512 0.2903

40 0.067723 0.292120 0.139615 0.077415 0.059310 0.047611 0.057012 0.2653

45 0.061923 0.278320 0.097815 0.063915 0.047911 0.054011 0.053512 0.2513

50 0.067423 0.221820 0.077116 0.071814 0.044111 0.051311 0.048012 0.2453

55 0.067023 0.144620 0.062716 0.071014 0.042311 0.045811 0.039612 0.2224

60 0.092123 0.110820 0.064416 0.061615 0.048411 0.040012 0.035313 0.2254

65 0.122423 0.111220 0.074316 0.050215 0.044911 0.037411 0.038512 0.2144

70 0.171823 0.086920 0.069617 0.050916 0.040912 0.036413 0.041014 0.2245

75 0.218823 0.072320 0.064217 0.044017 0.042712 0.038413 0.039414 0.1975

80 0.293123 0.070521 0.053617 0.040418 0.042412 0.040013 0.040214 0.2225

85 0.277723 0.057421 0.043118 0.040518 0.041312 0.039513 0.039314 0.2016

90 0.238423 0.049021 0.038619 0.037618 0.037413 0.039313 0.025916 0.1996

95 0.196523 0.052222 0.035020 0.045918 0.036813 0.038314 0.030116 0.1887

100 0.168423 0.055622 0.037721 0.055319 0.035214 0.025717 0.027619 0.1948

105 0.110524 0.062522 0.024925 0.037622 0.023817 0.028218 0.024821 0.15610

110 0.125126 0.054327 0.030931 0.022137 0.032321 0.026825 0.027428 0.22510

Nylon30 0.024824 0.053121 0.222816 0.362314 0.083411 0.066312 0.059413 0.3273

35 0.019624 0.064220 0.232415 0.069715 0.065111 0.051211 0.050412 0.2913

40 0.021924 0.088520 0.139915 0.061015 0.056210 0.049011 0.059012 0.2633

45 0.026623 0.131320 0.068616 0.059615 0.048510 0.050411 0.045812 0.2463

50 0.033723 0.341020 0.059616 0.078414 0.041211 0.045811 0.044712 0.2383

55 0.040923 0.244320 0.058516 0.068215 0.041111 0.044511 0.042212 0.2294

60 0.045623 0.252420 0.081616 0.066515 0.045911 0.036212 0.039813 0.2343

65 0.056223 0.082120 0.074016 0.056115 0.042211 0.035512 0.039912 0.2244

70 0.061023 0.054921 0.068317 0.034718 0.037712 0.039713 0.041014 0.2344

75 0.084423 0.056321 0.056117 0.047717 0.036612 0.048613 0.038614 0.2135

80 0.096323 0.054021 0.056317 0.045717 0.034112 0.039013 0.042714 0.2085

85 0.124823 0.060821 0.051118 0.042718 0.038912 0.044813 0.034015 0.2066

90 0.172223 0.066821 0.038319 0.039118 0.033313 0.034414 0.035915 0.2096

95 0.213123 0.068421 0.040919 0.048618 0.031413 0.035514 0.033516 0.1937

100 0.228223 0.057022 0.038821 0.033621 0.028215 0.025117 0.040516 0.1938

105 0.233924 0.064623 0.032725 0.038324 0.026617 0.033618 0.038419 0.2168

110 0.195825 0.069226 0.050226 0.040030 0.028422 0.025926 0.052322 0.22910

Fat30 0.06593 0.04478 0.254410 0.14026 0.070010 0.053814 0.040313 0.2823

35 0.03989 0.05419 0.21132 0.07724 0.05964 0.04027 0.04102 0.2423

40 0.02917 0.08885 0.08844 0.06077 0.04841 0.03813 0.03661 0.2193

45 0.02849 0.14975 0.06582 0.05866 0.03772 0.03932 0.03203 0.2013

50 0.031512 0.21832 0.06126 0.05907 0.03703 0.03528 0.03024 0.1893

Energy-dispersive measurements of x-ray coherent scattering form factors 4389

Table A1. (Continued.)

Energy Differential linear scattering coefficient μs (cm−1 sr−1) μt

(keV) 1.7◦ 3.2◦ 5.0◦ 6.3◦ 10.1◦ 12.6◦ 15.1◦ (cm−1)

55 0.02962 0.18035 0.05841 0.05901 0.03853 0.03174 0.02967 0.1833

60 0.03644 0.11235 0.06727 0.044812 0.038511 0.030912 0.02892 0.1933

65 0.03998 0.07148 0.06832 0.04191 0.03531 0.03066 0.031813 0.1824

70 0.04977 0.061718 0.05571 0.038511 0.02854 0.02883 0.02677 0.1834

75 0.07195 0.05752 0.04598 0.03498 0.03204 0.02941 0.02757 0.1695

80 0.08334 0.05485 0.03669 0.033617 0.03311 0.02677 0.02955 0.1685

85 0.13136 0.05152 0.039512 0.03272 0.03012 0.02697 0.02586 0.1725

90 0.14594 0.05041 0.034021 0.02621 0.02853 0.02433 0.02434 0.1646

95 0.16847 0.05103 0.03037 0.031016 0.02626 0.02266 0.02185 0.1626

100 0.17062 0.05682 0.03131 0.02921 0.02504 0.02498 0.02432 0.1677

105 0.150417 0.05185 0.03742 0.025228 0.02627 0.025022 0.023628 0.1718

110 0.142012 0.04226 0.03291 0.023412 0.019025 0.02076 0.023316 0.17311

Muscle30 0.049913 0.04258 0.10098 0.15404 0.12413 0.07073 0.06189 0.4072

35 0.046510 0.04192 0.11894 0.16858 0.09594 0.05957 0.06027 0.3412

40 0.045313 0.05003 0.14908 0.14895 0.06156 0.05803 0.04738 0.3002

45 0.042610 0.07075 0.17245 0.12539 0.05703 0.05436 0.04417 0.2842

50 0.038912 0.08767 0.14855 0.11688 0.05535 0.04432 0.04239 0.2632

55 0.03477 0.10367 0.12106 0.098011 0.04966 0.04176 0.04087 0.2513

60 0.03478 0.11473 0.11769 0.07493 0.044710 0.03595 0.036711 0.2423

65 0.03364 0.13967 0.10904 0.05767 0.03795 0.03558 0.034812 0.2303

70 0.04006 0.155811 0.09116 0.05868 0.041610 0.040013 0.04256 0.2353

75 0.04923 0.14726 0.07577 0.06338 0.042511 0.03979 0.039814 0.2413

80 0.04813 0.12185 0.05678 0.05637 0.038412 0.03619 0.03638 0.2183

85 0.05976 0.11458 0.051112 0.04828 0.03819 0.034611 0.035813 0.2114

90 0.05877 0.10508 0.053911 0.04818 0.035013 0.02917 0.032111 0.2224

95 0.067411 0.102111 0.050211 0.042911 0.032010 0.028712 0.03296 0.2104

100 0.07694 0.094012 0.051311 0.048112 0.035513 0.033417 0.029217 0.2224

105 0.098214 0.096912 0.05508 0.04569 0.034317 0.027516 0.031318 0.2475

110 0.103714 0.08017 0.054623 0.037412 0.03545 0.024125 0.022917 0.2397

Liver30 0.05356 0.05206 0.09666 0.15027 0.130310 0.080711 0.068714 0.4102

35 0.04886 0.049514 0.119910 0.17388 0.10128 0.06019 0.05918 0.3462

40 0.04896 0.05839 0.15608 0.151610 0.06318 0.05636 0.050910 0.2992

45 0.04676 0.07459 0.16839 0.125011 0.059210 0.051911 0.044312 0.2752

50 0.04335 0.08786 0.14848 0.11726 0.05787 0.04549 0.041410 0.2563

55 0.04075 0.10398 0.12556 0.09748 0.05267 0.04117 0.04286 0.2433

60 0.03696 0.12097 0.12397 0.06386 0.04317 0.03767 0.035411 0.2333

65 0.03896 0.14174 0.11498 0.05526 0.04194 0.03628 0.03378 0.2293

70 0.04078 0.13978 0.09099 0.055212 0.039111 0.03707 0.034110 0.2273

75 0.05185 0.13316 0.06418 0.05317 0.03863 0.03525 0.03657 0.2124

80 0.049011 0.11399 0.05448 0.05166 0.03697 0.033210 0.035915 0.2084

85 0.06109 0.10565 0.05299 0.04117 0.03485 0.035415 0.034610 0.2034

4390 B W King et al

Table A1. (Continued.)

Energy Differential linear scattering coefficient μs (cm−1 sr−1) μt

(keV) 1.7◦ 3.2◦ 5.0◦ 6.3◦ 10.1◦ 12.6◦ 15.1◦ (cm−1)

90 0.05798 0.09819 0.05075 0.04518 0.03106 0.02615 0.030015 0.2024

95 0.07109 0.10055 0.04769 0.04594 0.03197 0.03065 0.032312 0.2084

100 0.073511 0.09637 0.046410 0.034615 0.029216 0.02695 0.027617 0.2005

105 0.076010 0.08817 0.055312 0.03422 0.03118 0.025521 0.02268 0.2006

110 0.08097 0.068411 0.049610 0.02794 0.028316 0.014317 0.019216 0.1659

Kidney30 0.039724 0.030922 0.063617 0.137815 0.115511 0.065712 0.046513 0.3893

35 0.032424 0.029122 0.085316 0.169614 0.093011 0.050911 0.037413 0.3203

40 0.034224 0.040221 0.127815 0.132114 0.055311 0.048211 0.040612 0.2793

45 0.031024 0.049121 0.141715 0.123214 0.053811 0.045111 0.035712 0.2573

50 0.029024 0.069120 0.132215 0.110314 0.049711 0.037511 0.029612 0.2464

55 0.025724 0.077620 0.110515 0.088614 0.046711 0.033111 0.029412 0.2324

60 0.029124 0.099720 0.098416 0.056915 0.042412 0.030012 0.026514 0.2184

65 0.028024 0.117020 0.095215 0.041415 0.036411 0.030411 0.025712 0.2074

70 0.030725 0.121120 0.080416 0.043816 0.037612 0.027813 0.025714 0.1995

75 0.033225 0.132820 0.066617 0.047716 0.036912 0.035213 0.026815 0.2205

80 0.038225 0.103321 0.048117 0.046117 0.035212 0.029813 0.021615 0.2015

85 0.049025 0.108521 0.042018 0.026820 0.031513 0.029513 0.018416 0.1826

90 0.050624 0.093621 0.038518 0.039418 0.028813 0.024114 0.017117 0.1936

95 0.054124 0.092921 0.044318 0.039519 0.030814 0.025915 0.021417 0.1897

100 0.053025 0.079421 0.041419 0.031221 0.025615 0.019517 0.019919 0.1758

105 0.070525 0.064423 0.037422 0.026824 0.028216 0.015121 0.025919 0.16610

110 0.081926 0.064425 0.048025 0.019735 0.019923 0.019026 0.023326 0.19711

Bone30 0.242025 0.075927 0.193926 0.225626 0.281723 0.247323 0.149429 2.2734

35 0.238524 0.097525 0.205825 0.237926 0.209422 0.131323 0.120928 1.6964

40 0.131924 0.111024 0.223125 0.744625 0.226622 0.126623 0.123728 1.1935

45 0.097724 0.113724 0.259624 0.155425 0.148522 0.141923 0.099028 0.8876

50 0.110824 0.110624 0.690524 0.233925 0.133322 0.116723 0.082928 0.7386

55 0.073924 0.149824 0.209224 0.177825 0.156422 0.114323 0.075728 0.7006

60 0.060026 0.177024 0.232425 0.249525 0.128222 0.094223 0.068429 0.6137

65 0.069525 0.157924 0.180725 0.131125 0.110022 0.079623 0.067828 0.4659

70 0.066428 0.236624 0.173825 0.140526 0.107523 0.081523 0.057229 0.5999

75 0.091728 0.420023 0.223725 0.152226 0.094623 0.090623 0.069429 0.43012

80 0.097130 0.525123 0.201625 0.104626 0.087023 0.088223 0.055629 0.35815

85 0.124429 0.225024 0.127526 0.100327 0.086023 0.072124 0.055329 0.40914

90 0.104726 0.141125 0.147826 0.108327 0.075423 0.066224 0.058329 0.35117

95 0.116426 0.171524 0.151526 0.105727 0.074623 0.065024 0.053430 0.29422

100 0.135326 0.186425 0.106027 0.118628 0.079624 0.059825 0.053130 0.42218

105 0.111028 0.127526 0.084429 0.062031 0.049125 0.059226 0.040532 0.27131

110 0.069037 0.127729 0.161629 0.138531 0.047928 0.036031 0.046834 0.17766

Energy-dispersive measurements of x-ray coherent scattering form factors 4391

Table A2. Measured coherent form factors for water and plastics. The subscripts give an upperbound on the percentage uncertainty for each measurement. The values of Finc in this table wereobtained from the composition data of table 2 and the data of Hubbell and Øverbø (1979). Theyhave been used to extract the Fcoh values and so must also be used to construct scatter cross sectionsat arbitrary θ . The water and polyethylene data include measurements subsequent to King (2009).The other three plastics are from the same experiments reported in King (2009) but here are on amore detailed x-grid.

x Water Polyethylene Polystyrene Polycarbonate Nylon

(nm−1) Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc

0.363 0.8134 0.026 0.7004 0.035 1.42212 0.032 1.30612 0.029 0.82513 0.0310.378 0.7824 0.028 0.6786 0.038 1.44912 0.035 1.36512 0.032 0.82413 0.0340.386 0.7765 0.029 0.6745 0.039 1.42312 0.036 1.33312 0.033 0.82413 0.0350.402 0.7654 0.031 0.6704 0.043 1.55512 0.039 1.39412 0.035 0.77313 0.0380.410 0.7744 0.033 0.6086 0.044 1.57512 0.040 1.41712 0.037 0.80713 0.0400.419 0.7743 0.034 0.6586 0.046 1.63012 0.042 1.46812 0.038 0.79613 0.0410.436 0.7574 0.037 0.6504 0.050 1.69012 0.045 1.41612 0.041 0.78613 0.0440.445 0.7503 0.038 0.6155 0.052 1.72112 0.047 1.45912 0.043 0.77113 0.0460.463 0.7634 0.041 0.6175 0.056 1.81012 0.051 1.43212 0.046 0.81713 0.0500.473 0.7703 0.043 0.6346 0.058 1.82712 0.053 1.47412 0.048 0.82313 0.0520.492 0.7422 0.046 0.6335 0.063 1.93812 0.057 1.44712 0.052 0.82613 0.0560.502 0.7423 0.048 0.6325 0.065 1.95712 0.059 1.43712 0.054 0.84513 0.0580.513 0.7772 0.050 0.6516 0.067 1.96712 0.062 1.45412 0.056 0.88413 0.0600.523 0.7533 0.052 0.6245 0.070 2.00512 0.064 1.41512 0.058 0.88213 0.0630.545 0.7544 0.056 0.6344 0.076 2.02312 0.069 1.47812 0.063 0.94013 0.0680.556 0.7543 0.058 0.6544 0.078 1.99112 0.072 1.44112 0.065 0.93413 0.0700.579 0.7545 0.063 0.6435 0.084 2.04812 0.077 1.46812 0.071 1.00612 0.0760.591 0.7256 0.065 0.6414 0.088 2.04212 0.080 1.48812 0.073 0.98913 0.0780.615 0.7394 0.071 0.6582 0.094 2.00812 0.086 1.46912 0.079 1.05512 0.0840.628 0.7294 0.073 0.6633 0.098 1.98612 0.089 1.52112 0.082 1.07612 0.0880.653 0.7423 0.079 0.6753 0.105 1.97012 0.096 1.58512 0.088 1.07813 0.0940.667 0.7694 0.082 0.6884 0.109 1.88012 0.100 1.62012 0.091 1.14213 0.0980.694 0.9283 0.088 0.8415 0.117 2.06110 0.107 1.97711 0.098 1.29611 0.1050.709 0.8954 0.091 0.8213 0.121 2.05810 0.111 2.01911 0.101 1.28811 0.1090.738 0.9135 0.098 0.8704 0.130 2.02810 0.119 2.04310 0.109 1.41911 0.1170.753 0.9632 0.102 0.8913 0.135 2.01110 0.123 2.18710 0.113 1.49011 0.1210.784 0.9613 0.110 0.9063 0.145 2.12410 0.132 2.39710 0.121 1.47211 0.1300.800 0.9753 0.114 0.9402 0.150 2.13210 0.137 2.52610 0.125 1.43411 0.1340.833 1.0062 0.122 0.9773 0.160 2.13610 0.146 2.74210 0.134 1.56111 0.1440.850 0.9973 0.126 1.0143 0.166 2.26810 0.152 2.90710 0.139 1.57211 0.1490.885 1.0212 0.135 1.0863 0.177 2.35910 0.162 3.11110 0.149 1.66811 0.1590.903 1.0693 0.140 1.1904 0.183 2.42910 0.168 3.17010 0.154 1.70711 0.1650.940 1.0802 0.150 1.2815 0.196 2.62710 0.179 3.22710 0.165 1.74411 0.1760.960 1.1122 0.155 1.3646 0.202 2.71110 0.185 3.22010 0.170 1.83111 0.1821.00 1.1602 0.166 1.4797 0.215 2.88310 0.197 3.08010 0.182 2.03611 0.1941.02 1.1773 0.172 1.5448 0.222 2.96110 0.203 3.05410 0.187 2.22210 0.2011.06 1.2652 0.183 1.7098 0.236 2.97810 0.216 2.89610 0.199 2.94910 0.2141.08 1.3613 0.189 1.8648 0.243 3.1988 0.223 3.0208 0.206 2.8708 0.2201.11 1.4854 0.196 1.8608 0.251 3.1498 0.230 2.9458 0.212 3.5158 0.2271.13 1.4362 0.202 2.35511 0.258 3.0878 0.237 2.8488 0.219 3.26810 0.234

4392 B W King et al

Table A2. (Continued.)

x Water Polyethylene Polystyrene Polycarbonate Nylon

(nm−1) Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc

1.15 1.5233 0.208 2.36710 0.266 2.9778 0.244 2.7298 0.225 3.5828 0.2411.20 1.6263 0.222 4.78118 0.282 2.7728 0.258 2.4738 0.239 2.9918 0.2561.22 1.6783 0.229 3.50715 0.290 2.6848 0.266 2.3238 0.246 2.7208 0.2631.27 1.7635 0.243 2.3379 0.307 2.3868 0.282 2.2468 0.261 2.9618 0.2791.30 1.9142 0.250 2.3186 0.316 2.2598 0.290 2.2398 0.269 3.2178 0.2871.33 2.0172 0.258 2.9339 0.325 2.1358 0.298 2.0828 0.277 3.4728 0.2951.35 2.1072 0.266 2.6437 0.334 1.8678 0.307 2.1908 0.285 3.1147 0.3041.38 2.2192 0.273 1.8096 0.343 1.9118 0.315 2.1198 0.293 2.6178 0.3121.44 2.4342 0.289 1.22923 0.361 1.6918 0.332 2.0658 0.309 1.7148 0.3291.47 2.3832 0.297 1.23324 0.370 1.7698 0.341 1.9078 0.317 1.6728 0.3381.53 2.5051 0.314 1.15629 0.389 1.6058 0.359 1.8118 0.334 1.4619 0.3561.56 2.5281 0.323 1.12631 0.399 1.5329 0.368 1.8608 0.343 1.3969 0.3651.62 2.5142 0.340 1.17428 0.418 1.4059 0.386 1.7138 0.361 1.3669 0.3831.66 2.4231 0.349 1.25823 0.428 1.3559 0.395 1.6788 0.369 1.3729 0.3921.73 2.3812 0.366 1.03232 0.447 1.3129 0.414 1.5429 0.387 1.2949 0.4111.76 2.3461 0.375 0.96835 0.457 1.3479 0.423 1.5189 0.396 1.2299 0.4201.80 2.2761 0.385 0.96833 0.466 1.3679 0.432 1.4959 0.405 1.2409 0.4301.83 2.2142 0.394 0.97630 0.476 1.2459 0.442 1.5359 0.414 1.2849 0.4391.87 2.1892 0.403 0.95832 0.486 1.2359 0.451 1.4329 0.424 1.16510 0.4491.95 2.0931 0.422 1.10622 0.505 1.2789 0.470 1.4469 0.442 1.22110 0.4671.99 2.0821 0.431 1.34013 0.515 1.2599 0.479 1.3369 0.451 1.3029 0.4772.07 2.0302 0.450 1.05922 0.534 1.24610 0.498 1.3259 0.470 1.5189 0.4962.11 1.9932 0.459 1.06719 0.544 1.2819 0.507 1.3579 0.479 1.5539 0.5052.20 2.0313 0.478 1.6055 0.562 1.3617 0.526 1.4807 0.497 1.5047 0.5242.24 1.9833 0.487 1.6145 0.572 1.4177 0.535 1.5487 0.507 1.5217 0.5332.34 1.9333 0.506 1.4976 0.590 1.3287 0.553 1.5097 0.525 1.4827 0.5512.38 1.8873 0.515 1.3219 0.599 1.2747 0.562 1.5077 0.534 1.4347 0.5602.48 1.7133 0.534 1.06812 0.616 1.3387 0.579 1.4887 0.552 1.3477 0.5782.53 1.6523 0.543 1.1299 0.625 1.3067 0.588 1.5187 0.560 1.3507 0.5872.64 1.4443 0.561 1.0169 0.641 1.1398 0.605 1.3877 0.577 1.2217 0.6042.69 1.3923 0.570 1.0239 0.649 1.0929 0.613 1.3348 0.586 1.2888 0.6122.80 1.3182 0.588 1.0536 0.665 1.0699 0.629 1.2588 0.602 1.2128 0.6282.86 1.2814 0.597 1.2454 0.673 0.95110 0.637 1.2438 0.611 1.2288 0.6362.98 1.2212 0.614 1.0995 0.687 0.95010 0.652 1.1418 0.626 1.1938 0.6523.04 1.2512 0.622 0.9599 0.694 0.9809 0.659 1.0659 0.634 1.1518 0.6593.17 1.2253 0.638 0.8459 0.706 0.92910 0.672 1.1089 0.648 1.0219 0.6723.23 1.1955 0.646 0.9298 0.712 1.04410 0.678 1.02511 0.655 1.08210 0.6793.36 1.2326 0.662 0.96610 0.724 1.04910 0.691 1.10810 0.668 1.13410 0.6923.43 1.2196 0.669 0.9645 0.730 1.00010 0.697 1.06710 0.675 1.03910 0.6983.57 1.2244 0.684 0.9447 0.741 0.93011 0.708 1.1129 0.687 1.1419 0.7103.65 1.1815 0.691 0.8507 0.746 1.00210 0.714 1.1119 0.694 0.97711 0.7163.80 1.1213 0.705 0.9245 0.757 0.98810 0.725 1.1069 0.706 1.03610 0.7283.88 1.0666 0.712 0.9225 0.762 0.99210 0.730 1.01010 0.711 0.98610 0.7333.95 1.0424 0.719 0.9703 0.766 0.87911 0.735 1.04310 0.717 1.07010 0.7394.04 1.0666 0.725 1.0073 0.771 0.95111 0.740 1.1279 0.722 1.2109 0.743

Energy-dispersive measurements of x-ray coherent scattering form factors 4393

Table A2. (Continued.)

x Water Polyethylene Polystyrene Polycarbonate Nylon

(nm−1) Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc

4.12 1.0225 0.731 0.9922 0.774 1.07310 0.744 1.1729 0.727 1.1159 0.7484.29 0.9536 0.742 0.8022 0.782 1.01010 0.752 1.06610 0.736 1.0889 0.7574.38 0.9433 0.748 0.8223 0.785 0.99710 0.756 1.1079 0.741 1.01910 0.7614.56 0.8916 0.759 0.8643 0.793 0.95211 0.764 1.1009 0.750 1.04810 0.7694.65 0.8806 0.764 0.7706 0.796 0.92011 0.768 1.0789 0.754 0.97810 0.7734.84 0.8116 0.774 0.7343 0.803 0.83512 0.776 0.99810 0.763 0.96710 0.7814.94 0.7896 0.779 0.7412 0.807 0.85112 0.780 1.00310 0.767 0.88311 0.7855.14 0.7637 0.788 0.7183 0.813 0.78513 0.787 0.90511 0.775 0.84712 0.7935.25 0.7847 0.792 0.6745 0.816 0.72014 0.790 0.90711 0.778 0.83912 0.7965.47 0.7275 0.800 0.6604 0.822 0.74214 0.797 0.85312 0.786 0.85612 0.8035.58 0.7395 0.804 0.6715 0.825 0.70315 0.800 0.83912 0.789 0.82712 0.8065.81 0.7078 0.811 0.6144 0.831 0.74914 0.807 0.74414 0.797 0.81313 0.8135.93 0.7097 0.815 0.6283 0.834 0.65616 0.811 0.78513 0.800 0.74714 0.8166.17 0.5338 0.822 0.48813 0.840 0.59617 0.817 0.70114 0.808 0.63116 0.8236.30 0.6335 0.825 0.5984 0.843 0.66615 0.821 0.68215 0.811 0.61917 0.8266.56 0.56510 0.831 0.5933 0.849 0.71915 0.828 0.66916 0.818 0.71615 0.8336.69 0.57311 0.834 0.6262 0.852 0.68016 0.832 0.69215 0.822 0.78213 0.8366.97 0.6219 0.840 0.6323 0.859 0.72315 0.839 0.71615 0.830 0.63722 0.8437.11 0.6035 0.843 0.5075 0.862 0.72114 0.842 0.67015 0.833 0.74314 0.8467.40 0.5895 0.849 0.5597 0.869 0.72014 0.850 0.54834 0.841 0.68715 0.8537.87 0.6157 0.857 0.6005 0.879 0.72916 0.861 0.70316 0.852 0.71816 0.8638.03 0.56010 0.860 0.5695 0.882 0.66325 0.865 0.78214 0.856 0.75115 0.8678.70 0.5579 0.871 0.5624 0.895 0.69017 0.880 0.72615 0.871 0.65618 0.8819.25 0.4538 0.879 0.5058 0.906 0.65018 0.892 0.66917 0.883 0.66618 0.891

Table A3. Measured coherent form factors for tissues. The subscripts and values for Finc have themeanings given in table A2. The fat data include measurements subsequent to King (2009).

x Fat Muscle Liver Kidney Bone

(nm−1) Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc

0.363 1.5794 0.032 1.2908 0.026 1.4003 0.027 1.17713 0.027 2.18812 0.0280.378 1.5255 0.035 1.2985 0.029 1.3594 0.029 1.15212 0.029 2.44812 0.0300.386 1.4594 0.036 1.2745 0.030 1.3324 0.030 1.03813 0.030 2.42412 0.0320.402 1.2973 0.039 1.3246 0.032 1.3502 0.033 1.14912 0.032 2.26312 0.0340.410 1.2375 0.040 1.2876 0.033 1.3302 0.034 1.09012 0.034 2.25212 0.0350.419 1.1544 0.042 1.2886 0.035 1.3205 0.036 1.08212 0.035 2.16412 0.0370.436 1.1514 0.045 1.2507 0.037 1.3474 0.038 1.09812 0.038 1.99812 0.0390.445 1.1104 0.047 1.2487 0.039 1.3254 0.040 1.07712 0.039 1.92712 0.0410.463 1.0875 0.051 1.2546 0.042 1.2954 0.043 1.06412 0.042 1.76812 0.0440.473 1.0845 0.053 1.2787 0.044 1.3124 0.045 1.06812 0.044 1.72912 0.0460.492 1.0766 0.057 1.2227 0.047 1.3022 0.049 1.04312 0.048 1.50412 0.0490.502 1.0777 0.059 1.2216 0.049 1.3184 0.050 1.04312 0.050 1.54112 0.0510.513 1.0614 0.062 1.2316 0.051 1.2622 0.052 1.07212 0.051 1.48112 0.053

4394 B W King et al

Table A3. (Continued.)

x Fat Muscle Liver Kidney Bone

(nm−1) Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc

0.523 1.0565 0.064 1.2317 0.053 1.2735 0.054 1.06712 0.053 1.45412 0.0550.545 1.0433 0.069 1.1986 0.057 1.2663 0.059 1.02113 0.058 1.41912 0.0590.556 1.0704 0.072 1.2255 0.059 1.2562 0.061 1.05212 0.060 1.37413 0.0610.579 1.1074 0.077 1.1656 0.064 1.2353 0.066 1.02413 0.065 1.43812 0.0650.591 1.0792 0.080 1.1387 0.067 1.2384 0.068 0.97313 0.067 1.48812 0.0670.615 1.0964 0.086 1.1284 0.072 1.1932 0.074 0.99513 0.072 1.41212 0.0720.628 1.0592 0.089 1.1255 0.074 1.1694 0.076 0.96413 0.075 1.36912 0.0750.653 1.1016 0.096 1.0894 0.080 1.1652 0.082 0.92213 0.081 1.26113 0.0800.667 1.1095 0.100 1.1094 0.083 1.1514 0.085 0.97513 0.084 1.19213 0.0830.694 1.2707 0.107 1.2256 0.089 1.3105 0.092 1.02112 0.090 1.37414 0.0880.709 1.3456 0.111 1.2073 0.093 1.2892 0.095 1.06212 0.094 1.43914 0.0910.738 1.3829 0.119 1.1444 0.100 1.2647 0.102 1.05012 0.101 1.31914 0.0970.753 1.3937 0.123 1.2164 0.103 1.2935 0.106 1.03312 0.104 1.40713 0.1010.784 1.4386 0.132 1.2054 0.111 1.3073 0.114 0.97812 0.112 1.43313 0.1070.800 1.4697 0.137 1.2584 0.115 1.3044 0.118 1.02212 0.116 1.49213 0.1110.833 1.5623 0.147 1.2576 0.124 1.3134 0.127 1.07212 0.125 1.49413 0.1180.850 1.6305 0.152 1.2674 0.128 1.3505 0.131 1.01812 0.129 1.50313 0.1220.885 1.8143 0.163 1.3163 0.137 1.3924 0.141 1.12112 0.138 1.56313 0.1290.903 1.9304 0.168 1.3593 0.142 1.3956 0.146 1.08712 0.143 1.60813 0.1330.940 2.1123 0.180 1.4283 0.152 1.4635 0.156 1.15312 0.153 1.55813 0.1420.960 2.1992 0.186 1.4492 0.157 1.5174 0.161 1.23112 0.158 1.57613 0.1461.00 2.3773 0.198 1.5513 0.168 1.5685 0.172 1.33411 0.169 1.61813 0.1551.02 2.5342 0.204 1.5523 0.174 1.5975 0.178 1.31611 0.175 1.54413 0.1601.06 2.9091 0.218 1.6833 0.185 1.6634 0.190 1.36211 0.187 1.64613 0.1691.08 3.2432 0.224 1.8764 0.191 1.8803 0.196 1.6459 0.193 2.07514 0.1741.11 3.2831 0.231 1.8633 0.198 1.8634 0.203 1.5829 0.199 1.99314 0.1791.13 3.2074 0.238 1.9406 0.204 1.8533 0.209 1.6329 0.206 2.02014 0.1841.15 3.1472 0.246 1.9434 0.210 1.9316 0.216 1.6039 0.212 1.95614 0.1901.20 2.9663 0.261 1.9984 0.224 1.9705 0.229 1.6839 0.226 2.04413 0.2001.22 2.8661 0.268 1.9953 0.231 1.9905 0.236 1.6999 0.233 1.99713 0.2061.27 2.8352 0.284 2.1022 0.245 2.0986 0.251 1.8308 0.247 2.09013 0.2181.30 2.7132 0.292 2.1534 0.252 2.1364 0.259 1.8208 0.254 2.04313 0.2241.33 2.4473 0.301 2.1474 0.260 2.1444 0.266 1.9838 0.262 2.00313 0.2301.35 2.1475 0.309 2.2942 0.268 2.2835 0.274 2.0508 0.270 2.13414 0.2361.38 1.9965 0.318 2.3762 0.275 2.2635 0.282 2.1848 0.277 2.29914 0.2421.44 1.8363 0.335 2.3564 0.291 2.3905 0.298 2.2818 0.293 2.45814 0.2551.47 1.7823 0.344 2.3764 0.299 2.3995 0.306 2.2128 0.302 2.26514 0.2621.53 1.6961 0.362 2.4295 0.316 2.4554 0.323 2.2798 0.318 2.22414 0.2751.56 1.6464 0.371 2.4414 0.325 2.4365 0.332 2.4448 0.327 2.31714 0.2821.62 1.5613 0.390 2.4586 0.342 2.4104 0.349 2.3178 0.344 2.50313 0.2971.66 1.4334 0.399 2.3745 0.351 2.3645 0.358 2.3658 0.353 2.58413 0.3041.73 1.4444 0.418 2.2803 0.368 2.2604 0.376 2.2238 0.371 3.53913 0.318

Energy-dispersive measurements of x-ray coherent scattering form factors 4395

Table A3. (Continued.)

x Fat Muscle Liver Kidney Bone

(nm−1) Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc

1.76 1.4294 0.427 2.2963 0.377 2.2795 0.385 2.0788 0.380 4.08713 0.3261.80 1.3755 0.437 2.2194 0.386 2.2225 0.394 2.0358 0.389 3.98913 0.3331.83 1.3654 0.446 2.2034 0.396 2.1275 0.404 2.0578 0.398 3.38213 0.3411.87 1.4323 0.456 2.1544 0.405 2.0974 0.413 1.9678 0.407 2.81213 0.3491.95 1.4053 0.475 2.1034 0.423 2.0414 0.431 1.9518 0.426 2.06514 0.3641.99 1.3804 0.484 2.0704 0.433 2.0305 0.441 2.0208 0.435 1.86214 0.3722.07 1.4173 0.503 2.0264 0.451 2.0323 0.460 1.8588 0.454 1.86514 0.3882.11 1.3833 0.513 2.0104 0.461 1.9964 0.469 1.8788 0.463 1.94014 0.3962.20 1.5133 0.531 2.0214 0.479 2.0845 0.488 2.0026 0.482 2.56012 0.4122.24 1.5333 0.541 2.0594 0.489 2.0364 0.497 1.8926 0.491 2.46412 0.4202.34 1.5413 0.559 1.9134 0.507 1.9136 0.516 1.8426 0.510 2.07913 0.4362.38 1.4355 0.568 1.9023 0.516 1.8895 0.525 1.8006 0.519 2.01813 0.4442.48 1.3652 0.586 1.7143 0.535 1.7544 0.543 1.6777 0.537 2.09612 0.4602.53 1.3571 0.594 1.6823 0.544 1.6805 0.552 1.5997 0.546 2.29612 0.4682.64 1.2631 0.611 1.4614 0.562 1.5127 0.570 1.5557 0.564 2.41612 0.4842.69 1.2626 0.620 1.4383 0.571 1.5178 0.579 1.3118 0.573 2.36613 0.4922.80 1.1189 0.636 1.3786 0.588 1.3723 0.596 1.2418 0.591 2.33313 0.5082.86 1.1197 0.644 1.2843 0.597 1.3795 0.605 1.2128 0.599 1.97313 0.5162.98 1.1264 0.659 1.2942 0.614 1.2846 0.622 1.1568 0.616 1.54014 0.5313.04 1.0743 0.667 1.3023 0.622 1.2157 0.630 1.1658 0.624 1.57214 0.5393.17 0.9457 0.680 1.2865 0.638 1.2685 0.645 1.1488 0.640 1.62514 0.5533.23 1.0041 0.687 1.2595 0.645 1.3369 0.653 1.03211 0.647 1.66818 0.5613.36 0.9891 0.700 1.2884 0.660 1.2738 0.668 0.99311 0.663 1.91317 0.5753.43 1.0153 0.706 1.2257 0.668 1.2208 0.675 0.98611 0.670 1.92917 0.5823.57 1.0332 0.718 1.3458 0.682 1.25911 0.689 1.07010 0.684 1.44119 0.5963.65 0.9891 0.724 1.2165 0.689 1.21410 0.696 0.98810 0.691 1.35019 0.6033.80 0.9621 0.735 1.2555 0.703 1.1888 0.710 0.93811 0.705 1.56518 0.6163.88 0.9596 0.741 1.2416 0.710 1.08710 0.716 0.93811 0.712 1.55818 0.6233.95 0.9321 0.746 1.1608 0.716 1.1977 0.722 1.02010 0.718 1.70317 0.6304.04 0.9421 0.751 1.1633 0.722 1.1269 0.728 0.90311 0.724 1.55818 0.6364.12 0.9944 0.756 1.1397 0.728 1.1318 0.734 0.84812 0.730 1.44019 0.6424.29 0.9058 0.764 1.0386 0.739 1.0829 0.745 0.78013 0.741 1.49318 0.6534.38 0.8681 0.768 1.0567 0.745 1.0388 0.750 0.77713 0.746 1.39819 0.6594.56 0.8573 0.777 0.9806 0.755 1.0019 0.760 0.73213 0.757 1.37519 0.6714.65 0.8411 0.781 0.9605 0.760 0.93410 0.765 0.62816 0.762 1.22421 0.6764.84 0.7195 0.789 0.8856 0.770 0.8897 0.775 0.65316 0.771 1.34520 0.6874.94 0.7524 0.793 0.8925 0.775 0.90010 0.780 0.64116 0.776 1.27820 0.6925.14 0.7109 0.800 0.8774 0.784 0.90811 0.788 0.66415 0.785 1.23321 0.7025.25 0.6679 0.803 0.8898 0.788 0.85812 0.792 0.60817 0.789 1.08023 0.7075.47 0.6964 0.810 0.8808 0.796 0.82514 0.800 0.59518 0.797 1.09223 0.7175.58 0.6962 0.813 0.8515 0.800 0.85613 0.803 0.53620 0.801 1.03424 0.7215.81 0.6491 0.820 0.8096 0.807 0.8458 0.811 0.51422 0.808 1.05024 0.731

4396 B W King et al

Table A3. (Contiuned.)

x Fat Muscle Liver Kidney Bone

(nm−1) Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc Fcoh Finc

5.93 0.6701 0.823 0.8149 0.811 0.78014 0.814 0.50423 0.812 1.03524 0.7356.17 0.60512 0.830 0.68310 0.818 0.6439 0.821 0.60717 0.819 1.08718 0.7446.30 0.5919 0.833 0.7669 0.821 0.7176 0.824 0.50222 0.822 1.04919 0.7486.56 0.56810 0.839 0.71012 0.827 0.65413 0.831 0.41431 0.828 1.05424 0.7576.69 0.6146 0.842 0.7039 0.830 0.65612 0.834 0.31550 0.831 0.97225 0.7616.97 0.66211 0.849 0.72411 0.837 0.66016 0.840 0.35343 0.838 0.97026 0.7697.11 0.54916 0.852 0.72312 0.840 0.65015 0.843 0.46825 0.841 1.01620 0.7747.40 0.5821 0.859 0.67914 0.846 0.59110 0.849 0.50822 0.847 0.95821 0.7827.87 0.5203 0.869 0.7579 0.855 0.6595 0.858 0.46029 0.856 0.83930 0.7948.03 0.5863 0.872 0.74315 0.858 0.61618 0.861 0.43132 0.859 0.92327 0.7988.70 0.5352 0.885 0.71513 0.869 0.64015 0.873 0.32851 0.870 0.75734 0.8159.25 0.53413 0.895 0.63816 0.878 0.48416 0.882 0.36942 0.879 0.76234 0.827

References

Berger M J, Hubbell J H, Seltzer S M, Chang J, Coursey J S, Sukumar R, Zucker D S and Olsen K 2010 XCOM:Photon Cross Section Database version 1.5 (Gaitherbsurg, MD: National Institute of Standards and Technology)http://physics.nist.gov/xcom

Castro C R F, Barroso R C, Anjos M J, Lopes R T and Braz D 2004 Coherent scattering characteristics of normal andpathological breast human tissues Radiat. Phys. Chem. 71 649–51

Desouky O S, Elshemey W M, Selim N S and Ashour A H 2001 Analysis of low-angle x-ray scattering peaks fromlyophilized biological samples Phys. Med. Biol. 46 2099–106

Elshemey W M, Desouky O S, Fekry M M, Talaat S M and Elsayed A A 2010 The diagnostic capability of x-rayscattering parameters for the characterization of breast cancer Med. Phys. 37 4257–65

Evans S H, Bradley D A, Dance D R, Bateman J E and Jones C H 1991 Measurement of small-angle photon scatteringfor some breast tissues and tissue substitute materials Phys. Med. Biol. 36 7–18

Fernandez M, Keyrilainen J, Serimaa R, Torkkeli M, Karjalainen-Lindsberg M-L, Tenhunen M, Thomlinson W,Urban V and Suortti P 2002 Small-angle x-ray scattering studies of human breast tissue samples Phys. Med.Biol. 47 577–92

Griffiths J A, Royle G J, Hanby A M, Horrocks J A, Bohndiek S E and Speller R D 2007 Correlation of energydispersive diffraction signatures and microCT of small breast tissue samples with pathological analysis Phys.Med. Biol. 52 6151–64

Hasan M Z 2003 Measurement of x-ray scattering form factors over a wide momentum transfer range MSc ThesisDepartment of Physics, Carleton University, Ottawa, Ontario, Canada

Hasan M Z and Johns P C 2004 Energy-dispersive technique to measure x-ray scattering form factors over a widemomentum transfer range Phys. Canada 60 145

Hubbell J H and Øverbø I 1979 Relativistic atomic form factors and photon coherent scattering cross sections J. Phys.Chem. Ref. Data 8 69–105

ICRP (International Commission on Radiological Protection) 1975 Report of the Task Group on Reference Man ICRPReport 23 (Oxford: Pergamon)

Johns H E and Cunningham J R 1983 The Physics of Radiology 4th edn (Springfield, IL: Charles C Thomas)Johns P C and Wismayer M P 2004 Measurement of coherent x-ray scatter form factors for amorphous materials

using diffractometers Phys. Med. Biol. 49 5233–50Kidane G, Speller R D, Royle G J and Hanby A M 1999 X-ray scatter signatures for normal and neoplastic breast

tissues Phys. Med. Biol. 44 1791–802King B W 2009 Accurate measurement of the x-ray coherent scattering form factors of tissues PhD Thesis Department

of Physics, Carleton University, Ottawa, Ontario, CanadaKing B W and Johns P C 2008 Measurement of coherent scattering form factors using an image plate Phys. Med.

Biol. 53 5977–90 (corrigendum 54 6437)

Energy-dispersive measurements of x-ray coherent scattering form factors 4397

King B W and Johns P C 2010 An energy-dispersive technique to measure x-ray coherent scattering form factors ofamorphous materials Phys. Med. Biol. 55 855–71

Kosanetzky J, Knoerr B, Harding G and Neitzel U 1987 X-ray diffraction measurements of some plastic materialsand body tissues Med. Phys. 14 526–32

Leclair R J, Boileau M M and Wang Y 2006 A semianalytic model to extract differential linear scattering coefficientsof breast tissue from energy dispersive x-ray diffraction measurements Med. Phys. 33 959–67

Leclair R J and Johns P C 2001 X-ray forward-scatter imaging: experimental validation of model Med. Phys. 28 210–9Leclair R J and Johns P C 2002 Optimum momentum transfer arguments for x-ray forward scatter imaging Med.

Phys. 29 2881–90Lehmann L A, Alvarez R E, Macovski A, Brody W R, Pelc N J, Riederer S J and Hall A L 1981 Generalized image

combinations in dual kVp digital radiography Med. Phys. 8 659–67Lewis R A et al 2000 Breast cancer diagnosis using scattered x-rays J. Synchrotron Radiat. 7 348–52Narten A H 1970 X-ray diffraction data on liquid water in the temperature range 4 ◦C–200 ◦C Technical Report ORNL

4578 (Oak Ridge, TN: Oak Ridge National Laboratory)Peplow D E and Verghese K 1998 Measured molecular coherent scattering form factors of animal tissues, plastics

and human breast tissue Phys. Med. Biol. 43 2431–52Plechaty E F, Cullen D E and Howerton R J 1975 Tables and graphs of photon interaction cross sections from

1.0 keV to 100 MeV derived from the LLL evaluated nuclear data library Technical Report UCRL-50400vol 6 revision 1 (Livermore, CA: Lawrence Livermore Laboratory)

Poletti M E, Goncalves O D and Mazzaro I 2002 X-ray scattering from human breast tissues and breast-equivalentmaterials Phys. Med. Biol. 47 47–63

Royle G J and Speller R D 1995 Quantitative x-ray diffraction analysis of bone and marrow volumes in excisedfemoral head samples Phys. Med. Biol. 40 1487–98

Tartari A, Casnati E, Bonifazzi C and Baraldi C 1997 Molecular differential cross sections for x-ray coherent scatteringin fat and polymethyl methacrylate Phys. Med. Biol. 42 2551–60

Westmore M S, Fenster A and Cunningham I A 1996 Angular-dependent coherent scatter measured with a diagnosticx-ray image intensifier-based imaging system Med. Phys. 23 723–33

Woodard H Q 1962 The elementary composition of human cortical bone Health Phys. 8 513–7