Energy-dispersive measurements of x-ray coherent scattering...
Transcript of Energy-dispersive measurements of x-ray coherent scattering...
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: [email protected]
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