Flat and curved crystal spectrography for mammographic X ...chantler/opticshome/... · X=2d sine S=2Ltan9 Figure 1. Flat crystal, symmetrical Laue spectrograph. A point source is

1996, The British Journal of Radiology, 69, 636-649

Flat and curved crystal spectrography formammographic X-ray sources1'2C T CHANTLER, BSc, DPhil, 2R D DESLATTES, BSc, PhD, 2A HENINS, BSc, PhD and 2L T HUDSON,BSc, MSc, PhD

1School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia, and 2National Institute ofStandards and Technology, Gaithersburg, Maryland 20899, USA

AbstractThe demand for improved spectral understanding of mammographic X-ray sources and non-invasive voltagecalibration of such sources has led to research into applications using curved crystal spectroscopy. Recentdevelopments and the promise of improved precision and control are described. Analytical equations arepresented to indicate effects of errors and alignment problems in the flat and curved crystal systems. Theseare appropriate for all detection systems. Application to and testing of spectrographic detection (usingstandard X-ray film) is presented. Suitable arrangements exist which can be used to measure X-ray tubevoltages well below 1 kV precision in the operating range of 20-35 kV.

Department of Health (UK) guidelines require thatthe indicated voltage on X-ray generating equipment formammography should be correct to within 1 kV [1, 2].Smaller changes have been demonstrated to have effectson image quality (and, of course, on patient dosage). Toavoid adverse effects, control and calibration proceduresshould determine any such operating parameter to sig-nificantly better than the nominal requirement. In thiscase, a tolerance of better than 0.5 kV in absolute accu-racy, across the operating range of 20-35 kV, might beappropriate. Requirements relate to precision and repro-ducibility as well as to absolute accuracy. The precisionof a result will normally be significantly better thanthe accuracy.

Although the applied high voltage can be measuredby traditional (invasive) high voltage (HV) divider tech-niques with more than adequate precision for the caseof well-filtered DC potentials, sources used in mammog-raphy are normally not readily or appropriately investi-gated by such means. Consequently, there has been aproliferation of non-invasive techniques, which seek toestablish a voltage estimate on the basis of penetrationof the generated radiation through two or more absor-bers. Such devices are dependent on any prior filtrationof the source by target windows and normally must becalibrated using a potential divider. Equally, they areunable to reach the measurement criteria given above.

In response to this problem, the Quantum MetrologyDivision at the National Institute of Standards andTechnology (NIST) has proposed, developed, tested andpatented a flat crystal spectrograph [3] . This is a Lauespectrograph with photographic detection, adapted fromthe earlier design by Rutherford and Andrade [4] . In

Received 25 September 1995 and in final form 29 February1996, accepted 28 March 1996.

principle and experiment, ideal sources can be dispersedand focused with adequate precision to fulfil readily thesub-kV requirements. In practice, the source widthbroadens and obscures the required information, cre-ating a potential problem. However, use of a curvedcrystal is able to overcome this limitation.

The flat crystal spectrographThe flat crystal spectrograph is illustrated sche-

matically in Figure 1. X-rays from a point source willundergo attenuation and transmission through the crys-tal, but these direct lines of sight are shielded from thedetection region. Hence only scattered X-rays can reachthe detector. Incoherent scattering and specular reflec-tion from surfaces are shielded from the detector by useof a collimating slit in front of the detection region. Theslit location is chosen so that the distance from the pointsource to the crystal centre is equal to the distance fromthe crystal centre to the slit. The crystal is aligned in theLaue geometry, so that diffracting planes of interest arenormal to the crystal surface as indicated in Figure 1.

Then the Bragg condition for coherent scattering(diffraction) is given by

= qeV=hc/X

(la)

(lb)

where A is a wavelength in the spectrum from the source,qe is the charge on the electron, d is the interplanarspacing of the crystal, E is the corresponding energy ofthe radiation, and 6 is the diffracting angle. With align-ment of the diffracting planes as suggested, radiationdiffracted from any central region of the crystal (normalto the surface) is diffracted through the slit to reach thedetector. Each wavelength of the source spectrumproduces two images on the detector separated by a

636 The British Journal of Radiology, July 1996

Spectrography for mammographic X-ray sources

X-rayTube

Crystal Si (220)2d = 0.38403 nm

Laue diffraction

\

\' eX=2d sine

S=2Ltan9

Figure 1. Flat crystal, symmetrical Lauespectrograph. A point source is imaged byLaue diffraction through a slit and onto adetector plane. The Bragg condition forLaue diffraction is illustrated on the side.

distance

S = 2(X-Xo) = 2\ZP-BZ|tan 6 (2)

where X is the distance of the image offset from the axisdenned by the densitometer (the measured distance) andXo is the central axis denned by the source and thediffracting planes. B is the source location, Z lies on thecentral plane of the crystal, and P is the point on thedetector and the source-diffraction plane axis.

For a given operating voltage V of the X-ray tubesource, the maximum photon energy E = qeV corre-sponds to photon emission with the full energy of theincident electrons. This high energy limit of the continu-ous spectrum (the end-point energy) may be used todetermine the effective operating voltage, given thenumerical constant hc/e (known to high precision [5]),the interplanar spacing d, and the length ratio(X-X0)/\ZP-BZ\. Rather than direct metrology, thisis determined by calibration from accurately knowncharacteristic emission lines of the tube source. Twounknowns would require two emission or edge energies,or three observed lines to determine the scaling.However, the diffracting crystal is chosen so that the

interplanar spacing is well defined, so that only oneenergy or two lines are sufficient to provide the cali-bration. There are usually two resolved emission lines inthe range of energies imaged on the detector, so thesystem becomes suitably overdetermined.

The fiat crystal prototype used a silicon crystal with220 diffracting planes aligned as described, with anunstressed crystal spacing of ^ = 0.19201547(2) nm atT=22.5°C [6, 7]. Some experimental details are pro-vided elsewhere [3] . Detection used standard KodakDEF-392 X-ray photographic emulsion with routinedevelopment and imaging using an LKB scanning densi-tometer (see Appendix 3). Although effective, neither ofthese devices was critical to the results. Each densitome-try trace used an aperture of 0.05 mm in the dispersiondirection, integrating over 0.8 mm in image length, withstep lengths in the dispersion direction of 0.02 mm (over-lapping), and in the transverse direction of 0.8 mm (non-overlapping). Up to 10 traces of any given film wouldbe taken. Image analysis allows each scan to be registeredto a common channel prior to summation or averaging,to improve overall statistics and reduce noise due tobackground, dust or other (imaging) defects. For a

Vol. 69, No. 823 637

C T Chantler, R D Deslattes, A Henins and LT Hudson

typical \ZP — BZ\ distance of 100 mm, a voltage changefrom 20 kV to 30 kV leads to a contraction of the intervalbetween left and right images of the end-point energy by10 mm. Hence measurement of this distance and thescaling to within 0.1 mm establishes the X-ray tube volt-age with an uncertainty of approximately 0.1 kV.

A typical spectrum is given in Figure 2. The molyb-denum Kal i2 line on each side, together with the

0O

cCO_Q

oCO

.a<0

.j^"CO0

DC

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

V

i ' ' ' ' i ' ' ' ' i i ' ' ' • i

Nominally 24 keV

MoKcc

MoKp

Mo Kp and K-edge

i i i I i i i V V i ' i V

5 10 15 20 25 30 35 40 45Y position, mm

Figure 2. Typical densitometered spectrum from photographicemulsion located on the detector plane of the fiat crystalprototype. Nominal tube voltage = 24 kV. Si 220 lattice usedfor Laue planes, of thickness 0.24 mm. The energy scale neededcan be established either by length metrology or by referenceto characteristic X-ray lines.

molybdenum K/^ 3 line and the high-energy limit, formthe basic data used to calibrate the spectrum and derivethe high-energy limit. Because of the simple plate func-tion, the axis may be recast directly into energy units, asindicated in Figure 3, where the end-point energy isspecified with an appropriate 0.1-0.2 kV precision.

Errors and uncertainties explicitly include uncertaintyin the location of the calibration lines and that of theedge. The assignment of densitometer or photographicerror bars in the figure follows a simple overestimatebased on the non-smoothness of the data (assuming anabsence of structure).

Assumptions and difficulties of the flat crystal methodRay tracing

A simple ray tracing routine or a direct geometricexpression (as given above) can be used to model the flatcrystal profile. These may be good approximations butare never exact and do not follow from detailed dynamicdiffraction. Laue diffraction through an optically thincrystal does not have the angular profile of a delta func-tion. Instead, the profile will display features given inFigure 4, from more detailed theory [8-10]. The oscil-lations are dramatic and the phase of interaction dependscritically on the thickness. The thickness also allowsdiffraction loosely from Bragg planes near the front orrear surface of the crystal. These paths will interfere, butwill also "trace" through to different locations on theimage plane.

Detector angle misalignmentPerhaps more importantly, possible aberration or mis-

alignment has not been considered. The simplest type isa misalignment of the detector plate at an angle ocy tothe normal for the diffracting planes. This results in acompression of features on one side of the plate(X+—Xo), with an expansion towards the other side(X _ —Xo). Here the paired line separation is

= (X++X_-2X0)

= |ZP-BZ|s in0[ l /cos(0-a 1 )+l /cos(0 + a1 (3)

Q

'ban

ce\b

soi

itive

>R

elc

0.5

0.45

0.4

0.35

0.3

: \ i , 24 kVp nominal :

: \ :

\ :

\ t ^ .... w

23 23.5 24keV

24.5 25 25.5

Figure 3. An expanded view of one imageof the end-point region obtained using anominal tube voltage of 24 kV. Datapoints are densitometry readings; the errorbars represent the standard deviation ofthe distribution of readings from theregion of uniform density. The straightline plus quadratic results from a fit of theedge location and coefficients in a least-squares sense for the photometric data inthe background and continuum regions.The intersection of these two lines corre-sponds to the end-point energy and hencethe X-ray tube voltage.



10°

Figure 4. Actual Laue diffraction patternfor point source, monochromatic incidentradiation (24.8 keV energy diffracting fromSi 220 planes through a 0.1 mm thick

1.44018 1.4402 1.44022 1.44024 1.44026 crystal), as a function of diffracting anglein output profile, as would be observed atthe image plate.

The offsets are not equal, so measurement of the end-point energy from the midpoint of two calibration linesis not a robust procedure.

Detector angle normal to dispersion axisDeviation of the detector angle or alignment normal

to this plane results in lines of features sloping towardsthe point of closest approach of the detector plate planeto the source. The relation for ideal alignment is stillhyperbolic, but with zero slope. For a lateral offset XY

from this ideal, corresponding to an angle subtended atthe source B of ay, given by cosar =l/Jll+XY

2/(BZ + ZP)2] the relation is

S = 2(X-X0)

(4a)

For the non-ideal case where the plate and crystal planessubtend an angle a2, this becomes

S =2(X-X0)

= 2 a y -25Z/cos (a y -a 2 ) ] (4b)

While the modification represented by Equation (3) isexact, Equation (4) neglects possible effects of varyingthe azimuthal angle and the changing crystal boundaryconditions. These effects are minor perturbations of thediffraction profile compared with the larger geometriceffects discussed here.

The correct midpoint is obtained from the average ofpaired lines or at infinite energy. This misalignment onlyaffects the overall scaling, so can be subsumed within aneffective parameter L = S/(2 tan 9) for the mean (or scan)separation. The modified equation introduces noadditional dependence upon 9, so a least-squares fittingof the data and calibration lines will be insensitive tothis error.

Diffraction plane misalignmentAnother minor issue relates to crystal imperfections,

and particularly the planes of interest being not normal

to the crystal surface but having an additional angle a.Assuming that the plate is aligned with the diffractingplanes, the first order effect yields a lateral shift of theplate images, and a scaling

(5a)

)](5b)

where M is the new slit location and the point wheremonoenergetic X-rays are refocused. The scale is affectedin the calibration process, but the zero location remainswell defined.

Source locationThe source location is not an issue if all line and

bremstrahlung radiation arise from the same region ofthe anode. It can be a problem for old sources or non-uniform filters, however. The relation given shows thatif the source for one energy range is located 0.2 mmfurther from the crystal (BZ and BP increase by 0.2 mm)then the image will be contracted by 0.4 tan 9 mm. Thiswould lead to a direct error in the calibration and voltagedetermination. Equally, a lateral shift of 0.2 mm of thesource location for one energy range would lead to a0.2 mm shift sideways for the corresponding image. S =2(X — X0) would be unchanged, however, so that a useof corresponding pairs of calibration and end-pointenergy separations should resolve the difficulty.

Source and crystal sizeA more serious difficulty relates to the source size and

crystal thickness for a given monochromatic wavelengthcontribution. This resolution is linear with the size ofthe source image. A lateral size of 0.2 mm or a longitudi-nal shift by 0.2/tan 9 mm will broaden the image andspectrum by 0.2 mm. This has a serious effect on theprecision of the voltage determination. A similar diffi-culty arises from crystal thickness. Neglecting the fulldiffraction profile, the location of Z is broadened by

Vol. 69, No. 823 639

C T Chantler, R D Deslattes, A Henins and L T Hudson

approximately this thickness so that a 0.1/tan 6 mmthickness will lead to a 0.2 mm broadening. For thickercrystals, anomalous transmission leads to a fairly sharplypeaked function which reduces this effect.

Design criteria for the curved crystal spectrographA curved crystal Laue spectrograph is illustrated in

Figure 5. The main design criteria include a highreflectivity to focus enough X-rays across the spectraldistribution to allow good statistics to be obtained forthe calibration in a reasonable time. The detector plateseparation should be small enough for the calibrationlines to appear on both sides at reasonable productioncost. The resolution should be high and the dependenceof image location on source position should be small.Optimization can proceed by detailed dynamic diffrac-tion computation for selected cases (following [8, 9]),but many of the most important parameters may beaddressed using relatively straightforward ray tracing, orpreferably using approximate analytic formulae (follow-ing [10]).

DiffractionThis design uses curved crystal "backward" or

"double-diffracted" Laue diffraction to produce thefocused spectrum, while the "forward diffracted" beamis explicitly blocked by collimating slits. Further, thecrystals operate in the "thick crystal" limit, so these twobeams are well defined and well separated. This avoidsunnecessary complication of the spectral pattern.

ResolutionThe minimum width and hence maximum resolving

power with a monochromatic source increases withBragg angle or wavelength due to off-axis aberrations.Near the optimum focus, the widths increase almostlinearly with crystal thickness Tc, indicating the inexactfocus and a similar effect to that for the flat crystaldependence. Within the trends and near the focus, sig-nificant fluctuations follow oscillation of the Laueboundary conditions as a function of thickness and pola-rization. However, the magnitude of widths are dramati-cally reduced in all cases, compared with the flat crystalimaging. This is indicated in Table I for dynamic diffrac-tion from perfect Si 220 planes with a large 0.4 mm fullwidth half maximum (fwhm) source truncated at+ 0.3 mm, at a distance oiBZ— 150 mm from the crystal.

Any choice of crystal and geometry must be temperedby possible broadening and structure widths in thesource or final spectrum, and by limiting detector reso-lution. Although X-ray film is capable of 0.003 mm reso-lution, natural linewidths are typically 0.01 mm formolybdenum Kax (the most important structure for mostmammographic calibrations), corresponding to a 7 eVwidth; while densitometry currently used proceeds in0.02 mm steps with a slit broadening of up to 0.05 mm.Electronic detection (in readily available forms of anadequate size) is usually broader still. Hence the use ofapproximately focused curved crystals is quite adequateto achieve the qualitative improvement in resolution

Tc Curved crystal geometry

Rowland Circle (focus)Figure 5. Laue spectrograph with curvedcrystal focusing, indicating Rowland Circlefor focusing condition.

Table I. Simulated effect of crystal thickness on monochromatic full width half maximum linewidth (i.e. on resolution) for a rangeof energies, using Si 220 planes, PZ = 300 mm

Energy

Flat crystalCurved crystals Rc = 400 mmRC = PZ, Tc = 0.5mmTc = 0A mmTc = 0.05mm

41.3 keV

0.4 mm0.058 mm0.022 mm0.009 mm0.005 mm

31.0 keV

0.4 mm0.053 mm0.028 mm0.012 mm0.007 mm

24.8 keV

0.4 mm0.060 mm0.031 mm0.018 mm0.010 mm

17.7 keV

0.4 mm0.064 mm0.042 mm0.027 mm0.016 mm

Limitation

Source widthDefocused, source limitedDiffraction or attenuationThickness limitedThickness limited



required, and there is no demand to make the crystalsunstably thin. The overall problem of flat crystal imagingwith large sources of thick crystals can therefore beavoided.

ReflectivityThe second priority in this development is to increase

the reflectivity uniformly over the energy range of inter-est, so that shorter detection times may be used in thecalibration procedure. Curvature normally increasesintegrated reflectivity by allowing more crystal planesand regions to interact and diffract, and by reducing thelamellar thickness of the coherent unit (this is similar tothe reflectivity increase for increasing crystal mosaicity).This is indicated in Figures 6-8. Peak reflectivity willusually decrease, at least in the local surface region ofthe crystal, although the geometric focusing can oftenreverse this effect at the focal plane. Curvature to anoptimum radius typically increases reflectivities by afactor of 3 to 6. The reflectivity is weakly dependent onthe source and detector locations, but the main concern

is the efficiency across energy with different radii.Figure 6 indicates that radii smaller than 200 mm arenot suited to energies below 20-23 keV, for example.

A similar trend is obtained with increasing thickness.The optimum is determined by the increasing attenu-ation of thicker curved crystals, and the incoherence ofhighly curved lamellae (that is, the peak reflectivity con-tinues to decrease but no additional regions becomesignificantly diffracting). While low energy photons aregreatly attenuated at large thicknesses (Figure 7, Si 220diffracting crystal), this should not eliminate the pre-cision of determination of calibration line centroids.Significant differences in reflectivity arise for the samethickness, and a focused detector location, but withdifferent radii. The optimum parameterization and func-tional dependence vary significantly with crystal type ordiffraction planes, as shown by comparison of Figures 7and 8 (Si 111 diffracting planes). These integratedreflectivities include geometric losses for a cylindricalintegration, following standard convention [8, 9] .

£ 3.5 1 0 5

co

15

—-a— Flat Si 220— -B - R = 1 m

c

- <^ - R =40cmc

- - x - - R =30cm

R =20cmc

-R =10cmc

25 30 35

E, keV40

Figure 6. Integrated reflectivity versuscurved crystal radius for diffraction of Si220 planes with source-crystal distanceBZ=150mm, crystal-detector distancePZ = 200 mm, and crystal thickness 7 =0.1 mm.

—e— T =0.05mm, R =30cmc c

—B -T =0.1 mm, R =30cmc c

- $- - T =0.1 mm, R =20cmc c

-x - - T =0.2mm, R =20cmc c

+ - - T =0.5mm, R =30cm

20 25 30 35 40E, keV

Figure 7. Integrated reflectivity versuscurved crystal thickness Tc for diffractionoff Si 220 planes with source-crystal dis-tance BZ=150mm, crystal-detector dis-tance PZ = RC = 200 mm or 300 mm.

Vol. 69, No. 823 641

C TChantler, R D Deslattes, A Henins and LT Hudson

COd

idia

i

CO

<D

CO

CO

&

o_CD

^

CD

f—I—

2.5

2

1.5

1

5

10"5

io-5

10"5

io-5

10"6

0

i I i i —I—I—\—I— 1 1 1 1 | M ! | M M |

s 9 - - ^ :

\> / X ' ' \ ^

/ \ - n- \ ^ ^ " ^ \ -

: / x :^ / -- /

/i1 1 1 1 t ,. , - i - , - , - , \ r r

15 20 25 30 35 40E, keV

-e— T =0.05mmc

-a -T =0.1 mmc

o- - T =0.2mmc

•x - -T =0.4mmc

+ • - T =1.0mm

Figure 8. Integrated reflectivity versuscurved crystal thickness Tc for diffractionoff Si 111 planes with source-crystal dis-tance BZ=150mm, crystal-detector dis-tance PZ = R= 200 mm.

So far as the reflectivities and widths are concernedfor the specific application of interest here, use of Si 220crystal planes suggests crystal thicknesses Tc in the range0.2 mm > Tc > 0.1 mm, and radii of curvature Rc in therange 300 mm> Rc>200 mm. Conversely, use of Si 111diffracting planes would suggest 0.4 mm > Tc > 0.2 mmand 200mm>i?c>100mm. Technical difficulties ofstressing crystals to these small radii can yield practicallimits. The prime distinction between Si 220 and Si 111lies in the relative 2D spacings of the lattice planes,leading to larger offsets and widths for Si 220.

Curvature and detector plate locationWith ideal alignment of crystal planes, surface, collim-

ating slit and detector plate, and negligible crystal thick-ness, with radius of curvature Rc, and assuming that thegrazing Bragg angle is small so sin 9 x tan 9, then thecurved crystal plate factor is approximately

= 2{X-XQ)w2[{PZ-BZ)

+ BZ{BZ + PZ)/(BZ + Rcy\ sin 9 (6)

If, in addition, the location of the plate is at the focus,so that PZ = RC, then

S = 2(X-X0)&2PZsm9 (7)

This focal location is preferred and provides minimalfwhm on the detector plate for a given monochromaticradiation; and hence maximum resolving power. HenceRc can be chosen simply, for a given spectrograph lengthand crystal.

Flat crystal spectrographs could be designed with theplate closer than the image slit location (BZ>PZ orMP<0). This gives compression of the plate image (fin-ancially desirable for electronic detection) but yieldsstrong background and so is to be avoided. An equival-ent curved crystal device (now involving Rc, so PZ«RC)would not be focused. In both crystal types the designspictured are prefered. This implies that for a given PZlength, the offset X — Xo for a given energy will necessar-ily be larger in a curved crystal configuration. This can

enhance resolving power, but requires a larger spectro-graph to image the same low energy structure.

Equations 6 and 7 indicate this focusing and scaling,but are inappropriate for use in spectrographic dataanalysis. Errors of these relations arise at the 3% levelfor Si 220, for calibrating lines, due to the inequivalenceof sin 9 and tan 9.

Conversely, dynamic diffraction and detailed ray trac-ing are both unnecessarily complex for the current appli-cation. Neither is readily invertible, as desired for efficienton-site analysis and calibration. Hence it is importantto identify and criticise suitable approximate solutionsto the problem.

Defining equations for the curved crystal spectrographThe first step is to follow ray tracing assumptions and

to assume that the crystal is negligibly thin and ideallycylindrical (noting particularly that increasing thicknessdid not have a dominant effect on the detected widthsin the current regime in Table I). The alignment of planes,source and detector will be considered ideal. The resultof this approach is given in Appendix 1, with parametersrepresented in Figure 9.

Response of the spectrograph to defectsThe absolute accuracy attainable is a critical issue in

this development. It is not adequate to test the device ina typical situation, it is also necessary to understand the

Rc

Q

Figure 9. Curved crystal equation parameterization assumingpoint source and negligible crystal thickness.



type of errors or effects which may arise in less carefulimplementation, and to quantify these details. As someof the ideas have been presented above, this derivationwill be discussed in Appendix 2 to allow readers to reachthe conclusions more smoothly, and to then return tothe necessary underlying details.

Experimental testing with X-ray sources: uncertainty andcalibration

Focusing from Si 111 and Si 220 crystal planes hasbeen investigated in detail for the prototypical devicesillustrated in Figure 5. Significant uncertainty arises fromphotographic exposure, development and densitometry.For low optical densities, oscillations, aliasing and incon-sistencies from these sources can be maintained to lowerthan 0.01 in optical density units (or relative absorbance).For densities above 1, and approaching the useful maxi-mum of the densitometer around 3.6, these effectsincrease to around 0.1 in optical density. This uncertaintyrepresents a limitation from statistics and reproduc-ibility, but need not affect located centroids or calibrationresults at a severe level.

Estimates of limiting density precision can be basedon theory [11] but in the current context two alternatesimple empirical procedures were used to derive inputchannel error estimates for subsequent fitting. Each den-sitometer scan used a slit height of 0.8 mm. The centralregion of the exposed region of film was typically5-10 mm in height. Hence a given photographic expo-sure may yield from five to 12 scans of independent data.Each densitometer scan yields a profile of peaks and theend-point energy, and these may be combined using thestrong calibration lines to determine any variation ofoffset or scaling [3] . Statistical imprecision is then esti-mated by considering variation either between adjacentscans or along an individual scan. These estimates takeaccount of random noise, emulsion holes and relativelyrapid fluctuations, but also include variation fromnarrow line or edge structure. Hence they will generallyoverestimate the uncertainty in the data.

Linewidths and ^-coordinateThe current analysis was based on fitting peaks of

optical density versus energy, as opposed to transformingthe data to intensity profiles. The latter is more rigorousbut less convenient for a simple field exercise (as withroutine mammographic voltage determination). Thedifferences for the emulsion used and density-energyrange observed are unlikely to yield significant errorfrom this simplification. Linewidths in intensity spacecan correspond to values of density reduced from thepeak by 0.3 (in the logarithmic range of the relation) andthis width agrees with the slit width of 0.05 mm. Thiswidth is two to three times the natural linewidths andthe ideal diffraction widths of monochromated radiation,so the contributions to widths and lineshapes appearwell understood and in good agreement with predictions.

ResolutionOne of the major improvements of the curved crystal

approach is seen in Figure 10, taken with photographic

Den

sca

lO

pti

ive

«->(0oc

3.0

2.5

2.0

1.5

1.0

0.0

: MoKa 17374

• \

•

! L

eV

MoK^ 17479 eV

MoKp2

MoKp 19602 e V -1,3

1 ;

-

19965 eV "j

& K-edge

(I

-A!5 6

X, mm

Figure 10. Curved crystal image of one set of the four molyb-denum K calibration lines for a nominal tube potential energyof 26 keV, described in the text. The Koq and Kcc2 componentsare well separated and the K/?2 peak is clearly defined, evenwith broad sources, as opposed to flat crystal diffraction.

detection. The molybdenum Koq and molybdenum Ka2

lines are now clearly separated, as opposed to flat crystaldiffraction. Another qualitative improvement is theidentification and separation of the molybdenum K/?2

peak as compared with the molybdenum K-edge atslightly higher energies. The optical density contrast ofthe former increases with increasing energy or exposure,and is typically 0.7-0.9 above a bremstrahlung back-ground of 0.5 (below the K-edge) and 0.4 (above theedge).

This allows eight calibration points to be used todetermine the energy scale, instead of just four. The flatcrystal function only required two points for the twoparameters (neglecting defect variables), but the curvedcrystal formulae require four points, of which only onepaired set provides independent information regardingXo and an overall scale. Hence use of just a pair of Ka1>2

unresolved doublets and another pair of K/?t 3 peaks isnot adequate to determine a curved crystal scale. Havingthe additional well resolved and well defined peaks deter-mines Xo much more precisely, is able to estimate magni-tudes of misalignment variables from the overallconsistency, and is able to determine (and overdetermine)the independent parameters.

Within these formulae, assumptions may be maderegarding the alignment of crystal and detector (PZ andRc), in which case the formulae simplify, and three pointsare required [12].

The main calibration data discussed here used a 5 raAbeam for 1-4min exposure on a Si 111 crystal, with a0.3 mm x 0.3 mm X-ray source at nominal energies from22 keV to 40 keV in 2 keV steps. Fitted parameters usedeither quartic or quadratic formulae, with calibrationlines and a peak-finding formula optimized for photo-graphic data. The peak-finding routine is relatively crudeand uses simple background subtraction (to be addressedin a subsequent paper).

Vol. 69, No. 823 643


Line location accuracyThe calibration line locations are estimated to have

an accuracy of about 0.15 channels for the Ka doubletand about 0.6 channels for the K/? lines. In particular,K/?1?3 is unresolved and K/?2 is almost coincident withthe K-edge, so that the K-edge line and backgroundstructure have greater uncertainty. Each channel has astep-size of 0.02 mm or about 20 eV, 31 eV, 49 eV or67 eV at 22 keV, 28 keV, 34 keV or 40 keV energies. Analternate estimate of 0.3 channel uncertainty for all linesyields similar reduced x2 values of 1-5. A modified esti-mate of Kfi uncertainty of 0.75 channels was found togive significant variation in fitted parameters and indi-vidual end-point energy evaluations, but to have littleeffect on the overall average and generally to produceless reliable data. Hence the estimated accuracies areconfirmed by the resulting fits and statistics.

End-point locationsEnd-point locations are fitted separately with a linear

background and a quadratic dependence near the end-point. Reduced x2 values of those particular fits are alsotypically 1-2 (confirming the densitometry channel errorestimates) and yield correlation coefficients of R = 0.99over 200-400 channels of data, and end-point locationuncertainties of 0.25-1.0 channels, increasing to 1.75channels for poor fits with significant noise and weakend-points at high energies (38 keV-40keV). The end-point location is much harder to quantify than the cali-bration line locations, as is reflected in the relativeuncertainties.

Fits of the end-point location are not particularlyrobust (in this sense) and depend significantly on therange used. For example, fits of the first end-point of the28 keV data with 5.5 mm or 6 mm ranges and uncertaint-ies based on consistency between scans (H) or within ascan (V) yielded values of 15.817 + 0.011 mm,15.784 + 0.005 mm, 15.797±0.012 mm and 15.761+ 0.005 mm, thereby displaying variation up to fourtimes the derived fitted uncertainty. In other cases, thelocal scatter agreed with the fitting uncertainty. A morereliable estimate of the final uncertainty is to use thederived energies (including uncertainties of calibrationline energies) from a range of such reasonable schemesand compare the scatter observed.

Uncertainties are propagated to estimate the precisionof determined energies. Derived uncertainty for an indi-vidual end-point with a particular channel weightingscheme is usually dominated by the scatter betweendifferent schemes or between the pair of end-pointdeterminations.

Nature of fitting resultsEquation (8) in Appendix 1 was used to fit the derived

calibration line locations and energies in two majormodes. The first used a modified Levenberg-Marquardtroutine, while the second included a more extensive grid-search and singular value decomposition approach (inaddition). Reduced x2 values, parameter errors andderived energy uncertainties of both methods were usu-ally very similar. The second method always improvedthe x2 value by a small but significant factor. In either

method, correlation is minimized by expressing Rc interms of the more independent BZ/(BZ + RC). Bothmethods yielded typical uncertainties in Rc, BZ and PZof 20 mm, 5 mm and 5 mm, respectively. Conversely, theoffset Xo is well defined to 0.002 mm or better. Thisresult is a partial consequence of the function arid param-eter correlation with noise.

In the simpler mode of analysis, results returned valuesof Rc, BZ and PZ in very good agreement with themechanically measured parameters Rc = 250 mm, BZ =150 mm, PZ = 250 mm. This was convenient to establishthe effect of an ideal relation. It was not the result of theimproved fit, which typically shifted parameters by morethan the returned uncertainty. This appears a systematiceffect in that results of the simpler analysis yield a meanend-point energy some 100eV-150eV lower than themore extensive minimization. Both results appear sys-tematically and significantly lower than results usingquartic formulae.

Quartic evaluationUse of Equations (9-13) required the more extensive

fitting process to disclose a reliable minimum. In thiscase, correlation was reduced by expressing BZ as themore independent parameter l+BZ/Rc. Uncertaintiesof Rc were reduced to approximately 0.05 mm, whileuncertainties in PZ were reduced to around 0.2 mm.Values of both parameters are in good agreement withthe measured values, even after full minimization.

The returned uncertainty of BZ (and the value itself)via the parameter indicated above is very imprecise,corresponding to + 20 mm or more. This arises from thenature of the curved crystal arrangement, which isexplicitly insensitive to the source location B. Hence thisresult is a confirmation of the utility of the equations.The X-ray source is not a point source, whereas theformulae make that assumption. The inadequacy of thatassumption has negligible effect on the crystal curvatureRc, the detector location PZ, or the film offset Xo, buthas a large effect on the perceived source location BZ.

Summary of uncertaintiesThe uncertainty of measured parameters (especially

BZ) has a minor effect on derived energies. Derivationsusing measured parameters directly (without fitting), orusing uniform 0.3 channel weights in the fitting process,shift the located energies by less than one standard devi-ation. Unlike the use of Equation (8), the quartic fittingprocedure appears robust. Additionally, we would expectthe quartic approach to be improved on theoreticalgrounds and to yield physical parameters (as observed).In light of this confirmation, only results of quartic fittingshall be presented.

Uncertainties of intensity and density measurementdo not provide a serious constraint of overall precision,but a great utility of the curved crystal approach lies inthe enhanced resolution and hence calibration of theend-point energies.

Analysis and resultsA consequence of the quartic equations having lower

parameter uncertainty than the quadratic approach is



that the derived end-point energies also have loweruncertainty. The scatter between end-point determi-nations using different error estimates (H between scans,V within each scan, or L for a linear estimate) is similarand dominates over final error estimates. The majorcontribution to the uncertainty of an individual end-point energy is the local fitting error, with a smallercontribution from the quartic fit and extrapolation fromcalibration lines, and minor contributions from energyuncertainties and background considerations in cali-bration lines. Figure 11 indicates the derived end-pointenergy for a single error scheme (V), the first edge (1)and the nominal 22 keV energy. Fitting precision is10 eV, although this increases with local noise and withenergy, and the scatter between fits is much larger thanthis.

The overall result may be seen in Figure 12. Thedifference in energy between measured end-point ener-gies and the nominal value from the dialed source voltageis plotted against the nominal voltage. Three errorschemes and two end-point energies provide five sets of

measurements, as indicated. The first three (1 H, 1 V and2 H) are given error bars to illustrate individual fittingprecision and the precision of the calibration. These errorbars increase with energy as the energy scale is com-pressed and the relative noise increases. Particularschemes fail occasionally but obviously—as exampledby £1V (30keV) and £2V (36keV). Inspection of fittedprofiles for these points proves that the fit in questionhas failed. For low energies (22 keV-28 keV) the Vscheme yields more consistent results; while for higherenergies the H and L schemes are more robust. Twomeasurements were plotted for 30 keV; the second useda different crystal but produced a very similar averageresult (neglecting the outlier). Overall scatter is below100 eV for lower energies, but rises above this value athigh energies.

Results of these photographic trials indicate an errorof the nominal voltage of the source observed of approxi-mately 200 eV at low energies (22 keV) rising to about400 eV at high energies (40 keV), or a scale error ofapproximately 1%. Additionally, the first edge is

Curved crystal, Verr, first edge, nominally 22 keV

0.38

0.30

End-point energy22.26(1) keV

11 12 13 14 15 16X, mm

17 18 19

Figure 11. Detail of individual curvedcrystal end-point determination indicatingchannel uncertainties derived from scatterwith a scan, for the first end-point of thepair (£1V)> and a nominal source voltageof 22 kV. Each datum represents a stepwidth of 0.02 mm or 20 eV. Below the end-point, a quadratic dependence of densityupon energy is assumed; above the end-point, the background is assumed to belinear.

800

700 •

600 -

"5a.••oc(D

•aa>

a>T3

4)

ca>"5

1oc'

500

400

300

200

10024000 28000 32000 36000

nominal voltage, V40000

Figure 12. Overall display of derived end-point energies from the set of nominalenergies with the curved crystal prototype.Error bars are indicated on the first threesets. The first or second derived end-pointenergies are labelled (£x or E2) with asubscript indicating the source of the chan-nel uncertainty used (H for the scatterbetween scans, V for the scatter within ascan, or L for a uniform linear estimate).Occasional errors are observable, but thestrong trends have a small uncertainty.

Vol. 69, No. 823 645

C T Chantler, R D Deslattes, A Henins and L T Hudson

predominantly higher in derived energy than the second,in all schemes, indicating the magnitude of a misalign-ment following Equations (14) or (15). These are errorsof the nominal voltage and alignment, and not of theX-ray method.

Conclusions and applicationThe mean results at each energy follow a reliable trend

with an uncertainty at or below 100 eV. To this 100 eVuncertainty should be added 100-200 eV in quadraturefor the sum of (unquantified) misalignment defects of aparticular crystal. Even so, this is well within the cali-bration accuracy required for screening centres. Hencea precision of at or below 100 eV is achievable, with anaccuracy of better than 500 eV.

This paper has demonstrated a number of develop-ments and modifications, but detector technology andcrystal modifications have been neglected and are thesubject of a separate study [12]. Observed noise androutine exposures are not unreasonable in the curvedcrystal devices discussed in this paper. Important devel-opments are, however, required towards more routinenon-invasive calibration of voltage supplies in situ. Theseare discussed elsewhere, particularly with a view to clini-cal conditions and filtering [12]. Two basic routesappear open to further investigation and trials. Oneinvolves the type of device indicated here as a low-costapproach in countries with limited budgets or facilities;while a more attractive alternative for developed sitesand countries would involve superior and automateddetection and processing. These devices would be applieddirectly to clinical units for in situ non-invasive tubepotential measurement, following initial calibration,without any other intermediate.

Tests of these devices under clinical conditions withscreening X-ray sets in screening centres have been made[12] and the results are fully in agreement with theuncertainties and errors discussed above. The methodhas been proven to be a successful new non-invasivetechnique for mammographic voltage determination tohigh precision and accuracy. It should be clear that theresults of this paper lay the groundwork theoretically forthis new class of device. It also experimentally demon-strates an appropriate form for the low-cost approach.A more advanced detection method using the results andtechnique demonstrated herein, together with furtherdetails of the screening centre tests, will be presentedseparately.

AcknowledgmentsDr Chantler would like to thank other members of

the Quantum Metrology Division for helpful discussionsand interaction, and the Division itself for the VisitingResearcher position during much of the period of thisresearch.

References1. LAW, J, The measurement and routine checking of mam-

mography X-ray tube kV, Phys. Med. Bioi, 36, 1133-1139(1991).

2. DEPARTMENT OF HEALTH, Revised Guidance Notes forHealth Authorities on Mammographic X-ray EquipmentRequirements for Breast Cancer Screening, PublicationSTD/90/46 (HMSO, London) (1990).

3. DESLATTES, R D, LEVIN, J C, WALKER, M D andHENINS, A, Non-invasive high-voltage measurement inmammography by crystal diffraction spectrometry, Med.Phys., 21,123-126 (1994).

4. RUTHERFORD, E DA C and ANDRADE, E N, The spec-trum of the penetrating g-rays from Radium B and RadiumC, Phil. Mag., 28, 263-273 (1914).

5. COHEN, E R and TAYLOR, B N, The 1986 adjustment ofthe fundamental physical constants, Rev. Mod. Phys., 59,1121-1148(1987).

6. DESLATTES, R D and KESSLER, E G JR, Status of asilicon lattice measurement and dissemination exercise,IEEE Trans. Instr. Meas., 40, 92-97 (1991).

7. SCHWEPPE, J, DESLATTES, R D, MOONEY, T andPOWELL, C J, Accurate measurement of Mg and Al Ka1-2

X-ray energy profiles, J. Elect. Sped., 67, 463-478 (1994).8. CHANTLER, C T, X-ray diffraction of bent crystals in

Bragg geometry. I. Perfect-crystal modelling, J. Appl. Cryst.,25,674-693(1992).

9. CHANTLER, C T, X-ray diffraction of bent crystals inBragg geometry. II. Non-ideally imperfect crystals, model-ling and results, J. Appl. Cryst., 25, 694-713 (1992).

10. CHANTLER, C T and DESLATTES, R D, Systematic cor-rections in Bragg X-ray diffraction of flat and curved crys-tals, Rev. Sci. Inst., 66, 5123-5147 (1995).

11. CHANTLER, C T, Photographic response to X-rayirradiation. I: Estimation of the photographic error statisticand development of analytic density-intensity equations,Appl. Optics, 32, 2371-2397 (1993).

12. HUDSON, L T, DESLATTES, R D, HENINS, A ET AL,Current status and development of a curved crystal spec-trometer for energy calibration and spectral characteriz-ation of mammographic X-ray sources, Med. Phys.(submitted).

Appendix 1: Analytic defining equations for the curvedcrystal spectrographFollowing the introduction in the section "Definingequations for the curved crystal spectrograph" we pre-sent the solution using nomenclature introduced aboveand in the figures.

Quadratic approximationThe next step is to retain tan 6 terms initially, but to

assume that cot26»2RcBZ/(Rc + BZf. Then the priorsmall-angle assumption is relaxed, to yield

X - Xo ~ | PZ tan [9 + (BZ tan 6)/(Rc + BZ)~\

-BZtan[0-(BZtan0)/(Kc + BZ)]| (8)

This neglects the shift along the vector BP from thecrystal pole Z to the actual off-axis diffracting surface.For small Bragg angles, this is adequate. This solutioncorresponds to an analytical quadratic relation and isconvenient. To address the question of whether this solu-tion is adequate in the current voltage calibration, the"quadratic approximation" may be relaxed to a quarticapproximation. Within the assumptions above, this solu-tion is exact; relaxing the assumptions further is possiblebut would then have no analytic solution.



Quartic approximationFollowing Figure 9, this yields the defining equations:

ex=6-s, e2=e+s,PZ = Rc( 1-cos d) + QA cos 02,

X-XO = PQ = \RC sin<5-QAsin02|,

(9)

or

p = C-3B2/$, q = D + B3/S-BC/2,

r = E + B2C/16- BD/4 - 3B4/256,

(cos 4(S + B cos 3(5 + C cos 2<5 + D cos 5 + E = 0),

£ = - 2 ( c o s 0+1+.BZ/.RJ,

C = sin20 + (1 + BZ/RC)2 + 2 cos 0( 1 + 5Z/flc),

£ = 2[cos0 + (l+JBZ/.Rc)cos20],

£ = - 1 - 2 ( 1 + 5Z/.Rc)cos 0 - ( 1 + BZ/RC)2 cos 20(10)

Then solving for (5 and matching the sign with that ofX — Xo provides the image location relative to an offsetXo as

X - Xo = \PZ - Rc( 1 - cos <S)] tan(0 + 3) - RC sin S(11)

SolutionThe parametric form of Equation (9) is simple but

provides no analytic determination of X from 0 or of 0from X, therefore invoking numerical derivatives to solvethe problem. This is simpler than the full dynamic com-putation, but we shall bypass this method in favour ofan analytic determination of X from 0. The solutioninvolves defining

p / p / ) g f, then <z,p=f±0.5V#; *o =

If g<0, then oc = Q.5j(f2-g); £ = acos(//a);

to = p/3 + 2 3x /acosy;

whence £ is given by

(12a)

(12b)

(13)

A least-squares fitting procedure can derive <5n andhence PZ, BZ, Rc and Xo given values for Xn and 0n forthe calibration lines. Then fitted end-point positionsXE+, XE_ in image plane units (mm) can be used in aninverted iterative relation to provide SE and 0£, andfinally the end-point energy.

Comparison of formsAssumptions are still being made here about the uni-

formity of surface, source and diffraction plane orien-tation (to be partially addressed later). The crystalthickness and intrinsic diffraction profile primarilybroaden the result without confusing the calibration. Acomparison of simple scaling formulae (Equation (2) forflat crystals and Equation (7) for curved crystals) is givenin Figure 13 to indicate the image size and energy depen-dence for typical configurations. Compared to thesevalues, Figure 14 indicates the inadequacy of using sin 0to approximate tan 0 for the flat crystal relation at lowenergies. The analogous simplification for the curvedcrystals is much more significant because the plate func-tion X (or the dispersion function dX(A)/d/.) is muchlarger and the relation is less well approximated.

However, the quadratic relation (Equation (8))appears well behaved and parallel to detailed compu-tations of a or n polarized radiation including sourcewidths and crystal thicknesses. It was then hoped thatthe simpler form would fit the molybdenum K spectrato adequate precision. However, the errors of order 1 %appear sufficient to couple with the small set of cali-bration lines and distort the determination of the inde-pendent parameters. This could be due to correlationswith noise, defects of the alignment, or the greater robust-ness of the quartic relation (Equations (9-13)).

Appendix 2: The effect of spectrograph or alignmentdefects on calibration relations and possible inaccuracyFollowing the section "Response of the spectrograph todefects" and Appendix 1, we present the relations for thefollowing defects.

Detector angle misalignmentThe misalignment of photographic plate angle has a

similar dependence to the flat crystal relation:

EE

£o.

£"53•a

coco

Crystal plate functions

40

30

20

.t; 6oa .

- Curved Si220, Eqn 7•Curved SM11, Eqn 7• Rat Si220, Eqn 2• Flat Si 111, Eqn 2

20 30Energy (keV)

Figure 13. Crystal plate function X = PQ, mm, versus E, forBZ = 150mm, PZ = 250 mm, and curved or flat crystals of Si111 and Si 220 (using Equations (7) and (2)). The curvatureincreases the feature separation for any energy, and the scalingdecreases with increasing lattice spacing.

Vol. 69, No. 823 647


Curved Eqn 8 vs Eqn 7, Si 220Curved Eqn 8 vs Eqn 7, Si 111Flat Eqn 2 vs sin approx, Si 220

- Flat Eqn 2 vs sin approx, Si 111Curved crystal a peak, SM11Curved crystal n peak, Si111Curved crystal n midpoint, Si111

20Energy (keV)

30

Figure 14. Fractional shift of lateral pos-ition versus energy as compared withsimple Equations (7) and (2). While flatcrystal corrections from the use of sin ortan functions remain small and followparallel lines on this log-log plot, thecurved crystal relations show larger effectsand a stronger variation with energy. Rc =240 mm is used.

S = X++X_-2X0

= ( X - X 0 ) c o s # 2 [ l / c o s ^ -(14)

As for the flat crystal, this is an asymmetric scalingwith angle, so that compression and expansion areunequal and the midpoint of a given pair of lines doesnot yield Xo. An overall fit can determine o as theparameter is largely independent of other scaling param-eters. To first order and for small angles, any anglenormal to the dispersion plane is treated similarly to theflat crystal Equation (4b), once again yielding a relationindependent of 9 and d and hence only affecting theoverall scaling parameter. Detailed diffraction compu-tations show that this is not precisely true but is here anadequate approximation. Magnitudes of o and cc2 canusually be constrained below 7' corresponding toXY<0.\ mm at the centre of the detector plate.Diffracting angles vary from 10.6° for molybdenum Karadiation diffracting off Si 220 planes, to 2.8° for 40 keVX-rays diffracting off Si 111 planes. In the examplegeometry, this corresponds to values of d from 4° to 1°.

Diffraction plane misalignmentMisalignment of diffracting planes leads to a sym-

metric scaling to

91 = 0 — S + a, 92 = 6 + S — oc (15)

The remainder of Equation (9) is unchanged, but thismodification requires rederivation of coefficients ofEquations (10-11) for each side of the crystal (not givenhere). Further, a is correlated with S and the scalingparameters and hence less amenable to derivation. Thiseffect is energy dependent as for the flat crystal, but theinformation content in the calibration lines is usuallyunable to determine the value of a. However, a is typi-cally aligned to better than one arcminute and oftenbetter than a few arcseconds for good silicon and ger-manium crystals.

Table II illustrates the shift of X+ — Xo locations forprimary calibration lines and typical edge energies forthe case i?c = 240mm, PZ = 250mm, 5Z=150mm.Defects of a1 = T, a2 = 70', Xy = 10mm, and a = 2' arecompared with the quadratic and quartic formulae toindicate typical magnitudes. While the shift of image

Table II. Effect of defects on Si 220 plate function (value of \X — Xo\ in mm versus calibration or edge energy) for selected energies,with Rc = 240 mm, BZ = 150 mm, PZ = 250 mm

Line

E (keV)\X-X0\ (mm)Eqn (14), a1 = 7', X.«i = 7', X +Eqn (4b), a2 = 70', XY = 0Eqn (4a), XY= 10 mmEqn (15), a = 2', X.Eqn (15), a = 2', X +

MoKa2

17.37448.78348.81048.75748.75348.79948.93348.633

MoKa,

17.47948.47048.49648.44447.44048.48548.62048.320

MoKft.,

19.60242.92242.94242.90242.89542.93543.07142.773

MoKfc

19.96542.10142.12142.08242.07542.11542.25041.953

Edge?

25.00033.31233.32433.30033.29133.32233.45833.165

Edge?

30.00027.62327.63227.61527.60627.63227.76827.478

Edge?

35.00023.60723.61323.60123.59223.61423.75223.462

Mo, molybdenum.



location is significant at the level of energy uncertainties, to correlated errors in the fits. Equation (6) is muchthe largest shift from the diffraction plane misalignment worse, and produces errors of 1-2 keV without introduc-is still less than O.bkeV for calibration lines and less tion of noise. Hence the use of Equations (9-13) appearsthan 0.21 keV for possible edge energies. This suggests indicated.that the tolerances indicated are necessary (within afactor of two) to achieve the desired performance, and Appendix 3that typical crystal alignments will be adequate. This identification does not imply endorsement norAlthough the quadratic formula gave similar results and should it be taken to suggest that the identified itemsdependences of X on energy to Equations (9-13), the are necessarily best suited for the applications in whichoptimized parameters become unphysical and will lead they are used.

Vol. 69, No. 823 649

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Flat and curved crystal spectrography for mammographic X ...chantler/opticshome/... · X=2d sine S=2Ltan9 Figure 1. Flat crystal, symmetrical Laue spectrograph. A point source is