Photographic response to x-ray irradiation. Ill: Photographic …chantler/opticshome/... · 2015....

Photographic response to x-ray irradiation.Ill: Photographic linearization of beam-foil spectra

C. T. Chantler, J. D. Silver, and D. D. Dietrich

In this paper models for the relation of specular density to incident (x-ray) intensity with uncertainties areapplied to experimental data, indicating methods for the correction of additional effects.Linearization and error calculations are simplified by double linear interpolation, and the effect of this isquantified. Relative first-order intensities are determined directly. Secondary linearization or calcula-tion for higher-order lines gives correction factors that yield absolute and relative higher-order intensityratios. The effects of energy and angle on linearization are included. Densitometry uncertainty isestimated and quantified.

1. Introduction

Models developed in the first two papers"2 (denotedherein CI and CII) for smooth-surface, thick emul-sions are applied to photographic data collected fromtwo beam-foil spectroscopy experiments. In Table 1some of the more important parameters referred to inthis paper and discussed in the earlier papers aresummarized. A dual-arm Johann focusing x-rayspectrometer with curved diffracting crystal focused xrays onto DEF-392 film on the Rowland circle of150-mm radius (Fig. 1).3-6 Hydrogenic iron spectrallines in first order (Balmer-,3, X 7.12 A) and fourth-order (Lyman-a, X 1.78 A) diffraction from PET002 and ADP 101 crystals are compared. In thesecond experiment we compare hydrogenic germa-nium spectra (Balmer-,B, X 4.64-4.74 A; Lyman-a,X 1.167-1.172 A) similarly, using Si 111 and Si 444diffracting planes. Secondary peaks in the detectionrange, of other energies and at different angles,include third-, fourth-, and fifth-order diffraction ofLyman series radiation, first-order diffraction ofBalmer series, and heliumlike satellite peaks. Thisfollows earlier research67 in pursuing quantum elec-

When this work was performed C. T. Chantler and J. D. Silverwere with the Clarendon Laboratory, University of Oxford, ParksRoad, Oxford OX1 3PU, England. C. T. Chantler is now with theQuantum Metrology Division, National Institute of Standards andTechnology, Gaithersburg, Maryland 20899. D. D. Dietrich iswith the Lawrence Livermore National Laboratory, University ofCalifornia, P.O. Box 808, Livermore, California 94550.

Received 26 November 1991.0003-6935/93/132411-11$05.00/0.C 1993 Optical Society of America.

trodynamical tests in hydrogenic medium-Z systems.The germanium experiment was reported earlier,78while new data are presented for hydrogenic iron.

Experimental windows (Section 2) and densitome-try arrangements (Sections 3 and 4) affect the conver-sion of measured densities to source intensities.Problems of fog levels (Section 5) and developmentprocedures (Section 6) are addressed theoreticallyand with respect to observed data (Section 7).Spectrometer collimation contributes no error inmany arrangements, but in the current setup it hassome significant well-defined effects, for which allow-ance may be made (Section 8). Use of nonnormalincident angles and spectral-order overlap can beserious and are discussed in Sections 9, and 10, and11, respectively. We present linearization of typicalspectra with estimated error contributions to illus-trate successful implementation and inversion ofmodel formulas for varying energies. A summary ofqualitative and quantitative corrections for theseeffects is given for these experimental energies andangles.

2. Windows

In this and other typical spectrometers, thin0.0508-mm polypropylene windows with, e.g., 300-nmcoatings of aluminum, or 0.0127-mm beryllium win-dows, are used to cover film holders and reduce straylight, background fogging, and scattered electrons.These windows may lie immediately in front of andparallel to the film. We may allow for this bymodifying ,3, z, and J in Sections 2 and 15 of CI to

P = [1 - exp(-pjd')]exp[-(1ioto + j±t)/sin 0]. (1)

1 May 1993 / Vol. 32, No. 13 / APPLIED OPTICS 2411

Table 1. Summary of Primary Variables in Cl and Cl

Symbol Definition Equation/Section CI CII

I = Ip/Ad Exposing radiation intensity (photons/ [m2)D Specular optical density Eqs. (6), (25) Eq. (12)T, T Optical transmission of film or layer Eq. (3) Eqs. (9), (11)T, to, t Emulsion, supercoat, and substrate thicknesses Section 1 Eqs. (1), (6)MO Grains per unit area, monolayer packing density Eq. (2) Section 2d Mean AgBr grain diameter in emulsion Eq. (3) Section 2d' Mean path length through AgBr grain Eq. (1) Eq. (1)plo, Pl,1' p.Attenuation coefficients for gelatin, AgBr, emulsion Sections (12)-(14) Eq. (1)

- Geometric grain cross section Section 4 Section 2

a Effective cross-sectional area per grain Eq. (1) Section 2S Effective cross section of developed silver cluster Eq. (3) Section 2

0. SCf =- S = - Cluster factor, cross-section increase on development Section 17 Section 20g cg%v/v Mean volume fraction of AgBr grains in emulsion Section 2 Section 2

V=40 3 Grain volume Section 3 Section 3

a, do, a, b Semiempirical coefficients of earlier models Section 2, Eq. (25b)J, p, Zm' Integral and coefficients for absorption of x ray in grain Eqs. (25), (30) Section 4z, ZO, Zj Exposure probability for depth x, surface or half-layerj Section 4

3. Oxford Modified Joyce-Loebl Densitometer

Microdensitometry scans the film with a small-anglecone of illuminating light focused at and transmittedthrough the film. For specular densitometry thetransmitted beam is received by a nearly matchedaperture of an objective lens, imaged at a fixed slit,and delivered to a photocell. The height and spotsize are measured to 10%. The photomultiplier en-trance slit width is set to a factor of 2-5 smaller thanthe illumination source collimation, which is adjust-able from > 100 to 20-jm)shifts and motion over a period of 20-30 min as thetemperature rises, after which a steady state isestablished. This period was permitted to elapsebefore any scan was began. The main additionalsource of discrepancy was a diurnal temperaturevariation that had a potential steady linear variationof 10 lm over several hours or a slow oscillationover a period of a day. Since each scan takes from 40min to 12 h (depending on the density and the lengthof each scan), these effects are subsumed within alinear term. In any local (10-mm) region they arenegligible.

A serious hysteresis occurs during tracking fromthe end of one scan to the beginning of another. TheHeidenhain reading shifts an amount that is depen-dent on the program and motor-dependent stepping

2412 APPLIED OPTICS / Vol. 32, No. 13 / 1 May 1993

to the correct location. The stage relaxes over thefirst 0.5 mm, during which irregular step sizes mayresult from loose contacts between drive and stage.One avoids this to first order by overshooting thebeginning of the scan and tracking forward slowlyand in uniform steps to the desired beginning. Oneminimizes it by measuring the position with theHeidenhain, and a third feature remeasures anycount during which a significant (> 0.5-jim) shift orrelaxation has occurred. If the step width is notproperly performed by the motor (as measured by theHeidenhain), a second attempt may be made, afterwhich the program will halt if unsuccessful. A pe-riod of 0.5 s is allowed to elapse between counts sothat there is sufficient time to step and to allow thestage to stabilize. A final correction for this lies infurther analysis of profiles.

DEF is a double-emulsion film, but the nonnormalincidence angle of x rays makes it a source of doublelines and conveys no further useful information; thedensitometer will also focus on only one of the twoemulsions, so all densitometry uses films with thereverse emulsion removed with 0.1 M of NaOHfollowed by cleaning with water. The sides andedges are taped to glass during this process so that anattack of the remaining emulsion by NaOH or wateris avoided. 8 1 2 The emulsion (on the lower surface) ispressed flat on the stage and imaged on a viewingscreen so that the grain structure can be observed.Traversing a 100-mm length shows minor loss offocus, but there is a small curvature of the film in atransverse direction. This will average the densityover a slightly larger region than the 20-jim slitwidth, but the effect is small and introduces nosignificant shifts.

Before densitometry the exposed edge of each filmwas aligned to the longitudinal drive axis to betterthan 0.1 mm over 90 mm. This edge gave a goodestimate of the central plane of the spectrometer.

Variation of the illumination focus on the emulsionsurface provides another source of imprecision. Theillumination itself uses a stabilized power supply anda constant-current supply of 2-4 A to the source.This is measured as stable within statistics overhours of operation. The method for obtaining (uni-form) photon statistics is discussed in CI, Section 5.This enables glitches (caused by power surges) to beeliminated when the counts in the first 0.5 s arecompared with those in the remaining counting pe-riod and discrepancies outside two standard devia-tions are rejected.

4. Absolute and Relative Densitometry

Experimental densities relate to a background orunexposed region of the film with an offset added.Relative densities are reliable, but absolute densityestimates are poor. Attempts to compare differentfilms or scans with these estimates can involve discrep-ancies of density of 0.1 or more, unless calibrationwedges are used. They are reliable relative densityestimators over restricted ranges of D, but we are

concerned with absolute densities and with densitiesvarying from 0 to 4.

An improved method used the density of the mosttransparent channel as the zero estimate with a smalloffset for the fog level.8 Evaluation of the minimumover a local (10-mm) scan of 500 channels can stillyield significant variance.1 Long automated scanswith 5000 channels reduce this scatter markedlyand permit the remaining intensity variation to bemeaningfully discussed. The current arrangementalso established the zero as the minimum channeldensity of 9-20 parallel scans of a given film whenbasing observations at a common level.

Variations from the mean emulsion thickness oc-cur, especially at edges and emulsion holes, or wherecracks may develop.13 They are significant only forhigh exposures and may give a slight shift to thebackground level or yield a glitch in our spectra, butthey are otherwise insignificant. The definition ofthe minimum density as an average over three to fiveadjacent channels averages over most glitches (emul-sion holes) and improves the estimate. Any scans ofa different densitometer run (with a different initialzero setting) use this offset to yield a smooth andcontinuous background. Absolute densities are thenreliably estimated, and relative densities and intensi-ties within each film may be determined.

5. Allowance for Fog

Densities produced by densitometers are relative tosome such zero level. In the equations in CI and CIIthey are compared with unexposed but developedemulsions. Hypersensitivity, fog, and backgroundexposure need to be minimized or allowed for if theeffects of (additional) monochromatic x-ray irradia-tion are to be considered. These effects may exposegrains uniformly throughout the emulsion, preferen-tially at the surface or declining exponentially as aconsequence of absorption. Allowance can be madeby redefining the zero level, reducing the number offree grains per unit volume, identifying the back-ground density Db, and defining I (x rays) = I(D) -

For an observed density D, an observed fog densityDbf at an unexposed region of the film, and a maxi-mum density Dmax obtainable with the given film anddeveloper, the density caused by fog on an image isestimated14 as

Df = Dmx D Dbf. (2)max

Fog is less near high densities because availablegrains are reduced and soluble bromides generatedduring development restrain fog development. Equa-tions (3), (4), (7), et seq. in CI could use D - Df insteadof D, although variation in the fog level with positioncan invalidate this.13

Background fog levels from the sensitization ofgrains in DEF emulsion preparation in these condi-tions yield Dbf < 0.12. The variation in Df with D isnegligible for DEF emulsions, since Dmax >> 4; thus


only a constant shift caused by fog is required.Aerial fogging, reticulation, and pressure marks areeliminated by a correct development and drying tech-nique.

Dichroic fogging Ddf, from the development ofsilver halides dissolved by developer solvent, is great-est in regions of maximum development. It shouldlie on the surface layer only (randomly distributed),so that a constant spatially varying density compo-nent may be added to the true density. This can giveDdf -- 0.5, which could preclude further analysis inaffected regions. Where Ddf changes rapidly no anal-ysis is sensible, but for regions where it is nearlyconstant the effects on centroids is random and mayaverage to zero, while intensities of weak lines will beincreased and distorted compared with stronger lines.For line images on emulsions, this kind of fogging isreadily identified, and it is observed to be Ddf < 0.1 inregions of interest.

Ddf reduces the number of available grains (in thesurface layer) without affecting absorption coefficients.Effective densities are slightly reduced for a givenintensity, while errors are slightly greater. Thebackground exposure on the film dominates the foglevel in our case, and the effective increase in ob-served density should be < 0.12 for the most transpar-ent background region and negligible for all peaks(and tails of peaks).

6. Reciprocity and Development

The intermittency effect, where a series of discreteexposures gives a larger or smaller effect than acontinuous exposure with the same number of inci-dent photons, is a manifestation of reciprocity lawfailure. The experimental technique uses a 30-Hzbeam cycle, and these effects are important for visibleand UV light. However, x rays give a chain reactionthat tends to create a developable speck with a singlephoton, and hence both low- and high-intensity fail-ures are evaded.

Solarization is connected with both exposure anddevelopment; it decreases density when exposure isincreased at high exposure levels. Exposure is thena nonmonotonic two-valued function of density; thuslinearization becomes impossible. Standard linear-izations would show a symmetrical hollow at thecenter of strong peaks. Effects associated with vary-ing densities along the emulsion (as with Lyman-Balmer peaks) include Eberhard and Kostinsky ef-fects, where densities in, near, or between denseimages are reduced by soluble bromides releasedduring development. These effects are negligible forlow densities, small for strong developers (hydroqui-none) and longer development times, and reduced bybackground exposure or fog (Ref. 14, p. 875).

Following the recommended 5-min duration forD-19 developer implies that the density should haveapproached its limiting value by this time.15-18

Evenness of developmental time is attained by use ofa developing tank system with periodic agitationwhere the addition of each solution is completedwithin 10 s for typical film. No flattening, hollow-

ing, or repulsion of peaks was observed with DEFemulsion, D-19 development, and standard condi-tions at densities up to D = 4.0.

Migration of grains during swelling, drying, anddevelopment can yield shifts of centroids and relativeintensities. Silver clusters may be displaced by agrain diameter (with a maximum error of 1.6 jim,typically 0.6 Lm in a random direction, with the meanerror vanishing). Drying may introduce errors of 2jim, especially if the base suffers from shrinkageeffects, but this is mainly near the edge and in ourcase should be uniform along the film. Our filmsappear to be free from serious errors of those sorts.

7. Features of Experimental Data

The variation in density across and along the exposedportion of the films typically includes a smoothvariation of peak location caused by the fast beamDoppler shift (and a broadening from angular diver-gence) and a smooth component along the film fromscattered electrons and background radiation (Fig. 2).This variation dominates over the base fog level formost of the film. Superimposed on some films is asharp feature caused by direct radiation from thesource not prevented from reaching the film. This isdominant at the film edge closest to the source and isnegligible in regions of importance. Sometimes adiffuse shadow or cloud which arises from dichroicfogging, is displayed over some portion of the film.Dichroic fog on some emulsions is serious with re-spect to relative intensities but occurs only near thesecondary (third-order Lyman-.) lines; thus it doesnot show significant effects in this study. It does notoccur in any Lyman-a-Balmer-3 region.

8. Collimation and Errors

Contributions to uncertainties and variances fromx-ray counting, grain counting, and densitometerphoton-counting statistics, and experimental contri-butions from inadequate conversion to specular densi-ties and allowance for background errors, were dis-cussed in CI and CII. We assume that the currentprocedure gives negligible contributions to back-ground uncertainty and intrinsic linearization formu-las, so that statistical variance estimates provideuncertainties of linearized spectra.

A larger source of error concerning the photonintensity on the film relates to the collimating wiresused for alignment. These occlude 0.1 mm of thefilm for the length of the film and are roughly parallelwith 1-mm spacing. The density observed normallyincludes a 10% transmission from these wires. Insome scans two wires (or no wires) may occlude theregion, giving rise to (spurious) intensity variationbetween scans.

We could correct for this problem on average bydefining the area and number of available grains (Mo)to be reduced by 10%. Within the assumptions inEqs. (3) and (4) of CI, this is equivalent to multiplyingthe density by 1/0.9 (to give a mean density for theexposed region) and scaling the incident x-ray countIp = I x Ad by 0.9. (The estimate of the number of


--secondary lines --direct , W 1

radiation 0 . ,< X *zE -0 i 2 n- Lyman-a-

;0 = =; U Balmer-P r egion

pressure marks

Lyman- - Balmer-fi region

images of alignment wires

Fig. 2. Prints of a pair of typical films in the iron experiment, diffracted by PET crystals. The Lyman-a-Balmer-3 region is expandedbelow; images of alignment wires for curved (Doppler-shifted) hydrogenic transitions, compared with those for stationary source calibrationlines (in this case SiKa), are shown. The images are shifted since the sources are not coincident. Secondary lines, clearer in A120, arelabeled. On the smooth background variation is superimposed strong exposure from direct radiation in the short-wavelength region(A120). Clamping (pressure) marks lie at the end of the film from the drying process and at the side edges resulting from the use of a tanksystem in the development. A120 shows two faint regions of (dichroic) fog; B220 appears to be seriously affected by this.

photons for the 20 x 1000 jim2 region would notrequire scaling by 0.9, but error estimates should usethe reduced value.) Variation from scan to scanwould involve scaling densities by 1.00-1.20. Thiscan shift intensities by a factor of 2. Within eachscan, however, the error is constant; so it is notincluded in linearization or error estimates. Theymay be accounted for best by observation of relativex-ray counts for given peaks in different scans, withscaling that corrects for this. Symmetric distortionof channel intensities will occur for rapid changes indensity, tending to make widths broader and thepeaks of slightly more Gaussian character. (Tailswill have a smaller shift to a Lorentzian character.)This typically represents a < 3% increase of widthswith up to a 26% shift of the (peak) profile character.

Errors of Ad have no effect on relative intensitieswithin scans and do not distort noise or asymmetry;so they are not propagated to the peak fitting routinein the error field. They also do not affect calcula-tions of x-ray production cross sections since the areafactor cancels.

9. Effect of Angle on the Profile

In preceding equations we calculated the exposure fora region of uniform mean density with statisticalfluctuations. Increasing exposure is assumed todarken deeper layers as viewed by the densitometer.However, spectral lines are of finite and narrow widthand may show a large variation across the slit width

in the densitometer; lines are curved or locally slantedwith respect to the densitometer drive and rectangu-lar slit region; and the emulsion is observed at normalincidence while irradiation takes place at angle 0.

Lyman (and Balmer) linewidths are of the order of230 jim (full width at half-maximum). This is muchlarger than longitudinal step sizes and slit widths of20 jim, a grain diameter of 1.6 jim, and graininess atdensities of 0.3 of 5-6 jim. If spectral lines wereparallel to the slit, each peak would be divided into 11or 12 sections: near an optimum division with re-gard to statistical noise. Peaks correspond to densi-ties of 2 (for example); so the channel with amaximum range of density is that at half-maximum,which may typically have a density of 0.5-1.2 with avariation of 0.15-0.6 in density from one side of theslit to the other. These densities correspond to theend of the toe or the linear region of the D - n Icurve; so Eq. (33) of CI holds to some approximation:

D = -log 0 = C3(10glo Zma + 0.5772) - C4Zmin,

Zmax = c3I.

The transmission, averaged by the densitometer, isthen T AI-C3, C3 MS(sin0/jL'd). Relevantenergies, absorption, and scaling coefficients are givenin Table 2 for the parameter values from CI and CIIand for Lyman-CL and Balmier-P energies of iron andgermanium. Linearized exposures will be undis-


dichroic fog

Table 2. Absorption and Scaling Factors for Lyman-c - Balmer-p Radiation, Fe2'5 and Ge3l+ a

Charge State

Fe 25+ Fe 25+ Ge31 +

Transition Ba-g Ly-a Ba-p Ly-a Ba-P Ly-a

E (keV) 1.7 7.0 1.7 7.0 2.8 10.6

Spectrometer, diffraction in first or fourth orderDiffraction plane PET 002 PET 008 ADP 101 ADP 404 Si 111 Si 444O)R 0.955 0.955 0.734 0.734 0.842 0.842

0.099 0.0015 0.099 0.0015 0.0232 0.00044

Pl 2.192 0.148 2.192 0.148 0.636 0.050

0.50 0.057 0.48 0.057 0.22 0.020

Zma=exp(-pL'T/sin 0) 35-4 4.0-1 9.0-5 3.3-1 2.2-2 7.1-'ZminC 3 (Section 9) 1.0 8.2 0.9 7.4 2.1 23.5

ad = 1.6 pLm, T = 13.0 pum, MoS 0.94; the exponential notation is an abbreviation and should read in full as x 10 to the power indicated.

torted for Balmer radiation. Averages of I-8, I-24over ranges of ± 7.5% and ± 3.0% for Lyman radiationdecrease mean intensities by only 0.9% and 0.3% foriron and germanium data, respectively. The varia-tion of exposure in the densitized region for thisparticular agreement would not yield a major system-atic correction.

Lines are slanted because of the Doppler shift, andalthough it is possible to scan normal to the localcurvature over a short range of the film, the result isdifficult to analyze. The mean angle to the spectrom-eter plane is 12.25 + 0.1° with the lower edge typically11.20 a id the upper edge 13.4° because of the curva-ture of the lines. The angle of the spectrometerplane to the slit is a" 0.285°, so that the densitizedregion lies at 10.9-13.1° to the line curvature. Thevariation in density is then dominated by that fromthe top to the bottom of the slit. This will cross thepeak, lowering (peak) densities substantially. Theaverage density is a relatively reliable measure of themean exposure, as indicated by the I-8 averagesabove.

The most serious effect will be on the linewidths,which are broadened by this convolution with the slitfunction. For 1-mm heights Lyman and Balmerwidths increase because of this 200-jim broadeningby from 30% to 100%. These estimates are uncer-tain because of the alignment wires and other effects."Conversely, densitometry with a slit height of 0.4 mmgave negligible shifts of centroids or intensities, whilethe slit broadening was reduced to 89-107 j.Lm. Thenarrow height increases individual scan variance, butthe convolution problem is nearly eliminated, and thegreater number of scans gives improved estimates ofspectrometer parameters and reduced final uncer-tainty. This is a way of dealing with some of themore dominant effects of slanted lines and rectangu-lar densitometry. A comparison of results with nar-row and wide slit heights confirms spectral resolutionand intensities and gives information on deconvolvedprofiles.

The photon incidence angle on the emulsion isfundamental and alters both the centroid determina-

tion and the linearization formula. For a narrowbeam of x rays incident at an angle of 560 (the ironLyman-Balmer region using PET crystals), the re-gion of emulsion exposed in the first monolayer isdisplaced around the film from the region in the lastlayer of emulsion. This is different for varying ener-gies and angles. Iron Lyman radiation (assumingnegligible absorption) may give a mean shift of theobserved line by up to (T tan 0)/2 9.6 jim comparedwith a negligible shift for weakly penetrating Balmerradiation. The shift is nearly half of a channelwidth, so must be accounted for. It is simple toinclude a centroid correction in profile analysis" byusing the mean attenuation coefficient. The profilewill include a skew component of up to 9.6 jim inwidth convolved with the profile at the surface of theemulsion, which is also readily included in profileanalysis or deconvolution.

The more fundamental effect on intensities ofirradiation at angle 0 with densitometry at normalincidence is discussed in part in CII, Section 5. Thecomplexities of this model will require explicit evalua-tion of the angular dependence (under the givenassumptions) for a range of energies and angles.This is not pursued in this paper. However, themajor ideal, broad-source dependence follows thatexplained in CI, Section 19.

This discussion assumed that any narrow photonflux was incident normally, so that the density at agiven point corresponds to transmission of the sameincident light through that section of film; whereasthe density at one section will be solely due to theexposure of the surface monolayer, while that at thesection 9 jim along will be due only to exposure of thedeepest layer. Fortunately unconvolved widths areat least six channels (115 jm) so that any 20-jim slitwidth experiences the same incident flux across this4-9-jLm length. In other experimental arrange-ments this issue must be reassessed.

10. Comparison of Linearizations

An important difficulty in linearization of Lyman-Balmer regions arises from the overlap of different


orders and energies. In Fig. 3 we emphasize this byshowing spectral profiles in the Lyman-a.-Balmer-,3region for particular scans of three different DEFfilms, one for iron spectra diffracted with a PETcrystal, one diffracted with an ADP crystal, and one

60000

40000

.W

S

20000

30 3230 30 303130 3230 3330 3430

Channel Number(a)

400000

300000

a.In

200000

100000

o ;3275 3300 3350 3400 3450

Channel Number(b)

for germanium spectra diffracted with a silicon crystal.They illustrate the effects of large ranges in density of1.2, 3.0, and 2.8, respectively. Circles with barsindicate the best estimates of absolute incident pho-ton counts with total statistical (grain, x-ray, and

10000

0q

5000 C

1000 l l l

deviation' b/g error-1000I I I I I I I I I I I 1

3900 4000 4100 4200 4300Channel Number(c)

Fig. 3. Comparison of linearizations and errors for Lyman-a(ls-2p)-Balmer-p (21-41') spectral regions with resultant fits withVoigt profiles. Linearizations of Ref. 11 (*) with error field (---)are compared with typical fits (-), with linearizations of model 6 ofCII ( with error bars, and error field -) when first-order(Balmer) energies are assumed, and with corresponding lineariza-tions (x) when fourth-order (Lyman) radiation is assumed.

Intensities (photons per channel with a typical height of 0.4 mmand width of 20 pLm) are plotted against channel number. (a)Linearization A120T of exposure A120, scan 1, for PET diffractionof Fe25+ radiation. (b) Linearization A317A of exposure A317,scan 2, for ADP diffraction of Fe25 + radiation. (c) LinearizationBX25A of exposure X25A, scan 6, for silicon diffraction of Ge3 l+radiation. Model 6 linearization is similar to that of Ref. 11; thusit is omitted for clarity, while we include the discrepancy of fits (-)to compare with error fields. The b/g error is indicated by thedeviations between well-separated peaks, far from profile tails.


detector sources) uncertainties, following CII andmodel 6. Stars indicate a similar model, described indetail in Ref. 11, fitted by use of a smaller data set andwith minor modifications below, with the (dashed)error field repeated as the solid curve near the zerolevel. This indicates variation resulting from model-ing.

These linearizations agree at estimated one- ortwo-standard-deviation levels, although quoted inten-sity error estimates differ significantly. Contribu-tions from detector statistics are unchanged. Theearlier model underestimated the background (low-D)uncertainty from x-ray statistics by considering onlyNine (CI, Section 3 and CII, Section 7) and overesti-mated high-D uncertainty with a different form ofcorrelation parameters and grain uncertainty. Inthe three examples plotted the uncertainty for D =0.1 was underestimated by 22%, 45%, and 47%,respectively; that for D = 0.5 was overestimated by12%, -8%, -13%; and that forD = 1.3 was overesti-mated by a factor of 2.3, 1.7, 1.7 for the threeLyman-a-Balmer-3 regions. For peak densities onthe latter two films (D = 2.75, D = 3.05) thisoverestimation reached factors of 5.2 and 4.1.

These factors lead to different Xr2 values for fits butwith similar areas and profiles. The scatter in Fig.3(a) is typically twice the error field, which suggeststhat the smooth background is not dominated bylow-energy photons as linearized but by electrons orhigher-energy x rays. Asymmetry is distortedslightly, but symmetric profiles have undistortedcentroids. The x2 values are dominated by the inad-equacy of fitting complex profiles with simple Voigtfunctions, which are compounded by the inadequacyof the first-order linearization for Lyman-a peaks.Figure 3(b) provides an extreme case of this, wherethe profiles are poorly fitted because of peak distortion.In such cases one may prefer to linearize, assumingfourth-order (higher-energy) radiation, and to correctBalmer radiation after fitting the profiles. Despiteequally high densities, the precise fitting of profiles inFig. 3(c) is achieved with deviations in agreementwith error estimates, which indicates that radiationin this energy range may be responsible for thebackground levels observed. The experimental agree-ment of iron spectral relative intensities" from differ-ent films and after diffraction by PET or ADP crystalslies at the 25% level, which is the same order asexperimental fitting uncertainties and provides someconfirmation of the linearization and analysis proce-dure.

Lyman-a intensities incident on the film are muchweaker for the iron data with PET diffraction, aboutthe same for ADP diffraction, and much stronger forthe germanium data with silicon diffraction. Thisfollows from source strengths and diffraction efficien-cies. Experimental ratios may thus be used fortesting models for reflectivity from curved crystals.'9

The linearization for fourth-order-versus first-orderradiation reveals an optimum sensitivity for DEFemulsion of 7 keV and confirms the difficulty of

fitting overlapping profiles with different energiesand photographic sensitivities.

From Eq. (6) of CI we can estimate standard photonfluxes, from which the more exact but not readilyinvertible formulas of CII may be compared, by usingtabular interpolation. The parameters from Ref. 15with the assumptions therein were not plotted in thefigures. Discrepancies between this and the currentplots are -42%, -17%, -39% for D = 0.1; -25%,+5%, -28%forD = 0.5; -9%, +23%, -14% forD =1.3; and +68%, +23% for the peaks ofthe latter films.This indicates the different effect of angle on thelinearization between the method of Ref. 15 and thatof CII. The lower Xr 2 for the current model suggeststhat these discrepancies are inadequacies of the sim-pler model, particularly for densities below two, whichwill lead to errors of relative intensities if used. Thesimpler model is much better than linear or exponen-tial assumptions for linearization, which have beenused previously.8"12 Relative Balmer-3 intensitiesgiven by Henke's formulas appear to be accurate to afactor of 2, compared with the current model, butlarger errors arise at angles that are far from normalincidence or at low energies.

11. Order Overlap

The energy and angle dependence of linearizationalong the film is slow but becomes important over the100-mm lengths scanned in these experiments.Linearization must in any case make an assumptionabout the order of the dominant irradiation or thefraction of each kind at any given point. This as-sumption varies the expected x-ray energy by a factorof 4. The procedure adopted in the analysis ofcurrent experimental data is to initially linearize byassuming first-order radiation, which yields accuratevalues for the Balmer series and calibration sourcelines. Background exposure from scattered elec-trons from the beam and from direct or scatteredlight reaching the emulsion has a complex energydistribution but is smooth, is of low density in regionsof interest, and may be neglected. Incorrect linear-ization of the Lyman series and heliumlike spectramust be considered. This arises from the variationof E and hence ji', jio, and uji (absorption coefficients,see CI) for the same angle, focused onto the film in adifferent order of diffraction for a given crystal.

Linearized peaks are fitted in a self-consistent waythroughout scans of a given film so that correlation isminimized. The resulting Lyman intensities or ar-eas must then be corrected for the amount by whichthey have been overestimated in the linearization.We may use Eq. (6) of CI to estimate correctionfactors of this sort, particularly for relative intensityratios within a given order. They confirm the trendsand relative corrections in the table. It is less precisefor comparing radiation from different orders andgreatly differing densities. In Table 3 we presentcomparisons of correction factors using Eq. (6) com-pared with those using model 6 of CII for the range ofwindow materials and thicknesses and main primary


Table 3. Linearization Correction Factors I(Lyman)/I(First Order)

Window (Section 11)Fe Radiation,

PET Diffraction Al B4

Transition 3Ly-a 4Ly-,y 4Ly-P 4Ly-a 4Ly-oxE (keV) 6.97 8.70 8.25 6.97 6.97E (linearized) 2.32 2.175 2.063 1.743 1.743OR 0.657 0.711 0.759 0.955 0.955D = 0 Eq. (6), CI 0.359 0.424 0.349 0.157 0.155D = 1 Eq. (6), CI 0.286 0.328 0.268 0.123 0.120D = 2 Eq. (6), CI 0.221 0.250 0.198 0.0938 0.0907

D = 0.0 0.350 0.402 0.333 0.1618 0.1592D = 0.5 0.350 0.402 0.327 0.1628 0.1585D = 1.0 0.316 0.359 0.294 0.1447 0.1411D = 1.5 0.290 0.324 0.261 0.1294 0.1264D = 2.0 0.250 0.277 0.220 0.1090 0.1060D = 2.5 0.215 0.233 0.179 0.0884 0.0861

Be None

D = 0 Eq. (6), CI 0.828 1.120 1.010 0.679 0.937D = 0.00 0.808 1.057 0.966 0.697 0.965D = 0.25 0.816 1.058 0.958 0.701 0.968D = 0.50 0.812 1.056 0.953 0.696 0.964D = 0.75 0.793 1.034 0.929 0.682 0.938D = 1.00 0.734 0.955 0.853 0.620 0.857D = 1.50 0.669 0.851 0.757 0.555 0.766

Fe Radiation,ADP Diffraction Al

OR 0.525 0.565 0.600 0.734D = 0 Eq. (6), CI 0.261 0.278 0.228 0.0956D = 1 Eq. (6), CI 0.201 0.202 0.158 0.0670D = 2 Eq. (6), CI 0.136 0.141 0.106 0.0455D = 0.0 0.251 0.259 0.216 0.0958D = 0.5 0.244 0.256 0.206 0.0923D = 1.0 0.211 0.217 0.177 0.0782D = 1.5 0.184 0.187 0.147 0.0646D = 2.0 0.151 0.150 0.114 0.0493D = 2.5 0.121 0.118 0.0886 0.0362D = 3.0 0.101 0.0882 0.0648 0.0233D = 3.5 0.0756 0.0647 0.0454 0.0164

and secondary spectral regions for hydrogenic ironexperiments. In Table 4 we give corresponding re-sults for the hydrogenic germanium experiment.

This shows that linearity of photographic responsefails at D = 0.5 and quantifies the distortion ofprofiles by first-order linearization. It also indicatesdiffering correction factors for strong and weak linesbut shows that additional background density is ofminor effect in this context. In most films from ironexperiments, first-order linearization overestimatesLyman contributions for tails and more so for peaks.Peaks will appear narrower than they should be.

For PET diffusion at angles corresponding to Ly-man-a, 83.7% of first-order radiation is absorbed bywindow Al (334-nm aluminum coating on 50.8-jimpolypropylene), while negligible (2.7%) fourth-order(Lyman) radiation is absorbed. Differential windowabsorption between windows Al through B4 (withdifferent, measured aluminum thicknesses) is 1-2%.

Table 4. Linearization Correction Factors I(Lyman)/I(First Order)

WindowGe Radiation,Si Diffraction Be

E (keV) 10.6 13.24 12.55 10.6E (linearized) 3.533 3.31 3.138 2.65OR 0.594 0.640 0.681 0.842D = 0, Eq. (6), CI 2.02 3.48 3.13 2.29D = 1, Eq. (6), CI 1.63 3.21 2.88 2.02D = 2, Eq. (6), CI 1.32 2.91 2.58 1.75

D = 0.0 1.809 2.661 2.470 1.981D = 0.5 1.807 2.664 2.492 1.974D = 1.0 1.683 2.664 2.483 1.962D = 1.5 1.535 2.566 2.394 1.826D = 2.0 1.400 2.502 2.298 1.756D = 2.5 1.228 2.425 2.222 1.618

Use of 12.7-jim beryllium windows reduces the formerloss to 28%, and changing the angle to that for ADPdiffraction increases the loss to 91.5%. Thus win-dow absorption dominates iron data corrections, whilegermanium corrections are dominated by transmis-sion factors of radiation through the depth of theemulsion.

For the A120 film densitized with 1-mm slit heights(A120M), the strongest Lyman component gave adensity maximum of 0.55-1.1 (which varied betweenscans because of the source distribution and align-ment wires), the strongest Balmer component gavedensities of 0.7-0.85 at the peak, and Lyman-a2 gavea peak density of 0.22-0.55. Correction factorsfor particular peaks depend primarily on the densityfor the half-maximum intensity value (i.e., at the fullwidth at half-maximum). This mean density is-DM = 0.5(Dpk + Db/g) for Dpk < 1.4 or DM Dpk -

(0.6 -- 0.9) for higher densities. Typical values forthe different films used in our experiments are illus-trated in Table 5 for the fourth-order Lyman-acomponents. The Lyman-a, (1s-2p 3 /2 ) componentin A120M with densities at half-maximum of 0.65 or0.3 for the strongest and weakest scans may becompared with that for Lyman-a2 of 0.275 or 0.1.[I(Lyotj)/I(Lyot2)] is then overestimated by 4%. ForA317A, it is overestimated by 48%.

Table 5. Peak 1 (First-Order), Dpk and DM Estimates for Typical Films

Film

A120M A120T BX11A A317A B417A A322S BX25A

Crystal PET PET PET ADP ADP ADP SiWindow Al Al Be A3 B4 A3 BeSource Fe Fe Fe Fe Fe Fe Ge

I(4Ly-aj) 125k 52k 10k 640k 450k 200k 18kDpk 1.1 2.0 1.1 3.77 3.36 2.0 3.0DM 0.65 1.3 0.65 2.9 2.41 1.4 1.8

I(4Ly-a 2 ) 60k 19k 5k 280k 176k 110k 10kDpk 0.55 1.1 0.6 2.8 2.07 1.32 2.0DM 0.28 0.70 0.35 2.0 1.35 0.9 1.2


The accuracy of these intensity corrections is at the2% level. Tabulated corrections have significant un-certainties where Doppler shifts cause the transitionto cross a first- or higher-order absorption edge.The variation with energy is then discontinuous.There is a discontinuity in first-order iron radiationat 1.78 keV caused by Br L1. The interpolation ofcorrection factors should be made from either side ofthe tables. The Br K edge affects germanium Lyman-y/6 radiation; thus extrapolation from the table tohigher energies is not reliable. Germanium Ly-man-8 may show profile discontinuity in some scansas a result.

12. Densitometry Corrections

First-order linearization allows for most effects withinthe quoted output uncertainty. In this section wesummarize those systematic and random error sourcesthat are not so included.

(a) Scans containing alignment wires should, bescaled up in intensity to those lines of adjacent scans(so that we can estimate photon fluxes and distribu-tion). These scans are weak and may show largevariances of centroids compared with adjacent scansbecause of reduced statistics or the nonparallelism ofthe wires with the drive. Such scans are given lowweighting in the overall fits. Widths are normallysmaller, but if the wire image is centered on the scan,a flat-topped or double peak may result.

First and last scans may cover part of the unex-posed region and may be abnormally weak. If thespectrometer plane is poorly aligned, some peaks(usually third-order lines) may be absent from thefirst or last scans. Uncertainty in the area observedby the densitometer is at the 5-10% level but has anegligible effect on intensities (per unit area).

(b) Background electron scattering on the high-energy end of the film, and the effects of fogging thatusually arise in the same region (a consequence ofdeveloping imprecision), has no effect at first order.Where this varies rapidly or quadratically, foggingmay prevent weak lines from being fitted or measuredto better than 40 lm or more, and fits are givenappropriate uncertainty.

(c) Slanted lines yield average densities convolvedwith a slit function. They generally make lineseasier to fit but decrease resolution. Areas andcentroids remain constant, and heights of 0.4 mmminimize this.

(d) Linearization depends on energy; so energiesand angles for two channels must be given initially.Several films have calibration lines for this purpose,but in other films corresponding channels may needto be specified aposteriori by comparison with Lymanor Balmer lines, which requires a test linearizationfirst. Lyman and Balmer lines are Doppler shiftedand appear slanted; so identification of these chan-nels with particular energies yields an error. Anerror of 560 channels (which is 10-fold larger than a

typical error from this source) gives intensity errorsat the 1% level, because linearization is a slowfunction of angle and energy.

(e) The linearization here uses a double interpola-tion for intensity versus density, between D andbetween two energies/angles, beyond the extremesgiven in Table 3. The D interpolation is accurate to1%, especially at low densities. It may be comparedwith first-order shifts for PET iron data from valuesfollowing Ref. 8 by, e.g., 30% at low densities to 9% atD = 1, 8% at D = 2, and 12% at D = 2.5.

However, E interpolation overestimates intensitiesat intermediate energies by up to 14% with relativeintensities that are accurate to 3%. Absolute intensi-ties in the Lyman-a-Balmer-f3 region are typicallyaccurate to 4% (with a relative intensity accuracy of1%). This may be corrected for by scaling Lyman-a-Balmer-,B intensities by 0.96, Balmer-y by 0.86, Ly-man-1-Balmer-8 by 0.9, Balmer-e by 0.95, and third-order Lyman-a intensities by 0.99. At importantregions (Lyman-a) these shifts are the same size asthe uncertainty (1-3%).

(f) Error estimates for each channel use the sameinterpolation method. This underestimates errorsfor densities by up to 10% for intermediate angles.For densities over 0.2 (i.e., where lines exist) this hasa relative error of 2-3%. This increases x2 of fitsslightly but has a negligible effect on parameters.

(g) Corrections of relative intensities for orderoverlap in iron spectra with PET diffraction are givenin Section 11 and dominate most other effects.

(h) The fits of linearized profiles may underesti-mate or overestimate the peak areas because of thedifferences between fitting and linearized profiles,low background errors, or statistics. An example isprovided by fourth-order Lyman-a. radiation dif-fracted from ADP crystals over long exposures [Fig.3(b)]. These are very strong (D 3.7) lines withweak Balmer lines nearby. The effect of lineariza-tion errors (g) is to increase tails by factors of 10 andpeak intensities by factors of 60! Low backgroundand large peak errors force the fit to underestimatethe peak height and area by 25-29% for strongLyman-a, peaks (films A317A, B417A). B422S andA322S have a 10-15% error from this source.Lyman-ax2 components are affected at the 1-8% level,which is at the level of the fitting uncertainty. Noother lines of films need this correction.

Lyman radiation diffracted in third order may bestrongly asymmetric in profile for germanium data' 9

but is fitted with symmetric functions. These linesare strong, and intensities may be underestimated by20-25%. Centroids may also need correction, if thesepeaks are used in a subsequent analysis.

(i) To gain relative or absolute intensities pro-duced by the source, results are scaled by integratedreflectivities of the diffracting crystal. The observedcomponent of the film is usually mr-polarized; thusscaling gives the r component of absolute sourceintensities and a 50% estimate of the total. This


estimate can be improved with detailed investiga-tions."l

Potential centroid errors include asymmetric depthpenetration for nonnormal incident angles, smallershifts of < 0.2 jim for any channel because of grainextrusion and migration from adjacent channels, 1.0jim for any channel from the Heidenhain readingerror, stage relaxation and the drive periodicity, 10jim/60 mm for a linear thermal drift, and 20-40-jim shift from one scan to the next. The totalposition uncertainty is dominated by the effects ofprofile asymmetry, diffraction shifts, and other sys-tematics."

13. Conclusions

We have inverted the models from previous papersand applied them to specific experiments and densi-tometry procedures, and we have discussed the effectsand corrections relevant in the derivation of cen-troids, profile widths, assymmetry, and areas, allow-ing an accurate estimation of relative and absoluteintensities and profile widths with statistical uncer-tainties.

These procedures may be applied to other energies,emulsions, and spectrometers, as noted in CII forKodak 101-07. The optimized correlated model usedhere appears to be more reliable for extension to otherdensities, angles, and energies than earlier models,but further photographic experiments, particularly athigh densities, varying angles and ranges of energy,appear to be necessary for further investigation ofthese findings.

The authors thank Shell Australia, the HasselbladFoundation, and the Draper's Company for supportof this research and St. Anne's College, Oxford, for aJunior Research Fellowship covering part of theperiod of research. This work was partially sup-ported under the auspices of the U.S. Department ofEnergy by the Lawrence Livermore National Labora-tory under contract W-7405-ENG-48. Acknowledg-ments go to J. M. Laming, W. A. Hallett, and J. S.Brown for assistance and support and also to A. J.Varney for helpful comments.

References1. C. T. Chantler, "Photographic response to x-ray irradiation I:

estimation of the photographic error statistic and developmentof analytic density-intensity equations," Appl. Opt. 32, 2371-2397 (1991).

2. C. T. Chantler, "Photographic response to x-ray irradiation II:correlated models," Appl. Opt. 32, 2398-2410 (1992).

3. C. J. Hailey, R. E. Stewart, G. A. Chandler, D. D. Dietrich, and

R. J. Fortner, "A precision measurement of the 2 P3/2-2P1 /2

fine-structure splitting in hydrogenic iron and zinc usingbeam-foil spectroscopy," J. Phys. B 18, 1443-1448 (1985).

4. D. D. Dietrich, G. A. Chandler, R. J. Fortner, C. J. Hailey, andR. E. Stewart, "Observation of electric quadrupole decay inXe45+ and Xe44+," Phys. Rev. Lett. 54, 1008-1011 (1985).

5. D. D. Dietrich, G. A. Chandler, R. J. Fortner, C. J. Hailey, andR. E. Stewart, "Precision x-ray spectroscopy on 8.5MeV/amuheavy ions," Nucl. Instrum. Methods B 9, 686-688 (1985).

6. J. D. Silver, A. F. McClelland, J. M. Laming, S. D. Rosner, G. C.Chandler, D. D. Dietrich, and P. 0. Egan, "Simultaneousobservation of Lyman-a and Balmer-P transitions in hydro-genic iron, Fe25+: a novel technique for s Lamb-shift mea-surement," Phys. Rev. A 36, 1515-1518 (1987).

7. J. M. Laming, C. T. Chantler, J. D. Silver, D. D. Dietrich, E. C.Finch, P. H. Mokler, and S. D. Rosner, "A differential measure-ment of the ground state Lamb shift in hydrogenic germa-nium, Ge31+, Nucl. Instrum. Methods B 31, 21-23 (1988).

8. J. M. Laming, "A differential measurement of the ground stateLamb shift in hydrogenic germanium, Ge31+," D.Phil. thesis(University of Oxford, Oxford, 1988).

9. W. A. Hallett, "Measurement of An = 0 transitions in helium-like neon and silicon," D.Phil. thesis (University of Oxford,Oxford, 1991).

10. J. S. Brown, C. W. Band, E. C. Finch, R. A. Holt, H. A. Klein, J.Laursen, A. F. McClelland, N. J. Peacock, J. D. Silver, M. F.Stamp, and J. Takacs, "Studies of 1s2s 3S-ls2p 3P transitionsin helium-like Ne8+ recoil ions using photographic spectrosco-py," Nucl. Instrum. Methods B 9, 682-685 (1985).

11. C. T. Chantler, "Beam-foil spectroscopy of highly-ionizedatoms, precision measurements of hydrogenic Lamb shifts andx-ray diffraction of curved crystals," D.Phil. thesis (Universityof Oxford, Oxford, 1990).

12. A. F. McClelland, "Development of a novel technique for themeasurement of the 1S 1 2 Lamb shift in high Z hydrogenic ionsby Lyman ct/Balmer wavelength intercomparison," D.Phil.thesis (University of Oxford, Oxford, 1989).

13. G. M. B. Dobson, I. 0. Griffith, and D. N. Harrison, Photo-graphic Photometry (Oxford U. Press, London, 1926).

14. C. E. K. Mees, The Theory of the Photographic Process(Macmillan, New York, 1946).

15. B. L. Henke, J. Y. Uejio, G. F. Stone, C. H. Dittmore, and F. G.Fujiwara, "High-energy x-ray response of photographic films:models and measurement," J. Opt. Soc. Am. B 3, 1540-1550(1986).

16. P. D. Rockett, C. R. Bird, C. J. Hailey, D. Sullivan, D. B.Brown, and P. G. Burkhalter, "X-ray calibration of Kodakdirect exposure film," Appl. Opt. 24, 2536-2542 (1985).

17. W. C. Phillips and G. N. Phillips, Jr., "Two new x-rayfilms: conditions for optimum development and calibrationof response," J. Appl. Crystallogr. 18, 3-7 (1985).

18. B. L. Henke, S. L. Kwok, J. Y. Uejio, H. T. Yamada, and G. C.Young, "Low-energy x-ray response of photographic films. I.mathematical models," J. Opt. Soc. Am. B 1, 818-827 (1984).

19. C. T. Chantler, "X-ray diffraction of bent crystals in Bragggeometry I: perfect crystal modelling," J. Appl. Crystallogr.25, 674-693 (1992).


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