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

C. T. Chantler; J. D. Silver; D. D. Dietrich

doi:10.1364/AO.32.002411

Applied Optics
Vol. 32,
Issue 13,
pp. 2411-2421
(1993)
•https://doi.org/10.1364/AO.32.002411

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

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

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C. T. Chantler, J. D. Silver, and D. D. Dietrich, "Photographic response to x-ray irradiation. lll: Photographic linearization of beam-foil spectra," Appl. Opt. 32, 2411-2421 (1993)

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Abstract

In this paper models for the relation of specular density to incident (x-ray) intensity with uncertainties are applied 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 is quantified. Relative first-order intensities are determined directly. Secondary linearization or calculation for higher-order lines gives correction factors that yield absolute and relative higher-order intensity ratios. The effects of energy and angle on linearization are included. Densitometry uncertainty is estimated and quantified.

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Symbol	Definition	Equation/Section CI	CII
I = I_P/A_d	Exposing radiation intensity (photons/μm²)
D	Specular optical density	Eqs. (6), (25)	Eq. (12)
τ, τ_l	Optical transmission of film or layer	Eq.(3)	Eqs. (9), (11)
T, t₀, t_b	Emulsion, supercoat, and substrate thicknesses	Section 1	Eqs. (1), (6)
M₀	Grains per unit area, monolayer packing density	Eq.(2)	Section 2
$\overset{̅}{d}$	Mean AgBr grain diameter in emulsion	Eq. (3)	Section 2
d′	Mean path length through AgBr grain	Eq. (1)	Eq. (1)
μ₀, μ₁, μ′	Attenuation coefficients for gelatin, AgBr, emulsion	Sections (12)–(14)	Eq. (1)
$σ_{g} = \frac{π}{4} {\overset{̅}{d}}^{2}$	Geometric grain cross section	Section 4	Section 2
σ	Effective cross-sectional area per grain	Eq. (1)	Section 2
S	Effective cross section of developed silver cluster	Eq.(3)	Section 2
$C_{f} = \frac{σ}{σ_{g}}, S_{f} = \frac{S}{σ_{g}}$	Cluster factor, cross-section increase on development	Section 17	Section 2
%υ/υ	Mean volume fraction of AgBr grains in emulsion	Section 2	Section 2
$V_{0} = \frac{4 π}{3} {\overset{̅}{d}}^{3}$	Grain volume	Section 3	Section 3
α,d₀, a, b	Semiempirical coefficients of earlier models	Section 2, Eq. (25b)
J, β, z_m′	Integral and coefficients for absorption of x ray in grain	Eqs. (25), (30)	Section 4
z, z₀, z_j	Exposure probability for depth x, surface or half-layer j		Section 4

	Charge State

	Fe²⁵⁺		Fe²⁵⁺		Ge³¹⁺
Transition	Ba-β	Ly-α	Ba-β	Ly-α	Ba-β	Ly-α
E (keV)	1.7	7.0	1.7	7.0	2.8	10.6
Spectrometer, diffraction in first or fourth order
Diffraction plane	PET 002	PET 008	ADP 101	ADP 404	Si 111	Si 444
θ^R	0.955	0.955	0.734	0.734	0.842	0.842
μ₀	0.099	0.0015	0.099	0.0015	0.0232	0.00044
μ₁	2.192	0.148	2.192	0.148	0.636	0.050
μ′	0.50	0.057	0.48	0.057	0.22	0.020
$\frac{z_{max}}{z_{min}} = exp (- μ^{'} T / sin θ)$	3.5⁻⁴	4.0⁻¹	9.0⁻⁵	3.3⁻¹	2.2⁻²	7.1⁻¹
C₃ (Section 9)	1.0	8.2	0.9	7.4	2.1	23.5

Fe Radiation, PET Diffraction	Window (Section 11)

	A1				B4
Transition	3Ly-α	4Ly-γ	4Ly-β	4Ly-α	4Ly-α
E (keV)	6.97	8.70	8.25	6.97	6.97
E (linearized)	2.32	2.175	2.063	1.743	1.743
θ^R	0.657	0.711	0.759	0.955	0.955
D = 0 Eq. (6), CI	0.359	0.424	0.349	0.157	0.155
D = 1 Eq. (6), CI	0.286	0.328	0.268	0.123	0.120
D = 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	0.1592
D = 0.5	0.350	0.402	0.327	0.1628	0.1585
D = 1.0	0.316	0.359	0.294	0.1447	0.1411
D= 1.5	0.290	0.324	0.261	0.1294	0.1264
D = 2.0	0.250	0.277	0.220	0.1090	0.1060
D = 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	0.937
D = 0.00	0.808	1.057	0.966	0.697	0.965
D = 0.25	0.816	1.058	0.958	0.701	0.968
D = 0.50	0.812	1.056	0.953	0.696	0.964
D = 0.75	0.793	1.034	0.929	0.682	0.938
D = 1.00	0.734	0.955	0.853	0.620	0.857
D = 1.50	0.669	0.851	0.757	0.555	0.766
Fe Radiation, ADP Diffraction		A1

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

Ge Radiation, Si Diffraction	Window

	Be
E (keV)	10.6	13.24	12.55	10.6
E (linearized)	3.533	3.31	3.138	2.65
θ^R	0.594	0.640	0.681	0.842
D = 0, Eq. (6), CI	2.02	3.48	3.13	2.29
D = 1, Eq. (6), CI	1.63	3.21	2.88	2.02
D = 2, Eq. (6), CI	1.32	2.91	2.58	1.75
D = 0.0	1.809	2.661	2.470	1.981
D = 0.5	1.807	2.664	2.492	1.974
D = 1.0	1.683	2.664	2.483	1.962
D = 1.5	1.535	2.566	2.394	1.826
D = 2.0	1.400	2.502	2.298	1.756
D = 2.5	1.228	2.425	2.222	1.618

	Film

	A120M	A120T	BX11A	A317A	B417A	A322S	BX25A
Crystal	PET	PET	PET	ADP	ADP	ADP	Si
Window	A1	A1	Be	A3	B4	A3	Be
Source	Fe	Fe	Fe	Fe	Fe	Fe	Ge
I(4Ly-α₁)	125k	52k	10k	640k	450k	200k	18k
D_pk	1.1	2.0	1.1	3.77	3.36	2.0	3.0
D_M	0.65	1.3	0.65	2.9	2.41	1.4	1.8
I(4Ly-α₂)	60k	19k	5k	280k	176k	110k	10k
D_pk	0.55	1.1	0.6	2.8	2.07	1.32	2.0
D_M	0.28	0.70	0.35	2.0	1.35	0.9	1.2

Abstract

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Applied Optics