Photographic response to x-ray irradiation. I: Estimation of the photographic error statistic and development of analytic density-intensity equations

C. T. Chantler

doi:10.1364/AO.32.002371

Applied Optics
Vol. 32,
Issue 13,
pp. 2371-2397
(1993)
•https://doi.org/10.1364/AO.32.002371

Photographic response to x-ray irradiation. I: Estimation of the photographic error statistic and development of analytic density-intensity equations

C. T. Chantler

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C. T. Chantler, "Photographic response to x-ray irradiation. I: Estimation of the photographic error statistic and development of analytic density-intensity equations," Appl. Opt. 32, 2371-2397 (1993)

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Abstract

Formulations for specular optical density as a function of incident x-ray intensity are shown to be inadequate, theoretically and compared with available data. Approximations assuming low intensities, grain densities, or energies yield significant error in typical emulsions. Unjustifiable simplifications limit analysis and consequent results. The avoidance of assumptions leads to models for rough and smooth emulsion surfaces, which correspond to Kodak 101-01 and DEF-392 emulsion types. The self-consistent use of spherical grains yields scaling that is dependent on emulsion roughness. We obtained improvement over standard formulations, avoiding the empirical character of earlier models and associated parameterization. The correlation of grain locations and occluded emulsion area is approximated within monolayer depths but neglected between layers. Effects of the incident angle from a broad source, scattering, and photoelectrons are considered. The models presented herein apply to the vacuum UV and x-ray energies from 9 eV to 20 keV and may be preferred over alternative models at lower energies, densities greater than unity, emulsions with high grain fractions, or where interpolation over energy ranges is desired. Error contributions may be dominated by intensity statistics or densitometry statistics. Both are inadequate in medium-density regimes. We have derived estimates including pseudo-binomial grain development statistics, using a summation over layers.

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M₀S	Terms	Powers of y

		$\frac{1}{y ln y}$	$\frac{1}{ln y}$	$\frac{y}{ln y}$	$\frac{y^{2}}{ln y}$	$\frac{y^{3}}{ln y}$
0.75	4	1.251	−2.05	1.1777	−0.457	0.0791
0.75	∞	1.386	−3.00	4.50	—	—
0.65	4	0.9974	−1.5256	0.7536	−0.2700	0.0446
0.65	∞	1.050	−1.857	1.724	—	—
0.55	4	0.799	−1.110	0.455	−0.147	0.0229
0.55	∞	0.799	−1.222	0.747	—	—
x	4	$≃ - ln (1 - x) \frac{x (x^{4} - 1)}{1 - x} \frac{x^{2}}{2} + x^{3} + \frac{3 x^{4}}{2}$			$\frac{- x^{3}}{3} - x^{4}$	$\frac{x^{4}}{4}$
x	∞	−ln(1 − x)	$\frac{- x}{1 - x}$	$\frac{x^{2}}{2 {(1 - x)}^{2}}$	—	—

Values of y	0.50	0.25	0.05	0.01	0.001	10⁻⁵
$\frac{1}{ln y}$	−1.443	−0.7215	−0.3338	−0.2172	−0.1448	−0.0869
Values of z = −ln y	0.693	1.386	2.996	4.605	6.908	11.51
First term (ln z)	−0.3665	+0.3266	+ 1.0972	+ 1.5272	+ 1.9327	+2.4432
Terms in z required^a	5–7	7–9	11–14	14–18	20–24	32–36
Expansion limit (∫^y)	−0.9559	−0.6959	−0.5903	−0.5791	−0.5774	−0.5772
Area $(\int_{0}^{y})$ upper bound	−0.3608	−0.0902	−8.35 × 10⁻³	−1.09 × 10⁻³	−7.24 × 10⁻⁵	−4.35 × 10⁻⁷
Area $(\int_{0}^{y})$ lower bound	−0.2010^b	−0.1153	−1.39 × 10⁻²	−1.94 × 10⁻³	−1.34 × 10⁻⁴	−8.31 × 10⁻⁷

y_max	0.990	0.905	0.50	0.25	0.05	0.01	0.001
y_min	0.905	0.500	0.25	0.05	0.01	0.001	10⁻⁵
z_max	0.100	0.693	1.386	2.996	4.605	6.908	11.51
z_min	0.010	0.100	0.693	1.386	2.996	4.605	6.908

$Integrals : 1 st = \int \frac{d y}{y ln y} = [ln z]; 2 nd = \int \frac{d y}{ln y} = [ln z - z + \dots]; \dots$
First	2.2958	1.9379	0.6931	0.7706	0.4300	0.4055	0.5108
Second	2.2084	1.4459	0.2600	0.1055	0.0113	0.0017	0.0001
Integrals^a	5	5	3	2	2	1	1
Terms required^a	3–4	4–6	6–8	8–12^b	12–16^b	16–22^b	22–34^b
$(1) Total : \int \frac{ln [1 - M_{0} S (1 - y) d y}{y ln y} [expression in parentheses in Eq . (24)]$
(M₀S = 0.75)	0.0669	0.4208	0.4358	0.7663	0.5152	0.5037	0.6390
(M₀S = 0.65)	0.0578	0.3576	0.3607	0.6196	0.4119	0.4018	0.5095
(M₀S = 0.55)	0.0488	0.2970	0.2921	0.4909	0.3228	0.3142	0.3982
(2) Estimated correction for M₀S_n terms, n < 4 (estimated error)
	(%)	(%)	(%)	(%)	(%)	(%)	(%)
(M₀S = 0.75)	+0.002	+0.13	+ 1.41	+5.3	+8.6	+9.6	+9.7
(M₀S = 0.65)	+0.001	+0.07	+0.76	+2.8	+4.4	+4.9	+5.0
(M₀S = 0.55)	+0.001	+0.03	+0.38	+ 1.3	+2.1	+2.3	+2.3
(3) Effect of truncation after z⁶ with limit = −0.5881 for z > 2.6
	(%)	(%)	(%)	(%)	(%)	(%)	(%)
(M₀S = 0.75)	+0.002	−0.2	+2.5	−3.0	+4.4	+0.7	+0.04
(M₀S = 0.65)	+0.001	−0.2	+ 1.9	−2.4	+4.1	+0.6	+0.04
(M₀S = 0.55)	+0.001	-0.1	+ 1.4	−2.0	+3.8	+0.6	+0.04
(4) Effect of truncation after z⁶, (M₀S)⁶ terms, versus exact result
	(%)	(%)	(%)	(%)	(%)	(%)	(%)
(M₀S = 0.75)	−0.00	−0.8	+ 1.5	−2.9	−2.8	−4.0	−4.2
(M₀S = 0.65)	−0.00	−0.5	+ 1.0	−1.6	−0.7	−1.4	−1.6
(M₀S = 0.55)	−0.00	−0.3	+0.6	−0.9	+0.1	−0.4	−0.5

Model			Parameter

%υ/υ	t₀ (μm)	T (μm)	t_b (μm)	d (μm)	d (μm)	C_f	S_f	χ_r²
Ref. 8, Eq. (6)	0.40	1.0	13.0	185.0	1.6	0.0	0.84	4.17	3.3
Ref. 11	0.40	1.3	13.0	177.6	1.6	—	1.51	2.00	—
Experiment	0.38	1.1	14.2 − t₀	177.6	1.6	—	>1	1–2	—
	±0.03	±0.7	±0.7	±7.4	±0.3	—	?	?	—
Smooth, Eq. (25a)	0.386	1.68	24.0	(185)	1.50	—	2.30	2.00	2.4
Smooth, Eq. (25a)	0.384	1.78	14.2	(185)	1.54	—	2.36	1.99	3.0
Smooth, Eq. (25b)	0.41	2.0	13.5	(185)	1.69	0.59	2.26	2.25	2.2
Rough, Eq. (25b)	0.41	2.0	13.5	(185)	1.56	0.38	2.35	2.35	2.5

Model^a	Parameter

	%υ / υ	t₀ (μm)	T(μm)	d (μm)	d₀ (μm)	C_f	S_f	χ_r²^b	χ_r²^c
Ref. 8, Eq. (6)	0.74	0.0	0.6	0.6	0.0	1.06	6.69	60	58
Ref. 8	0.73	0.0	0.6	0.65	0.035	0.95	6.47	28	23
Experiment	0.74	0.0	1–2	0.85	—	> 1	1–2	—	—
	±0.1	±0.1	±1.0	±0.25	—	?	?	—	—
Rough, Eq. (25c)	0.74	0.0495	Mono	0.7155	—	1.47	1.53	13.2	10.0
Rough, Eq. (25c)	0.74	0.0524	Mono	0.850	—	1.50	1.535	—	7.4
Rough, Eq. (25c)	0.74	0.052	Mono	0.915	—	1.27	1.535	—	7.4
Rough, Eq. (25b)	0.74	0.050	1.87	1.20	0.64	1.31	1.40	—	5.5

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Applied Optics