The formula for the absorption of light in the simplest case starts with a beam of light of intensity (or flux of photons) I that has just penetrated

Incident tensity of light.

into a material with a uniform concentration of absorbing pigments. We ignore reflection and refraction. The photons penetrate into the material and are absorbed at different depths (Figure C.1).

The mechanism for absorption is that a photon transfers all its energy to an electron in the absorbing pigment. The photon is "lost" from the light beam by being absorbed in a single event. The electron is excited by the gain in energy to a higher energy state in the electron configuration around the atom in the pigment. This single event, the transfer of photon energy to an electron and the disappearance of the photon, is closely related to the photoelectric effect described in Appendix A, although here the electron is not ejected from the material. The electron is ejected from the atom in the pigment by the absorption of an x-ray. The atom is left in an excited state. As we will see in Appendix F, the de-excitation of the atom leads to the emission of x-rays.

The decrease in intensity of the light (or photon flux) as the beam penetrates into the material can be visualized if we separate the absorbing material into sets of thin slices, all of a thickness t (Figure C.2). The material is homogeneous, and the thickness t is chosen so that only a small fraction of the light is absorbed in passing through each layer. The symbol A is used to denote the fraction absorbed. The fraction of light absorbed in the first layer is AI, where I is the incident intensity; the amount of light transmitted through this layer is (1 — A) times the incident intensity. For example, if the fraction of light absorbed per layer is 10% of the incident flux, then for 100 incident photons incident on the first layer, AI = 0.1 X 100, or 10 photons will be absorbed and 90 transmitted to the second layer.

The light intensity incident on the second layer is (1 - A)I, and A(1 - A)I will be absorbed on the second layer. The amount transmitted through the second layer is (1 - A)2I. To continue our example, with 90 photons incident on the second layer, A X 90 = 0.1 X 90 = 9 photons will be absorbed and 81 transmitted. That is,

Light

Fig. C.2. Light incident on a light-absorbing material that is represented by a stack of equivalent layers of thickness t. The lengths of the arrows decrease as the light penetrates, showing the decreasing in-

Layer 1: I photons incident, AI = 0.1 X 100 = 10 absorbed, and

(1 - A)I = (1 - 0.1)100 = 90 transmitted; Layer 2: (1 - A)I = 90 photons incident, A(1 - A)I

= 0.1(0.9)100 = 9 absorbed, and (1 - A)(1 - A)I = (0.9)(0.9)100 = 81 transmitted.

This process continues, with fewer photons being transmitted into layer 3, layer 4, and so on, and fewer photons being absorbed in each successive layer, although the fraction A absorbed remains constant.

The absorption of photons is given in more general form in Table C.1 for an absorbed fraction A.

For n layers, the total absorption TA in all layers for absorbed fraction A will be

Ta = A + A(1 - A) + A(1 - A)2 + • • • + A(1 - A)"-1.

The first term in equation (C.1) represents the absorption by the first layer. The second term represents the absorption by the second layer, and so forth.

The total transmitted light Tt through " layers is

Equation (C.1) is more conveniently written using an exponential function. Using an absorption coefficient a, the intensity of light at depth t, I(t) for an incident intensity I0, is given by

where a is given as a fraction per distance such as 0.1 per mm. Figure C.3 shows the intensity of light as a function of penetration distance into materials with two different absorption coefficients. The exponential decrease is illustrated on a linear scale by the curves in Figure C.3. An increase in a leads to a decrease in penetration. At the depth where the product at equals unity, the intensity has dropped by 1/e, or a factor of about 0.37, times the incident intensity.

These relations rely on the fact that the fraction of light that is absorbed is independent of the intensity of the incident light. For a given

TABLE C.1 | |||

Transmission for Absorbed Fraction A | |||

Incident |
Absorbed |
Transmitted | |

on layer |
in layer |
through layer | |

layer 1 |
I |
AI |
(1 - A) I |

layer 2 |
(1 - A) I |
A (1 - A) I |
(1 - A)2 I |

layer 3 |
(1 - A)2 I |
A (1 - A)2 I |
(1 - A)3 I |

layer n |
(1 - A)n-1 I |
A (1 - A)n-1 I |
(1 - A)n I |

Amount of light incident on, |
absorbed in, and transmitted through equiva- | ||

lent layers with absorbed fraction A and flux I incident on |
layer 1. |

wavelength of light, each successive equi-thick-ness of the medium absorbs an equal fraction of light passing through it, but the amount of light the layers receive is successively decreasing. Hence, the intensity of light passing through a medium will decrease with depth. The exponential attenuation coefficients apply for x-rays as well as photons in the visible portion of the electromagnetic spectrum. The exponential relation is derived in the following:

The number of A/ of photons that are absorbed in a given thickness Ax is proportional to the incident intensity /, or the number / incident on it, times the thickness Ax, which is proportional to the number of electrons available to absorb photons:

where the minus sign indicates that absorption decreases the number of photons.

If we use the symbol a to denote the absorption coefficient, where a is the fraction of photons absorbed per unit thickness, then (C.4) can be written:

or in standard differential notation as

This equation can be integrated from the surface at x = 0 to a depth x, which leads to the relation that the intensity /(x) at depth x is given by

where /0 is the intensity of photons at x = 0.

The absorption coefficient is a number per unit distance, such as 0.1 per cm, which can be written 0.1 cm-1. At a depth x where ax equals unity (in this example, x = 10 cm, since 0.1 X 10 = 1) the number of photons decreases by a factor of 1/e, or 0.37 (the value of e~2.718).

For small values of ax, for example ax = 0.01, equation (C.7) can be approximated by a linear function:

;-r 1 1 |
^10% absorption |

- \ |
per mm |

\ |
^50% absorption |

per mm | |

1 1 1 |
1 1 1 "l ------1_ |

4 6 8 Thickness, mm

4 6 8 Thickness, mm

Fig. C.3. The intensity of light transmitted through an absorbing material as a function of thickness d in mm for (1) 10% absorbance and (2) 50% absorbance per mm (from T.B. Brill, Light (Plenum Press, New York, 1980)).

The analytical section of almost every art museum contains an x-ray generating system for examining works of art. Such a system produces high-energy, short-wavelength photons (for example 17.4 keV x-rays have À = 0.071 nm = 0.71 angstrom, and 8.04 keV x-rays correspond to À = 0.154 nm = 1.54 angstrom). As shown in Figure C.4, a painting is placed over x-ray sensitive film and is irradiated with x-rays. The film after exposure and development reveals regions where the x-rays have been absorbed.

For x-rays, photoelectric absorption (where the x-ray gives up its energy in a single interaction with an electron) is the major cause of the attenuation of the photons penetrating the material.

The intensity I of x-rays transmitted through a thin foil of material for an incident intensity I0 follows the exponential attenuation relation of equation (C.3) with a change in the nomenclature from a to f to follow the convention in standard x-ray usage:

Fig. C.4. Procedure for x-radiography of paintings. An x-ray sensitive film is placed behind a painting. The x-rays incident on the painting are absorbed at different depths depending on the density and atomic number of the pigment particles. The transmitted x-rays expose the film emulsion. The amount of darkening of the developed film is a measure of the intensity of transmitted x-rays and hence the amount of x-ray absorbing pigments.

Fig. C.4. Procedure for x-radiography of paintings. An x-ray sensitive film is placed behind a painting. The x-rays incident on the painting are absorbed at different depths depending on the density and atomic number of the pigment particles. The transmitted x-rays expose the film emulsion. The amount of darkening of the developed film is a measure of the intensity of transmitted x-rays and hence the amount of x-ray absorbing pigments.

where p is the density of the element (g/cm3), f is the linear attenuation coefficient, and f/p is the mass attenuation coefficient given in cm2/g. Note that x-ray absorption is not influenced by the crystal structure of the pigment, but only by the number of atoms per unit volume and the thickness of the pigment layer.

Table C.2 gives the density and mass absorption coefficient at two wavelengths of several elements found in artists' pigments. One notes from the table that the mass absorption coefficient tends to increase with increase in atomic number (Z) and that the absorption coefficient is greater (in some cases a factor of ten greater) for the longer-wavelength (lower-energy) photons. There are apparent anomalies with the absorption of Cd at 0.711Â, less than that of arsenic. This, as we will see in Appendix E, is due to the electronic structure and binding energies of

TABLE C.2

Atomic Number (Z), element, density p in grams/cm3 (pigment) and mass attenuation coefficient (p/p) in cm2/gram at wavelengths of 0.711 and 1.54A,

Mass attenuation Coeff. (cm2/gm)

Z |
Element |
Density (gm/cm3) |
17.4 keV 0.711À |
8.04 keV 1.542À |
Pigment |

22 |
Titanium |
4.51 |
23.25 |
202.4 |
titanium white |

26 |
Iron |
7.87 |
37.4 |
304.4 |
ochre, sienna |

29 |
Copper |
8.93 |
49.3 |
51.5 |
malachite |

33 |
Arsenic |
5.78 |
65.9 |
75.6 |
king's yellow |

48 |
Cadmium |
8.65 |
27.3 |
229.3 |
cadmium red |

80 |
Mercury |
13.55 |
114.7 |
216.2 |
vermilion |

82 |
Lead |
11.34 |
122.8 |
232.1 |
Naples yellow |

electrons. Figure C.5 gives the mass attenuation coefficient as a function of energy for several elements. This shows both the strong energy dependence of the absorption coefficient (y/p decreases with increasing energy) and the five- to tenfold jumps in absorption at specific energies. These jumps occur when the energy of the photon exceeds the binding energy of a set electron, thus leading to an increase in absorption. For example, at 8 keV incident photon energies the absorption of iron (Fe,

X-ray Absorption

Fig. C.5. The x-ray mass-absorption coefficient (y/p) in cm2/gram versus x-ray energy in kilo electron volts (keV) for carbon (C), aluminum (Al), and tin (Sn). The sharp increase in absorption for Sn at 29.2 keV corresponds in energy to the K-shell, binding energy (Taken from Feldman and Mayer, Fundamentals of Surface and Thin Film Analysis, (North Holland, New York, 1986)).

Z = 26) can exceed that of tin (Sn, Z = 50). A factor-of-two difference in absorption coefficient can make a factor-of-ten difference in x-ray attenuation due to the exponential nature of the absorption process (equation (C.9)) because e23 = 10.

Therefore, details of underlying drawings can be revealed in an x-ray radiograph if the underlying pigments have a higher absorption coefficient than the covering layer; for example, vermilion covered by titanium white.

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