Cross section as a target

The concept of cross section is illustrated in Fig. 5 a. The atom is visualized as having a target area, or "bull's-eye", which a photon may or may not strike. This target area is called the cross section a with units of m2. The value of the cross section for a given atom is tiny, e.g., 10—24 m2. In this simplistic picture, the photon is taken

Figure 10.5. (a) Photon beam encountering a small target (atom). Individual photons may (hypothetically) "strike the target" and be absorbed or scattered, or they may proceed on undisturbed if they "miss" the target. (b) Beam of N photons impinging on targets of cross section a in a thin slab of area A and thickness dx. The fractional change in the number of atoms, dN/N, is obtained from this geometry. This expression can be integrated to obtain the exponential attenuation of the number of atoms in the beam as it passes through matter.

Figure 10.5. (a) Photon beam encountering a small target (atom). Individual photons may (hypothetically) "strike the target" and be absorbed or scattered, or they may proceed on undisturbed if they "miss" the target. (b) Beam of N photons impinging on targets of cross section a in a thin slab of area A and thickness dx. The fractional change in the number of atoms, dN/N, is obtained from this geometry. This expression can be integrated to obtain the exponential attenuation of the number of atoms in the beam as it passes through matter.

to be a point-like object (with no physical size). Whenever a photon strikes one of the targets, the photon is taken to be absorbed or scattered by the target atom. If it misses, it is taken to continue on without an interaction. A cross section, like a probability, is meaningful only if a well defined process is specified, e.g., "the cross section for the absorption of a photon of frequency v by a hydrogen atom through the ejection of an electron in the n = 1 shell".

The size of the target area (cross section) reflects the probability for the absorption (or scattering) to occur. If the process is quite likely to occur, the cross section is relatively large; conversely, unlikely processes are described with a small cross section. The absorption process is actually a quantum-mechanical probabilistic effect. A photon passing very close to the atom may have a high probability of interacting and being absorbed while one passing farther away has a low probability.

However, all that one measures is how many photons of a beam are absorbed, not the details of each interaction. The simplistic all-or-nothing target interpretation is a way to visualize the meaning of cross section, and it can be used in quantitative expressions. The cross section for Thomson scattering quoted earlier (1) is an example of this target-area interpretation.

A conceptual measurement that would determine the cross section is the following. Place one atom in a photon beam of transverse area 1 m2 (Fig. 5a). The cross section is the number of scattered (or absorbed) photons divided by the total number flux in the beam (Fig. 5a),

2 Number of scatters per second (s-1)

Number of photons in 1 m beam per second (m-2 s-1)

or equivalently

+ a (m2) = ^-—^-- (Cross section defined) (10.16)

Consider a beam with 100 photons traversing 1 m2 in one second (flux = 100 m—2 s—1) that yields 2 scatters each second (rate = 2s—1) from a single target. We would infer that the cross section of the target must be 0.02 m2, in accord with (16).

Mean propagation distance

The relation between the cross section and the mean (average) distance that the photon will travel is derived here. It takes on several forms.

Exponential absorption

A beam of photons impinges on a thin slab of material (Fig. 5b). The slab has thickness dx and area A (m2). It contains many scattering atoms, each with cross section a (m2) and a random position in the slab. The slab is taken to be sufficiently thin so that the sum of all the individual cross sections in area A is much less than area A. In the limit of an infinitesimal slice, there are no overlapping target areas.

Let the number of target atoms per unit volume be n (particles/m3). The number of targets in our slab is therefore nA dx, and the total area blocked by all of the target areas is a nA dx. Divide this by the slab area A to obtain the fractional area blocked, an dx. Now a uniform beam of N photons impinges on the slab in some fixed time. The fraction of these photons absorbed by the slab, dN / N, will be equal to the fractional area blocked, an dx. Thus, dN

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