N

where the minus sign represents the fact that the number N decreases as the beam passes through the slab in the positive x direction; dx > 0 and dN < 0.

The absorption in material of substantial thickness x, is obtained by adding the contributions of many thin slabs, by integration,

Rate of scattering (s

rN (x > d n _ r Ai N "Jo where Ni and N(x) are the initial and final (at position x) number of photons in the beam, and x is the thickness traversed by the beam. For constant a and n along the path, integration yields

+ N (x) = Ni exp(—a nx) (Exponential attenuation) (10.20)

Thus one finds that, for a uniform medium (fixed a and n), the beam intensity N (x) decreases exponentially as a function of the distance traversed, x. If the cross section a or the density of atoms n is increased, the number of surviving photons at a given x decreases, as expected. At distance x = (an)-1, the number has been reduced to 1 /e of its initial value.

Mean free path

A calculation of the average propagation distance, known as the mean free path xm, turns out, fortuitously, to be equal to the 1 /e distance. Thus, xm = (an)-1 (m; mean free path) (10.21)

and (20) may be written,

The intensity N (x) decreases to Ni/e after traversing the thickness x = xm. At this distance, 63% (1 - e-1) of the photons are absorbed. One can think of the mean free path as a typical absorption distance. Note that (20) and (22) are identical except that different parameters are used to describe the characteristics of the material being traversed.

Mass units and opacity

Yet another form of (20) makes use of the matter density p (kg/m3) of the absorbing material. Multiply and divide the exponent in (20) by p,

Introduce the definition of opacity, an

+ k = — (m2/kg; opacity) (10.24) p and the column mass density %,

% = px (kg/m2; column mass density) (10.25) and rewrite (23)

The opacity k (m2/kg) is another commonly used version of the cross section; note from (24) it may be written as k = a/m where m is the mass of the target particle. Its units (m2/kg) indicate that it is the blocked area (sum of all the atomic cross sections) per kilogram of material.

The column mass density % (kg/m2) describes the distance traversed by the photons in terms of the amount of matter encountered along the way. Consider a column of matter 1 m2 in cross section; for a given density of matter, its length can be expressed as the number of kilograms it contains, in units of kg/m2.

In the upper expression of (26), k and p are assumed to be constant. In the lower, changes in density p are subsumed into % (kg/m2) which tells us how much material has been traversed, independent of its distribution along the path. In this case, though, the exponential relation is valid only if k is constant along the path, as it would be if the mix of types of atoms along the path were fixed so that the cross section per kilogram (k) does not change.

These expressions lead to other versions of the mean free path. From the last term of (26), the flux is reduced to 1/e its initial value in the distance % = 1/k, giving the mean free path,

in units of the column mass density, kg/m2. Also, the central term of (26) yields xm = 1/kp (m; mean free path) (10.28)

which is the relation between opacity k and mean free path xm.

The use of the mass units % (kg/m2) to measure thickness, as in (26), is quite sensible since most absorption processes depend on the number of atoms or electrons encountered by the photon in its travels, and the number encountered is roughly proportional to the mass traversed (kg/m2). A path through 1 kg/m2 of lead contains approximately the same number of electrons and nucleons (protons and neutrons) as does 1 kg/m2 of cotton. Thus one might expect equal absorption in the two materials. In fact, they can be quite different because cross sections depend on the atomic number, the ionization state, and the molecular structure of the target atoms. Mass units thus highlight the physical character of the interactions themselves.

The matter thickness to the center of the Galaxy is ~0.2 kg/m2, the thickness of 1 m of water is 1000 kg/m2, and the thickness of the standard earth atmosphere is ~10 300 kg/m2. The atmosphere is equivalent to 10 m of water or ~0.9 m of lead; the density of lead is 11 300 kg m-3. It is this material that protects us from the energetic cosmic rays that permeate interstellar space and have energies sufficient to not be deflected by the earth's magnetic field.

Optical depth

Astronomers use yet another form of notation to describe absorption, the optical depth, t = x/xm = kpx, which is simply the thickness in units of the mean free path, a dimensionless quantity. The formal definition, allowing for variation of k

10.4 Cross sections Table 10.1. Absorption parametersa

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