Propagation of gamma rays and production of free electrons in air

A new concept of remote detection of concealed radioactive materials has been recently proposed [1]. This concept is based on breakdown in free air at the focal point of a high-power beam of electromagnetic waves. To realize such breakdown, the wave amplitude in a focal region should exceed the breakdown threshold. To initiate the avalanche breakdown process, seed free electrons should be present in this focal region during the electromagnetic pulse. When the wavelength is short enough (sub-THz or THz frequency range) the wave beam can be focused in a spot with dimensions on the order of a wavelength and, then, the total volume where the wave amplitude exceeds the breakdown threshold can be rather small. This fact allows one to realize conditions [1] when the breakdown rate in the case of the ambient electron density is rather low. So in the cases when the breakdown rate is much higher than the expected one can conclude that in the vicinity of a focused wave beam there is a hidden source of radioactive material which ionizes the air. Some issues important for realizing this concept are discussed in Refs. [2] and [3].

In the present paper, we analyze another issue important for realizing this concept which was only briefly discussed in previous references, viz. propagation of $\gamma$-rays in air and production of free electrons by these quanta, mainly due to inelastic Compton scattering. It should be noted that, in principle, due to the atomic reaction radioactive materials emit both MeV-scale $\gamma$-rays and $\beta$-electrons. For example, a disintegrating atom of ${}_{27}^{60}$Co produces one $2.505$ MeV $\beta$-electron and two $\gamma$-quanta with the energies $1.173$ and $1.332$ MeV [5]. Metal shielding and container walls largely stop $\beta$-electrons, while absorbing only a fraction of $\gamma$-rays. All unabsorbed $\gamma$-rays, even scattered ones, will eventually leak through the walls and propagate in air. An example of energy distribution of $\gamma$-rays and electrons outside the container wall is shown in Fig. 1 reproduced from Ref. [2].

The energetic $\gamma$-quanta escaped from the container will occasionally collide with air molecules, experiencing Compton scattering and photoelectric absorption, as discussed below. Their propagation distance can be estimated based on the interaction cross-section $\sigma$ given, e.g., in Ref. [4] on Figures 2.50 (d) and (e) for nitrogen and oxygen, respectively. (One of these very similar figures was reproduced in Ref. [2].) From these figures one can draw, at least, two conclusions. First, for energies between 0.1 MeV and 1.2 MeV shown in Fig. 1, the dominant effect in the production of free electrons is the Compton scattering (this conclusion agrees with Figures 2.42, 2.43 of Ref. [4]). Second, the interaction cross-section $\sigma$, as the function of energy, gradually increases from about 0.055 cm²/g at 1.2 MeV to about 0.16 cm²/g at 0.1 MeV. Correspondingly, the mean free pass of $\gamma$-rays in air ($n_{a}=2n_{m}\approx 6\cdot 10^{19}\$atoms/cm³) defined as $l=1/(n_{a}\sigma)$ varies from $\sim$120 m for 1.2 MeV quanta down to about 40 m for 100 keV quanta. When the energy of $\gamma$-quanta decreases down to 30 keV and below, the photoelectric absorption becomes the dominant effect.

It is important that both the Compton scattering and photoelectric absorption ionize air molecules, producing high-energy electrons within the MeV range. These primary electrons, which should not be confused with the $\beta$-electrons directly from radioactivity, cannot propagate too far: they will be absorbed by air within a few meters [4]. During their short lifetime, however, most of these primary electrons produce thousands of secondary ones. It is this process that determines the local rate of total electron production required for the avalanche breakdown [3]. As discussed above, all unabsorbed $\gamma$-quanta propagate away from the radioactive source, continuing to produce free electrons. Hence to estimate the total rate of free-electron production at various distances from the radioactive source, one needs to analyze the collisional propagation of $\gamma$-quanta in air. In this paper, we analyze the photon propagation in the near-source zone, i.e., at distances less or of order of the mean free pass $l$ estimated above.

As propagating $\gamma$-quanta collide with air molecules, producing high-energy primary electrons, they experience a strong angular scatter and energy losses. The major electron production takes place when the photon energy $\mathcal{E}_{\gamma}$ remains between 30 keV and 1.3 MeV, i.e., in the energy range where the Compton scattering dominates. In this range, one can successfully apply the Klein-Nishina (K-N) theory that describes Compton scattering of photons by free electrons [4]. Indeed, after a typical collision between the incident photon and an electron bound within an O₂ or N₂ molecule, the originally bound electron becomes free and acquires a significant momentum (energy). If the acquired energy $\mathcal{E}_{\gamma}$ greatly exceeds the electron binding energy then both the scattered photons and released electrons will have momentum distributions fairly close to those if the originally bound electron was free and at rest. Furthermore, if the incident-photon energy is well above the maximum electron binding energy (the K-shell edge), $\mathcal{E}_{K}\simeq 500$ eV for the air molecules, then the K-N theory should be applicable to all bound electrons with the exception of small-angle scattering when the released electrons acquire a small energy, $\mathcal{E}_{e}\lesssim\mathcal{E}_{K}$ [4, Section 2.6.2.1].

As described in numerous textbooks, see, e.g., Ref. [4, Section 2.6], in the frameworks of the K-N theory, the kinetic energy, $T^{\prime}\equiv c\left(p_{e}^{2}+m_{e}^{2}c^{2}\right)^{1/2}-m_{e}c^{2}=\mathcal{E}_{\gamma}-\mathcal{E}_{\gamma}^{\prime}$, of the outgoing electron and the scalar momentum of the scattered photon, $p_{\gamma}^{\prime}=|\vec{p}_{\gamma}^{\prime}|=\mathcal{E}_{\gamma}^{\prime}/c$ (for the incident photon with $p_{\gamma}=|\vec{p}_{\gamma}|=\mathcal{E}_{0}/c$) are given by

In (1)–(2), $\Theta_{\gamma}$ is the scattering angle, i.e., the angle between the momenta of the incident and scattered photons, $\vec{p}_{\gamma}$ and $\vec{p}_{\gamma}^{\prime}$, which is rigidly related to $p_{\gamma}$ and $p_{\gamma}^{\prime}$,

$\Omega_{\gamma}=(\Theta_{\gamma},\Phi_{\gamma})$ is the solid scattering angle, $m_{e}$ is the electron rest mass, $c$ is the vacuum speed of light, $\alpha\equiv\mathcal{E}_{\gamma}/(m_{e}c^{2})=p_{\gamma}/(m_{e}c)$ is the energy of the incident photon normalized to the electron rest energy and $r_{0}\approx 2.818\times 10^{-13}$ $\operatorname{cm}$ is the classical electron radius.

The K-N formula (2) allows one to calculate the total cross-section and, therefore, the production rate of the primary electrons, $2Z_{\mathrm{Air}}n_{m}c\sigma_{\mathrm{KN}}$, where $n_{m}=n_{a}/2\approx 3\cdot 10^{19}\$cm^-3 is the total density of air molecules and $Z_{\mathrm{Air}}$ is the mean air atomic number (for the standard air composition of 79% N₂ with $Z_{\text{N}}=7$ and 21% O₂ with $Z_{\text{O}}=8$, we have $Z_{\mathrm{Air}}\approx 7.21$). Of prime interest to us is the production rate, $dN_{e}/dt$, of secondary (knock-on) electrons which will inevitably end up attaching to neutral molecules, i.e., forming negative ions [3]. Each primary electron with the kinetic energy $T$ produces on average $T/(34\operatorname{eV})$ secondary electrons [5]–[7]. Then, to determine the total average rate of secondary electron production by photons of a given initial energy $\alpha$, we should first integrate the K-N differential cross-section over all solid angles with the weight factor $T^{\prime}$ given by (1a) and (3),

and then divide the result by $34\operatorname{eV}$. The exact calculation of the last integral of (4) yields a function

shown in Fig. 3 by the black solid line. Note that the function $K(\alpha)$ is proportional to the average “absorption cross section” [9, Eq. (8e-28)],

In the energy range of interest, $\mathcal{E}_{\gamma}\leq 1.332\ \operatorname{MeV}$, i.e., $\alpha\leq\mathcal{\alpha}_{\max}\approx 2.61$, the complex function $K(\alpha)$ can be quite accurately approximated by the dependence $0.37\alpha$ ($\sigma_{a}=0.37\pi r_{0}^{2}\approx 9.23\times 10^{-26}\operatorname{cm}^{2}$) shown in Fig. 3 by the red dashed line. This approximation allows us to describe the total average rate of secondary electron production by a simple formula:

where $F_{\gamma}(\alpha)$ is the photon energy distribution normalized to the total photon density, $n_{\gamma}=\int F_{\gamma}(\alpha)d\alpha$, and expressed in terms of the full momentum distribution function $f_{\gamma}(\vec{p}_{\gamma})$ (Sect. 3) as $F_{\gamma}(\alpha)=m_{e}^{3}c^{3}\alpha^{2}\int f_{\gamma}(p_{\gamma},\Omega_{\gamma})d\Omega_{\gamma}$. Thus, to evaluate the average rate of secondary electron production as a function of the distance from the radiation source, one needs to determine the photon energy distribution along with its distance variations. This requres a collisional kinetic theory for propagating photons.

Radiated and scattered $\gamma$-quanta can be treated as a gas of incoherent and unpolarized photons. For photon energies above 30 keV, this gas is prone mainly to Compton scattering, while at lower energies the photoelectric absorption starts dominating. Both processes result in air molecule ionization, i.e., production of free electrons. The coherent Rayleigh scattering is at least an order of magnitude smaller and can be neglected. Also, the process inverse to photon absorption, viz., photon production during collisions of free electrons with air molecules, in the range of electron energies we are dealing with ($\leq 1.3$ MeV), is negligibly small. Therefore, we will assume the spatial and momentum distribution of photons to be independent of any feedback from the free electrons.

We will treat individual $\gamma$-quanta kinetically as classical particles localized in both real and momentum spaces. The photon momentum distribution, $f_{\gamma}(\vec{p}_{\gamma})$, may have a temporal ($t$) and spatial ($\vec{r}$) dependences. Since photons propagate and scatter practically instantaneously we will ingore any temporal dependence. Being concerned mainly with distances from the radioactive material much longer than the container size $a$, we will treat this material as a point-like source of $\gamma$-quanta. Ignoring possible effects of the ground, air inhomogeneity and container anisotropy, we will assume spherically symmetric propagation of photons.

In the spherically symmetric geometry, the photon momentum distribution depends on one spatial variable, viz., the distance $r$ of a localized photon from the point-like source, and two momentum components: the absolute value of the momentum, $p_{\gamma}$, and the momentum polar angle with respect to the radius vector at the given location, $\theta_{\gamma}$ (the latter should not be confused with the angle between the scattered and incident photons, $\Theta_{\gamma}$). Thus the photon momentum distribution is a function of three variables, $f_{\gamma}\left(r,p_{\gamma},\theta_{\gamma}\right)$. Since the angle $\theta_{\gamma}$ is not invariant with respect to the collisionless motion, it is more convenient to pass from $\theta_{\gamma}$ to the “impact parameter” $\rho$,

equal to the shortest distance from the radioactive source ($r=0$) to the straight-line trajectory of a freely propagating photon. The parameter $\rho$ is proportional to the photon angular momentum and remains invariant during the free photon propagation. It can be changed only during brief collisions that lead to Compton scattering. For any given positive $\rho$, we distinguish between two groups of photons with different signs of $\cos\theta_{\gamma}(\rho)$ in (7): the “outgoing” photons with the ‘$+$’ sign (corresponding to $dr/dt>0$) and “ingoing” photons with the ‘$-$’ sign ($dr/dt<0$), as illustrated by Figure 4. We will denote the two corresponding distribution functions by $f_{\gamma}^{\pm}$.

In terms of $f_{\gamma}^{\pm}\left(r,p_{\gamma},\rho\right)$, the total photon number density, $n_{\gamma}$, and radial flux density, $g_{\gamma}\left(r\right)$, are given by

The momentum distribution function of freely propagating $\gamma$-quanta with occasional spiky collisions satisfies the Boltzmann kinetic equation derived in Appendix A,

where the left-hand side describes the collisionless propagation of photons, while the right-hand side (RHS) describes their collisions with air molecules.

The first term in the RHS of (9) describes the kinetic arrival of photons to the elementary phase volume $d^{3}p_{\gamma}$ around given $\vec{p}_{\gamma}$ due to collisional scattering of incident photons from $d^{3}p_{\gamma}^{\prime}$ around $\vec{p}_{\gamma}^{\prime}$,

where

and $\Theta_{\gamma}$ is the angle of Compton scattering from $\vec{p}_{\gamma}^{\prime}$ to $\vec{p}_{\gamma}$,

(here $p_{\gamma}\leq p_{\gamma}^{\prime}$). In (10), we keep $\theta_{\gamma}$, $\theta_{\gamma}^{\prime}$ for easier integration over the angular variables, but before appplying $\partial/\partial r$ in the LHS of (9), $\cos\theta_{\gamma}$ should be expressed in terms of $\rho$ using (7).

The last term in the RHS of (9) describes the total kinetic departure of $\vec{p}_{\gamma}$-photons due to their scattering with the total collision frequency, $\nu(p_{\gamma}=m_{e}c\alpha)$, proportional to the total K-N cross-section [4],

Equation (9) should be supplemented by the boundary conditions. In accord with (8b), at a small distance $r$ that satisfies $a\ll r\ll l=c/\nu$, the photon distribution function radiated by the point-like radioactive source can be written as

where $I_{\gamma}^{(0)}=3.7\times 10^{10}\operatorname{s}^{-1}Q\left(\mathrm{\operatorname{Ci}}\right)$ is the total number of radioactive decays per unit time, $\sum_{s=1}^{k}$ denotes summation over all kinds of primary photons radiated during one disintegration of the radioactive material (usually, $k=1$,$2$,$3$) and $\delta\left(p_{\gamma}-p_{s}\right)$ expresses the fact that all radiated photons have discrete energies $\mathcal{E}_{s}=cp_{s}$ (e.g., for ${}_{27}^{60}$Co: $k=2$, $\mathcal{E}_{1}=1.332$ $\operatorname{MeV}$ and $\mathcal{E}_{2}=1.173$ $\operatorname{MeV}$). Equation (14) represents the inner boundary condition at $r\rightarrow 0$ applicable to the outgoing photons. For the ingoing photons, we use the natural condition of $f_{\gamma}^{-}\left(p_{\gamma},\rho\right)\rightarrow 0$ at $r\rightarrow\infty$. Between consecutive collisions, some ingoing photons pass the minimum distance $r=\rho$ and become outgoing. This yields the matching condition between the two groups:

Further, all collisional processes lead only to decreasing photon energies, so there are no photons with energies exceeding the initial ones, $f_{\gamma}^{+}\left(p>\max(p_{s})\right)=0$. Lastly, we assume that at $\rho=0$ ($\cos\theta_{\gamma}=\pm 1$) all momentum distributions behave regularly.

The RHS of (9) satisfies automtically the condition $\int_{0}^{\infty}p_{\gamma}^{2}dp_{\gamma}\int_{0}^{r}(\hat{L}_{r}f_{\gamma}^{\prime\pm}-\nu f_{\gamma}^{\prime\pm})\left(r^{2}-\rho^{2}\right)^{-1/2}\rho d\rho=0$, which provides in accord with (8b) conservation of the total photon flux through any surface surrounding the radiation source. This flux conservation is due to the fact that kinetic equation (9) includes only photon scattering. Photoelectric absorption and other possible photon losses can also be included in $\nu(p_{\gamma})$ by a simple addition to the RHS of (13). Those effects would break the total flux conservation.

Kinetic equation (9) is solved analytically in Appendix B. The complete solution is given by an infinite series of multiply scattered photons (B.2), where $f_{0}^{+}=\left(f_{\gamma}^{+}\right)_{\mathrm{prim}}$ is given by (B.1), $f_{n\geq 1}^{\pm}$ is determined by recurrence relations (B.4) with the linear integral operator $\hat{L}_{r}$ defined by (10). This general solution, however, involves an increasing number of nested integrations which for $n\geq 2$ are difficult to calculate. Below we restrict our treatment to distances $r$ from the radiation source located within the mean free path for 1.2 MeV $\gamma$-quanta, $r\lesssim l_{s}\equiv c/\nu(p_{s})\sim 120\ \operatorname{m}$ (the “near zone”). At these distances, one might expect two dominant photon groups: the primary $\gamma$-quanta beams ($n=0$) and singly scattered photons ($n=1$). The momentum distribution for singly scattered photons is obtained in Appendix B.2. However, a mere cutoff of the infinite series (B.2) to its first two terms, $f_{0}^{+}$ and $f_{1}^{\pm}$ given by (B.17)–(B.22), becomes less satisfactory at distances $r\sim l_{s}$ because further scattering of singly scattered photons breaks the total flux conservation. Below we modify the two-group description of photons to make it exactly flux conserving.

To calculate the total electron production rate $dN_{e}/dt$ given by (6), we need to determine the spatially dependent photon energy distribution at various distances from the radioactive source, $F_{\gamma}(\alpha)$, where $\alpha=p_{\gamma}/(m_{e}c)$ is the normalized energy. In this section, however, it is more convenient to use

which differs from by the normalization factor following from $dn_{\gamma}=F(p_{\gamma})dp_{\gamma}=F(\alpha)d\alpha$: $F_{\gamma}(\alpha)=m_{e}cF(p_{\gamma})$. In terms of $F(p_{\gamma})$, (6) becomes