Radiative transfer theory

The study of the passage of electromagnetic radiation, gamma rays, neutrons, etc., through matter, examined by means of a linear kinetic equation or transport equation (see Kinetic equation).

The problem of the determination of the radiation field in the atmosphere and the scattering of light in accordance with known physical laws was first considered in the 1880's in connection with research on the illumination of the daylight sky. The kinetic equation for radiation transfer was derived at the start of the 20th century for radiative equilibrium in stellar atmospheres. The physical significance of the equation lies in the balances for the energy, number of quanta, and number of particles in an element of the phase space in terms of the particle's coordinates and velocities:

$$ \tag{* } \frac{d \Phi }{dt} = \left ( \frac{\partial \Phi }{\partial t } \right ) _ { \textrm{ coll } } + S, $$

where $ \Phi ( \mathbf r , \mathbf v , t) $ is the distribution function of the particles at the point $ \mathbf r $; $ \mathbf v $ is the velocity; $ t $ the time; $ d/dt $ is the total derivative along the path of a particle; $ ( \partial \Phi / \partial t) _ { \mathop{\rm coll} } $ is the rate of change in the distribution due to collisions with matter (of neutrons with nuclei or of quanta with atoms); and $ S $ is the output of the particle source. The independent variables in the distribution for electromagnetic radiation, which define the mean intensity, are the direction vector and its frequency. The same equations are used to describe the propagation of particles and quanta because these kinetic equations have the same physical significance, namely energy balance in phase space.

The collision term is an integral:

$$ \left ( \frac{\partial \Phi }{\partial t } \right ) _ { \mathop{\rm coll} } = \lambda \int\limits \Sigma _ {S} ( \mathbf r , \mathbf v ^ \prime \rightarrow \mathbf v ) \Phi ( \mathbf r , \mathbf v ^ \prime , t) d \mathbf v ^ \prime - \Sigma ( \mathbf r , \mathbf v ) \Phi ( \mathbf r , \mathbf v , t), $$

so that the transport equation (the kinetic equation) is an integro-differential equation. Here $ \Sigma $ is the total cross-section for the interaction of the particles with matter in an elementary collision, while $ \Sigma _ {S} ( \mathbf r , \mathbf v ^ \prime \rightarrow \mathbf v ) $ is the transition cross-section (transition probability), namely the probability of a transition from velocity $ \mathbf v ^ \prime $ before scattering to a velocity $ \mathbf v $ afterwards (with allowance for the probability of a collision occurring). The following is the total derivative of the distribution function along the path for the free motion:

$$ \frac{d \Phi }{dt} = \frac{\partial \Phi }{\partial t } + \mathbf v \mathop{\rm grad} \ \Phi . $$

To define the solution completely, it is necessary to specify the initial condition

$$ \Phi \mid_{t=0} = f( \mathbf r , \mathbf v ) $$

and a boundary condition. At the boundary of the body (the region of space within which the equation has to be solved) one can specify, for example, the condition for absolute particle absorption

$$ \Phi ( \mathbf r , \mathbf v , t) = 0 \ \textrm{ for } ( \mathbf v \cdot \mathbf n ) < 0, $$

where $ \mathbf n $ is the outward normal to the surface (boundary) of the body. More general boundary conditions can also be used, describing reflection from the boundary or passage through vacuum (for a non-convex body bounding with vacuum, this is a region in which there are no collisions). V.S. Vladimirov [1] has made a mathematical study of these equations for the case $ \partial \Phi / \partial t = 0 $ in the single-velocity case, i.e. on the assumption that only the direction of propagation changes, while the quantum or particle energy is constant. For isotropic scattering this reduces to an integral equation with a positive kernel, to which one can apply the theory of completely-continuous operators with invariant cones in a Banach space. Here the homogeneous problem has a positive eigen value (derived from the factor in the collision integral) that is not larger than the modulus of any other eigen value $ \lambda _ {j} $, and that corresponds to at least one non-negative eigen function (corresponding to $ \lambda _ {1} $). This theorem can be extended to anisotropic scattering. In many cases, the first eigen value is simple, while the corresponding eigen function is positive almost-everywhere in the coordinate and direction phase space.

Such, for example, is the case where $ \Sigma _ {S} ( \mathbf r , \mathbf v ^ \prime \rightarrow \mathbf v ) > 0 $ almost-everywhere. Conditions have been determined under which the Hilbert–Schmidt theory (cf. Hilbert–Schmidt integral operator) applies for transport with anisotropic scattering, and a new variational functional has been constructed for transport equations with even probabilities of transition in terms of the variable $ \mu _ {0} = ( \mathbf v \cdot \mathbf v ^ \prime ) $. The new variational method has been applied to equations in the method of spherical harmonics (cf. Spherical harmonics, method of), which are derived together with the boundary conditions by applying Galerkin's direct variational method to the transport equation if one takes linear combinations of spherical functions dependent on the propagation direction as test functions, these being multiplied by unknown functions of the spatial coordinates. The variational principle allows one to select the best boundary conditions for the method of spherical harmonics; these have previously been derived empirically, from a set of possible linearly independent conditions at the boundary of the body that was twice as large (for planar geometry) [1].

In the non-stationary case [2], in the examination of the spectrum, the eigen value appears non-linearly in the kernel of the integral equation (for isotropic scattering), which means that the number of points in the discrete spectrum is finite (and in certain cases, such as the non-stationary problem for thermal neutrons in a small block moderator, there are no points at all), while in addition there is a continuous spectrum of eigen values.

In some cases, analytic solutions have been obtained to the transport equations. For example, the Wiener–Hopf method has been used to solve the Milne problem, and expansion in terms of singular eigen functions for the transport operator enables one to solve various one-dimensional problems [3].

Requirements of engineering have led to the development of numerical methods for solving neutron-transport equations in order to calculate critical states in nuclear reactions (an eigen value problem for equation (*) with $ \partial \Phi / \partial t = 0 $). One of the basic methods is that of spherical harmonics, which however is complicated to computer implement, as are other methods that converge slowly (this is due to singularities in the kernel of the integral equation). See Transport equations, numerical methods.

References

[1]	V.S. Vladimirov, "Mathematical problems of uniform velocity transport theory" Trudy Mat. Inst. Steklov. , 61 (1061) (In Russian)
[2]	S.V. Shikov, "Aspects of the mathematical theory of reactors: linear analysis" , Moscow (1973) (In Russian)
[3]	K.M. Case, P.F. Zweifel, "Linear transport theory" , Addison-Wesley (1967) MR0225547 Zbl 0162.58903
[4]	V.V. Sobolev, "Transfer of radiation energy in the atmospheres of stars and planets" , Moscow (1956) (In Russian)
[a1]	S. Chandrasekhar, "Radiative transfer" , Dover, reprint (1960) MR0111583 Zbl 0749.01014 Zbl 0037.43201