Milne problem

A problem in radiative transfer theory concerning the single-velocity kinetic transport equation of quanta or particles in a half-space. The integral equation of the Milne problem with a source at infinity under zero incident flux of the radiation was first introduced by E. Milne [1] for the case of isotropic scattering of quanta, diffusing without absorption in a stellar atmosphere.

The Milne equation takes the form

$$ \tag{1 } B ( x) = \frac{1}{2} \int\limits _ { 0 } ^ \infty B ( t) E _ {1} ( | x - t | ) d t . $$

Here $ B ( x) $ is the radiation (or particle) density and

$$ E _ {1} ( x) = \ \int\limits _ { 0 } ^ { 1 } \frac{e ^ {- x / \mu } } \mu d \mu $$

is the exponential integral function $ ( E _ {1} ( x) = - \mathop{\rm Ei} ( - x ) ) $.

In neutron physics the Milne problem is used for the formulation of approximate boundary conditions for solutions of the equations of a diffusion approximation in a bounded domain; in this connection one takes into account neutron capture by the medium, anisotropic scattering and the curvature of the boundary.

Here the Milne problem is to solve the integro-differential equation

$$ \mu \frac{\partial \psi }{\partial x } + \psi ( x , \mu ) = \frac{c}{2} \int\limits _ { -1 } ^ { +1 } p( \mu , \mu ^ \prime ) \psi( x , \mu ^ \prime ) d \mu ^ \prime $$

with boundary conditions on the boundary of the half-space occupied by the matter, with a vacuum

$$ \tag{2 } \psi ( 0 , \mu ) = 0 \ \ \textrm{ for } \ 0 < \mu \leq 1 , $$

where $ c $ is the mean number of secondary neutrons, colliding once with a nucleus ( $ c < 1 $ for scattering and absorbing neutrons by the medium), and $ p ( \mu , \mu ^ \prime ) $ is the indicatrix of the scattering ( $ p ( \mu , \mu ^ \prime ) = 1 $ for isotropic scattering). The spherical or cylindrical Milne problem on the distribution of neutrons in the space outside absorbing spheres or cylinders is stated analogously.

The solution of the Milne problem is conveniently given by applying the Laplace transform to the integro-differential transfer equation (see [3]) and using the Wiener–Hopf method to solve the functional equations thus obtained.

In order to solve the Milne problem it was suggested that expansion relative to generalized eigen functions and methods for solving singular integral equations be used (see [4]). The solution of the Milne problem for $ p ( \mu , \mu ^ \prime ) = 1 $, $ c < 1 $ is sought in the form

$$ \psi ( x , \mu ) = \ \psi _ {0-} ( x , \mu ) + a _ {0+} \psi _ {0+} ( x , \mu ) + \int\limits _ { 0 } ^ { 1 } A ( \nu ) \psi _ \nu ( x , \mu ) d \nu , $$

where

$$ \psi _ \nu ( x , \mu ) = \ \phi _ \nu ( \mu ) e ^ {- x / \nu } , $$

$$ \phi _ \nu ( \mu ) = \frac{c \nu }{2} P \frac{1}{\nu - \mu } + \lambda ( \nu ) \delta ( \nu - \mu ) $$

are the eigen functions of the continuous spectrum, $ P $ denotes the Cauchy principal value, $ \delta ( \nu - \mu ) $ is the Dirac $ \delta $- function, and

$$ \lambda ( \nu ) = 1 - \nu c \ \mathop{\rm Artanh} \nu ,\ \ \psi _ {0 \pm } = \ \phi _ {0 \pm } ( \mu ) e ^ {\mp x / \nu _ {0} } . $$

The discrete eigen values $ \pm \nu _ {0} $ are the roots of the characteristic equation

$$ \nu c \mathop{\rm Artanh} \frac{1} \nu = 1 . $$

The eigen functions of the discrete spectrum take the form

$$ \phi _ {0 \pm } ( \mu ) = \ \pm \frac{c \nu _ {0} }{2} \frac{1}{\pm \nu _ {0} - \mu } . $$

The system of eigen functions $ \phi _ {0 + } ( \mu ) $ and $ \phi _ \nu ( \mu ) $, $ 0 \leq \nu \leq 1 $, turns out to be complete in the space of generalized functions on the interval $ 0 \leq \mu \leq 1 $ and they are orthogonal with respect to a weight $ W ( \mu ) $ which is the solution of a singular integral equation (see [4]).

The boundary condition (2) of the Milne problem gives $ ( \mu \geq 0 ) $:

$$ - \phi _ {0-} ( \mu ) = \ a _ {0+} \phi _ {0+} ( \mu ) + \int\limits _ { 0 } ^ { 1 } A ( \nu ) \phi _ \nu ( \mu ) d \nu , $$

that is, $ a _ {0+} $ and $ A ( \nu ) $ are defined as the coefficients in the expansion of the function

$$ \phi _ {0-} ( \mu ) = \ \frac{c \nu _ {0} }{2} \frac{1}{\nu _ {0} + \mu } . $$

The asymptotic density of the neutrons,

$$ B _ {ac} ( x) = 2 \pi \int\limits _ { -1} ^ { +1 } [ \psi _ {0-} ( x , \mu ) + a _ {0+} \psi _ {0+} ( x , \mu ) ] d \mu = $$

$$ = \ 4 \pi e ^ {- x _ {0} / \nu _ {0} } \sinh \frac{x + x _ {0} }{\nu _ {0} } , $$

vanishes for

$$ x = - x _ {0} = \ \frac{\nu _ {0} }{2} ( \mathop{\rm ln} a _ {0+} - i \pi ) . $$

For $ c = 0 $, $ p ( \mu , \mu ^ \prime ) = 1 $, the Hopf constant $ x _ {0} = 0.710 446 $.

References

[1]	E.A. Milne, Mon. Notices Roy. Astron. Soc. , 81 (1921) pp. 361–375
[2]	E. Hopf, "Mathematical problems of radiative equilibrium" , Cambridge Univ. Press (1934)
[3]	I. Sneddon, "Fourier transforms" , McGraw-Hill (1951)
[4]	K.M. Case, P.F. Zweifel, "Linear transport theory" , Addison-Wesley (1967)

References

[a1]	W. Greenberg, C. van der Mee, V. Protopopescu, "Boundary value problems in abstract kinetic theory" , Birkhäuser (1987)
[a2]	C. Cercignani, "The Boltzmann equation and its applications" , Springer (1988)
[a3]	B. Davison, J.B. Sykes, "Neutron transport theory" , Clarendon Press (1957)
[a4]	M.M.R. Williams, "The slowing down and thermalization of neutrons" , North-Holland (1966)