Transport equations in moving material Part I: Neutrons and photons

ProgressmNuclearEnergy,Vol.46, No. 1, pp. 13-55,2005 Aval[ab[e orl]lne at w w w sclenced[rect,com

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TRANSPORT PART

EQUATIONS NEUTRONS

I:

IN MOVING MATERIAL AND PHOTONS

B.R. Wienke Los Alamos National Laboratory Applied Physics Division Group X-7, Materials Science Los Alamos, N.M. 87545

ABSTRACT

Transport equations are usually formulated against a static material background in transport applications for neutral (neutrons, photons, mesons) and charged (electrons, protons, ions) particles. When the background is moving, the equations need be modified and corrected. We discuss these modifications and corrections, in both relativistic and nonrelativistic regimes, and contrast equations across applications. Neutron, photon, electron, neutrino, and meson transport are treated in a two part series. This review collects, collates, and expands published and unpublished developments and equations for moving material transport in different particle arenas. Hopefully, such provides a useful reference and source for readers across disciplines, linking basic underlying dynamics and representations. Part I centers on neutrons and photons, and Part II deals with electrons and neutrinos, almost an historical split. While tranpsort equations for these particles share much in common, there are fundamental differences between particles that interact pointwise versus continuously, particles of zero rest mass versus nonzero rest mass, particles urith relativistic energy versus nonrelativistic energy, and particles with short range versus long range. Our focuses are the fundamental equations and underlying dynamics of transport in moving backgrounds. The impacts in applications, important and legion, are detailed in the literature as cited. Accordingly, and invoking topic closure, applications are not recounted. Also, numerical techniques, developed with the emergence of supercomputing, are not treated herein. Both applications and numerical techniques are probably subjects of a separate communication Some modern transport codes dealing with moving material issues, however, are cited for reader reference. Many codes have been in existence for 25+ years, undergoing continuous updating, modification, improvement, and physical extension. For Part I, the main points of discussion and development are seen in the following Table of Contents, divided into neutron and photon moving material transport. An attempt at coverage of moving material transport with brevity is sought, and hopefully references provide details minimized, or not recounted, herein. © 2004 P u b l i s h e d b y E l s e v i e r L t d

K E Y W O R D S - neutron transport, photon transport, moving material, effective cross sections, transport approximations, relativistic dynamics, fundamental equations frame invariance, coupled transport and hydrodynamics, diffusion

13

B. R. Wienke

14

T A B L E OF C O N T E N T S OVERVIEW NEUTRONS AND HEAVY PARTICLES Neutron Properties B o l t z m a n n E q u a t i o n / Fixed Material / Moving Material Effective Cross Sections / Asymptotic Limits / Multiple Legendre Moments N e u t r o n - T h e r m a l B a c k g r o u n d S c a t t e r i n g / Effective q2 Kernel / Effective q Kernel

/ Effective Constant Kernel / Effective q-1 Kernel / Effective q-2 Kernel / Resonance Absorption H y d r o d y n a m i c s A n d T r a n s p o r t Coupling//Eulerian Hydrodynamics//Modified Eulerian Hydrodynamics / Material Transport Coupling Differential A c c e l e r at i on T e r m s / I n Plane - N(x, q, O) / 1D Sphere - N(r, q, O)

/ 219 Cylinder - N(h, z, qh, qz, X) / 219 Sphere - N(r, O, q, w, 7) / 319 Plane - N(x, y, z, qx, qy, qz) N e u t r o n T r a n s p o r t A p p r o x i m a t i o n s / Legendre Expansion / Diffusion Expansion PHOTONS Photon Properties R a d i a t i v e Transfer E q u a t i o n / EuIerian Frame//Lagrangian Frame//Stimulated Emission

/ Thermodynamic Equilibrium L o r e n t z T r a n s f o r m a t i o n s / Local Material Transformations / Exact (u/c) 4 Expansions P h o t o n - E l e c t r o n Scattering//Effective Low Energy Kernel / Effective Isotropic Kernel

//Effective Klein-Nishina Kernel / Effective Thomson Kernel//Effective Bjorken Kernel //Total Scattering Cross Sections R a d i a t i o n H y d r o d y n a m i c s / / F i c k Diffusion//Flux Limited Diffusion / Scaling//Eddington Generalized Approximation P h o t o n T r a n s p o r t A p p r o x i m a t i o n s / Source-Absorption Expansion//Diffusion Expansion

//Asymptotic Expansion / Equilibrium Expansion SUMMARY ACKNOWLEDGMENTS REFERENCES

OVERVIEW Transport equations have a wide range of applicability in diverse areas of physics, chemistry, medicine, biology, engineering, and applied mathematics. 0-33) Most applications employ the static equation, treating the material background as fixed. Although the static approximation is a very good starting point, background material motion effects become pronounced as local background media accelerations and velocities increase, especially with respect to the lower energy components of the transport particle spectrum. The effects can be very important, (34-77) even replacing the solution space of the static equation. Even if physical corrections to the static equation are straightforward conceptually, they are often difficult to treat numerically. Material motion terms induce additional degrees of freedom onto solution algorithms based on the static transport equation. When fluid and background speeds are very high, relativistic corrections also enter the moving material transport equations, further complicating analysis. Photon transport is intrinsically relativistic because photons move with the speed of light, c. Because material motion corrections to the static transport equation can be very complex, initial efforts and advances track with increases in computer power and coupled numerical methods. Such is the case for the 1970s to 1990s time frames, when very powerful and sophisticated production codes came online for transport simulations and applications. These codes and their predecessors now usefully span stellar burn to inertial confined fusion, particle beams to medical radiotherapy, climate modeling to vapor film deposition, and high energy particle interactions to imaging, just to mention a few. Code references reflect the same, (Ts-s5) with state of the art numerical recipes and solution schemes. (25'27'32)

Transportequattonsin movingmaterialI

15

Certain analyses require coupling of hydrodynamic and transport equations. Hydrodynamics is posed in eulerian (fixed mesh or reference frame) and lagrangian (moving mesh or reference frame) representations, as well as hybrids. To maximize the efficiency of coupled calculations, it is sensible to pose both the transport and hydrodynamic equations in the same picture. Motion corrections to the transport equation take different functional forms in eulerian and iagrangian frames, though the physical content is the same. Another approach is to correct cross sections by virtue of relative material motion, while still working in the fixed reference frame through which particles stream and interact, We consider both in the following, and the two approaches are dubbed the dynamical moving frame (DMF) and static effective interaction (SEI) pictures, respectively. The two frames are called the comoving and fixed frames. Excepting photons, neutrinos, high energy electrons and fast mesons, material motion effects are mostly nonrelativistic, involving classical Galilean (velocity) frame transformations. The treatment of photons (zero rest mass) and ultrahigh energy particles (nonzero rest mass) is patently relativistic, requiring full Lorentz transformations between rest, fluid, and fixed laboratory frames. (3,14,19,23,53) Relativistic effects start at the 1% level with E/Eo >_ 1/100 roughly, with E the particle energy and Eo the rest energy (mass), or when v/c > 1/10, with v the particle velocity, and c the speed of light. Photons, of course, are patently relativistic because of zero rest mass. High energy electrons, neutrinos~ other leptons and mesons are also relativistic because of their small rest masses. The field variable convention used herein is standard. Scalar quantities are lightface type, vectors are boldface type, and dyads (tensors of rank 2) are calligraphic (upper case) type. Higher order tensors are neither necessary, nor introduced. Lorentz four vectors, convenient for relativistic application, carry either covariant or contravariant indices, #, to distinguish them from ordinary three (boldface) vectors. NEUTRONS

AND HEAVY PARTICLES

Neutron P r o p e r t i e s Neutrons are spin 1/2 particles obeying Fermi-Dirac statistics. (14,31) They have rest mass of 938 Mev, and zero charge. Neutron interactions with matter are short ranged, and cross sections are pointwise. Their fundamental interactions with matter are strong, in the category of strong, weak, electromagnetic, and gravitational forces. Because of heavy mass, observable neutron energies are mainly nonrelativistic, and transport equations linked to neutrons are classical in dynamics. With the advent of nuclear reactors in the mid 1950s, neutron transport equations, and their solutions, became a focal point for study, application, and analysis. In weapons physics and astrophysics, neutron transport in dense media remains a topic of interest. Neutrons were discovered in 1932 as fragments knocked out of nuclei. Protons and neutrons were first proposed as nucleons in 1911, meaning both are stable constituents of the nucleus, with slight mass difference due to Coulomb repulsion of like protons.

Boltzmann Equation The linear Boltzmann equation (5'13,17'2°'25'3°) is a useful (linear collision kernel) expression for nonrelativistic particle transport (neutrons, heavy mesons, slow electrons) applications. The usual formulation assumes a fixed background against which particles scatter, absorb, and interact. When the background is moving, the Boltzmann equation can be modified in a number of ways, as mentioned. Moving material transport in a relative velocity representation is one useful approach, and we consider it after first formulating the general transport equation with external forces (producing accelerations) in a fixed material frame (no material motion).

Fixed Material In some inertial frame, consider the motion of a packet of particles which suffer collisions and reactions with other particles, and are possibly subject to external (noncollisional) forces. (36) In six dimensional velocityconfiguration space, v and r, in time, t, the packet particle density, N, moves under internal collisions and

16

B. R. Wienke

external forces, F. At time, t, the density is N(r, v, t), while at t + A t , density is N ( r + Ar, v + A v , t + A t ) , so that, expanding in a Taylor series about At, N ( r + Ar, v + Av, t + At) = N(r, v, t) + v . V N A t + a. V v N A t + ~ON A t

(1)

with, Ar=vAt,

Av=aAt

(2)

given particle acceleration, a, dv a =

--

dt and for V~ and V velocity and spatial gradients, Taking the limit, lim N ( r + Ar, v + A v , t + At) -- N(r, v, t) = ON - - + v . X7N + a . V v N /,,~o At Ot

(3)

(4)

If a(r) denotes the invariant probability (per unit distance) for all collisions and reactions (total cross section), then the probability, ~, that a particle will not be lost from the packet, N, in time, At, is, = 1 - ~(v)vAt

(5)

The cross section depends on relative particle velocity as always. But here, since the background material is fixed, the relative particle velocity, q, and frame particle velocity, v, are the same. Denoting all sources (collisional and external) by, Q, then by conservation of particles, N ( r + A r , v + Av, t + At) = N(r, v, t)[1 - a(v)At] + Q(r, v, t ) A t

(6)

Taking the limit again, lira N ( r + Ar, v + Av, t + At) -- N(r, v, t) = Q _ vcr(v)N At

(7)

At--*0

Collecting terms, the Boltzmann equation is recovered in the fixed frame picture, ON 0---t- + v . V N + a . V v N + v a ( v ) N = Q

(8)

The source term, Q, is sum of collisional inscatter plus external sources, S, and is specifically written, Q = f a~(v,v') 03 v, v ,N d,3 v , + S t

(9)

as a function of r, v, v , and t. By Galilean invariance, (3,4'9) one expects the kinetic terms of the transport equation to remain frame invariant (same form) in all systems moving with constant velocity with respect to one another, as well known classically. This occurs in the relative velocity picture with a moving material, but the physical content of each term is diffrent in frames. While calculations are easily effected in one or another frame, one has to be careful to always transform variables back to a fixed (laboratory) frame to make measurements and apply particle diagnostics in same. Accelerations, of course, are a special consideration, because they are not Galilean invariant. Galilean invariance only applies to inertial frames moving at constant velocity with respect to each other. So that brings us back to formulating the transport equation in terms of relative velocities, and with accelerations related not only to external forces on the transport particle, but also with pseudo-forces induced by material accelerations in the relative velocity picture. (36,39'41) This is more complicated, and needs proper definition. A relative frame is not an inertial frame because of overall frame accelerations. But it is still possible to write a conservation statement in that relative frame. (4,19'29) In moving materials calculations, the relative frame is the material rest frame.

Transport equations in movingmaterialI

17

Movin9 Material In a moving frame, consider the motion of a packet of particles which suffer collisions and reactions with other particles, and are possibly subject to external (noncollisional) forces. (36) In six dimensional velocityconfiguration space, q and r, in time, t, the packet particle density, N, moves under internal collisions and external forces, F. At time, t, the density is N(r, q, t), while at t + A t , density is N ( r + Ar, q + Z~q, t + A t ) , so that, expanding in a Taylor series about At, N ( r + A r , q + Aq, t + At) = N(r, q, t) + q . ~TNAt + a . V q N A t + -ON -~ At

(10)

for relative velocity, q, as always, q = v - u

(11)

dq c - - dt = a - b

(12)

and relative acceleration, c, given by,

and, coupled to the above, A r ----vAt,

A q ----aAt -- bAt

(13)

where the acceleration, b, is the material acceleration, du b =- d--/

(14)

and for ~q and V relative velocity and position gradients, assuming material velocity, u. If there are no external forces, a = 0, and then, dq dt

c . . . .

b

In such case, the relative acceleration, c, is aptly a pseudo-acceleration, caused by the frame representation in the moving material. While this is a complication for some kinetics, the moving frame picture allows simple representation of material sources and interaction dynamics. Taking the limit, lim N ( r + A r , q + Aq, t + At) -- N(r, q, t) ON At--,0 At ---- 0--t + q" V N + c . XYqN

(16)

If a(q) denotes the invariant probability (per unit distance) for all collisions and reactions (total cross section), then the probability, ~, that a particle will not be lost from the packet, N, in time, At, is again, ¢ = 1 - o(q)qAt

(17)

Denoting all sources (collisional and external) by, Q, then by conservation of particles, N ( r + Ar, q + Aq, t + At) ----N(r, q, t)[1 - a(q) At] + Q(r, q, t) A t

(18)

Taking the limit once more, lim N ( r + Ar, q + Aq, t + At) -- N(r, q, t) = Q - qa(q)N At

At--*0

(19)

Collecting terms, the Boltzmann equation is recovered in the relative velocity picture, ON 0---[-+ v . ~TN + c . V q N + q~r(q)N --- Q

(20)

Again, the source, Q, is the sum of inscatter plus external sources, S, ,.T,.3 Q - - f 0 a~( q , q ' ) qiv aq ' +S

(21)

B. R. Wienke

18 t

but, as a function of r, q, q , and t here. There is much in the above equations needing additional attention. For instance, the density, N, in a moving frame changes as volume, V, changes under hydrodynamic compressions and expansions, (9) so that, dN

ON

O N OV

d~- - 0~- + 0V 0t

(22)

Additionally, coupling to hydrodynamics is often desirable, and taken up later.

Effective Cross Sections Without background motion, the transport equation was cast, ON

0--t- + v - V N + a- V v N + w ( v ) N = Q

(23)

With background motion, the cross section must conserve interaction probability, so that, ON Ot

+ v. VN + a. VvN + qa(v)N = Q

(24)

which requires, in the (r, v, t) picture, (7 e

(25)

and, 0cr(v, v ' ) ( q ) O o ' ( q , q ' ) O 3 q 03v,

=

Oaq,

' Octe 03v~ = OSv---W

(26)

in both eulerian representations. The expressions above define the effective differential and total cross section, and offer a useful picture to move from fixed frame cross section representations (standard cross section sets) to moving material representations.(3a,35) When there is no background motion, q = v, and ae = a(q) = ~r(v). Customarily, the most general form is now given by, ON

0---t- + v . V N + a . V v N + vc%N = Qe

(27)

for, Q, =

/

0~7e , , , / . 3 ,

~v,

v2v

a v +S

(28)

and represents a fixed frame transport equation with modified cross sections. Because of the dependence on q and v, additional anisotropies are introduced into the scattering. They can be very complicated for arbitrary motion.(Ss,a0,a2-44) A s y m p t o t i c Limits A n d Anisotropy

The effective cross sections conserve interaction probability, but the scaling factors, q / v , q ' / v ' , impart additional directional anisotropy to the terms. While differential cross sections already depend on angle, the total cross is only a function of energy (or magnitude of relative velocity). The relative importance of added anisotropy depends on the magnitude of the material and particle velocities. Since, q = (v 2 + u 2 - 2v. u) 1/2

(29)

we have in the small material velocity limit, u < < v, -q = 1 - - - u ' v v

v2

(30)

while in the large material velocity limit, u > > v, U'V

with similar expansions for q ' / v ' . The anisotropy, measured by the u . v term, occurs in 1*h order in the low material velocity limit, but only in 2n~ order in the high material velocity case. For u ~ v, the anisotropy competes with other terms, to all orders, with the scaling factor, q / v , ranging from 0 to 2. This anisotropy scaling maintains within all effective cross section expansions. (a2,aS)

Transportequahons in moving matertalI

19

Multiple Legendre Moments The effective cross sections can be expressed as Legendre polynomials by first expanding the scaling terms, ~= (q)=

(v2 +U2-v 2uv#)1/2 = ~

( 2 n + 1)~npn(#)

(32)

rt----0

taking one-dimensional geometry with,

u " v = uv#

(33)

Employing,

(a2 + b2 _ 2abT)l/2co s [A(a2 + b2 _ 2ab~.)U2]pn(~.)d T _

/

(ab)Ir1/2 Jn+ 1/2 (Aa) Nn+ l/2 ( Ab)

1

0
(34)

for J and N half integer Bessel and Neumann functions, the moments take the form, ~r

02

~r

02

v>u

-

=

v< u

(35)

The first two moments are given in the symmetrical (u and v) picture, ~0=

2(3v 2 + u 2) 3v 2 ,

~1--

2u(u 2 5v2) 15v3

v>u

(36)

~0 -

2(3u 2 + v2) 3u 2

~1 --

2v(v 2 -- 5u 2) 15U3

V< U

(37)

,

-

In the large u and v limits, only the 0th moments survive. Physically, the relative symmetry of the Legendre expansion in the large material and particle velocity limits is to be expected. The relative speed approaches either the material speed, u, or the particle speed, v in the large velocity limit, and the interaction probability, qa(q), depends only on the relative speed. The differential scattering kernel can also be expanded in a multiple Legendre series, and then reduced through orthogonality of the Legendre polynomials. Expanding the scaling factor, density, and cross section, including the Jacobian in the cross section definition, it can be written, (46)

Oae ' N ' = 1 ~ f i 03v 'v ~ E /=0

!

!

fi(21+l)(2m+l)(2n+l) 2 2 2

. . . . . ~za'~N'~Pl(# )P'~(~)P=(# )

(3s)

ram0 n=0

t

where ~l, am, and N~ are moments, for, v

•

V r

(39)

= vv'¢

with Pro(#) and Pro(#') related to Pm(~) by Legendre addition. Using the addition theorem for Pm(~), and integrating, we obtain, (4o) 1

l=O J

with 5m the Kroneker delta. In the large u and v' regimes, only the ~0 terms survive, and the usual Legendre expansion of the scattering kernel is recovered•

20

B. R. Wienke

Neutron-Thermal

Background

Scattering

Neutrons often scatter from backgrounds in thermal motion. (5'17'24) Effective cross sections obtain from integrating the thermal background over full energy dependent cross sections. One normalized appropriate background is an isotropic, nonrelativistic, Maxwellian ensemble, which, in velocity space, takes the form,

/ M

=

for k Boltzmann's constant, T the temperature, w the scattering particle velocity, ¢ the azimuthal angle, # the scattering angle cosine between incident neutron with velocity, v, and scattering particle, M the scattering particle mass, and v . w = vw# (42) M 2kT'

a-

e = o~v2

(43)

The scatterers are mostly atoms, so that denoting their atomic weight, A, the mass is just, M = A m , with m the neutron mass. The relative velocity, q, is as usual, q = v - w,

(44)

q2 = v 2 + w 2 _ 2 w v #

so that the effective differential cross section reduces to the form,

o°.

03v I

_S

OSv'

and the total effective cross section follows,

~

=

0~-3 ' jf b %~~ ,v

(46)

The integrations over the background motion can be performed when specific representation is given to the differential cross section, and we consider a number of separate cases for sake of generality. For simplicity, a velocity picture is assigned to the differential cross sections. In the following, 7 and ~? are the incomplete Euler functions, and F is the complete Euler function, ~/(n,e) =

ezp (-t)t~-ldt,

~(n,e) =

~/(n, e) + r](n, e) = F(n) =

/]

ezp(-t)t~-ldt

(47)

exp(-t)t'~-ldt

where, for n integer, F(n)

(~

1)!

r ( n + 1/2 ) = ~V2 [ ( 2 ~ - - 1 ) ! j~ L 2~

'

(48)

Effective q2 K e r n e l

Taking a differential cross section dependency on energy, with ~ constant 0~(v, v') 03v '

_ ~q2

(49)

we obtain, 0-e

(

M

~3/2 (27r=~ k5v 2 1

a3v' - \ 2 - j ~ ) +

I"

[/o"

(10v4w+20v2=3 + 2 w = ) w e x p \

(2v 5 + 20v3w 2 + l O v w 4 ) w e x p \

2kT

) dw

I

2kT

)d"

21

Transport equations m moving material I

-

[5v~(3/2, ~) + 10v~-1~(5/2, ~) + ~-~(7/2, ~)

~

+v5o~1/2r](1, e) ~- lOv3o~-l/2r](2, e) + 5vo~-3/2r](3, e)]

(go)

In the low velocity limit, v --* O,

(51)

0%' ~

In the high velocity regime, v --* co, O0"e ~ EV 2

(52)

03v' Effective q Kernel

Assuming a differential cross section dependency on velocity, for constant ~,

0z(v,v') 03 v,

(53)

-- ~q

we find, Oae Oavs - \ ~ /

-=

if0 (2v2 + 2w2)w2 exp

~

\ 2-~-~--/

dw

[v2r(3/2)+"-lr(5/2)]

(54)

With v --* O, ~O"e

(55)

Similarly, for v -~ co, Oare 0av'

(56)

Effective Constant Kernel

With the differential cross section constant, for fixed a again,

0~(v,v') 03V s

(57)

we see, 03V-----7--

-

-

(

~

\-~V2 ]

+i

E/0

(6v2w + 2w3)w exp \ - - ~ - ~ ] dw

2~ (3~rl-~v2) [3V2V(3/2' c) + o~-1~/(5/2, e ) + V30~1/2~1(1, e)~- 3va-1/2~](2, e)]

(58)

As before, for v --+ 0, (59)

03v ' ~

while, with v -~ co, 03 v'

(6o)

B R. Wienke

22

Effective q-1 Kernel Assigning a differential cross section dependency on inverse velocity, for n constant,

v') Oav'

(61)

q

we trivially recover,

(62) Thus, for all values of incident velocity, v, the effective cross section remains, 0G e 03V ~

(63) V

Effective q-2 Kernel W i t h a differential cross section dependency on inverse energy, n constant,

O (v, v') 03v' -- q2

(64)

we find,

03V-----'-- ~

~-~-]

0

2w2

exp \ 2kT ] dw

oo (_MW2~ dw] +J Iv 2vw exp k" 2kT ] =

(

2~ )[.y(3/2, c)+val/2~(1,c)]

(65)

For small incident neutron velocity, v -+ 0, age

(66)

03V t

and at the other end of the spectrum, v --* oo, Oo-~

~

(67)

03v----7~ v-~

The above results are simple for small and large incident neutron velocity, but, in between, effective cross sections are more complex. At large velocity, the incident neutron sees the scattering Maxwellian background as fixed. At low velocity, the background thermalizes the neutron and modulates the cross section by factors of M/2kT. At high incident velocities, the relative velocity and incident velocity are much the same, and the integrals over the Maxwellian background collapse to unity.

Resonance Absorption In the thermal region, neutrons can be absorbed in moderators at various resonant energies. The cross sections for resonance absorption are sharply peaked at the neutron resonance energy, e0. The absorption cross section takes the well known form, (5'1°'17'2a)

]

- 0)2 + r 2,

(68)

with F the energy width of the resonance peak at half maximum, e the neutron energy in the center of mass system, and ~0 the probability of compound nucleus formation after neutron absorption. Denoting the neutron mass, m, and the atomic mass of the background scatters, M, we have,

M = Am

(69)

Transport equattons in moving matertal I

23

with A the atomic number. If the background of nuclei is moving with velocity, w, the relative velocity, q, is the usual, q =

v -

w

(70)

with v the neutron velocity. Defining the reduced mass, #,

Mm

Am

# -- M +------~= A +------~

(71)

the neutron energy, e, in the center of mass system becomes, =

1

2

(72)

and obviously, 1 ~m < # _< m

(73)

If the background is distributed in velocity with distribution function, ~5, the effective resonance cross section, crcrew, takes familiar form,

~'e~ = 4~rf a r ~ ( w ) w 2 d w = 4~r f c~re~(q)q2dq

(74)

O"e

for an isotropic distribution function. Changing integration variables to neutron relative energy, e, we find,

de = #qdq

(75)

and therefore,

kT

i721 de

(76)

taking a thermal Maxwellian background, ~,

o(q =

exp

- A mq "~ •j

(77)

The absorption capture probability can be cast, 04'31) cr0 = noTr~2

(78)

with no the number of target nuclei per unit volume, and/3 the reduced de Broglie wavelength, h / 3 - 2~r#v

(79)

with h the Planck constant. For heavy target nuclei, obviously the center of mass and laboratory frames are roughly the same. For light target nuclei, the frames are different, and effects are enhanced in the effective resonance cross section, on top of the thermal background motion. The above intergral does not reduce to quadratures as in preceeding cases, but can be numerically integrated fairly easily for implementation in codes. All effective cross sections described can be computed numerically from data libraries at the National Nuclear Data Center (47-5°) amd imbedded in neutron transport code modules. In general, this is often the best, and only, way to fold motion corrections into transport analyses. Collecting and building neutron cross section libraries is a tedious task, and the format nicely presents itself for postprocessing by users.

24

B. R. Wienke

Hydrodynamics And Transport Couplimg Hydrodynamics is posed in three different standard representations: 1. eulerian (fixed frame) 2. modified eulerian (comoving frame) 3. lagrangian (moving frame) with the difference between eulerian and lagrangian one of viewpoint. Eulerian hydrodynamics focuses on fixed positions in some inertial frame, and asks for the state of a fluid moving through the fixed frame. Lagrangian hydrodynamics asks for positions, plus states, of initially tagged moving fluid cells as a function of time. The modified eulerian picture contains elements of both, and is the frame described above for the transport equation, that is, moving in relative velocity space. The modified eulerian frame is often called lagrangian in the literature of applications. It is the frame many computer codes employ, but we hereforth avoid that convention. The pure lagrangian frame is complicated because initial values of coordinates need be tagged and followed as the system evolves. The modified eulerian frame uses fixed laboratory spatial coordinates, and relative velocities, and the modified eulerian frame is basically fixed in space. A pure lagrangian frame is complex and less useful for most transport applications. Details can be found in appended references, (<1°'12,16'19'22) but simply stated, field variables are not functions of fixed space and time. Rather, field variables are attached to moving fluid mass points, and we follow these mass points in pure lagrangian dynamics. This is not only a complex frame for transport analysis by itself, but also an extremely convoluted frame for coupling of field variables in both the transport and hydrodynamic equations. Hydrodynamics is nothing more than a statement of mass, momentum, and energy conservation in a moving fluid. The statements can be made in the fixed laboratory (eulerian) frame, and in the comoving or moving fluid (modified eulerian or lagrangian) frame. Eulerian Hydrodynamics. Conservation of mass for fluid flow is written,

ap + V.(pu) = 0 Ot

(80)

for p local fluid density, and u local fluid velocity. Momentum conservation is more complicated, taking the form, a(pu) + + vp = r (Sl) Ot denoting the velocity dyadic, u u = b/, with pressure, p, and total force, f. The energy conservation statement is,

with internal energy, e, and external energy source, w. A relationship connecting pressure, p, internal energy, e, and density, p, the equation of state (EOS), closes the above flow relationships, permitting exact numerical solution for arbitray boundary conditions and flow regimes.(<9,27) Modified Eulerian Hydrodynamics The above set are posed in the fixed (eulerian) reference frame, through which the fluid moves. The other frame, moving with the fluid (comoving), is often more suitable for numerical application, particularly when vortices, subscale disturbances, and turbulence are present. (15,29) Dynamics in high speed flow, nozzle deflection, and eddying fall into the latter category, and numerical simulations often rely on modified eulerian analysis in the moving fluid stream. Transformation to the modified eulerian frame in the above set is most simply accomplished using the advective derivative, D / D t , related to the eulerian time derivative, O/Ot, via,

D 0 + u. v Dt -- Ot

(sa)

Transportequattonsin movingmaterialI

25

aS the temporal operator in the moving Dame. In the moving frame, we also have by virtue of the above definitions, Dr Or

Dt

-

+ (u. V)r

Ot

(84)

as a vector statement, and, D q _ 0 ( v - u) Dt ot + (u. V ) ( v - u)

(85)

as a dyadic statement. That D/Dt represents the time rate of change in the comoving frame is seen in the relationship(a,9) with p the mass density, 0 / A v d V [P~t l = -~ [f, vdV(pQ)l (86) with Q any field variable, and the integration over any convenient volume, V, whose boundary points move with the local fluid velocity, u. The above eulerian set transform in the modified eulerian frame,

Dp

-~

9--7 + p V - u = 0

(87)

Du P D 7 + Vp = f

(88)

t

[

1

pu 2 + e + u. V lpu2 + e 4- V.(pu) = w

(89)

The full set of hydrodynamic equations are now conveniently posed in the same frame as energy sources, pressure gradients, and forces, that is, observables linked to internal properties of moving fluid or material rest frames. Similarly, transport cross sections, opacities, and sources naturally link to moving fluid or material rest frames, so such frames are convenient for transport calculations. Hydrodynamics and transport can be explicitly linked as follows, using the moving frame transport equation and advective derivative.

Material Transport Coupling Employing the advective operator, D

a

Dt

Ot

+ u- V

(90)

the moving material transport equation takes the form,

DN

D--T 4- q" V N + e- V q N + q~r(q)Y = Q

(91)

Form invariance with the fixed frame equation is evident when time derivatives are referenced in their separate frames (Galilean invariance). The equivalence is strictly classical, and nonrelativistic. Time and space are absolute in classical dynamics, but not so relativistically, as would be the case for radiation transport. The acceleration, c, and velocity, q, are linked through the material velocity, u. If there are no external forces, a = 0, and v is constant in time and space,05-19) and, denoting local material velocity, ~', and acceleration, i~, we obtain du c = - d--~-= -i~ - (q" V)[" (92) where, i~ = ~-~ + (i'. V)~

(93)

The transport equation can be linked to the hydrodynamic continuity equation by noting in the modified eulerian frame,

1 Dp pdt

-- V. ÷

(94)

26

B. R Wtenke

so, tagging mass, m, as a constant of motion for density, p, and volume V, P=V

m

(95)

reduces to the well know form, 1 Dp p Dt

1 DV V Dt

(96)

For the particular form of the acceleration, c, above, the operator identity is useful in coupling material divergences, (i: -6 (q. V)f') .VqN = Vq. IN (i: -6 (q. V)I-)] - N V . ? (97) so that the comoving transport equation is coupled generally, 1 D(VN) - -

V

Dt

-6 q . V N -6 q a N = Q -6 V q .

[N(i: -6 (q. V)f-)]

(98)

Consider, next, specific representations of the acceleration terms in the moving material picture.

Differential Acceleration Terms The above operator forms reduce to various frame representations. Five are detailed in the absence of any external forces a = 0, namely, one dimensional plane and spherical geometries, two dimensional cylindrical and spherical geometries, and 3D planar geometry, (36'37'39) The standard approach in expanding all operators is to first use their cartesian representations, followed by transformation to the appropriate sets specified by the density functions. The angular relationship between position vectors and relative velocity vectors close the set, and transformations. (25) In curvilinear bases, direction vectors are not fixed in particle streaming space (as in rectangular bases), and change with particle directions. Consequently, transport operators like gradients, divergences, vector products, etc must also track these changes. This results in angular derivatives with respect to streaming directions. So, operators are first expanded in rectangular (Cartesian) basis, taking the gradient, V, as an example,

v = E1

/=1,3

'

0 0%~]

mE l .

0z j,

q. V = ,=1,3 ~ qz m

(99/

07,~ 0-/ j

(i00)

with l unit vectors in the x, y, m directions,~m approriate density variables in the desired representation,7Vf the number of variables in the basis~ and with I denoting x, N, z, and ql components of the vector, q. The transformations are more complicated in 3 D and 2D than ID obviously. 1D Plane - N ( x , q, #)

In 1D plane geometry, density, N, is a function of position, x, relative velocity, q, and scatttering angle, #, such that, q . x = qx cos 0 = qx~t = qxx (101) with material velocity, :~, and acceleration, ~, so that,

u--i,

c---~-(q.V)i

(102)

The material divergence takes the form, V.~=--

05

(103)

Ox

Consequently, we obtain, DN Dt

ON ON (1 - #2) + # q ~ x + qcrN = Q + ~ # ~ - - ÷ - q ~

" ON + q#2 - - - Oq + t ~ ( 1 -

O~ O N tL2)Ox OtL

(104)

27

Transport equations in moving material I 1D Sphere - N ( r , q, O)

In 1D spherical geometry, density, N, is a function of position, r, relative velocity, q, and scattering angle, #~ a~ in the 1D planar ca~e~

(105)

q . r = qr co~ 0 = qrt~ = q x x H- qyy + qzz

and material velocity, ~, and acceleration, i:,

u=e,

¢=-e-

(q.V)e

(lO6)

Material divergence reduces to, 0~

÷

(lo7)

Collecting terms, we get, DN D--T+q

ON (1 #~r +

(1 +qaN=Q+i:

ON

~-~q +

-

+q

_

(1o8)

a,

219 C y l i n d e r - N ( h , z, qu, qz, X)

In 2D cylindrical geometry, density, N, depends on position, r, r=h+z

(109)

with h and z, radial and axial cylinder vectors, and # the projection of q onto z, (110)

q . z = qz cos 0 = qztz = qzz

and for X the scattering angle in the h plane,

(111)

q . h = qhh cos X = qxx + qyy

The material velocity,/', and acceleration, i:, have components in the h and z directions, u = 1~+ ~,,

(112)

c = - ( h + ~) - (q. V)(fi + ~)

The divergence of the material velocity takes the form, oh

V'÷=Or+2r=--+0h

h

(113)

h + --az

Using all of the above, we find, DN -

Dt -

[

ON + q s i n 0 cos X Oh

__s i n

0 s i n X O N + cos h OX

00N l + qaN --~z J

..ON Oh O N + Z ~ q z + qhcoS X Oh Oqh

sin

Oh O N

[

aN

1

= Q + h cos X-~q h - -qh -sin

02 O N

X--~-~X+ qz Oz Oqz

xO-~x]

(114)

2 D Sphere - N ( r , O, q, w, 7)

In 2D spherical geometry, density, N, is a function of position, r, the projection of r onto z, r . z = r z cos 0 = r z ~

(115)

the relative velocity, q, and scattering angles, w and ~/, q . z = qz cos w = qz~ = qzz

(116)

28

B. R. Wienke

q.r=qr

(117)

cos ~ = q r # = qxx +quy+qzZ

Material velocit~ ~, and acceleration, ~, remain, u=~,

c=-~-(q.V)~

(118)

As before, v.÷=N+2;

(119)

Substituting terms,

[

DN ON sin w t t ON sin 2w ON D---t + q #-~r + - - r --00 + r Ow

sin w sin ~ cot 0 r

+ q~N = Q

ON ( 1 - #2) 0N ~sinOON 1 20i'ON tz2.0i'ON Oi'ON +iz / ~ q + - - - - q O# + - - ~ - j q +q/~ ~rr~q + i t ( l ) ~ r - ~ + ~ c ° s O o r 0--( +-r ( 1 -

)-~+cos~cos

q#~q +(1-~t)-~+sinO-~

Note we have used mixed notation above in the derivatives, that is, # = cos ~, ~ = cos w, for simplicity. This is also done to eliminate writing square roots of angular cosine terms.

3D Plane - N(x, y, z, qz, qy, qz) In 3D plane geometry, the differential acceleration terms are straightforward and easy to obtain without extensive variable transformation. Density, N, is a function of position, r, relative velocity, q, and the projection of q onto r, r = x + y + z (121) with,

q. r = q~x + qyy + qzz = qx cos 0 + qy cos ¢ + qz cos X = qxa + qy~ + qz5 = qr cos 7

(122)

where, of course,

a 2 +/32 +52 = 1

(123)

Material velocity and acceleration in the x~ y and z directions are denoted, (124) Similarly, 05

09

O~

(12 )

Collecting all terms, there results,

__DNDt+ q f cos oONox ..ONoqu+5--ONoqz -x-- + cos ¢ONoy --z- + cos 50N]ozj - z - , + qaN = Q + x..ONoo. w - + y_z-;-_ qx 05 ON 09 O_NN+ qz ~z ON +q~ ox-~qx + qY oy Oqy Oqz

(126)

Cartesian bases are always least complex whenever evaluating forms of the differential acceleration terms, because bases vectors are fixed. No angular derivatives appear.

Transportequationsm movingmaterialI

29

Neutron Transport Approximations Legendre Expansion Focusing on 1D plane and spherical geometeries only, it's possible to expand the comoving frame transport equations using standard techniques of Legendre expansion of density, scattering, and external source terms. (5,17'45-46) The expansion recovers all the well known results, plus gives the explicit form of material motion corrections in the Legendre picture. To facilitate moment expansions, orthonormality of Legendre functions, P1, is expressed by,

f

_~(.)P~(~)aa = 2~ f l-~ Pz(~)Pt(.)d~ = (214~r + 1)

(127)

with useful recursion relationships listed alongside, (2/4- 1)#P~(#) = lP/_l(/_t) 4- (Z 4- 1)Pl+l (#) 2, dPz (~) (2/+ 1)(1 - # ) ~ -- l(l + 1)(Pl-1 - P/+l)

(12s)

First taking 1D plane geometry, N(x, q, 0), as before, with,

(129)

q. x = qx cos 0 = qxt~ we expand the density, cross section, interaction kernel, and external source in the Legendre basis,

N(x,q,O) = f i

2 m + 1 Nm(x, q)Pm(#)

vn~O

S(x,q,O) = ~

2k 4-1Sk(x,~

q)Pk(#)

k=O Oa(q, q') _ ~-, 21 4- 10gz(q, q')

03q,

Z_..,z=o 4~r

03q'

Pz(#)

(130)

Inserting the above into the hydrotransport equation, multiplying by Pn, using the addition theorem for Legendre functions, integrating over df~, and employing recursion in standard fashion, we find,

. DNn

(2n÷l)---b-~4-q

[( n 4 - 1 ) ~ONn+l 4- n ~ONn-I ] 4-(2n4-1)q~rN~ j

= (2n + 1)

4-~ I (n 4- 1) ~ ÷ n O N n _ l 0q

-~q,q ~,~aq + (2~ + 1 ) &

4- ( n + l ) ( n + 2) Nn+l q

O~ [ (2n + l)(2n2 4- 2 n - 1 ) ONn

n(n-1)Nn_ll q

(n 4-1)(n 4- 2) ONn+2

~(~ - 1) OY._2 ~(~ + 1)(2~ + 1) g . +q~-n---1) Oq + ( 2 n - - 1 ) ( 2 n + 3 ) + (n + 1)(n + 2)(n + 3)gn+ 2 _ n(n - 1 ) ( n - Z) g~_2]

(2~ + 3)

( ~ - : 15)

J

(131)

Similarly, in 1D spherical geometry, N(r, q, 0), repeating the expansions and steps in the planar case, with

q. r = qr cos 0 = qr#

(132)

B. R. Wienke

3O we recover,

(2n+ l)-D-~ +q I(n+ l ) ~ r + l +nONn-ll

f Octn , ,

,

= (2n + 1) j ~q,q N~q + (2n + 1)S~ +r

+

[L(n+ 1) ----w----oq ONn+l + nONn-1 ~

(o÷ !) r ~-

Lq

+

(n + 1)(n + 2) q

(-2n--~-~ ) (2 - ~ 3~

Nn+I

O~-

n(n-

1)Nn-ll

q

J

( 2 n + 3)

n(n - 1) ONn-2 n(n + 1)(2n + 1) +q (--~n-- 1) 0----~ + ( 2 n - 1 ) ( 2 n + 3 ) Nn

~(~- 1)(~- 2) + (~+ 1)(n+2)(n+3) (2n + 3) Nn+2~-~n -- i i U,-2 +(2n+

1

0q

1)qSON~ Oq

_

(133)

In both of the above 1D geometries, it is further stipulated that N - i = N-2 = 0. All terms on the right hand side, excepting the first two, result from the velocitygradient contribution to the static transport equation. Remaining moment terms are well known for the corresponding geometries.

Diffusion Expansion The above sets of Legendre moment equations can be closed in a diffusion approximation in low order truncation, that is, the so called P1 truncation. More generally, for n = 0, 1, 2, 3 .... , the above equations are coupled to n i 1, n =t=2, rendering exact solution difficult at best. For a given value of n, there are n + 1 moment equations involving n + 3 unknowns, No, N1, N2, ...Nn+l, Nn+2. To reduce the number of unknowns to the number of equations, the Pn scheme can be invoked by writing for given order of approximation, n,

ONt ONl . . . . Ox Or

0

l>n

ONt --=0 Oq

l>n

Nz = 0

l >_ n + 1

(134)

The above can then be used to truncate the coupled Legendre moving material set of equations. The particular case, n ---- 1, called the diffusion approximation, (5,17,31) is well known, and widely exercised in transport applications. In such truncation two coupled moment equations result. The first charts the spatial variation, while the second consistently defines a diffusion coefficient for the first. Consider both in the 1D planar and spherical pictures as above. In the P1 scheme, in 1D plane geometry, we obtain,

DNo ON1 /Ocro , , , D---~-+ q-~x + qo'No = -- -~-q Nodq + So -+q2

O(q2N~) Oq

1 0:~ ONo -3q-~x Oq

(135)

DN + q--~-x ONo + 3qcrN1 = 3 f O ~q, ~ , q Nld , q, + 3S1 3---~

+x~

Ok (30No 2 gl + 3q~x \-~--~-q + g ~ - ]

(136)

An important ingredient in a diffusion approximation is Fiek's law, (1'4'6'1°,17'19'23) which relates the current, J = qN1, to the scalar flux, ¢ = qNo, via J = - D - ~ _9_ OX

(137)

Transportequationsin movingmatermlI

31

for 1D plane geometry. Accordingly, it follows,

Oq

Ok

0x

0x

( ONo Ok) J = - D kq--~x - No-~x

(138)

and, dropping velocity derivatives of N1, the coupled set can be recast,

Dt

q~x D \ -~x =

q -~x

+ qaNo

f Oao , , , 1 Ok ONo J ~q,q Nodq + So - 3 q'ox Ox

(139)

3DN1 qCON° ./" Oal , , , Dt + -~x + 3qaN1 = 3 _ -~-q,q Nldq + 3S1

(14o)

From Fick's law and the second equation, and taking $1 and N1 small, D = (3a - 3#al) -1

(141)

for p the average scattering angle, and,

qNl#0.1 =

f Ocrz , , 1N, Ok -5-(q,q Nldq + ~ o ~

(142)

Similarly, in 1D spherical geometry, there results,

-DNo -+ Dt

q O(rN1) t - + q0.No = f ~00.0 - q t Nodq + Bo r Or -t q2

O(q2N1) aq

1 O÷ONo 4 ~ ONo 5q~r-~q + -3q-~ Oq

(143)

00"1 q' Nldq t ' + 3S1 3 DN1 - ~ ÷ q-~rON°+ 3qo-N1 = 3 .,'f -~-q,

+i~ONo+3q~r(3ONo -~q

2NI~+3qf(ON1

\g--~q +g-q)

k ~qq

30No 5 aq

2N1~

(144)

5 q )

Retracking the steps in the diffusion approximation in plane geometry, for 1D spherical geometry, it follows, for J = qN1 and ¢ - qNo as preceeding, J = -D-~-r~

(145)

with,

Oq O÷ Or Or / OYo

0÷)

J = - D kq---~r - No N

(146)

so that the spherical set is recast,

DNo Dt

q O [rD(ONo r Or \ -~r

N_o 0÷'~ ]

q Or ] ] + q0.No

f00. 0 , , , 1 0~0No 4 ~0No = !J 3-~¢q N o ~ + So - ~qK~--~-~ + 5 q ; 0q

(147)

B. R. Wienke

32

DN1 + q--fir-tON°÷ 3q~N1 ----3 ,Jf 3~

Oal q ,~l'qla , ,q , -~ -~q,

3S1

(148)

Again, from Fick's law, the second equation, and taking $1 and N1 small, D ----(3(7 - 3~al) -1

(149)

, , + -~ 1 No-~r 0~ qNlflt(rl = f 0 ~- ~l - q , Nldq

(150)

with,

In a moving material picture, the diffusion approximation gains additional motion terms, and the average scattering angle is modulated by the material drift derivative. The fixed background result is recovered in standard form when setting all material velocity and acceleration terms to zero. As in all diffusion approximations, though, for any particle, highly anisotropic material motion, sources, and scattering cross sections render the diffusion approximation inadequate. Full transport solutions are requisite,(27,~1,4°,42) basically numerical solutions. Neutron transport codes (Ts-sl) date their origins to the early days of high perfromance computing. Codes are standalone and embedded in multiphysics packages, coupled to hydrodynamics, radiation transport, thermonuclear burn, and shock driven phenomena. Moving material effects discussed are important transport issues resolved numerically in 1D, 2D, and 3D applications.

PHOTONS

Photon Properties

Photons are spin 1 particles obeying Bose-Einstein statistics.O4,al) They move with the speed of light, c, and have zero rest mass and charge. Photon interactions are short range, and cross sections are pointwise, as with neutrons. Photon interactions with matter are electromagnetic in the schema of strong, weak, electromagnetic and gravitational forces. Because photons have zero rest mass, dynamics are totally relativistic, and the photon transport equation is nonclassical. This holds for high or low energy photons. Photon transport equations, and solutions, were a focus of scientific activity in the early 1900s, spanning the development of quantum mechanics to astrophysics in scale. Photons weren't discovered as such, but their particle nature was postulated in the early 1900s. Light has been around since the dawn of creation, big bang or otherwise. Prior to 1900, light was considered a continuous electromagnetic wave. Corpuscular particle attributes awaited later experiments based on the fundamental nature of particle interactions, something well beyond wave behavior only. In the early 1900s, it was also shown that a purely wave nature of light would lead to a universe that was incinerated by high frequency radiation - the so called ultraviolet catastrophe. The particle and wave nature of light is also termed complementarity. Radiative Transfer Equation Moving material effects for photons are important in high velocity regimes, that is, when v/c > 0.10. But, since the speed of light is constant, relative velocity treatments do not apply to photon transport. Effective cross sections, requiring relativistic Lorentz transformations, are needed for moving material radiative transfer applications. Plus the situation becomes more complicated because Lorentz transformations need be applied consistently to all terms in the transport equation, not just the cross sections as seen in the neutron case. Before taking up Lorentz transformations and application to radiative transfer, we first consider the form of the transfer equation, and frame invariance.

33

Transport equations m moving material I

Eulerian Frame

We take another tack in deriving the radiative transfer equation, (2'6's'19'23'31) looking at temporal, streaming, absorption, outscatter, inscatter, and emission terms in a balance statement. The radiative intensity, I, is an energy angular flux, a function of position, r, photon frequency, v, and direction, ft, as in the neutron case. With photon intensity, I, the time rate of change of photons, A T T , in a volume element, AV, is, OI A ATT = -~ V

(151)

with the volume element outside of the time derivative, and fixed in space. In same element, AV, the streaming rate of photons in all spatial and angular dimensions and directions, A S S , is simply, ASS = c. VIAV

(152)

= c~. VIAV

since photons stream with speed c in all directions. The absorption rate, A A A , depends on the intensity, I, and the absorption coefficient, an, A A A -- c a a I A V (153) With differential scattering coefficient, #s, the outscattering, AOO, and inscattering, A I I , are given by,

AOO=cfddfda'.s±AV=casZAV AII=c

du'

dft'

(154)

-~7 # s I A V

Differential scattering coefficients, #s, and total scattering coefficients, as, are simply connected, I

or8 --

t

( 155 )

#sdv d ~

The term u / J in the inscatter, A I I , conserves photons, since the intensity, I, is proportional to the photon density times the frequency, v. For photon source, S, including external and internal emission processes, A E E , is, AEE = SAV (156) Thus, adding up all terms, (157)

ATT + ASS + AAA = AII - AO0 + AEE

we obtain, dropping the volume element, AV, common to all terms, 10, -

cot + ~'VI+aI-=S+ -

-

dr'

dn'

Vv

m,1 ,

(158)

for quantities defined in the fixed frame, and interaction coefficient, a, the sum of absorption and scattering coefficients, cr = aa + a~ (159) Lagrangian Frame

In a lagrangian frame, we adopt the same approach, and obtain the same equation. This comes as no surprise, since the speed of light is constant in all frames, and interaction terms are thus inwriant. In the moving frame, the volume element changes in time, so we note, along the flow line, A F F , the coupled temporal and streaming derivatives, AFF=

d(IAV) OI OAV OI --~ = - ~ A V + I -~ -- - ~ A V + c n .

VIAV

(160)

All other terms are unchanged, as in the fixed eulerian ease. Accordingly, as before, AAA = CaaIAV

(161)

34

B. R. Wienke AOO=c/du'/d~2'#JAV=c~IAV -~ #sI A V a~ =

#

'

~dv d ~

,

(162) (163)

AEE = SAV

(164)

AFF + ASS + AAA = AII - AO0 + AEE

(165)

Again, adding terms, we obtain, dropping the volume element, AV, common to all terms,

-cl O -o -tI + f t . V I + a a I = S +

/

du'

/d~,¢u,

[ ~ ) , , ,± ,

(166)

with all quantities now given in the moving frame. Stimulated Emission The radiative equation is a classical equation of transfer, based on classical physics concepts. However, there is a more to a complete representation. Quantum mechanics requires that the scattering and emission sources are enhanced by the presence of photons in the final state. (13-14'1s-19). The modification to the radiative tranfer equation is easily written down, with the stimulated emission terms simple multipliers of the source and scattering integrands,

O,(u)~ -1 0OtI + ~ " V I + aa I = / f ddu'

#8I' ' ( 1 + 2--G~v3/c2I

-fdu'fd.Q'#sI(l@~)+S(l@2~u~)

(167)

In most applications, the induced contributions are neglected. But we carry them along for completeness of development. At high photon energy or low temperature, effects of stimulated emission become more important, as seen in the exponential term of the Planckian distribution.

The manifestation of the induced terms can be seen more simply in terms of the Bose-Einstein statistics satisfied by photons. If ~r represents the probability of single photon emission or scattering, and n represents the number of photons in the final state, the actual probability induced by photons in the final state, H, is n =

+

(16s)

For electrons, neutrons, and neutrinos, obeying Fermi-Dirac statistics, the exclusion principle (13) holds and a minus sign appears in the above, with n = 0, 1, only, H = ~r(1 - n)

(169)

Additionally, with elastic scattering for fermions and bosons, where energies and frequencies do not change, the induced terms all vanish. Thermodynamic Equilibrium The source, S, usually denotes photons arising from spontaneous atomic emission, and is treated by a local thermodynamic equilibrium (LTE) assumption, (6-7,10-12,19,23,29) firSt introducing a corrected absorption coefficient, ha, Y c2B = (1 +

S -- naB

(170)

Transport equations in moving material I

35

whereby, at complete thermodynamic equilibrium,

.

-all

B=

62

exp

~-~

-1

(171)

The exponential term multiplying aa measures the effective decrease in absorption due to induced emission, and thus corrected absorption coefficient for T the absolute temperature, Above, B is the Planck blackbody distribution and at equilibrium, the radiation field, I, must equal the Planck field, B. Consequently, the number of photons inscattered must then equal the number of photons outscattered, yielding a detailed balance condition for the scattering kernels, 1

c2B

Furher, at complete thermodynamic equilibrium, ~ ( B - X) = 0

(173)

But even if local thermodynamic equilibrium is not invoked, the form of the radiative transfer equation is written using the foregoing, 10I 7 0-7 + ~

+

du'

V I = ,~o ( B - 0

dFl' ( 7 ) # s I

1+2hu3 j

in which case B is not the Planckian blackbody function. However, the L T E assumption is widespread in radiation hydrodynamics analysis because of the vast simplification it introduces, that is, thermodynamics enters the analysis. The form is useful for making further approximations. Lorentz

Transformations

Since the speed of light is constant in all frames, the radiation transport equation is invariant (same) in all frames (eulerian, modifed eulerian, and lagrangian). Obviously, the relative velocity of light, and the absolute velocity are always the same, that is, c, the speed of light. Space and time are thus expanded or contracted by this fact, via Lorentz transformations. It's useful to list them at this point, because they will be employed later. Two four vectors, Vu, and, Uu, moving at speed, u, with respect to each other (in coordinate rest frames attached to the vectors) along their mutual z-axis, transform covariantly, (3,18 - 20,23,52,53) V. = L.~.U,.

(175)

that is, Vu = (171,V) = (171,V2, V3,174) and Uu = (U1, U) = (U1, U2, 0"3,U4), where Lu~ is the Lorentz transformation matrix, so specifically,

v2

0

1

0

0

u2

V3

=

0

0

1

0

U3

V4

-sinh ¢

0

0

cosh ¢

U4

for, U

tanh ¢ =/3 = C

(176)

36

B.R. Wienke 1 cosh ¢ -- -y = (1 - u2/c2)1/2 -

1

(1 -/3u)1/2

(177)

and c the speed of light. The four vector components span time-space in the usual fashion across the manifold. The scalar product of two four vectors is a Lorentz invariant in all frames, taking particle energy-momentum, p~ = ( E / c , p), as example in its rest frame, p.p.

= p.g.vp.

_

=

+p. 2

=

( 0 0 0)

(178)

for rest mass, m, and metric tensor, g ~ , defined,

g~" =

0 0 0

-1 0 0

0 -1 0

0 0 -1

(179)

as standard t i m e l i k e metric, and g ~ = g,,~. Covariant and contravariant four vectors differ only in the sign of the space components. The time-space four vectors, (ct, r) and (c¢o, ro), transform under L as, t=~(to-UZo/C2),

x=xo,

Y=Yo,

z=~/(Zo-Uto)

(180)

Similarly, the energy-momentum four vectors, ( E / c , p) and ( E o / c , Po), are related by, E ---- ~(Eo - Upoz),

Px -- Pox,

py = Poy,

Pz = ~/~Voz - u E o / c 2)

(181)

The power-force four vectors analogously transform, that is, ( Q / c , f) and (Qo/c, fo), Q = 7(Qo - Ufoz),

f~ = fox,

f y = for,

f z = 7 ( f o z - u Q o / c 2)

(182)

Four vectors depending on derivatives in time are constructed from the proper time, ~-, according to the prescription, d d d-7 = ~ 7 (183) with t the local time in the rest frame. Hence four velocity, v**, is defined in terms of the four position, r~ ----(ct, r), above, dr~ = ~ ( c , v ) vl* = dT

(184)

dvu -- 72 ('y2va/c2, 7 2 v a v / c 2 - a) a~ = dT

(185)

vlzv p" = c 2,

(186)

and acceleration, au, follows,

We see, v , a v" = 0 ,

a , a "~ = ,74a 2

so four velocity and acceleration are orthogonal. The four force, f~, written above, is generalized, f , --_ dP,d~ - m-~rdvu

(187)

with components, f~ = ( Q / c , f). Coupled to these transformations are differential products of energy-momentum, and time-space, in different frames. Some useful products follow from the identities below, using the four vector transformations above in time-space and energy-momentum. dpodro -- 0(po, rO)gpd r = J d p d r

0(p,r)

(188)

37

Transport e q u a t i o n s m m o v i n g m a t e r i a l I

with ff denoting the relatisvistic Jacobian. Since p and r are independent, we obtain from the Jacobian determinant, f f = Opo Oro Opo Or__ 2 _ Eo E _ 1 (189) Op Or

Or Op

E Eo

since, cqpo __ Eo Op

Cqro

--

E '

Or

--

E

(190)

Eo

Consequently, any momentum distribution, ¢, is also invariant across the Lorentz frames, ¢(Po, ro) = fl~b(p, r) --- ¢(p, r)

(191)

Evoking similar transformations, it is seen that, drodto = drdt,

(192)

dpodEo = dpdE

using the four vectors, (E/c, p), and, (ct, r), plus the Lorentz transformation, n. Local Material Transformations

The equations of physics are invariant under Lorentz transformations. (]s-19,23,Ss,Ss) We saw that above. In words, the equations are form invariant in arbitrary, nonaccelerated frames provided all quantities appearing in the equations are local and measured in the frame. In many applications, however, quantities are know in one Lorentz frame (usually the rest frame of a moving material or fluid), and observers in another fixed Lorentz frame (spectator frame) measure field variables and related quantities in terms of their local frame equations. It is then necessary to transform rest frame quantities to the spectator frame using Lorentz transformations. In the transformed equations, both variables and operators are affected. (19,23) Accelerations pose problems for Lorentz frames, unless one works in the framework of general relativity. (is) General relativity is obviously a very complicated, but unnecessary, approach for radiation transport and radiation hydrodynamics. In continuous media, the assumption is made that local frames (within mesh spacings, say) are Lorentz frames, and that very abrupt changes in accelerations and velocities do not occur across local frame boundaries. (12'52'53) If accelerations are small and continuous, this assumption of local invariance is a good one, plus a good low order approximation to general relativistic dynamics, as dynamics smoothly evolve in time-space. If the background material is moving with velocity, u, the observer in the spectator frame moves with velocity, - u , relative to the material rest frame. Defining as before, u /3 = c '

1 V = (1

u2/c2)

-

1/2 =

1 (1 - ~2)1/2

(193)

plus the angles, 0 and 0', ~'~ • U ~

I$C08 0 = O~U,

~'~ • U ~ U C 0 8 0 j ~

0{~

(194)

with, = (1 - ~ . u / c ) = (1 - ~ / c )

~'

= (1 - n ' . u / c ) = (1 - J ~ / ~ )

= (1 - ~ e ) = (1

-

a'/S)

(195)

we can detail the transformation properties of all field variables. Denoting rest frame quantities with subscript o, spectator and rest frame terms in the radiation transport equation are related by, frequencies first, u = (eV)-lVo,

u' = (e'V)-lu'o

(196)

for scattering angles, Ft = ( e 7 ) - 1 [ ~ o

+ V(V + 1)(e? + 1)u/c]

~'~' = (£'"~)-1 [~~o -~- ')'(V - ~ 1)(c'V + 1)u/c]

(197)

and specific intensities,

I = (~;)-~Io,

I ' = (~'~,)-~g

(198)

B.R. Wienke

38 with sources,

B = (e,,/)-3Bo

(199)

oo = (~)~oo,

~ = (~)~o

(200)

#s ="7#so,e

#s ="-~#so

(201)

S = (~)-~So,

and cross sections for scattering and absorption,

with scattering kernels,

and differential scattering angles, df~ = (e~,)2d~o,

t

2

dgt' ---- (e 7) df2o

(202)

The full transport operators transform as, (203) and the scalar products of scattering angles are related by, (1 - Ft. Ft') = (ce'72)(1 - ~o" ~'o)

(204)

Substituting the above into the spectator frame equation yields the rest frame equation as required, and as seen earlier in the eulerian and modified eulerian pictures.

Exact (u/c) 4 Expansions The material velocity terms, c and % and their products can be expanded to order, (u/c) 4, in each term as follows,

(eT) 2 = (1 - 2aZ + a2f~2) + (1 - 2afi~ + a2Z2)Z 2 + (1 - 2aZ + a2/~2)Z4

(206)

f~2 (~,)3 = (1 - 3aZ -[- 3a2f~2 - aaf~3) + [~ - ~9 a ~ -t- 9a2~ 2 2 - -3a;~31 -

(207)

+

c =(l+a~

+a

s

/~ + a

~-

~

+ ---g--

÷a'4f~'4)

--

-

(1+

~

+a

(2os)

~

+c~ ~

-

+

+

c~

/~ )~

~ ' ~ = (1- a Z - J Z ' + ~ZJZ') + (1- ~ Z - J Z ' + ~ZJZ')Z2 + (1 - ~ Z - J Z ' + ~ZJZ')Z 4 ( ~ ) - 1 = (1 + ~Z + ~2Z2 + ~3Z3 + a4Z~) _ 0~2~2

+

+ --~

(209) (210)

(211)

+ --Y- +

a3~ 3

(212)

(ET)-2 = (1 + 2a~ + 30/2~2 --~4a3~ 3 + 5o~4~ 4) - (1 -~- 2a~ + 3a2f~2 + 4a3~ 3 + 5a4/~4)f~2

+

[3 + 9~_ + - 9~2~ 2 + 15oz3~ 3 + 4514~4.1~4 -y~

(213)

(214)

Transportequationsin movingmaterialI

39

where, -I
0_<;~_<1 Of particular interest are the terms S, B, as, and aa since they are tagged in the fluid Crest) frame. To order u2/e 2, we find, for the sources, s = (~/)-2So = so

1 + 2~

+ (3o~2 - I ) ~ -

B = ( ~ / ) - ~ B o = Bo

1 + 3~

+

2

~

(225)

and also for the absorption and scattering cross sections,

~rs=

(e~,)aso = ~ o

tTa = ( ~ ) ~ o

1 - a u- +

c 2 c2J

= ~o [1 - ~ u +

-2"~1u2 t

(2i~)

Similarly, the relativistic Doppler shift takes the form, v - vo = Vo [ ~ cu + (2o~2-1)u~1

(2i7)

Customarily, Bo and So are expanded in a Taylor series about the Doppler shift, and Lorentz transformations carried out to order u/c, for applications. This is a standard low order approximation that is employed widely. (6'i1'19'2°'2s'75) While the frame transformations given above are Lorentz invariant, Taylor series expansion bridging rest and moving frames are not generally Lorentz invariant, with inconsistencies increasing with material speed and anisotropy. Photon-Electron Scattering Photon scattering off an ensemble of moving electrons(52-5s) is an important ingredient in transport calculations, so called Compton and inverse Compton scattering in fundamental dynamics and basic interaction picture. The ensemble can be at rest, moving uniformly, or distributed statistically in a Maxwell-Boltzmann or Fermi-Dirac temperature correlation, plus combinations of all. Consider some as follows. A relativistic Maxwell-Boltzmann distribution (59) at temperature, T, takes the normalized isotropic form,

O(p)=(4~rm2kTK2(m/kT) ) exp

[-(P2+m2)V21#~ j

(218)

with momentum, p, electron rest mass, m, and temperature, T, in the same energy units (not carrying factors of e, the speed of light, k, Boltzmann's constant, and h, Planck's constant), standard in relativistic mechanics, and for K2 the modified Bessel function. Moments are written, < p~ > = 4~r

@(p)pn+2dp

(219)

and take general form,

>= L

(220)

)J

with K modified Bessel functions, and F the complete Euler function. Specifically,

< p >=

m \ v:m /

K2(rn/kT)

(22i)

B.R. Wienke

40

and,

(222)

K2(m/kT) In low temperature limits, m / k T > > 1, it follows that,

~rkT 1/2

m2kTK2 (rn/kT ) ~ m2 ( ~ )

lim

m/kT---*oo

m

15kT]

exp (- -ff-~) [l + --~-m-mj.

(223)

and, for p/m < < 1,

pm /2]im _O . (p

--F-m2)l/2,-~,m(l+~-~)

(224)

so that the nonrelativistic form obtains,

1 ~ 3/2 (I)(p) = \21rmkT]

_p2

exP(2--m~)

(22 )

Similarly, in the low temperature limit, the usual moments obtain,

p > ~ m \-~-m ]

+ -~-mJ

(226)

--~--mJ

(227)

and

A fixed ensemble of electrons is denoted, • (p) ----5(p)

(228)

and is typically the low order approximation used in radiative transport. If the electrons are moving uniformly in space, • (p) = 5(p - Q) (229) with Q drift momentum. If drift is superposed on thermal Maxwellian motion the distribution is skewed,

{

1

[-(p+qT]

(230)

but only about Q, and the normalization is the same as the nondrift case. Effective total and differential cross sections for radiation transport require folding over any electron distribution, O, in same fashion as seen for neutron transport. The effective differential cross section for photon-electron scattering is written, (25,26) !

#s

0a:

e - u)~)(p )K(u~, %, Yt,, %)d3p

OvOg~ fJ 5(e' +

with, as before, I

as =

f

!

J #sdudf~

(231)

2 (23)

for incident photon energy, u', background electron energy, e', final photon energy, u. and final electron energy,, e. Using four vector photon and electron invariants,04,53,ss), that is, the energy-momentum four vectors, % and eu for the initial and final electrons and uu and uu for the initial and final photons, we have, recalling that photons have equal energy and momentum apart from the factor c 2, t

I

t

u~ + % = u~ + %

(233)

Transport equations in moving material I

41

so that, t

i

t

t

m 2 ~ e~2 A- ~,2 -4- tj 2 -4- 2 p ' u ' c o s a' -- 2 p ' u c o s c~ -- 2 u ' u c o s 0

(234)

cos a -= cos a ' c o s 0 -4- s i n a ' s i n Ocos ¢

(235)

and with, where a ' and a are the angles between incident electron and incident and final photons, 0 is the angle between incident and final photons, and ¢ is the azimuthal orientation of the scattered photon. Taking the scalar product of incident electron and photon four vectors, and using four vector conservation, we also obtain, t

t

!

/~

__

e'u' - p ' u ' c o s a ' = uu'(1 - c o s 8) + e ' u - p ' u c o s

a

(236)

with the same angular definitions. In the above, in the electron rest and photon rest frames, respectively, %c

----e,e" ----m 2,

u',u'"

u, uu ----0

(237)

and, additionally, the set closes with, ,

I

t

i

i

r

i

v~e ~ = v~e ~ = e u - p u cos a

% u ' " ----uu' (1 -- cos 0) = e,e'" - m 2 t

t~

e , u ~ = e'u - p ' u c o s

(238)

a --- % u

The Lorentz invariant interaction term takes four vector form, (59-61)

K = -~-nr~(u)( ~ 2=y.m2 ~e. )[( =

--.4.1m2,12 -1.4.=+=u"c'" uue'uI

(239)

or, simplifying


it

4 1_os0 2

2m21 cos0

K = - 2 - Leu'(e' - p'cos a ' ) J [ (e' - p ' c o s a')2(e ' - - p ' c o s a) 2 -- ( e ' - - p ' c o s a ' ) ( d - - p ' c o s a) u'(e'-p'cosa')

-;'cos

u(e'-p'cos~7)l

(240)

+ J(c' -#cos

with ro the classical electron radius (2.8 x 10 -13 cra), and n the electron (spatial)number density. The 5 function permits one integration over du trivially, but remaining integrations over d3p and d~t are formidable. Numerical techniques are requisite to generate the total effective cross, ~r~. Computer programs (62,63) are available to produce momemtum integrated differential and total photon electron cross sections. Turning to the effective differential cross, a number of interesting cases result for various forms of the background electron distribution, ~. Integrating over the 5 function, the effective differential cross section presents a form that is convenient for numerical approximations,(3°,31) recovers standard Klein-Nishina, Thomson, and Bjorken results, (14,54-56) and also extends them with temperature dependences, Oa'e = Of~


~ e' ( e' - p cos a' )

Kd3p '

and, introducing the dimensionless parameters, a, al, and a2, t 1~777, ~

tzlm

--p'cos a = ~m

6.

(~2

1)U2mcos a

(241)

42

B.R. Wienke !

t

a2m = e -p

cos a = a m -

(a 2 - 1 ) l / S m c o s a

(242)

the Compton-like expression results from the earlier four vector identity, a l m a ' = uu'(1 - cos O) + a 2 m v

(243)

T h e equation above permits upscatter and downscatter, unlike the pure C o m p t o n case where only photon downscatter is kinematically possible. For al > as, upscatter occurs, while for al _< as, only downscatter is , p, possible. For a = al = a2 = 1 that is, e = m and = 0, the C o m p t o n scattering law is recovered. Useful forms below beyond the stationary Klein-Nishina expression have been detailed (6°-62) and employed in both deterministic and Monte Carlo transport applications. (64-a6) Effective L o w Energy K e r n e l

Rather than perform the three-dimens!onal integration over (I)(p'), we can replace the kernel, K , with appropriately averaged expressions for < p > and < cos a . Energy and m o m e n t u m follow easily, a m = < e > = [< p'S > +m211/2 =

+ 1

\K2(m/kT)J

l

(244)

so that, alto

:

am

"~ ( a 2

-- 1 ) l / 2 m c o s ce' >,

a2m = a m + (a 2

!

1)1/2tacos a

t

cos ~ = < cos a > cos O+ < s i n ~ > s i n 0cos ¢

(245)

Substituting into the full Lorentz kernel, yields,

'

0~_ 0~

~ 0~ 2an2

(5)

I

~ (l=cos_e) s [ (al~2) s

cose)+a~.

(1 -

ala2

'+

a2tJ

,] alv

(246)

Experience has shown that the above provides a relatively simple and accurate approximation for low energy photon-electron scattering. It has simple properties. In the (low energy) limit, (a 2 - 1) 1/2 --+ 0, we also have a ----- a I = a 2 , SO that,

o n = 2a~

-

~

-

a~

a ~-

~

+ -~ + 7

(247)

Effective Isotropic K e r n e l

Except for large differences between u' and T, we might reasonably expect t h a t the incident photon views the electron distribution isotropically as far as its average scattering angle, so t h a t we take, < cos

O/

>= 0

(248)

and, from the earlier relationships, al/~t =

a 2 m ----a m -

a?n,

(a 2 - 1 ) l / 2 m c o s a (249)

cos a = s i n O cos ¢

providing a useful and reasonable expression for the differential cross section, and subsequently, the total cross section after integration over d~. Effective K l e i n - N i s h i n a K e r n e l

Using the low energy expansions, the effective Klein-Nishina cross section can be t e m p e r a t u r e corrected taking al = a2 = a. Accordingly, from the above, 8at

2a 2

L

a4

a2

+ -v + ~

(250)

(1 - cos 0) 5 - 2(1 - cos o) + -~ + ~

(2~1)

If a = 1, the stationary Klein-Nishina expression results, on

=

2

Transportequationsin movingmaterialI

43

Effective Thomson Kernel For low incident photon energy, u' ~ 0, we have from the foregoing equations, (u) ~m lira --7 = lim = 1 ~'-~0 \ u / ~'-~0 u'(1 - cos 0) + ~rn

(252)

The temperature corrected Thomson cross section drops out dependences on photon energies in the KleinNishina case,

0~'~

nr~ [ ( 1 - cos O)2

O~ = ~

2(1-cos0)

t~4

~2

+2

1 (253)

Effective Bjorken Kernel In the high energy regime, u' --+ oc, only the second last term of the Klein-Nishina result survives, and the modified Bjorken cross section obtains, 0a'~ __ nr2o ( u ) 0~ 2~ 2 ~v

(254)

The Bjorken limit is well known in high energy physics applications. (14'31) High energy particles like photons, mesons, neutrinos, relativistic electrons, and neutrons in the Mev range all tend to pass through matter with minimal scattering and absorption. Neutrinos stream through galaxies, virtually unperturbed. High density, gravitationally collapsed stars, are needed to scatter and absorb neutrinos copiously.

Total Scattering Cross Sections All of the above representations can be integrated for the total cross section, a'e" For comparison, we note that at low temperature, T --* 0, and with a ~ gl ~ au,

'

('

which is the effective Klein-Nishina cross section, after defining a,

//' c~ = - m

(256)

Taking t~ = 1 gives the standard (stationary) Klein-Nishina result,

ae ~2~rnr2o 2 a 2 +

] In (l + 2a) + (~_-2-~)2j

(257)

The more detailed effective kernels can similarly be integrated directly. In these cases, ~1, a2, and ~ are independent, and give finer grain to the expressions. In the same limit with u' -+ 0, all sets approach the Thomson limit, (54) 8 2 a~ .w. ~nr o (258)

Radiation Hydrodynamics The term radiation hydrodynamics implies inclusion of a radiation field in the fluid equations (s,12Jg,52,53) For large radiation fields, contributions from the radiative momentum and energy are significant, and need be included in fluid balance equations. Radiative contributions enter as source terms. Furthermore, for large fluid velocities, relativistic corrections, and the attendant Lorentz transformation properties of radiative and hydrodynamical field variables and operators, need be considered. But, consider first the radiative source terms. Radiative terms impacting the hydrodynamical mass, momentum, and energy balance equations are the radiative energy density, H , radiative energy flux, J, radiative momentum flux (rank 2), 7~, and radiation stress tensor (rank 2). P. They are defined as follows, from the specific intensity, I,

B. R. Wienke

44

with the cross product a rank 2 tensor (dyadic) in Ft. All are energy integrated angular moments of I over f ~ , that is, scalar, vector, and rank 2 tensor for n = 0, 1, 2. These are added to the corresponding material terms in the balance equations. For fluid velocity, u (as before), mass density, p, material internal energy (above rest mass energy, pc2), e, pressure, p, and heat, w, the radiation hydrodynamics equations are written in the eulerian picture, mass continuity first, 0~ + v.(pu) = 0

(259)

Ot

while momentum conservation is written,

-~ pu + - j

+ v ; + v . ( p u + p) = f

(26o)

and the energy balance statement is,

0-~ -~pu + e + H

+V.

~pu + e + p

u+V-J=w

(261)

In the modified eulerian picture, the continuity equation is as before, DP-1 - V . u = 0

(262)

P Dt while momentum balance requires, p~-~ u +

+ Vp + v .

p - ~

= f

(263)

with dyadic notation, J u = .F, and energy conservation takes the form, pN

~2 +

+ v . O + pu - H u ) = w

(264)

Again, the picture is slightly different, but the physical content is identical in the two equation sets. Above, the continuity equation has been subtracted from the momentum and energy equations for simplification. Relativistic hydrodynamics is concerned with transformations of fluid rest frame quantities to the spectator frame, in which the fluid equations are posed and defined. Just as in the radiative transfer equation. Under Lorentz transformations, the eulerian radiation hydrodynamics equations can be posed, with all quantities defined in the fluid rest frame but the equations posed in the observer frame (or laboratory frame). (7°-73) The continuity equation in the observer frame is given by,

O(~'p) + V-('ypu) = 0 Ot

(265)

for momentum conservation, 0"-t .y2

p+N+~.

~

+Vp+V..y2

p+~5+~_ /,/+7) = f

(266)

and for energy conservation, 0__ [ ~ 2 ( p a + ~ + p) - p + H] + V. [ ~ ( p a 0t

+ ~ + p)~ + J] = w

(26~)

Transport equations in moving material I

45

In the modified eulerian picture, continuity is written,

7P

D('yP)-I Dt

V .u = 0

(268)

momentum balance requires, (269) so that energy conservation reduces to, D

'PN k

1)p

+p)-p + H1 +

+ (o-

(270)

J

In the limit, c -+ c~, we have in the observer frame, 1 u2

i u2 lim 7 --+ 1 + - c-~ 2 c 2' plus, u2

u2

]Lz'n V 2 ~ 1 + ~ - ,

lim(v

2 - 1) --~ c-~-

and the relativistic equations collapse to the classical (nonrelativistic) earlier forms. The radiation hydrodynamics equations above are usually solved in low order u/c expansion of rest frame field variables, Ho, Jo, and 7~o. These relationships (23'72-73) follow from the previous section,

H = Ho + ~ U . Jo J = Jo + uHo + 7~o.u

1

= po+

(271)

In the comoving frame (rest frame), the radiation hydrodynamics equations are the same, but with subscript o on all variables. Such is the manifestation of Lorentz invariance. Equations can be solved in one frame or the other, but mixed frame solutions require Lorentz transformations. Such transformations tag the moving material properties to some order of u/c in applications. For radiation transport, moving material corrections become very complicated in high speed flow regimes.

Fick Diffusion The complexity of the are most easily made in corrections. A commonly In the diffusion regime in

coupled equations often requires simplifying assumptions. Simplifying assumptions the comoving (rest) frame, where underlying dynamics are not clouded by u/c used simplification is the radiation diffusion regime, one useful in opaque material. the rest frame of the fluid, c

Jo--

3GoVHo

(272)

with ao a characteristic distance for photon interactions (free path) in the rest frame material. When this approximation is made, the temporal behavior of the current, Jo, is assumed to be zero in all coupled equations. The immediate effect is that radiation in the comoving frame propagates faster than the speed of light, c, and particularly so in optically thin (transparent) regions of the material. Radiation streams faster than c because a c20Jo/Ot term gets droped out of the momentum balance equation under simple Fick diffusion above. Dropping the current derivative also changes the hyperbolic nature of the mathematical system to parabolic. Radiation angular distributions are also affected adversely by both assumptions. In streaming applications, this is very undesirable, and physically unreal. (51' 70- 74) Additionally, the simple representation of diffusion by Fick's law in the comoving frame is not preserved under Lorentz transformation to the fixed frame. Indeed, equilibrium in the rest frame does not translate to equilibrium in the fixed frame.(3,1s) Under u/c expansions, the radiation distributions are skewed in both energy and angle. This remains in both diffusive and streaming regimes, that is, in both optically thick and thin moving material.

B.R. Wienke

46

Flux Limited Diffusion Some of the problems associated with Fick diffusion in a comoving frame can be addressed with a flux limiter which enforces casuality in propagation speed in the moving material. (74-77) Again, this is strictly applied in the rest frame of the moving fluid or material. While somewhat artificial, this prescription works well in many applications. (23,29,70-77) Fick's law is recast in terms of a flux limiter, A, which enforces the streaming limit, and collapses to the diffusion limit in proper circumstance, cA Jo = - 3no VHo

(273)

do <_ clio

(274)

enforcing the condition, always. For opaque material, A --* 1, while for transparent (thin) material, A < 1. This also imposes casuality on radiation propagation in the fixed (observer frame under Lorentz transformation. In general, 0 < A _< 1

(275)

A number of approaches are useful in applications, all preserving casuality and recovering limiting forms within moving material backgrounds. Defining the relative radiation gradient, p,

VHo P = aoHo

(276)

in the statistical mechanics approach, A---- 3(2 + p) 6 + 3p + p2

(277)

while in assuming piecewise linear variation in angular intensity, F, we have, 6

A-- 3 + ( 9 + I 2 p 2 ) U 2 ' 3 A = 1 + p + (1 + 2/92)1/2 '

0_
3/2 < p < c~

(278)

Obviously, in the diffusion limit, p -+ 0, and A --+ i. In the streaming limit, p -+ oc, so A -+ p-1. In applications, p is obtained and updated iteratively. In highly anisotropic radiation transport, both of the above diffusion linked approaches break down.

Scaling Hydrodynamics and radiation transport do evolve on different time scales with regard to characteristics and observables. Dimensional scale lengths are also different and important, particularly when numerical solutions to coupled radiation hydrodynamics equations are sought. Basically, there are three characteristic length and five time scales: (23'71) 1. particle mean free path, A, which is the distance over which a particle travels before collision with another particle; 2. structural flow length, l, determined by gradients in physical field variables (like temperature, density, and pressure); 3. photon mean free path, 7, which is the distance a photon travels before it gets absorbed or scattered. For a fluid, or continuum, description of flow, A/l < < 1. Otherwise, a free molecule picture must be assigned to the flow. In the fluid limit, nonideal effects such as conductivity and viscosity can be quantified. When ~/l _< 1, the flow is optically thick, and when ~/1 _> 1, the flow is optically thin. There is an associated hierarchy of time scales too, with some interesting ratios:

Transport equations in movmg material I

47

1. photon time, t 7 = ~//c, the photon flight time over mean free path; 2. radiation time, tl = 1/c, the radiation flow time over characteristic structure length; 3. fluid flow time, t~ = I/u, the fluid flow time over characteristic structure length; 4. diffusion time, td = (1/7)2(~//e) = lZ/Tc, the diffusion time for a photon to random walk over a characteristic structure length; 5. relaxation time, tr = pev/4~e~T 3, the radiation relaxation or cooling time of material at temperature, T, with ff the radiation constant defined in the next section. Important coupled physical ratios include: 1. radiation flow coupling ratio, t.y/tt = "y/l, measures how tightly coupled radiation is to hydrodynamic flow. If the ratio is small, local emission-absorption processes dominate. If the ratio is large, retardation effects in photon transport must be tracked because casuality requires photon emission and absorption at different times; 2. thermal flow equilibrium ratio, t.y/t~ = (~//l)(u/c), measures how closely equilibrium in the flow is achieved. If the ratio is large, photons will be created in the same physical environment they were destroyed. If the ratio is large, thermal equilibrium breaks down, and photons tend to stream; 3. radiation response ratio, tt/tu : u/c, measures ability of radiation field to adjust to flow changes. If the ratio is small, radiation is quasi-stationary, and reacts to changes instantaneously. If the ratio is large, the radiation field is not static, and changes with flow; 4. diffusion response ratio, td/tu : (u/c)(I/7), measures how rapidly radiation can diffuse in the flow compared to how rapidly energy can be advected by the flow. If the ratio is small, the radiation field propagates in the static diffusion regime. If the ratio is large, the radiation field propagates in the dynamic diffusion regime. In shocked fluid flow regimes, if the ratio is close to one, then the radiation front propagates upstream a distance, l = (c/u)% from the shock. 5. Boltzmann number, /3 = (u/c)(tr/tT) = 4pc~u/~cT 3, measures the ratio of energy flux in the fluid to the radiation flux from a free boundary. If the ratio is small, radiation transport dominates energy transport by the fluid. If the ratio is large, energy transport by the fluid dominates radiation transport.

Eddington Generalized Approximation In diffusive transport limits, the radiation pressure,P, is linked to the energy density, H, by the Eddington relationship, obviously a scalar statement, (279) P = g1H More generally, in terms of the flux limiter, A, the relationship is recast, P = ~H

(280)

in a coarse grain sense. Within a flux limited diffusion aproximation of the earlier form, eA J = -~:VH 3a

(281)

the most general representation of the Eddington realtionship takes tensor form, ~7.79 = ~TAH

(282)

and presents itself for direct introduction into the radiation hydrodynamics equations. The same functional forms of the limiter maintain as detailed.

B. R. Wienke

48

Photon Transport Approximations Source-Absorption Expansion In the transport equation, the absorption coefficient, aa, and source-emission term, S = t~aB, must be obtained from the fluid frame quantities using Lorentz transformations, as detailed previously. First, taking the combination, S

1-F~

-aaI=n~(B-I)

(283)

and transforming from the fluid rest frame (subscript o), we find,

na(B - I) = eTn,~o

- I

(284)

with frequency transformation (relativistic Doppler shift) noted before,

Uo

(285)

The intensity, I, and frequency, u, are measured in the observer frame. Next, we expand the frame source, Bo, and absorption coefficient, a~o, in a power series about the moving frame frequency, y, by writing and remembering that the expansion itself is not Lorentz invariant,

Bo = ~

~ [ &'2 J (Uo - u) n =

r~O

~:o =

~l[O~B°lun(eT-1)~ ~ [ Ou2 J

ro ,.
f~O

(286)

r~--'--O

~ L o~,2 l (Uo- ~,)'~=

~

~ " L o~,~ j

changing frequency from 1,,oto u according to the relativistic Doppler prescription The product naoBo, reduces somewhat at complete thermodynamic equilibrium,

~aoBo = Crao

exp ~ kT ]

but in general, such simplification does not occur, especially with high speed material motion. Accordingly, the Taylor expansion is used, even though Bo and nao are not equilibrium values, and thus approximations can be introduced for the Taylor coefficients. Taking separate representations to order (u/c) 2, there results,

e;aB = ~a

"~ n,~ Bo 1 -i- --c q- ~ c

OBou ( u2 +N2~o

~

au ~

2c2 J

3a~u 2) -

+ 0~o~

(289)

and,

n~B = (eTn~o)B ~ B nao 1 - a___?_u+

c

0n~o v

+ Ouo

u9

~

au +

c

+ --~-vo2u

-if-d-j

t,,---c-~--]]

(290)

or, taking product inside the expansion,

+

O(~ooBol ~, ~ ~ ~

c

2 ~ 2 . h 02(,~ooBo)~,2 ~ ) + o~,2o k ~.:~ )

(291)

Transport equations in moving material I

49

plus, expanding both Bo and n~ and multiplying the expansions together to order (u/e) 2, .oS

=

r..,.oo

[(.aoBo)]I (2.. L-~-~]~-j 0.o, (.. (n.oBo)

~

3.2.2 .2)

1 + --c

-~

+ ~c

(292)

These Taylor series expansions are not intrinsically Lorentz invariants themselves, but are employed (7°-~2) in various analytical and numerical code applications. For u/c < < 1, the Doppler shift is small, and the Taylor expansion is acceptable. Diffusion Expansion As in the neutron case, the radiative transport equation can also be expanded in a diffusion approximation. First, the intensity, I, is given as a two term expansion of energy density, H, and flux, J, noting that H and J here differ from earlier definition by a factor of c, J H ÷ 3-~-[1" 41r

(293)

1 3(aa ÷ a s ) V H

(294)

I= plus the Fick statement for radiative transfer, J =

called the Eddington expansion. 0) Multiplying the transfer equation by 1 and ~, integrating over d~, and expanding the scattering kernels in Legendre moments, we obtain the coupled set, 1 0H c - - ~ ÷ V . J = na(4~rS - H) - aaH

+

.: (7).o..

c O--/÷ V H + (ha + aa)J = C2

+ -~J.

.,.

f

~

~

\~,

C2

,.}

with moments, #Is = 2~r

#is i/2y r

lJ 3 }

(295)

i (') du' f

+ s-VfJje.

f

] du

-~7 # l , J

~

\~,

,.}

d~#sPt(w)

(296) (297)

1

For anisotropic elastic scattering,

~s = az~(~ - / )

(29s)

with al expansion constants, the induced terms in the two moment equations vanish. Introducing Fick's law, coupling energy density and flux, J = -DVH (299) yields the diffusion equation with source from the first, while the second defines the diffusion coefficient, D, 1 OH _ V . ( D V H ) = na(47rB - H) - aaH + aoH' c Ot 10J 1 c 0---t+ V H + (ha + ¢a)J = a l J

(300) "(301)

D ,~ (3aa + 3aa - 3al) -1

(302)

with, The form is reminiscent of the neutron results, but more complex due to emission-absorption. For nonelastic scattering, the more general case, the above set can still be treated in a diffusion framework, but with modified diffusion coefficients.

B. R. Wienke

50

Asymptotic Expansion Asymptotic diffusion theory(139) uses the first of the diffusion moment equations, but imposes asymptotic limits on the the diffusion coefficient, D. This is tantamount to assuming that the photon intensity and absorption coefficients are slowing varying functions in a nearly isotropic media. Such is the case for infinite media, and for optical distances far from any boundaries. The mathematical constraints require, cr = O'a -+- O's

(3o3)

V. _/d~t~t~I = a D V H plus the Helmholtz condition, V2H

(304)

= a2cr2H

with c~ an exponential interpolating factor for intensity, I, at large distances, r,

(305)

I = • ezp (-aar)

and • the field interpolant. With these stipulations, the diffusion coefficient, D, takes asymptotic form, in terms of the scattering fraction, s, and interpolant, a, D =

(1 - s) O"s

8 ~

aa + as

lln

1 s = 23

(lq-a'~ \ 1- a J

(306)

This approximation folds motion effects directly into o"a and as in the definition of D, that is, the u/c transformations listed earlier. Note, ~ is sometimes used in place of o"a in the asymptotic relationships.

Equilibrium Expansion Equilibrium diffusion theory (2,19'6s) is another extension of Eddington's law. Assuming that the radiation intensity is almost isotropic and close to thermodynamic equilibrium, from the first moment equation, and the Planck distribution, B,

H=4~rB-

c2

exp

~-fi

-1

(307)

Plugging the result into the Eddington approximation, it follows, j=

4rVB= 3¢

-

47rOB, 3--~a--TVT

(308)

and, I

=

B

-

10B_ a~-~-q. VT

(309)

Once the temperature distribution is known, the specific intensity, I, is explicitly represented. The equilibrium approximation converts the radiation equations to a coupled set of temperature relationships, certainly valid when thermodynamic equilibrium ensues. Additionally, recognizing that the integral over ~ gives the Stefan-Boltzmann l a w (27-29'32'70-74), w i t h radiation constant, ~, 8~rak4

f0 °~ dvB = 47r C~T4,

~=

15h3c 3

(310)

and noting,

f fd.~-~ f = ~T 3 7~

(311)

Transport equations in movingmaterialI

51

it is easy to define a mean diffusion coefficient, D, such that,

D foeC dU~ T = foCC du( 3cra + 3crs - 3al )-l ~ T

(312)

Then, the equilibrium approximation takes temperature form,

J = -~cDVT 4

(313)

As before, moving material effects enter through aa and as. that is,

aa = e~/aao ,

as = e~/aso

(314)

with all other transformations as listed. Computationally, moving material effects most often require numerical resolution, just as with neutron transport. Modern radiation transport codes (s2-sS) deal with relativistic effects to various orders of u/c in applications spanning astrophysics to particle physics, fission to fusion, and low energy phenomena as well. SUMMARY

We described in rudimentary detail transport equations for neutrons and photons against a moving ba~kground. Hydrodynamics and the coupling of transport equations were similarly described. Treatments for neutrons are typically nonrelativistic (Galilean invariant), while treatments for photons are relativistic (Lorentz invariant). Neutron and photon interactions are both pointwise and cross sections reflect the short range nature of these interactions with matter. However, neutron interactions are strong, while photon interactions are electromagnetic in the scheme of strong, weak, electromagnetic, and gravitational forces (in descending strength of interaction). Transport of heavy neutral particles moving at nonrelativistic (low) energies parallels neutron transport as detailed herein. Transport of lighter neutral particles moving at relativistic energies recovers features of photon transport. Charged particles, such as electrons, protons, and ions, whether low or high energy, impart additional complexities to transport analysis and equations. Moving material complexities remain in all cases, high or low energy, Material motion effects can be treated completely within existing solution algorithms in applications. They can also be treated perturbatively, particularly when the ratio of material speed to transport particle speed is small. For neutrons, this means q/v is small, while for photons, this imples u/c is small. Material motion imparts directional anisotropy to particle transport solutions, through both effective cross sections and material velocity dynamical terms in the transport equation. This can be a very severe complication to standard solution techniques, particularly iterative ones when signs of material motion correction terms flip across the solution domain. Direct inversion techniques are less impacted. The distributions of transport particles in moving frames axe also often very different from distributions in fixed frames. As material and fluid speeds increase, differences widen. Conservation laws remain the same, but temporal and spatial evolution are different. Transport equations can be solved in fixed or moving frames independently, and solutions then transformed to one or the other frame. This is difficult in practice, so mixed moving-fixed frame solutions are obtained instead. Such approach usually forces transformations of rest frame (known) physical observables to another laboratory frame, or possibly vice-versa. In the limit of zero material motion, of course, solutions in the rest and laboratory frames are the same. We have not detailed applications herein, as such would be a monumental undertaking on top of this analysis. Same pretty much said for codes and numerical methods, though some codes were referenced earlier and briefly described. In both these regards, the references cited provide useful information plus additional references themselves. Consider some specific pointing which may be useful to the reader, in addition to all cited earlier, References 34 - 48 focus on neutron moving material impacts for a wide range of applications involving high and low speed backgrounds. References 60 - 77 duplicate the analysis for photons, especially coupled to hydrodynamics. Modern neutron transport codes are detailed in References 78 - 81, and modern photon transport codes are described in References 82 - 85. Excellent treatments of numerical

52

B. R. Wienke

methods used to solve both neutron and transport problems are found in References 17, 21, 25, 27, and 32. Numerical solution schemes are the only viable means for real world problems and applications. References 1 - 33 contain smatterings of applications and numerical methods, plus the coupling of basic dynamics to quantitative representation. In a fol]owup, we will detail moving material transport for electrons and neutrinos. Electron interactions are also electromagnetic, but because electrons carry charge, their interactions with matter are long range and continuous. Neutrino interactions with matter are weak, and pointwise. Neutrinos are also totally relativistic, like photons, and carry zero rest mass, often streaming through a void universe without interaction. ACKNOWLEDGMENTS It is a pleasure to thank my colleagues and friends over the past 35 years at Los Alamos National Laboratory, Livermore National Laboratory, the United States Navy, and Colleges and Universities in the United States and abroad. Some include Bill Gibbs, Dick Thompson, Kaye Lathrop, Nilendra Deshpande, Stuart Meyer, Jack Uretsky, Lloyd Hyman, Kiuck Lee, JiStewart, Tom Seed, Gene Bosler, Don Dudziak, Wally Walters, Pete Miller, Paul Whalen, Bob Seamon, George Bell, Andy White, Milt Wing, Vance Faber, Bob Hiromoto, Ralph Brickner, Joe Devaney, Jim Morel, Ed Larson, John Hendricks, Dmitri Mihalas, Gerry Pomraning, Dick Kracjik, John Pedecini, Gary Wall, Brown Rogers, Doug Johnson, Duane Wallace, Dean Preston, Jon Dahl, Gordon Olson, Ray Alcouffe, Charlie Slocum, Conrad Longmire, Randy Baker, Bengt Carlson, Bill Krauser, Joe Downey, John Gustafson, Ari Nikkola, Giovanni DeStefano, Sten Stockmann, Peter Readey, Ron Russell, Elizabeth Russell, Gene Melton, Kees Hofwagen, Chris Parrett, Tim O'Leary, Mark Flahan, Jim Bram, Jed Livingstone, Tom Hemphill, Brett Gilliam, Sig Gerstl, Hassan Dayem, Bill Mudd, John Richter, Doug O'Dell, Tom Manteuffel, Pier Tang, Reid Worlton, Don Wade, Joe Mack, Dick Silbar, Leon Heller, Ray Nix, Barry Ganapol, Bill Fillipone, Brad Clark, Mike Sohn, Gordon JiG, Bill Gee, Don McCoy, Stephen Lee, Mary Alme, Jack Brownell, Mike Henderson, Charles Lehner, Dick Vann, Wayne Gerth, Peter Bennett, Mike Gittings, Mike Clover, Tom Betlach, Jose Aragones, Guillermo Velarde, Tom Hill, Don Burton, Bill Chandler, Thurman Talley, All Brubakk, Carson Mark, and many others of casual acquaintance. And it's also a pleasure to be affiiated with Los Alamos National Laboratory in areas of particle and nuclear theory, particle transport, weapons physics and applications, Nevada Test Site underground testing and experiments, code development, computational physics, numerical methods, and high performance computing. REFERENCES 1, A. Eddington, The Internal Constitution Of Stars, Dover Publications, New York, 1926.

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53

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