THE B O L T Z M A N N TRANSPORT EQUATION

THE B O L T Z M A N N TRANSPORT EQUATION

T h e purpose of this appendix is to develop the mathematical descrip- tion of the transport of particles through matter. T h e equation to be derived, sometimes called the Boltzmann equation, describes the macroscopic motion of particles in a medium with sufficient accuracy for most purposes. In the cases of interest to us, the particles are either neutrons or photons, which are electrically neutral, a fact which sim- plifies the problem considerably.

In general the particles will be moving in a medium having special nuclear properties; for example, the medium may contain fissionable material. T h e nuclear properties of the material are essential to our calcula- tion. T h e properties will be characterized phenomenologically in terms of theoretically or experimentally determined cross sections. W e shall assume all of the cross sections to be considered subsequently are known.

Newton's equations of motion might be used to compute the motion of each particle in a medium. Indeed, this is the approach taken in the M o n t e Carlo method described in Chapter 6. T h i s description is so detailed, however, that very large, very high-speed computers are required to solve most practical problems by this method. Further, such a detailed picture is difficult to comprehend and/or use and is very seldom of interest.

A description of the average behavior of the neutrons will be quite sufficient for very nearly all our needs. T h e fluctuations from average behavior are of interest chiefly in the very early stages of a nuclear reaction, such as the startup of a reactor. T h e Boltzmann equation to be derived here will take the dynamics of the average particle population into account, but not the behavior of each neutron or photon by itself.

T h e population will be treated as a statistical entity by considering the ways in which the particles may be born, move, and die.

281

In order to proceed further, it will be necessary to define the direc­

tional density and the directional flux of particles. First, we shall denote the position of the particle by the vector r and its velocity by the vector v . Each of these two vectors consists of three components, of course. T h e directional density w(r, v , t) of particles is then defined to be the number of those particles which are present on the average in a unit volume located at r at time t and which are traveling in a unit velocity volume at the velocity v .

A few remarks concerning this definition are needed by way of clari­

fication. It is to be emphasized that the seven variables r, v , and / are absolutely independent of each other. W e may always choose the point at which we examine the neutron density quite independently of the speeds or directions in which the neutrons travel, and conversely.

T h e possibility of choosing these two variables independently stems from the Newton equations of motion themselves. It requires a specifica­

tion of the initial position and velocity of a particle in order to determine its future position and velocity at some later time. T h e Boltzmann equation must, of course, be consistent with Newton's laws (Reference / ) . W e may next inquire into the possibility of treating r and ν as independent variables at some later time in the Boltzmann equation. T h e r e is really no conflict when it is remembered that the Boltzmann equation will describe the average behavior of the whole population rather than that of an individual particle. Consider some particular point r. A t this point we can measure the densities of particles over a whole range of velocities.

In doing so, we are, of course, looking at particles having a variety of initial velocities. Indeed, we may very well find that the density is zero for many velocities, Newton's equations and the relevant nuclear events prohibiting particles from achieving those velocities given the original initial velocities.

T h e second point that needs to be clarified with respect to the defini­

tion of particle density is the concept of specifying the density of neutrons per unit velocity volume. T o this end, we use the symbol dr to stand for an element of volume in ordinary space; we also use the symbol dv to stand for an element of volume in velocity space.

T h e symbol dv stands for dx dy dz in Cartesian coordinates for exam­

ple: dv stands for v² dv d(cos θ) άφ in spherical coordinates for example.

T h e components of the vector r may be referred to a different coordinate system from that to which the components of the vector ν are referred.

T h e number of particles in the element dr dv is then w(r, v , t) dv dv.

This is the number of particles whose spatial coordinates lie between χ and χ + dx, y and y + dy, and ζ and ζ + dz and whose velocity

coordinates lie between vx and vx + dvx , vy and vy + dvy , and vz and vz + dvz .

It is of interest to consider the number of particles that travel in a unit solid angle and in a unit range of speed. T o this end, let Ω denote a unit vector in the direction of the velocity vector v :

Ω = ν / ν , ( A . l ) the magnitude of a vector quantity being denoted by the symbol for

the vector in italic type, instead of bold face. T h e element dv of velocity volume can be expressed as follows:

dv = v^£dvdSl, (A.2)

where d£l is an element of solid angle. For example, in spherical coor­

dinates dSl = sin# άθ dcp, where θ is the colatitude angle and ψ is the azimuthal angle describing the direction in which the particle moves (not the direction of the radius vector r ) . N o w , the number of particles in an element of volume must not depend upon the mode of description that happens to be used, so

n ( r , v , t) dr dv = n(r, ν, Ω , t) dr dv d£l, (A.3) or

w(r, v , t) = *i(r, ν, Ω , t)jv². (A.4)

Here w(r, ν, Ω , t) is the number of particles in a unit volume of space at time t (as before) whose speed lies in a unit speed volume at ν and going in a unit solid angle centered on the direction Ω . T h e functions « ( r , v , t) and « ( r , v9 Ω , t) are not the same but are related by Eq. ( A . 4 ) .

T h e directional flux Φ(Γ, V , t) is a vector, is defined by

Φ(Γ, ν , t) = vn(r, v , t), ( A. 5 )

and is the number of particles at r of velocity ν per unit spatial volume and per unit velocity volume that in a unit time cross a unit surface whose normal lies along v . T h e flux or track length <£(r, v, t) is defined by

<£(r, v, t) = J dSl vn(r, ν, Ω , t) , ( A . 6 )

where the integral is to be computed over all solid angles.

T h e magnitude </>(r> ν, Ω, t) of the directional flux is related to the magnitude of the directional flux normalized per unit volume and per unit velocity by

φ(τ,ν1ή=φ(τ)ν,Ω,ήΙν*. (AJ)

From Eqs. ( A. 4 ) , ( A. 5 ) and ( A . 7 ) , w e find that

φ(Γ,ν,η,ί) = vn(r,v,n,0- (^Α·⁸)

T h e density «(r, v> t) of particles per unit volume per unit speed is defined by

w(r, v, t) =

j da

w(r, ν, Ω, t), (A.9)

and is related to the flux by

vn(r,

v, t) = <£(r, v, t), (A. 10) as can be seen from Eqs. ( A . 9 ) and ( A . 6 ) . From this result it is seen that the flux is the total distance traveled by all particles in a unit volume per unit time. T h e symbol w(r, v, t) for the directional density and that w(r, v, t) for the density are distinguished from each other by the presence of the bold face velocity symbol in the former and the italic symbol in the latter. Likewise for the distinction between the magnitude of the directional flux and the flux.

Various directional moments of the particle distribution may be defined. T h e only one that we shall need is that for the net current J(r, v, t) of particles

J(r, v, t) =

j

dSl vSl « ( r , v , * ) . (A.l 1) It is to be noted that this is a vector equation; it stands for three equations when written out in component form.

T h e Boltzmann equation may now be derived. T o find this equation, one needs only to account for the fate of all particles, i.e., to conserve all neutrons. Since the equation for photons is slightly different from that for neutrons, we shall write the equations for neutrons. T e r m s are to be expected that describe the rate of change of the number of neutrons, the numbers of particles scattered into the region of interest, and the loss of particles from the region by scattering and absorption. T h e Boltzmann equation is merely a particle balance among these terms.

T h e rate of change of the number of particles in an element dr dv with time is equal to the rate at which particles are added to this element minus the rate at which they are removed from this element.

T h e rate of change with time of the number of neutrons in the element dr dv is

(A.12) T h e number of particles added to the element dr dv by a source is

S(r, v, t) dr dv . (A. 13) T h e source as contemplated here is to consist of all particles arising from means not linearly proportional to the flux of the system. T h e source is not to consist of fission neutrons or those appearing from an (w, 2ri) event, for example.

T h e number of particles leaking out of the element dr dv over the bounding surface must be considered. In most problems of interest, there is a spatial gradient of the magnitude of the directional flux, which leads to the diffusion of particles across the bounding surfaces in con­

figuration space.¹ T o facilitate the calculation of this diffusion of particles, it will be convenient to introduce the notation d²r for a surface element;

since a surface element is a vector, d²r is to be a vector normal to the surface of the element oriented in the usual positive sense and of a magnitude equal to the area of the surface element. T h e net leakage from dr dv due to flux gradients is coordinate space is then given by

j d²r ·φ(Γ, ν, t) dv = j d²r · vw(r, v, t) dv , (A.14)

= J* dr Vr · vw(r, v, t) dv , (A. 15)

= Vr · vw(r, v, t) dr dv , (A. 16)

= ν · V^r w(r, v, t) dr dv . (A. 17) In Eq. (A.14) the integral is carried out only over the surfaces bounding dr. In Eq. (A.14) the definition ( A . 5 ) is used. Relation ( A . 15) follows by the divergence theorem of Gauss, the integral extending only over the

1 In most problems involving an external force field acting on the particles, the particles may diffuse across the surfaces bounding an element of velocity volume. However, such a circumstance does not normally arise with either photons or neutrons.

element dr\ Eq. ( A . 16) follows from the observation that w(r, v , t) is a physical quantity and therefore continuous in its r dependence so that over dr it varies only negligibly. Finally, Eq. ( A . 17) follows by noting that ν is quite independent of r and from the definition of Vr , which is

v' ^{= i}4 + i ' | ; ⁺ ' ^ '^{k ( Α}·^{1 8)}

ir, j r , and kr being mutually orthogonal unit vectors in configuration space.

A t a collision, elastic scattering, inelastic scattering, fission, an ( « , 2n) event, or the like may occur; in general, the number of neutrons leaving a collision will differ from the number entering. T h e probability that a collision occurs is characterized by the macroscopic cross section σ, which is a function of the number of nuclei per unit volume, the nuclear species constituting the target, the relative velocity between the target nucleus and the particle, and the energy with which any escaping par­

ticles emerge from the collision. T h e macroscopic cross section is the probability σ per unit length that any particle of the specified relative energy will suffer a collision in traveling a unit distance in the specified material.

T h e total macroscopic cross section is defined by

σ = σ8 + σΓ, (A.19)

where σ8 is the total macroscopic scattering cross section, and σΓ is the total macroscopic reaction cross section. T h e total scattering cross section is defined to be

CTs = ^σβ 8 + °\s > (A.20)

where σθβ and σ ι8 are the macroscopic elastic and inelastic scattering cross sections, respectively. T h e total reaction cross section is defined to be

σΓ = at + σ2η + σ0 , (A.21)

where at , σ2η and av are the macroscopic fission, (w, 2rc), and radiative capture cross sections, respectively. T h e total multiplicity, c, of an event is defined by

c = [^σο 8 + ^σ1β + ^{v c r}r + 2σ2„]/σ , (Α.22) where ν is the number of neutrons appearing from the fission process

per fission. T h e multiplicity c is from its definition just the number of

neutrons appearing as the result of an event. For each event caused by a neutron of velocity ν ' , there is a probability / ( ν ' —> ν, Ω ) dv dSl that one of the neutrons emerging from the reaction will have a speed ν in the range dv and a direction Ω in the range </Ω.

j ( ν' - ν, Ω ) φ / ) σ(ν') = /Μ( ν ' - > ν, Ω ) σθ 8Κ ) + /l s( v ' - ν, Ω ) σΐ 8(ι;')

-|- /Γ( ν ' — ν, Ω ) ν(ϋ') σΓ(ϋ') -h /2„ ( ν ' - > ν, Ω ) 2σ2 η(ι/) , (Α.23) where the partial probabilities σβ 8 / e s( v ' — Ω ) , σί 8 / i s ( ν ' - > ν, Ω ) , erf / f ( ν ' —> Ω ) , and σ2η/2η ( ν ' —• Ω ) are defined for elastic scattering, inelastic scattering, fission and an (w, 2w) event, respectively, similarly to the definition of σ / ( ν ' —• Ω ) for a reaction. These probabilities are not probabilities per unit velocity, of course. T h e y will be called collision transfer probabilities.

From the definition of the macroscopic cross section as a probability that a neutron suffer a collision per unit distance traveled in the material specified, it follows that the number of collisions suffered by all particles per unit volume per unit velocity in a unit time is given by

a(r, ν, V ) I ν -- V I w(r, v, t), (A.24) where V is the velocity of the target nucleus. Similarly, the number of

collisions suffered by all particles in a unit volume per unit time and per unit speed is given by

σ(ι·, ν ) wi(r, ν, t), (A.25) if the speeds of the target nuclei are negligible. Otherwise, the macro­

scopic cross section must be replaced by a suitable average. It is con­

venient to define the microscopic cross section as

<x(r, v, V)/yV(r, V ) (A.26) where 7V(r, V ) is the number of nuclei per unit volume per unit velocity,

all nuclei being assumed monoenergetic with the velocity V . T h e microscopic cross section has the dimensions of an area, is measured in units of 1 0^{- 2 4}c m², called the barn, and is independent of the density of the material. T h e microscopic cross section may be thought of as the effective area of a nucleus with respect to the specified type of collision for neutrons of the given relative velocity. T h i s interpretation is harmo­

nious with the definition (A.26) and the interpretation of the macroscopic

cross section for the probability that some collisions take place. T h i s probability then is simply the product of the number of nuclei present times the area presented by each.

T h e total number of neutrons entering the element dr dv as a result of collisions is then

j dv' j dV/(ν' - > ν, Ω ) c(v') a ( r , ν ' , V ) | v ' - V | w ( r , ν ' , t) dr ^ . ( A . 2 7 ) Likewise the total number of neutrons removed from the element dr dv is given by

j dVa{r, ν , V ) I ν - V I « ( r , v , t) dv dr . ( A . 2 8 ) T h e discontinuous nature of motion in velocity contrasts with the con­

tinuous character of the motion in space. Diffusion across velocity surfaces due to the continuous changes of velocity is incorporated into the theory by terms not discussed here, but readily incorporated into our equation; discontinuous changes are taken into account by expressions (A.27) and (A.28).

T h e neutron balance equation is then

t ] = S ( r , ν , 0 - ν · Vr W( r , v , t)

- j dV a ( r , ν , V )

I

ν - V

I

w ( r , v , / )

+ j dv' j dvJ^^^Q c(v') a ( r , ν ' , V ) I v ' - V I n ( r , ν ' , t). ( A . 2 9 ) T h i s important equation is the Boltzmann equation. It forms the basis of nearly all the calculations of particle transport. T h e equation can be specialized to the case that usually occurs in which the velocities of the target nuclei are negligible compared with the velocities of the neutrons.

In this case

A T (r, V ) = N(r) 8 ( V ) ,

where 8( V ) is the Dirac delta function. Consequently in this case Eq.

(A.29) becomes

dn(r^ v , t) = s^ v ^ _ v v^ ^ ^ _ ^ ^ ^ ^

+ j dv'^{f (y} ^ ⁾ c(v') a ( r , ν') v'n(r, ν ' , 0 . ( A . 3 0 ) ^Ω

T h e Boltzmann equation (A.30) is frequently expressed in terms of w(r, ν, Ω , t) instead of /z(r, v , t):

In the above equations it has been assumed that the macroscopic cross sections are independent of the flux. For certain problems, such as those involving xenon poisoning, temperature feedback, burn-up of fuel, and others, the coefficients will be functions of the flux. T h i s dependence can be immediately incorporated into the transport equation by merely inserting the appropriate dependence on the flux. ( W h i l e easy to incorporate, solving the resulting nonlinear transport equation is usually much more difficult than the linear one.)

A number of more basic assumptions have been made in deriving the Boltzmann equation. These are as follows:

1. T h e element of velocity and space of interest is sufficiently large that statistical fluctuations within these elements are negligible. Statistical fluctuations cannot be considered by the Boltzmann equation (see References 4 and 5 ) . T h e probability of a neutron, for example, inducing a chain reaction cannot be treated by the Boltzmann equation. L e t us estimate the fluctuations that might take place in a volume element, dr.

T h e r e might be some 10⁷ neutrons present per c m³, corresponding to a thermal flux of 2 Χ 10¹² neutrons per cm²-sec. T h e probable fractional error in such a case due to statistical fluctuations would be of the order of ( 1 0⁷ dr)'¹/². T h e statistical fluctuations could be of some importance at startup, for example.

2. T h e collision time of neutrons has been assumed to be zero. T h i s approximation is extremely good, since the compound nucleus lives less than 10~¹⁴ sec and since the ranges of nuclear forces are extremely small compared with the mean distances between neutrons. Although 10~¹⁴ sec is very long compared with characteristic nuclear times, the character­

istic periods involved in various nuclear applications are always very much longer. Consequently, only the binary encounters considered above in the derivation of the Boltzmann equation are important. Except for the delayed neutrons which emerge from fission fragments, the neutrons emerge from the point of the collision. Even in the case of delayed neutrons, the ranges of fission fragments are so very short in any solid or liquid that the delayed neutrons substantially originate at the site of the fission. However, if delayed neutron emitters be considered

dn(r, ν, Ω , t)

Ft = S ( r , υ, Ω , t) — νΩ · V,.w(r, ν, Ω , t) — a ( r , v) vn(r, υ, Ω , t) + dv'd&f(v' -> ν, Ω ) c(v') a ( r , ν') v'n(r, v\ Ω ' , t). ( A . 3 1 )

in the time-dependent Boltzmann equation, the decay of the delayed neutron emitters must be considered as a source term in the Boltzmann equation and appropriate time-dependent conservation equations be written for the delayed neutron emitters. (It is noted here that the ap­

proximation of binary collisions is not always justified when charged particles are present, because the relatively long range Coulomb fields frequently lead to cooperative effects, a fact that greatly complicates the theory.)

3. Collision of neutrons (or photons) with themselves have been neglected. A t the fluxes present in practical systems, the fraction of all collisions that are neutron-neutron (or photon-photon) collisions is extremely small indeed. T h e densities of the neutrons (photons) is very low compared with the density of target nuclei. For this reason the Boltzmann equation applied to neutrons (or photons) is linear. Although it is easy to incorporate the nonlinearities into the Boltzmann equation, as one must frequently do for plasmas, it is very difficult to solve the resulting equation.

4. In Eq. (A.31) the vibration energy of the atoms or molecules of the material in which the neutrons slow down is assumed negligible. T h i s approximation might be expected to be invalid when the neutron energy becomes only a few times that of the binding energy of the atoms or molecules of the crystallographic lattice. T o the degree that the results depend on the average energy loss only, the consequences will be accurate as low as an electron volt. For a few problems the more complicated Boltzmann equation (A.29) must be used.

5. Normally the force fields acting on the neutron are zero, except for the nuclear fields. T h e effects of the nuclear fields are phenomo- logically incorporated into the scattering and the absorption terms.

Gravitational forces are far too weak to influence the motion of neutrons or photons significantly in most problems. Inhomogeneous magnetic fields, which couple with the magnetic moment of the neutron, almost never exist and, even if they did, they would have to be enormously intense to sensibly influence the motion of the neutron. Homogeneous magnetic fields and electric fields are without influence on the motion of the neutron because of its lack of charge. Consequently the neutron travels in straight lines between collisions.

6. Effects that depend on the orientation of the neutron (or photon) are not incorporated into the Boltzmann equations above.

Even in its simplified forms, the Boltzmann equation is extremely difficult to solve exactly because of the integral term.

T h e term ν · V « ( r , v , t) must be calculated with some care. W e illustrate

the calculation by considering spherical geometry. T h e components of ν are then as follows:

V,. = ν cos θ = μν , (A.32)

vQ — ν sin θ — V l — u?v , (Α.33)

where μ is defined as cos Θ, and where the various angles and vectors are defined in Fig. A . l . In spherical coordinates

(A.34)

y

F I G . A . I. Position and velocity vectors for spherical geometry.

N o w the directional density can depend only upon r, ν, μ,, t: because of the spherical symmetry, the directional density must be the same at all points equally distant from the center and must be independent of spatial azimuth about or latitude with respect to the ζ axis. T h e direc­

tional density must also be independent of the azimuth of the velocity vector with respect to the radius vector because of the spherical geo­

metry. T h u s

ν · V « ( r , ν, ί) = ν [μ ^ + n(ry ν, μ, t). (A.35)

T h e Boltzmann equation for photons is very similar to that for neu­

trons. Since all photons travel with the speed of light, we must use the energy or wavelength of a photon as an independent variable instead of the speed. Photons are slowed down almost entirely by the Compton

process. If Λ' is the wavelength of the incident photon and Λ the wave­

length of the scattered photon, then by energy and momentum con­

servation we learn that

A = A' + (1 - μ), (A.36) where Λ is the wavelength in Compton units, i.e., Ah0/m0 cQ is the

physical wavelength of the photon, where m0 , cQ , and h0 are the mass of the electron, the speed of light, and Planck's constant. One photon emerges for each photon incident.

T h e probability that a photon going a unit distance be scattered through an angle θ into a unit compton wavelength range is given by (Reference 6)

Η ψ ί

5

(4f)"^)" ⁽⁴

⁺ "λ - " ') · ^{< Α}·^{3 7)} where pM is the mass density, N0 is Avogadro's number, Μ is the atomic

weight, and Z0 is the atomic number. T h i s probability is then equal to

φ ' (4)'

^{( w + 7 - * + - ^}

(A.38) by use of the Compton relation. N o w for the Boltzmann equation (A.31), we are interested in knowing the probability that the photon of wave­

length Λ' be scattered into an element dSl of solid angle and into an element^-dA of wavelength. In other words, the element of solid angle and the element of wavelength in which the scattered photon lies are regarded as independent variables in our formulation; yet, the Compton relation, i.e., the conservation of energy and momentum, relates these two entities quite uniquely. W e may still consider the variables Λ', θ, and Λ as independent if we restrict their possible values to only those satisfying the Compton relation. T o this end, we use the Dirac delta function δ[1 + (Λ' — Λ) — /χ], which will be zero unless the Compton relation is satisfied. Since the variables Λ and θ are now regarded as independent, the probability of scattering into dSl and into άΛ will then be the product of the probability of scattering into άΛ times the probabil­

ity of scattering into dR. N o w , as it stands [1 + (Λ' — Λ) — μ] is not the probability of a photon scattering into a unit solid angle because

άμ d<po[\ + (Λ' -Λ)-μ]=2<π.

—ι *Ό

F r o m this relation we see that

LTT

is the probability of scattering into an element dSl of solid angle centered at Ω . T h e probability that a photon of wavelength A' be scattered into a unit solid angle centered at θ and into a unit wavelength centered at A is then (Reference 6)

K(A, A') J - 8f 1 + {A' - A) - μ], ( A . 3 9 )

LIT

where

[4

^Γ+

^

^+2{Λ

' ~

^Λ)+{Λ

' -

^Λ)2

] ·

^(Α

·

^4Ο)

T h e Boltzmann equation for photons is then dn(r, Α, Ω , t)

dt = 5 ( r , Α, Ω , ί ) - c0& · Vn( r , Α, Ω , i ) - c0a ( r , A) w(r, Λ , Ω , *) + J - Γ* Λ Ι ' f ι/Ω' il') 8[1 + ( Λ ' - il) - /χ] c0w(r, il', Ω ' , ί). ( A . 4 1 )

T h e Boltzmann equation just derived can be recast into the form of an integral equation. T h e integral equation may be found by integrating the Boltzmann equation with respect to r. A rather considerable amount of mathematics ensues. For this reason and because the physics becomes clearer, we prefer to give a derivation of the integral equation from fundamental principles instead. T o this end we must first consider the streaming of neutrons in a vacuum. T h e neutrons in matter may be regarded as traveling in a vacuum between collisions with the nuclei that comprise the matter.

In streaming in a vacuum, the neutrons must move with constant energy, since there are no nuclear events to alter their energy. In the absence of any substantial force fields, as is very nearly always the case, the neutrons travel in straight lines between collisions. In such a cir­

cumstance, it is easy to show that the directional density is independent of position. Again, the matter can be proved mathematically or physically.

W e leave the mathematical proof for problems.

Consider Fig. A . 2 . T h e number of neutrons crossing an element ΔΑγ

of area located a t r that are headed so as to cross an element AA2 of area located at r + d at time t is

w(r, ν, Ω , t) AAxAA2\d². (A.42)

\. d \ F I G . A . 2 . Streaming of neutrons in a vacuum.

T h e solid angle of the neutrons crossing at any point of the surface element AA2 is

AAJd*.

Consequently, the density n(r + d , ν, Ω , t) of neutrons per unit speed per unit solid angle at time t is

~ χ / ~ d\ AA, AA2 d'²

n(r + d, v, < M ) = » (r, « , < M - - ) ^ ~AAX

= i i( r , i ; , n, i - ^ ) , (A.43) as claimed.

W e must now incorporate the effects of nuclear collisions into our thoughts; see Fig. A . 3 . T h e probability that a neutron at rx gets to a point r along its trajectory without suffering a collision is

exp (— J ² drza(v — r3 , v))

T h e number of neutrons created per unit time per unit volume per unit velocity at time t — r2jv at position r1 of velocity ν is

G (r i, ν, ί - - £ - ) = J dv' Μΐϊ^ΐϋι^ϊ c { r i, υ·) σ ( Γ ι, </) ψ (Γ ι, ν ' , ί ) , (Α.44)

as may be seen directly. T h e time required for a neutron to get from r1 to r is

r — r.

since the speed of the neutron is unaltered between the two points. A neutron at r1 at time t — r2/v arrives at r at time t. Consequently, the rate of arrival of neutrons of velocity ν at r at time t directly and without collisions from a unit volume at r1 is given by

G (r — r2, v, t — - ^ - J exp - Φ (ΓΧ , Γ, Ϊ Ι ) ,

where the optical depth Φ is defined by

Φ (Γχ , r, v) = \ dr3a(r - Q.r3 , v)

(A.45)

(A.46)

Position of neutron at time t

Location of neutron at an earlier time t

Trajectory of neutron

f Origin

F I G . A .3 . Position vectors associated with the integral equation.

T h e rate of arrival of neutrons of velocity ν at r at time t directly and without collisions from any point at all is given by

J dr2G(r — r2 , v, t ^ exp — Φ (Γχ , r, v).

By including all those and only those neutrons at r that have suffered their last collision in a unit volume at rx before reaching r , each neutron is counted once and only once.

T h e result

(£(r, v , t) = j ⁴ dr2G (r — r2, v , t — exp - Φ ( Γχ , r, v)

+ φ ( r - r4, v, t - exp - Φ ( Γ5 , r, ν) (A.47)

then merely states that the rate of arrival of neutrons per unit volume at r at time t of velocity ν equals the number that are born per unit time of velocity ν per unit velocity at a suitably earlier time between r — r4

and r and that get from the site of their birth to r with no collision plus the rate of arrival per unit volume and per unit time at r — r4

at the earlier time t — rjv of neutrons of velocity ν times the probability of a neutron getting f r o m r — r4 t o r with no collision.

References

Derivations and discussions of the neutron transport equation may be found in several references listed in Chapter V . For more general discussions of particle motion, References J and 4 and 5 are suggested. T h e Dirac delta function is discussed in References 2 and 3. T h e Compton relation and the scattering of photons in general is well covered in 6.

1. Rose, D . J., and Clark, Jr., M . , "Plasmas and Controlled Fusion." Wiley, N e w York, 1961, Chapters 4 and 6.

2. Dirac, P. A . M . , " T h e Principles of Quantum Mechanics." Oxford Univ.

Press (Clarendon), London and N e w York, 1947, pp. 58-61.

3. Schiff,L. I., "Quantum Mechanics." McGraw-Hill, N e w York, 1955, pp. 50-51.

4. Allis, W . P., Motions of ions and electrons, "Handbuch der Physik" (S. Flugge, ed.), Vol. 21. Springer, Berlin, 1957, pp. 383-444.

5. Chapman, S., and Cowling, T . G., " T h e Mathematical Theory of Nonuniform Gases." Cambridge Univ. Press, London and N e w York, 1952.

6. Evans, R. D., " T h e Atomic Nucleus." McGraw-Hill, N e w York, 1955, Chapter 23.

Problems

1. (a) Calculate the flux of thermal neutrons at which neutron-neutron colli­

sions become comparable to collisions with HJ. Assume the neutron-neutron cross section is the same as the neutron-proton cross section. Let the density of the Η atoms be that of Η in water.

(b) Calculate the pressure of thermal neutrons on a perfect reflector at a flux of 10¹⁴ neutrons/cm²-sec. Assume that neutrons are incident and reflected perpendicular to the plane reflector.

2. (a) Prove that j άΩφ(τ, ν, t) = 0 if the neutron density is an isotropic distribution of neutrons.

(b) Prove that w(r, v, t) = 4-nn(r, ν, Ω , t) if the latter corresponds to an isotropic distribution of neutrons.

(c) Prove that ^(r, v, t) = vn(r, v, t).

(d) W h y does | ν — V | w(r, v, t) appear in Eqs. (A.24) and (A.28) instead of (ν - V) w(r, v, t) or | ν - V | n(r, v, t) ?

(e) Write the Boltzmann equation (A.31) in terms of fluxes, instead of in terms of densities.

(f) Write the one-speed, time-independent Boltzmann equation.

3. (a) Show that the term in the Boltzmann equation for plane geometry representing neutron transport becomes

μ — tl(xd t V, μ, t).

ox

(b) Show that the term v Vn(r, v, t) in the Boltzmann equation for cylin­

drical geometry is

Γ / / d 1—Tj² a \ dl

K

¹

-

^{*· (" aT +}

^{τ) +}

^{* d x}> "· "·⁴⁾

where μ = cos 0, η — cos φ, and the angles and other variables are defined by the accompanying figure.

4. Write the time-dependent Boltzmann equation for a neutron in a gravitational field. Solve it for the case of the neutron in a vacuum in a uniform field.

5. Write the time-dependent Boltzmann equation in the case of pure absorption.

Solve it.

6. Write the time-dependent Boltzmann equation for neutrinos with an arbitrary source. Solve it.

7. Multiply the Boltzmann equation by g(r, v, t), an arbitrary function vanishing at ν = i °°, and integrate over all velocities and show that

[fi(r, t) g( r , v, t)] - n{r, t) ^ ( r , v, t)

+ Vr . n(r, t) vg(r, ν, t) - w(r, i) Vr · V£(r, ν, t) J r f v ^ ( r , ν, i) [-

Si

The averages are defined according to « ( r , i) g(r, t) = J d\ g(r, ν, i) /(r, ν, i).

8. From the results of problem 7, prove that the Boltzmann equation conserves particles.

9. From the results of problem 7, prove that the Boltzmann equation is consistent with Newton's second law relating a force to a rate of change of momentum.

T o this end, let g = rav. Interpret the result. What happens if there are several species of particles present ? In the result let vR be defined by

ν = v(r, t) + vR ,

so that VR is the velocity of random motion. Show that

Using the conservation law derived in problem 8 and by evaluating various derivatives, show that

m n ( r'^{t }}

N r

⁺

t c ^ ' i ) ^*'

^T)

coll

3 / ( r , v, <)]

Interpret this result.

10. From the results of problem 7, show that the Boltzmann equation is consistent with the conservation of energy. T o this end, let g = mv*/2. Show that

d ι mn -—\ „ in —-\ r , mv² / df \

Interpret the result. Next let

ν = v ( r , t) + vR .

Show that

1 1 _ 3 - nm v² = - nm v² + - p 2 2 2 where p is the pressure of the particles. Show that

v²\ = v²v Η . 5pv nm Show that

_ dv mv² dn 3 dp 1 _ nm ν · — 1 ^{h - + -} m v²V · (nx)

dt 2 dt 2 dt 2

+ \- nm ν · V v² -f \ ν * Vp + ^ />V · ν =

ί

^{dv ^ ma2} ( - ^ - ) 2 2 2 J 2 \ dt /co Using the conservation equations for particles and momentum, show that

Show next for particles of one type that p = p0n⁵'\

11. Prove that the directional density is independent of position by writing the Boltzmann equation for a vacuum. Hint: Compute the derivative of the directional density along the trajectory of the neutron and show that this derivative is zero.

12. Suppose only the flux were known in a system in which neutrons are diffusing.

Show how the directional flux may be determined from the flux in the case of an isotropic source.

13. Show that an infinite plane cavity in an infinite plane slab does not influence the neutron distribution.

14. Prove that the flux outside a spherical shell is equivalent everywhere to that arising from two appropriately placed planes.

15. Starting from the integro-differential form (A.30) of the Boltzmann equation, derive the integral equation (A.44). Show that

' V # ( re , ν, ί.) =

[ J

_ 1 Γ d<Kr + Slr7 , ν, t + r7/v) 1

vl d(t + r7/v) Jr+nr7.v Show that

r ^ ( re, v,i6)i

+ o(r6 , v')<Krt 6 , v, t6) = G(r6 , ν, ίβ).

t = t6 — r7/v; r7 = r6 — r, where r6 is a vector from the origin to a future point along the trajectory of the neutron. Next, deduce Eq. (A.44).