’ 2 cos x ’ sin x,

n1 (x) =

x x

3 1 3

’ 3’ cos x ’ 2 sin x.

n2 (x) =

x x x

Despite the appearance of negative powers of x, the jn (x) are all ¬nite at

x = 0. The nn (x) all diverge to ’∞ as x ’ 0. In general

jn (x) = fn (x) sin x + gn (x) cos(x), (8.146)

nn (x) = ’fn (x) cos(x) ’ gn (x) sin x, (8.147)

where fn (x) and g( x) are polynomials in 1/x.

We also de¬ne the spherical Hankel functions by

(1)

hl (x) = jl (x) + inl (x), (8.148)

(2)

hl (x) = jl (x) ’ inl (x). (8.149)

These behave like

1 i(x’[n+1]π/2)

(1)

hl (x) ∼ e , (8.150)

x

1 ’i(x’[n+1]π/2)

(2)

hl (x) ∼ e , (8.151)

x

at large x.

The solution to the wave equation regular at the origin is therefore a sum

of terms such as

•k,l,m (r, θ, φ, t) = jl (kr)Ym (θ, φ)e’iωt ,

l

(8.152)

where ω = ±ck, with k > 0. For example, the plane wave eikz has expansion

∞

ikz ikr cos θ

(2l + 1)il jl (kr)Pl (cos θ).

e =e = (8.153)

l=0

8.3. BESSEL FUNCTIONS 245

Example: Peierls™ Problem. Critical Mass. The core of a fast breeder reactor

consists of a sphere of ¬ssile 235 U of radius R. It is surrounded by a thick

shell of non-¬ssile material which acts as a neutron re¬‚ector, or tamper .

R

DF

DT

Fast breeder reactor.

In the core, the fast neutron density n(r, t) obeys

‚n 2

= ν n + DF n. (8.154)

‚t

Here the term with ν (≈ 108 sec’1 ) accounts for the production of additional

neutrons due to induced ¬ssion. The term with DF (≈ 6 — 109 cm2 sec’1 )

describes the di¬usion of the fast neutrons. In the tamper the neutron ¬‚ux

obeys

‚n

= DT 2 n. (8.155)

‚t

Both the neutron density n and ¬‚ux j ≡ DF,T n, are continuous across

the interface between the two materials. Find an equation determining the

critical radius Rc above which the neutron density grows without bound.

Show that the critical radius for an assembly with a tamper consisting of 238 U

(DT = DF ) is one-half of that for a core surrounded only by air (DT = ∞),

and so the use of a thick 238 U tamper reduces the critical mass by a factor

of eight.

Factorization and Recurrence

The equation obeyed by the spherical Bessel function is

d2 χl 2 dχl l(l + 1)

χl = k 2 χl ,

’ 2’ + (8.156)

x2

dx x dx

246 CHAPTER 8. SPECIAL FUNCTIONS I

or, in Sturm-Liouville form,

1d dχl l(l + 1)

x2 χl = k 2 χl .

’ + (8.157)

x2 dx 2

dx x

The corresponding di¬erential operator is formally self-adjoint with respect

to the inner product

(f — g)x2 dx.

f, g = (8.158)

Now, the operator

d2 2d l(l + 1)

Dl = ’ 2 ’ + (8.159)

x2

dx x dx

factorizes as

l’1

d d l+1

Dl = ’ + + , (8.160)

dx x dx x

or as

d l+2 d l

’

Dl = + + . (8.161)

dx x dx x

Since, with respect to the w = x2 inner product, we have

†

d 1d2 d 2

=’ x =’ ’ , (8.162)

x2 dx

dx dx x

we can write

Dl = A† Al = Al+1 A† , (8.163)

l l+1

where

d l+1

Al = + . (8.164)

dx x

From this we can deduce

Al jl ∝ jl’1 , (8.165)

A† jl ∝ jl+1 . (8.166)

l+1

Actually the constants of proportionality are in each case unity. The same

formul¦ hold with jl ’ nl .

8.4. SINGULAR ENDPOINTS 247

8.4 Singular Endpoints

In this section we will exploit our understanding of the Laplace eigenfunctions

in spherical and polar coordinates to explore Weyl™s theory of self adjoint

boundary conditions at singular endpoints. We also connect it with concepts

from scattering theory.

8.4.1 Weyl™s Theorem

Consider the Sturm-Liouville eigenvalue problem