# Neutron Flux Profile in a Spherical Moderator

#### Israakaizzy

Hello People
I need help with the following assignment:
It states:
Consider an ideal moderator with zero absorption cross section, Ʃa = 0, and a diffusion coefficient, D, which has a spherical shape with an extrapolated radius, R. If neutron sources emitting S neutrons/cm3sec are distributed uniformly throughout the moderator, the steady neutron diffusion equation is given by,
D∇2$\phi$ -Ʃa$\phi$=-S

a) Simplify the above neutron diffusion equation for this moderator in spherical coordinates and state the appropriate boundary conditions.

By solving the simplified diffusion equation, obtain the neutron flux profile, $\phi$(r).

I know I need to divided the neutron diffusion equation and cancel out the absorption cross section and end up with something like:
2$\phi$ = -S/D
and the particular solution would be something like S/Ʃa
but what's the general solution to:
D∇2$\phi$ =0
in spherical coordinates?

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#### hmeier

Israakaizzy,

I think it should be something like Asinh(λ.r)/r + Bcosh(λ.r)/r, applying the border conditions B=0.

hope it helps,

Hernán

#### Israakaizzy

ok
Just explain me what is $\lambda$ equal to? Is it 1/L ?

#### hmeier

Israakaizzy,

You are right, it should be 1/L if Ʃa were different than 0.

I did the maths for the homogeneous part:
∇$^{2}$$\phi$=0

saying that:

$\phi$=$\frac{\widehat{\phi}}{r}$

The Lapplacian inspherical coordintates turns:

∇$^{2}$$\phi$=$\frac{∂^{2}\widehat{\phi}}{∂r^{2}}$ + $\frac{2}{r}$$\frac{∂\widehat{\phi}}{∂r}$

proposing an exponential solution:

λ$^{2}$e$^{λr}$ + $\frac{2}{r}$λe$^{λr}$ = 0

So:

λ= -$\frac{2}{r}$

and finally:

$\phi$= $\frac{A}{r}$ + B

Don´t forget that for the inhomogeneous part you have to use the Lapplacian in sphericals.

Regards,

Hernán

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