Can Someone explain Why we integrate over 4[tex]\pi[/tex]? What allows

[candice_84]

Can Someone explain Why we integrate over 4$$\pi$$? What allows us to get rid of Omega?

 

[Astronuc]

One is simply integration over all 'directions'. 4π is just the total solid angle, which represents all directions/orientations.

 

[candice_84]

2pi = 360 which is enough.

 

[Astronuc]

2pi = 360 which is enough.

2 pi in 2D, not 3D.

In 3D, 2 pi is half the solid angle encompassed by a sphere, i.e. hemisphere.

Think - the area of a sphere is 4pi r², where r is the radius.

Note, when one refers to
$$\phi(r,E,\vec{\Omega})$$
one is referring to the angular flux in n/cm2-s-(unit E)-steradian.

Integrating over the solid angle gives the 'scalar' flux.

http://en.wikipedia.org/wiki/Neutron_transport

 

[candice_84]

solid angle is a volume?

 

[Astronuc]

solid angle is a volume?

No solid angle is the solid angle, like angle is angle in 2D. The 4π (steradians) solid angle is the 3D analog to 2π radians in 2D.

The total solid angle would be the area of a sphere divided by r², i.e. A/r² = 4πr²/r² = 4π, just like 2π = circumference (2π r) of the circle divided by r.

http://en.wikipedia.org/wiki/Steradian
http://en.wikipedia.org/wiki/Solid_angle

http://mathworld.wolfram.com/SolidAngle.html
http://mathworld.wolfram.com/Radian.html
http://mathworld.wolfram.com/Angle.html

 

[candice_84]

This integration is only correct if we assume neutrons are produce and move uniformly. Its better to not convert it to scalar format, am i right?