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

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Discussion Overview

The discussion centers around the concept of integrating over 4π in the context of solid angles in three-dimensional space. Participants explore the significance of this integration in relation to angular flux and neutron transport, examining the mathematical and conceptual implications of solid angles.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants explain that integrating over 4π represents integration over all directions or orientations in three-dimensional space, as 4π is the total solid angle.
  • Others assert that 2π is sufficient for two-dimensional contexts, while in three dimensions, 2π only accounts for half of the solid angle (hemisphere).
  • One participant notes the relationship between the area of a sphere and solid angles, stating that the area of a sphere is 4πr², which leads to the conclusion that the total solid angle is 4π steradians.
  • Another participant questions whether solid angle can be considered a volume, clarifying that solid angle is distinct from volume, similar to how angle in two dimensions is distinct from area.
  • A later reply suggests that the integration over solid angles is only valid under the assumption that neutrons are produced and move uniformly, proposing that converting to scalar format may not be advisable.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and implications of integrating over 4π, with some agreeing on its significance in three dimensions while others challenge the assumptions underlying this integration. The discussion remains unresolved regarding the appropriateness of scalar conversion in the context of neutron transport.

Contextual Notes

Participants highlight the dependence on assumptions about neutron behavior and the definitions of solid angles. The discussion includes references to mathematical relationships and the nature of solid angles, but does not resolve the implications of these factors.

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

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


2pi = 360 which is enough.
 
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candice_84 said:
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
 
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solid angle is a volume?
 


candice_84 said:
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
 


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?
 

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