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In basic scattering theory, $d\Omega$ is supposed to be an element of solid angle in the direction $\Omega$. Therefore, I assume that $\Omega$ is an angle, but what is this angle measured with respect to? None of the textbooks I am referring to give a clear indication of what this $\Omega$ is. The only relevant figure I could find is given below.

enter image description here

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I've nothing to add to Lubos' comments, except to say that the shaded bit in the diagram (between $\Theta$ and $\Theta + \mathrm{d}\Theta$ is where the solid angle at the drawn scattering angle (i.e. $\Omega(\Theta)$ intersect the drawn sphere). –  dmckee Aug 20 '12 at 15:23

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Sorry, a solid angle is something different than an ordinary angle, see

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

so it is not measured "with respect to anything". Solid angle $\Omega$ measures the size of a set of directions in the 3-dimensional space via the formula $$ \Omega = \frac{A}{R^2} $$ where $A$ is the area of the intersection of all these directions (semi-infinite lines) with a two-dimensional sphere of radius $R$. For example, in your picture, all the directions (semi-infinite lines) start at the interaction point and the relevant area is the annulus (a part of the sphere) that is dashed. The letter $\Omega$ (capital omega, the solid angle, an "O" with a hole at the bottom standing on two feet), shouldn't be confused with $\Theta$ (capital theta, an "O" with an "H" inside it) which is an ordinary angle, the polar angle in the spherical coordinates. If the solid angle has the shape of an annulus (going over all the azimuth angles $\phi$), then $d\Omega = 2\pi\sin\Theta\,d\Theta = 2\pi\cdot d(\cos\Theta)$. For an infinitesimal rectangle on the spherical surface in spherical coordinates, $d\Omega = \sin\Theta\,d\Theta\,d\phi $

You calculate its area $A$, divide it by the squared distance of the this area from the center (i.e. by the squared radius of the sphere), and you obtain the solid angle $\Omega$.

The "total" solid angle surrounding the point is $4\pi\approx 12.6$, much like the total angle in the plane is 360 degrees or $2\pi$ (in radians). The word "solid" indicates that the solid angle is a property of a solid (filled) three-dimensional body composed of all the semi-infinite lines (rays) starting from the center and going in the allowed directions; it is a generalization of the ordinary angle in which one spatial dimension is added but it is not the same thing.

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