Normalization of a pair-wise radial radial distribution function with solid angle dependence

Question

I have 3D position data of particles. I need to calculate g(r, $theta$). Where g(r, $theta$) is defined as:
$$
g(r, theta) = frac{H(r, theta)}{N cdot n(r,theta)}
$$
r is a Euclidean distance between particles and theta is the polar angle from particle moving direction to r (Explained in attached picture). H(r,theta) is the histogram for all N pairs.

Usually normalization will be performed by dividing H by N. But I also need to normalize by Volume around the particle. As further will go away from the particle more chances there are to find a particle there. I have found normalization function of the form of n(r, theta):
$$
n(r,theta) = frac{2}{3} pi [(r + delta r)^{3} - r^{3}][cos(theta) - cos(theta + deltatheta)] 
$$
First of all I do not understand how this function have come into be. Secondly, is this even correct?. Ideally it should be equal to 1 for particles moving in random direction.

GiorgioP · Accepted Answer

I would not call a function $g(r,theta)$ a radial distribution function. It does not depend only on the radial distance $r$.
I also think that your normalization is not fully consistent with the definition of this function. It should be such that $rho g(r,theta) dV$ counts the number of particles present in a system of average number density $rho$ in a volume $dV$ characterized by a distance $r$ and solid angle $2 pi sin(theta)mathrm{d}theta$ with respect to a particle fixed at the origin. Therefore, if the value of the histogram $H(r,theta)$ counts the number of particles in such a volume, it should be normalized by the number of particles that would be in the same volume if the density would be uniform. I.e. you should divide by $rho Delta V(r,theta)$.
Based on its meaning, the explicit  finite difference expression for
$Delta V(r,theta)$ can be obtained as the difference of differences of the volumes of four spherical sectors (a look at the picture of a spherical sector would help). If $V_s(r,theta)= frac{2 pi}{3}r^3(1-cos(theta))$ is the volume of the spherical sector of radius $r$ and angle $theta$,
$$
begin{align}
Delta V(r,theta)&= left[ V_s(r+delta r, theta +delta theta) - V_s(r+delta r, theta ) right] - left[ V_s(r, theta +delta theta) - V_s(r, theta ) right] &=frac{2 pi}{3}[(r + delta r)^{3} - r^{3}][cos(theta) - cos(theta + deltatheta)] 
end{align}
$$
which is precisely the formula you quote for your $n(r,theta)$. Notice however that (also for dimensional reasons) it should be multiplied by $rho$ and not by $N$.

Normalization of a pair-wise radial radial distribution function with solid angle dependence

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