- #1
Selveste
- 7
- 0
Since the integrand is spherically symmetric I use spherical coordinates
[tex] \int d\vec{r} = \Omega_d\int_{0}^{\infty}dr r^{d-1} [/tex]
where [itex] \Omega_d [/itex] is the solid angle i [itex] d [/itex] dimensions. Since I am to plot as a function of [itex] 1/\beta U_0 (= k_bT/U_0) [/itex], I introduce a new varible [itex] \Theta = 1/\beta U_0 = k_bT/U_0 [/itex]. This gives
[tex] B_2(T) = \frac{1}{2} \Omega_d \Bigg( \int_0^R dr r^{d-1} + \int_R^{\infty} dr r^{d-1}(1-e^{\frac{1}{\Theta}(\frac{R}{r})^{\alpha}}) \Bigg)[/tex]
Now I wonder if this is correct. And how to procede to compute the integrals numerically. Here [itex] R [/itex] is not given. Should I just set it to some value? I thought about integrating with respect to some varible [itex] Rr [/itex] or [itex] r/R [/itex], but I can't seem to get rid of [itex] R [/itex].
[tex] \int d\vec{r} = \Omega_d\int_{0}^{\infty}dr r^{d-1} [/tex]
where [itex] \Omega_d [/itex] is the solid angle i [itex] d [/itex] dimensions. Since I am to plot as a function of [itex] 1/\beta U_0 (= k_bT/U_0) [/itex], I introduce a new varible [itex] \Theta = 1/\beta U_0 = k_bT/U_0 [/itex]. This gives
[tex] B_2(T) = \frac{1}{2} \Omega_d \Bigg( \int_0^R dr r^{d-1} + \int_R^{\infty} dr r^{d-1}(1-e^{\frac{1}{\Theta}(\frac{R}{r})^{\alpha}}) \Bigg)[/tex]
Now I wonder if this is correct. And how to procede to compute the integrals numerically. Here [itex] R [/itex] is not given. Should I just set it to some value? I thought about integrating with respect to some varible [itex] Rr [/itex] or [itex] r/R [/itex], but I can't seem to get rid of [itex] R [/itex].