update solution 2.2

src/4_sommerfeld_model_solutions.md

@@ -39,10 +39,20 @@ Distance between nearest k-points is $\frac{2\pi}{L}$ and their density across n
 2.
 
 $$
-g(k)dk = \left(\frac{L}{2\pi}\right)^n\int_k d\bf k
+g_{1D}(k)dk = \left(\frac{L}{2\pi}\right) 2 .dk
 $$
 
-Check how to convert volume to surface intergral and surface to line integral.
+Factor 2 is due to positive and negative k-points having equal energy.
+
+$$
+g_{2D}(k)dk = \left(\frac{L}{2\pi}\right)^2 2\pi k . dk
+$$
+
+$$
+g_{3D}(k)dk = \left(\frac{L}{2\pi}\right)^3 4\pi k^2 . dk
+$$
+
+$4\pi k^2$ is the volume of spherical shell enclosed between k and k + dk.
 
 3, 4.
 
@@ -190,3 +200,4 @@ $$
 $$
 C_v(T) = \frac{\partial E}{\partial T} = \frac{3Ak_B^2T^2}{\pi c^2}
 $$
+