${\mathrm{m}}^{*}=\frac{{\hslash }^2}{{\displaystyle \frac{{\mathrm{d}}^2\mathrm{E}}{\mathrm{d} {\mathrm{k}}^2}}}$

${\mathrm{m}}^{*}=\frac{\hslash }{{\displaystyle \frac{{\mathrm{d}}^2\mathrm{E}}{\mathrm{d} {\mathrm{k}}^2}}}$

${\mathrm{m}}^{*}=-\frac{{\hslash }^2}{{\displaystyle \frac{{\mathrm{d}}^2\mathrm{E}}{\mathrm{d} {\mathrm{k}}^2}}}$

none

Classical free electron theory

Quantum free electron theory

Band theory

none

Free electron gas

Thermal motion of free electrons

Drift motion of free electrons

All

$\mathrm{E}=\frac{{\mathrm{h}}^2}{8{\mathrm{mL}}^2}({\mathrm{n}}_{\mathrm{x}}^2+{\mathrm{n}}_{\mathrm{y}}^2+{\mathrm{n}}_{\mathrm{z}}^2)$

$\mathrm{n}=({\mathrm{n}}_{\mathrm{x}}^2+{\mathrm{n}}_{\mathrm{y}}^2+{\mathrm{n}}_{\mathrm{z}}^2)$

$\mathrm{n}=\frac{1}{8}\times \frac{4\mathrm{\pi }}{3}{\mathrm{n}}^3$

None

Drude

Lorentz

Somerfield

None

Pauli Exclusive

Hund's 

aufbau

None

impurities and crystal defects

thermal vibration

non - periodicity of lattice potentials

all of above

electrical conductivity

specific heat and thermionic emission

paramagnetism

all

the difference between conductors, semiconductors, and insulators

positive Hall coefficient of metals

lower conductivities of divalent metals than monovalent metals

all

approximate model

real model

both a and b

none