ABSTRACT
In Sec. 6.4.2 we identified phonons in metals as plasma oscillations of the ionic lattice in a medium whose dielectric function is determined by the conduction electrons. Here we address the problem of collective plasma oscillations of the homogeneous electron gas. We have already seen that in the linear response approximation, the electric potential ϕ ( q , ω ) $ \phi (q, \omega ) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math7_1.tif"/> in a medium subjected to an external potential ϕ ( 0 ) ( q , ω ) $ \phi^{(0)} (q, \omega ) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math7_2.tif"/> is given by ϕ ( q , ω ) = ϕ ( 0 ) ( q , ω ) / ε ( q , ω ) $ \phi (q, \omega ) = \phi^{(0)} (q, \omega )/\varepsilon (q, \omega ) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math7_3.tif"/> . Since collective excitations sustain even when no external source is present (i.e. ϕ ( 0 ) ( q , ω ) = 0 ) $ \phi^{(0)} (q, \omega ) = 0) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/inline-math7_4.tif"/> , the condition for collective charge (or plasma) oscillations of the electron gas is ε ( q , ω ) ≡ 1 - 1 q 2 ε 0 Π ( q , ω ) = 0 . $$ \varepsilon (q,~\omega ) \equiv 1 - \frac{1}{{q^{2} \varepsilon _{0} }}\Pi (q,~\omega ) = 0. $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351121996/259a876e-1ad1-4598-839f-e9587e88d783/content/math7_1.tif"/>
