MIT OpenCourseWare http://ocw.mit.edu 8.512 Theory of Solids IISpring 2009For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Lecture 4 3.1 Thomas-Reiche-Kuhn or f-sum rule Motivation :- One can derive following equation for one partice energy levels �2m �n − ( �0)|⇒n|xˆ|0→|2= 1 (3.1) n Proof :-1 = −i⇒0|[x,ˆ pˆ]|0→ 1 = �⇒0|xˆ|n→⇒n|pˆ|0→ − ⇒0|pˆ|n→⇒n|xˆ|0→ n ˆpˆ = im[H, xˆ] � ⇒n|pˆ|0→ = im(�n − � )⇒0|xˆ|n→ 0� � 1 = 2m(�n − �0)|⇒n|xˆ|0→|2n One can extend the above formalism to condensed matter systems. As wave functions are extended, position operator becomes unbounded for an infinite system. Thus we use density operator instead of position operator. Final equation that can be obtained would be � (ωn − ω ˆ20)|⇒n|χ†q= Nq 2¯|0→| /2m (3.2) n One can see that for localised states, exp(iq¯.r¯) � 1 + iq¯.r¯ And two sides of above equation look reasonable. Consider, ⇒0| ˆ2 [[χ† †q¯ˆ , H], χˆq¯]|0→ =�(En − E0)|⇒n|χˆq¯|0→| +�(Em − E0)|⇒m|χˆ |0→|2q¯(3.3) n m By time reversal symmetry, eigenfunctions of hamiltonian πn, are real. Thus, ⇒n|χˆ† 3 � q¯ |0→ = �d rπn �exp(iq¯.r¯i)π0i ⇒n|χˆ†| d3q¯0→ �= �rπn �exp(−iq¯.r¯i)π0i = ⇒n|χˆq¯|0→ 2 � † � nq = 2m (En − E0)|⇒n|χˆ2q¯|0→|n 1� � � � By using the expression for the fluctuation function one can rewrite the above expression d��S(¯q, �) = Nq 2/2m (3.4) 0 In case of neutron scattering from phonons, ri represents the lattice position and m should be replaced by the ion mass M. 3.2 Longitudinal f sum rule Recall :-D(q, �) = ρ(q, �)E(q, �) � · D(q, �) = 4∂χext � · E(q, �) = 4∂χtot Define, eD = �Uext eE = �U Thus, Uext(q, �)U(q, �) = ρ(q, �) From linear responce n(q, �) = η(q, �)Uext(q, �) 1 E||(q, �) U ρ= = ||(q, �) D||(q, �) Uext 4∂e2 < n > = 1 + q2 Uext 1 4∂e2 = 1 + η(q, �) (3.5)ρ||(q, �) q2 Now, imaginary part of dielectric constant corresponds to imaginary part of responce function which is density correlation function. Thus from above equation we see that 1 −∂ �2d��Imρ||(q, �)= 2 pl (3.6) 0 2� This is the longitudinal f-sum rule. Define � that can be used for approximations, n(q, �) = �(q, �)U(q, �) (3.7) D||(q, �) Uextρ||(q, �) = = E||(q, �) U 4∂e2 < n > = 1 − q2 U 4∂e2 = 1 − �(q, �) q2 If we approximate � by �0 for free fermions, we recover the RPA for ρ(q, �). 3.3 Conductivity sum rule Ohm’s law and continuity equation can be written as, j||(q, �) = ε||(q, �)E||(q, �) (3.8) �n � · j − e = 0 �t q · j(q, �) + e�n(q, �) = 0 qj||(q, �) + e�n(q, �) = 0 Where E||(q, �) is internal electric field. 4∂e2 < n > ρ||(q, �) = 1 − q2 U 4∂e2 < n > = 1 − i q E|| 4∂e j||= 1 + i � E|| Hence we obtain the important relation : 4∂i ρ||(q, �) = 1 + ε||(q, �) (3.9) 3�  � ” � � � � � � � Thus imaginary part of conductivity is related to real part of dielectric constant which is in turn related to real part of response function. For large frequency limit one can approximate the responce function as (� � � + i�), � 1 1 η = |⇒0|χˆq¯|n→|2 + n � − En + E0 � + En − E0 � |⇒0|χˆq¯|n→|2 2(En − E0) �2 − (En − E0)2 n � − E0)� |⇒0|χˆq¯|n→|2 2(En �2 n Where we have taken the limit of large frequency in the third step. Using the sum rule 3.2 we get �2 η � pl (3.10)�2 Using equations 3.5 and 3.10 we conclude, 1 �2 � 1 + pl (3.11)ρ(q, �) �2 �2 lim ρ(q, �) = 1 − pl (3.12)��� �2 On the other hand from equation 3.9 we see that : −� ε (�) = (ρ − 1) (3.13)4∂ In large frequency limit Kramers-Kronig relation can be written as � ” d� ε (� )ε (�) = − (3.14)2∂ � Combining the last three equations one gets the desired conductivity sum rule, d�ε ||(�, q) = �2 (3.15)pl/8 0