Free Electron Model. Assume N electrons that are non-interacting (identical) fermions Assume they are confined to a box of V=L 3

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Cassandra Barber
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1 As we mentioned previously, the Pauli exclusion principle plays an important role in the behavior of solids The first quantum model we ll look at is called the free electron model or the ermi gas model It s simple but provides a reasonable description of the properties of the alkalai metals (Li, Na, ) and the nobel metals (Cu, Ag, Au) 1

2 Assume N electrons that are non-interacting (identical) fermions Assume they are confined to a box of VL 3 Hopefully you recall the energy levels of the 3D infinite potential well π h ml n n n x y z x y + ( n + n n ) Note as L grows large the discrete energy spectrum becomes continuous What is the ground state when N is large? Assume T0 to start with z

3 Consider a number space to help count the number of states 3

4 Then we can write π h x ml The number of 1 N r π 3 And at T ( n + n + n ) N r ( ) πr 8 3 We can rewrite this in terms of 1 3/ states 1 0 the ermi energy is highest occupied energy state N r 1 π 3 3/ y z r 1 up to radius r is energy the just 4

5 We can then solve for the ermi energy 1 3N π 3/ h π m 3N 3 πl Many metals have ~1 free electron/ion or nn/l 3 ~ 10 9 e/m 3 Thus the ermi energy for most metals is ~ 10 ev And the ermi temperature T /k B is ~ 10 5 K xtremely high and much greater than T for room temperature ~ 300 K But this is not the temperature of the electron gas, rather it is a measure of where the ermi energy is at / 3 5

6 N/L 3 6

7 and T 7

8 Comment Once you know the ermi energy you can also calculate the ermi momentum and the ermi velocity p v p m m m The ermi velocity is the same order of magnitude as the orbital velocity of the outer electrons in an atom and ~10 times the mean thermal velocity of a non-degenerate electron gas It s still non-relativistic however 8

9 The density of states g() is defined as dn π 3/ 1/ g( ) 1 d It can also be written in terms of g( ) 3N 3/ 1/ Thus we can calculate n()g() D Recall D Then n( ) 1for < g( ) for < and 0 for > and 0 for > 9

10 g() and n() 10

11 As we learned earlier, once you know n() you can calculate many thermodynamic quantities Sanity check N N 0 3 n( ) d N 3/ 3 o g( ) d 3/ N 11

12 Total energy U U n( ) d N 3/ 5 5/ g( ) d 3 5 N I used U here to indicate the total energy 1

13 Recall our discussion earlier in the semester on the molar heat capacity of solids f f NkBT nrt 1 d f CV R n dt or solids C 3R 5 J/molK V This is called the DuLong - Petit law But conduction electrons in a metal should contribute an additional (3/)R The observed electronic contribution is only ~0.0R 13

14 We can calculate the electronic contribution in the free electron model U C V T However up to this point we have been working at T0 in order to calculate C V so we have to calculate U(T) for T>0 U( T ) U( T ) 0 n( ) d 0 0 g( ) e β 1 ( ) g( ) + 1 D d 14

15 We could do the integral but it s more important to notice that in heating a ermi gas, we populate some states above and deplete some states below This modification is only important in a narrow energy range k B Taround k B T 15

16 The fraction of electrons transferred to higher energy is The energy increase of these electrons is So the increase in internal energy is Making the molar heat capacity k k B B N C V T T / ( k T ) B (for 1mole, N / U N T kbt N R Av ) T T 16

17 The correct proportionality constant is found doing the integral C V π N Av k B kbt π R T T The point is, that at room temperature, T<<T and the electronic contribution to the heat capacity is small This is because only a small fraction of electrons around can be thermally excited because of the Pauli exclusion principle This was one of the big successes of the free electron model and not predicted classically 17

18 The free electron model for metals can also be used to calculate lectrical conductivity Although to get agreement with experiment, long collision lengths (mean free paths) were required The source of how electrons can move ~ 1 centimeter (for a pure metal at low T) without scattering was unknown Thermal conductivity Although the classical calculation of thermal conductivity (fortuitously) gave good agreement with experiment also 18

N independent electrons in a volume V (assuming periodic boundary conditions) I] The system 3 V = ( ) k ( ) i k k k 1/2

N independent electrons in a volume V (assuming periodic boundary conditions) I] The system 3 V = ( ) k ( ) i k k k 1/2 Lecture #6. Understanding the properties of metals: the free electron model and the role of Pauli s exclusion principle.. Counting the states in the E model.. ermi energy, and momentum. 4. DOS 5. ermi-dirac