Solid Thermodynamics (1)

Transcription

1 Solid Thermodynamics (1) Class notes based on MIT OCW by KAN K.A.Nelson and MB M.Bawendi

2 Statistical Mechanics 2

3 1. Mathematics 1.1. Permutation: - Distinguishable balls (numbers on the surface of the balls): Number of distinguishable balls: n Total number of arrangements (or permutations): n(n-1)(n-2)(n-3) 1=n! - Distinguishable balls(n) into different sites(n): Number of sites: N Number of distinguishable balls: n (N>n) Total number of arrangements: 1.2. Combination: N! = P (N-n)! N(N-1))(N-2) (N-n+1)= N n - Not-distinguishable balls(n) into different sites(n): : n balls n 3P 2 : sites Ball: a,b,c. : N sites - Number of arrangements: N! 1 N! = = NC (N-n)! n! (N-n)!n! n 3C 2 Mathematics 3

4 1.2. Special Combinatorial Problems A. Putting distinguishable i obects(n) into boxes(g). ( ) (g>n) Condition: a limit of no more than one obect per box. : g boxes, N distinguishable obects. Number of ways= = g! gp= N (g-n)! B. Putting indistinguishable i obects with a limit it of no more than one obect per box. g! Number of ways= = g C N = (g-n)!n! C. Putting distinguishable obects. No limit on the number per box. : g boxes, N distinguishable obects. Each obect can have g choices. Number of ways = = g gggg.. g = g N No limit N Mathematics 4

5 D. Putting indistinguishable obects. No limit on the number per box. : g boxes, N indistinguishable obects. Assuming distinguishable obects first. O 1 O 2 O 3 O 4 O N No limit Reference position (g+n-1) B1 O 2 O 7 O 5 B5 B3 O 4 B6 O 1 O 9 B (g+n) symbols Rearrangement is possible! B1 O 2 O 7 O 5 B6 O 1 O 9 B3 O 4 B5 B Boxes: Possible number of arrangements: (g-1)! Balls: Possible number of arrangements: N! Number of ways (arrangements)= = (g+n-1)! (g-1)!n! Mathematics 5

6 1.3. Lagrange Multipliers A formal procedure for determining a maximum point in a continuous function subect to one or more constraints. : continuous function. f ( x, y, x) f f f df dx dy dz 0 x y z at maximum (or minimum) Since x, y, z are independent, changes dx, dy, dz are also independent. Hence at maximum, f f f 0 x y z But x, y, z are interrelated to each other, gxyx (,, ) 0 ---(*) () Then eq.(*) is no longer valid! How can we get the condition for maximum (minimum)? > Lagrange multiplier method. g g g dx dy dz 0 x y z f g f g f g dx dy dz 0 x x y y z z Then, the conditions for maximum (or minimum) f become, f g f g f g 0 0 x x y y z z 0 Lagrange multiplier, should be chosen. Mathematics 6

7 1.4. Stirling s formular ln N! N ln N N 1.5. Important integrals n ax xe 0 2 dx n=0, n=1, n=2, n=3, 1 2 a 1 2a a 1 2 2a Mathematics 7

8 2. Important Statistics Degeneracy No. of particles g N g 3 = 5 N 3 g 2 = 2 g 1 = 3 N 2 N Boltzmann statistics Particles are distinguishable. No limit on the number of particles per quantum state. N g for energy N N! N g Number of ways = = g N! N! N! Number of arrangements a e of N particles into groups N 1, N 2, N 3,.. N.. Important Statistics 8

9 2.2. Bose-Einstein Statistics Particles are indistinguishable. No limit on the number of particles per quantum state. ( g N 1)! ( g 1)! N! for energy Number of ways = = 2.3. Fermi-Dirac Statistics ( g N 1)! ( g 1)! N! Particles are indistinguishable. Limit i of no more than one particle per quantum state. g! ( g N )! N! for energy Number of ways = = g! ( g N )! N! Important Statistics 9

10 Example 1) Boltzmann statistics N 3 1 g 3 4 N! 4! 432 I N! 3! 1! II 4! 1296 III 4! 324 2! 1! 1! 1! 3! 2) Bose-Einstein statistics ( g N 1)! ( g 1)! N! 5! 4! I 40 2!3! 3!1! 4! 3! 3! II 72 2!2! 2!1! 2!1! 3! 5! III 30 2!1! 2!3! 3) Fermi-Dirac i statistics i g! ( g N )! N! 3! 4! I 4 0!3! 3!1! 3! 3! 4! II 36 1!2! 2!1! 3!1! 3! 3! III 3 1!2! 3!0! Important Statistics 10

11 2.4. Equilibrium Distribution N N 1 and U N 1 N 1. Boltzmann statistics g N! N! ln ln N! N ln g ln N! ln Nln N N ( N ln g N ln N N ) Nln N ( N ln g N ln N ) To get the most probable distribution, the maximum w, d(ln w)=0) d(ln ) (ln g dn ln N dn dn ) and dn 0 g d(ln ) (ln g dn ln N dn ) ln dn N And du dn 1 Using the method of Lagrange multipliers, ln g dn J ---- > N g ln 0 N N g e e Important Statistics 11

12 2) Bose-Einstein statistics Generally, g >> 1 ( g N 1)! ( g 1)! N! ( g N )! ln ln ln ( g N )! ln g! ln N! ( )!! g N ln ( g N ) ln g N ( g N ) g ln g g N ln N N ( g N ) ln g N g ln g N ln N To get the most probable distribution, the maximum w, d(ln w)=0 0 g N d(ln ) ln g N dn dn ln N dn dn ln dn 0 N Using the method of Lagrange multipliers, g N g ln ln 1 0 N > N N g ee 1 Important Statistics 12

13 3) Fermi-Dirac statistics g! ( g N )! N! g! ln ln ln g! ln ( g N )! ln N! ( g N )! N! ln g ln g g ( g N ) ln g N ( g N ) N ln N N g ln g ( g N ) ln g N N ln N To get the most probable distribution, the maximum w, d(ln w)=0 g N d(ln ) ln g N dn dn ln N dn dn ln dn 0 N Using the method of Lagrange multipliers, g N g ln ln 1 0 N > N N g ee 1 Important Statistics 13

14 In physical systems, the particles being considered d (electrons, atoms ) are indistinguishable and hence must obey either Bose-Einstein or Fermi-Dirac statistics. The Boltzmann model for a system of independent particles is not really a reasonable picture of anything existing in nature. However, most gases at low to moderate density, the number of quantum states available at any level is much larger than the number of particles in the level. N > ee 1 g ( g N 1)! Consider Bose-Einstein statistics w ( g 1)! N! ( g N 1)! N ( g N 1)( g N 2)( g N 3) g g ( g 1)! g! And Femi-Dirac statistics, ti ti w ( g N )! N! g! g ( g 1)( g 2) ( g N 1) g ( g N )! N Comparing to Boltzmann statistics, we can conclude; Bl Boltzmann N N Hence, Hence, N g N N g N i BEFD N! :effect of discounting the distinguishability of N! the N particles Important Statistics!! 14

15 3. Relation to Macroscopic Properties System n, E, Entropy: Boltzmann s hypothesis The entropy of the system (S) is given by where K is Boltzmann constant (R/N AVO ) Stirling s approximation: Relation to Macroscopic Properties 15

16 ni And Pi : probability of ni in N N 0 S kn Pln P nr Pln P 0 r i i i i i 1 i 1 r 0 Entropy 16

17 Entropy 17

18 Using Entropy 18

19 Evaluation of the isolation constraints Isolated System The constrained maximum in the entropy function (ds=0), Evaluation of the isolation constraints 19

20 or, Define the partition function by, Then, Evaluation of the isolation constraints 20

21 The volume term has no counterpart in the statistical expression for the entropy, because in this introductory development of statistical thermodynamics it is assumed that the average volume occupied by an atom is the same for all energy levels. - Evaluation of the isolation constraints 21

22 Evaluation of the isolation constraints 22

23 Macroscopic properties from the partition function A: Helmholtz Free Energy or, Macroscopic properties from partition function 23

24 P A V T G H TS U PV TS APV H U PV Macroscopic properties from partition function 24

25 Application of the algorithm A model with 2 energy levels Model with 2 energy levels 25

26 Translational Partition Function 26

27 Translational Partition Function 27

28 Configurational Partition Function 28

29 Separation of Partition Functions 29

30 Separation of Partition Functions 30

31 Separation of Partition Functions 31

32 Separation of Partition Functions 32

33 Calculation of Macroscopic Properties 33

34 Calculation of Macroscopic Properties 34

35 Calculation of Macroscopic Properties 35

36 Calculation of Macroscopic Properties 36

37 Calculation of Macroscopic Properties 37

38 Calculation of Macroscopic Properties 38

39 Calculation of Macroscopic Properties 39

40 Calculation of Macroscopic Properties 40