Thermodynamics of finite quantum systems

Transcription

1 Thermodynamics of finite quantum systems Shell structure in finite quantum systems Erice summer school, July 25-30, 2010 Klavs Hansen Department of Physics, University of Gothenburg

2 Topics Metal clusters valence electrons, quantum order, fermion systems Rare gas clusters classical equations of motion

3 What is a metal cluster : MN, N countable copper atoms Source: DESY Spherical, good conductor = free electrons in spherical mean field potential

4 Mean field spherical potential

5 Clusters have 1) Nuclear degrees of freedom (vibrations, phonons) 2) Electronic degrees of freedom Vibrations: N atoms, 3N-6 vibrational degrees of freedom, Quantum energies: He droplets ev Na 0.01 ev C 0.1 ev Several vibrational quanta in each mode, E (3N-6) kbt, heat capacities C (3N-6)kB

6 Electrons: N atoms, zn valence electrons (z small integer) Fermions, only top kbt energy layer thermally excited Li Na K EF 4.7 ev 3.2 ev 2.1 ev TF K K K EF Thermal excitation energy zn (kbt)2/ef Heat capacity znkb (kbt)/ef

7 Classical vs. quantum statistics of N electrons partition functions for equidistant spectrum at T = 5000 quantum mechanical Z classical Z 4000 ln(z cl ), ln(z qm ) N

8 Sodium clusters, abundance spectrum N=20 N=8 N=40 N=58 W.D.Knight et al., Phys. Rev. Lett. 52 (1984) 2141

9 Sodium clusters, abundance spectrum, larger clusters S.Bjørnholm et al., Phys.Rev.Lett. 65 (1990) 1627 N

10 M.Y.Chou et al., Solid state Comm. 52 (1984) 645

11 Level diagram for Saxon-Woods potential: Single particle density of states Nishioka et al. PRB 42, 9377 (1990) N=1000 N=3000 Lesson For large enough N, level spacing < kbt, thermal excitations of electrons unavoidable!

12 Shell energy of sodium clusters Calculated in mean-field single particle potential shell supershell Nishioka et al., PRB 42 (1990) 9377

13 This is what we were finite T

14 Single particle states in the Woods-Saxon potential

15 angular momentum number of nodes in radial wavefunction

16 Condition for degeneracy (shell structure) E E nr l =0 n r l nr and l quantum numbers, i.e. integers. If E E : = ratio of small integers n r l then bunching of levels, shell structure

17 Interpretation of degeneracy/shell structure Semiclassically: ωr/ωa = E = ℏ r n r 2/1 pendulating Electron orbit The cluster, confining electrons 3/1 triangular E = ℏ l l 4/1 square

18 Lengths of orbits pendulating L = 4 rn triangular L = 3.31/2 rn = rn square L = 4.21/2 rn = rn (L +L )/2.λF-1 = r1 N1/3 / r1 = N1/3 = N1/3 / Every time N1/3 is increased by a new shell appears

19 Experimental result

20 How do we do thermodynamics with these systems? Fermi-Dirac distributions (grand canonical ensemble) (independent particles, standard treatment) sounds promising But: particles not conserved, (even for free clusters) Ei n i= 2 i e E i 1 e = 1 1 e Ei 2 i =0 1 n i 1 ni n =ni 1 ni k B T 1

21 Calculation for equidistant spectrum w. spacing

22 The microcanonical ensemble then? (free, isolated particles, sounds promising) But: Electrons equilibrate with the vibrations which have a much higher heat capacity. Electrons in metals are NOT isolated systems (for very long)

23 Electron-vibration coupling time by femtosecond pump-probe experiments M.Maier et al. PRL (2006)

24 Canonical, then? The level density, E = derivative wrt. energy of the total number of quantum states below E for the system = the 'number of states' at energy E = the microcanonical partition function = the exponential of the entropy

25 Level density of valence electrons of C60 NB: femtosecond experiments level density (1/0.02eV) ke ( E, ε ) d ε = meσ (ε ) ρ d ( E Φ ε ) ε dε π 2 h3 ρ p ( E) Comparison of theoretical and experimental electronic level densities J/cm 2 2 J/cm J/cm 2 4 J/cm J/cm 3 J/cm Daughter level density Parent level density Daughter level density, theoretical input Excitation energy (ev) 100

26 Electron emission from hot molecules ln(yield) [arb. units] 8 C14H10 6 Y exp(-ε/kbt). C22H12 4 T (Electron temperatures) up to K. C24H12 C60 Laser parameters: λ=780 nm, τ=150 fs, Ι= W/cm2. C C70 ns-laser Kinetic Energy [ev] 6 7 8

27 Partitioning of energy between the vibrations and the electrons E = vib E E el el E el de el Motion on a Born-Oppenheimer surface Excitation onto a Born-Oppenheimer surface

28 Energy partitioning, continued 1) The energy stored in the vibrations is much bigger than that stored in the electronic excitations expand in Eel 2) The level density is a rapidly growing function of energy expand ln(ρ(ε)) d ln vib E ln vib E E el ln vib E E el de Eel vib E Eel vib E e

29 Finally E el E vib E el E el e de el = vib E Z el Zel is the canonical partition function of the electron system

30 An excursion into the zoo of computational methods and results Start with the simplest model (ladder levels, harmonic oscillator) E n=n ; n=0,..., ; g n = 1 EF ~ N Simplest model of Fermi gas

31 Calculation of the canonical partition function N 1 Z, N = Z, N 1 e N Z, N e

32 Solution to recurrence relation: Z, N = N j=1 1 j 1 e Curious facts: i) partition function of N harmonic oscillators with spectrum hωj = j ii) partition function of bosonic system with same spectrum

33 Thermal energy of the equidistant spectrum E = E = E E e j =1 de E E e de j j e 1 n=1 ln Z = 1 T2 2 T 2 = 2 6 n Technical note: first term in Euler-Maclaurin series

34 Calculation of level densities ( Reminder: Z = ~e S ) E E e de [incoherent mumbling ] 2 E = E 0 2 C T e 0 E 0 Z T e E0 Find T as solution to 1 2 CT 2 E T =E 0 k B T

35 Quality control: Helium droplets (ripplon) level density, comparison with exact count Phys. Rev. B 76 (2007)

36 Level densities of equidistant spectrum ln Z integrate E = E Z 6 to get 1 2 CT e E0 3

37 Let's try a quasirealistic spectrum E ( ) 20 E ( ) E ( ) add spin degeneracy N j= N N e e... j 2 j 2 j 2 j=1 j=1 1 e 1 e 1 e 2 2

38 Total partition functions convolution of all electron partitionings Z N =even = j =1 1 e 2 m= e j 2 m m 1 Z N =odd = j=1 1 e m= e j 2 low temperature limt easy m

39 High temperature limit S x = m= e m m x 2 x / 4 = e m = e (we want S(0), S(1)) Periodic in x: Fourier expansion 2 1 /2 T T / s 0 = 1 2 e /2 T s 1 = e / e T / m x/ 2

40 Comparison to exact calculation:

41 AuN+ dissociation energies w. L. Schweikhard

42 Laser excitation and evaporation of AuN+ in the Greifswald Penning trap

43 Numerical calculation of the thermal properties of mean field electrons define partition function of n electrons and nl levels

44 Finite temperature shell energies O.Genzken, M.Brack Phys.Rev.Lett. 67 (1991) 3286

45 Experiments, Niels Bohr Institute Copenhagen Theory, Regensburg M.Brack Rev. Mod. Phys. 65 (1993) 677

46 Calculation of level densities

47 Another recurrence relations Ez = E E E e de

48 One more for the heat capacity

49 Heat capacities

50 Heat capacities, small clusters

51 100K to 1000K in 100K steps

52 That was it