Physics of Condensed Matter I

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1 Physics of Condensed Matter I INZ`PC Faculty of Physics UW Jacek.Szczytko@fuw.edu.pl

2 Summary of the lecture 1. Quantum mechanics 2. Optical transitions 3. Lasers 4. Optics 5. Molecules 6. Properties of molecules 7. Crystals 8. Crystalography 9. Solid state 10. Electronic band structure 11. Effective mass approximation 12. Electrons and holes 13. Carriers 14. Transport 15. Optical properties of solids

3 Classical and quantum universe Hydrogen atom: Eigenstates of L:

4 Classical and quantum universe Hydrogen atom: Eigenstates of L: Real functions:

5 Classical and quantum universe Hydrogen atom: Eigenstates of L: Spherical harmonics Y lm :

6 Classical and quantum universe Eigenstate: E.g. hydrogen wavefunction Stan 1s Quantum numbers!

7 Electric field Stark effect of hydrogen atom H ' pe dipole moment p electric field E eze z Eigenfunctions of hydrogen atom for 2p state: ψ 200, ψ 21 1, ψ 210, ψ 211 Perturbation H ij = ψ i H ψ j

8 Electric field Stark effect of hydrogen atom H ' pe dipole moment p electric field E eze z Eigenfunctions of hydrogen atom for 2p state: ψ 200, ψ 21 1, ψ 210, ψ 211 Perturbation H ij = ψ i H ψ j

9 Electric field Stark effect of hydrogen atom H ' pe dipole moment p electric field E eze z Atom in ligand field

10 Electric field Stark effect of hydrogen atom H ' pe dipole moment p electric field E eze z Atom in ligand field Crystal Field Theory (CFT) (magnetic properties as well as color.)

11 Magnetic field and spin Magnetic field: H = mb clasically: m = I ԦS = e T πr2 = Here m is the magnetic moment e 2πr/v πr2 = e 2 rv [Am 2 ] m r L = Ԧr m 0 Ԧv v L = L x, L y, L z

12 Magnetic field and spin Magnetic field: H = mb Here m is the magnetic moment clasically: thus: m = I ԦS = e T πr2 = m = e 2m 0 L = μ B ħ L e 2πr/v πr2 = e 2 rv [Am 2 ] m r Bohr magneton μ B = ħe 2m 0 μ B = 9, (57) J/T L = Ԧr m 0 Ԧv v H = mb = μ B ħ LB μ B = ħe 2m 0 L = L x, L y, L z

13 Magnetic field and spin Magnetic field: for B = 0,0, B z H = mb = μ B ħ LB Here m is the magnetic moment we have: H = μ B ħ L z B z = μ B B z m where m = l, l + 1, l 1, l Here m is the quantum number n, l, m m r the base: l, m l = 1 m = 1 m = 0 L = Ԧr m 0 Ԧv v m = 1 B = 0 B

14 Magnetic field and spin Magnetic field: for B = 0,0, B z H = mb = μ B ħ LB Here m is the magnetic moment we have: H = μ B ħ L z B z = μ B B z m where m = l, l + 1, l 1, l Here m is the quantum number n, l, m m r the base: l, m l = 1 m = 1 m = 0 L = Ԧr m 0 Ԧv v m = 1 B = 0 B

15 Magnetic field and spin Stern-Gerlach experiment (1922 r.)

16 What is the spin? What is mass? Mariusz Pudzianowski

17 What is the spin? What is momentum?

18 What is the spin? What is charge?

19 What is the spin? Spin? Sebastian Münster, Cosmographia in 1544 Disney

20 The history

21 The history Magnetyt (z 1750 r.) typowy ferryt i magnes z ziem rzadkich. Każdy z nich o gęstości energii 1J. A lodestone magnet from the 1750's and typical ferrite and rare earth used in modern appliances. Each of these produce about 1J of energy

22 What is the spin? How magnets work?

23 What is the spin? How magnets work?

24 What is the spin? How magnets work?

25 What is the spin? How magnets work?

26 What is the spin? How magnets work?

27 What is the spin? How magnets work? Moving charges produce a magnetic field

28 What is the spin? How magnets work?

29 What is the spin? How magnets work?

30 What is the spin? How magnets work?

31 What is the spin? How magnets work?

32 What is the spin? How magnets work? These smal magnets are electrons

33 What is the spin? How magnets work? These smal magnets are electrons

34 What is the spin? Skąd się biorą magnesy? A więc płynie jakiś prąd?

35 What is the spin? How magnets work? The movement of electrons around the nucleus? The spin of electrons around it s own axis? So, there is a current? Internal property of electrons?

36 What is the spin? How magnets work? The movement of electrons around the nucleus? The spin of electrons around it s own axis? So, there is a current? Internal property of electrons?! internal angular momentum SPIN

37 What is the spin? internal angular momentum Albert Einstein - Johannes Wander de Haas, Berlin 1914, Internal property of electrons!

38 What is the spin? projections of the spin on axis

39 What is the spin? projections of the spin on ANY of axes has only TWO values:

40 Magnetic field and spin Stern-Gerlach experiment (1922 r.)

41 Magnetic field and spin Spin, spin-orbit interaction Spin operators መS x, መS y, መS z, መS 2 ψ Ԧr, S z = ψ Ԧr χ S z Spinor መS x, መS y = iħ መS z, etc. Pauli matrices: σ x, σ y, σ z መS x = 1 2 ħσ x = 1 2 ħ መS y = 1 2 ħσ y = 1 2 ħ 0 i i 0 መS x = 1 2 ħσ z = 1 2 ħ projections of the spin on the axis z χ = 1 0, χ =

42 Magnetic field and spin Spin, spin-orbit interaction Spin operators መS x, መS y, መS z, መS 2 H = μ B ħ L + g መS B መS x, መS y = iħ መS z, etc. g-factor for the agreement with experiments Pauli matrices: σ x, σ y, σ z መS x = 1 2 ħσ x = 1 2 ħ መS y = 1 2 ħσ y = 1 2 ħ 0 i i 0 መS x = 1 2 ħσ z = 1 2 ħ projections of the spin on the axis z χ = 1 0, χ =

43 Magnetic field and spin Spin, spin-orbit interaction Spin operators መS x, መS y, መS z, መS 2 H = μ B ħ L + g መS B መS x, መS y = iħ መS z, etc. g-factor for the agreement with experiments Pauli matrices: σ x, σ y, σ z መS x = 1 2 ħσ x = 1 2 ħ g = ± መS y = 1 2 ħσ y = 1 2 ħ 0 i i 0 መS x = 1 2 ħσ z = 1 2 ħ projections of the spin on the axis z χ = 1 0, χ =

44 QED Quantum ElectroDynamics g = ±

45 Magnetic field and spin Spin, spin-orbit interaction Spin operators መS x, መS y, መS z, መS 2 H = μ B ħ L + g መS B g-factor for the agreement with experiments Total angular momentum operator መJ = L + መS, the base j, m j ൿ μ Total magnetic moment M = M L + M S = g B μ L L g B ħ S ħ መ S =1 =2 M መJ - magnetic anomaly of spin

46 Magnetic field and spin Spin-orbit interaction H SO = λl መS with the base n, l, s, m l, m s For s-states L = 0 L መS = 0 Total angular momentum operator መJ = L + መS, the base j, m j ൿ H SO = λl መS = λ 1 2 J2 L 2 S 2 = λ L z S z L +S + L S + fine-structure constant λ = hc A = Zα2 2 α = 1 r 3 e2 4πε 0 ħc Ry = hcr R = m ee 4 8ε 0 2 h 3 c R = 1, m -1 E SO = න ψ H SO ψ dv = Z න L መS ψ r3 ψ dv

47 Magnetic field and spin Spin-orbit interaction H SO = λl መS with the base n, l, s, m l, m s For s-states L = 0 L መS = 0 Total angular momentum operator መJ = L + መS, the base j, m j ൿ e.g. for ψ 210 we get H SO = λl መS = λ 1 2 J2 L 2 S 2 = λ L z S z L +S + L S + E SO = 1 L መS = ħ2 2 r 3 = 1 24 = 1 Ze 2 g s L መS 2 4πε 0 2m 2 c 2 r 3 1 r 3 = Z3 1 n 3 3 a B l l l + 1 Z a 0 j j + 1 l l + 1 s(s + 1) 3 and for general n (principal quantum number) Z 4 j j + 1 l l + 1 s(s + 1) a 3 0 n 3 2l(l + 1/2)(l + 1)

48 Magnetic field and spin Spin-orbit interaction H SO = λl መS with the base n, l, s, m l, m s For s-states L = 0 L መS = 0 Total angular momentum operator መJ = L + መS, the base j, m j ൿ തL ҧ S = 1 2 ҧ J 2 തL 2 ҧ S 2 = L z S z L +S + L S + L = 1; መS = P 3/2 2 P 1/ the base: n, l, s, j, m j ൿ shortly: j, m j ൿ

49 Fine structure The fine structure means the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the Schrödinger equation. We got corrections to the value of energy levels. Kinetic energy relativistic correction Spin-orbit coupling Darwin term

50 Fine structure The fine structure means the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the Schrödinger equation. We got corrections to the value of energy levels. Kinetic energy relativistic correction Spin-orbit coupling Darwin term

51 Fine structure The fine structure means the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the Schrödinger equation. We got corrections to the value of energy levels. Kinetic energy relativistic correction E = Ԧp 2 c 2 + m 0 2 c 4 The correct description of the atom requires taking into account the relativistic effects which lead to the Dirac Hamiltonian. The square root can be expanded into a series: E = m 0 c p2 2m 0 2 c 2 p4 2m 0 4 c 4 + = m 0c 2 + p2 p4 2m 0 8m 3 0 c 2 + thus kinetic energy can be expressed as E K = E m 0 c 2 = p2 2m 0 p4 8m 0 3 c 2 + and the Hamiltonian is: iħ t ħ2 Ψ Ԧr, t = 2m 2 + V Ԧr ħ4 8m 3 0 c 2 4 Ψ Ԧr, t

52 Tadeusz Stacewicz Fine structure iħ t ħ2 Ψ Ԧr, t = 2m 2 + V Ԧr ħ4 8m 3 0 c 2 4 Ψ Ԧr, t Using perturbation theory one can find correction due to the relativistic mass change for each principal quantum number and corresponding energy E n. For instance the electron speed of 1s in gold 79 Au v = 53% c! m = m 0 1 v2 c 2 ΔE n = E n 2 2mc 2 4n l = α2 Z 2 2n 4 E n n l α = e2 4πε 0 ħc Correction from this perturbation: Small for light atoms Rapidly decreases with the principal quantum number n Significant for large Z E n = mc2 α 2 2 Z 2 n

53 Fine structure The fine structure means the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the Schrödinger equation. We got corrections to the value of energy levels. Kinetic energy relativistic correction Spin-orbit coupling Darwin term H SO = λl መS = λ 1 2 J2 L 2 S 2 = λ L z S z L +S + L S + = 1 2 Ze 2 4πε 0 g s 2m 2 c 2 L መS r 3 e.g. for ψ 210 we get E SO = 1 r 3 = 1 24 Z a 0 3 and for general n (principal quantum number) Z 4 j j + 1 l l + 1 s(s + 1) a 3 0 n 3 2l(l + 1/2)(l + 1)

54 Fine structure The fine structure means the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the Schrödinger equation. We got corrections to the value of energy levels. Kinetic energy relativistic correction Spin-orbit coupling Darwin term Darwin term is the non-relativistic expansion of the Dirac equation: βmc 2 + c 3 α n p n n ψ Ԧr, t = iħ ψ Ԧr, t t Negative E solutiuons to the equation antimatter

55 Fine structure The fine structure means the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the Schrödinger equation. We got corrections to the value of energy levels. Kinetic energy relativistic correction Spin-orbit coupling Darwin term Darwin term is the non-relativistic expansion of the Dirac equation: β = βmc 2 + c α x = α n p n n ψ Ԧr, t = iħ α y = ψ Ԧr, t t i 0 0 i 0 0 i 0 0 i α z = α, β matrices

56 Fine structure The fine structure means the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the Schrödinger equation. We got corrections to the value of energy levels. Kinetic energy relativistic correction Spin-orbit coupling Darwin term Darwin term is the non-relativistic expansion of the Dirac equation: H Darwin = ħ2 8m 2 c 2 4π Ze2 4πε 0 δ Ԧr H Darwin = ħ2 8m 2 c 2 4π Only for s-orbit, because: ψ Ԧr = 0 = 0 for l > 0 Ze2 4πε 0 ψ Ԧr =

57 Fine structure The fine structure means the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the Schrödinger equation. We got corrections to the value of energy levels. ΔE n = E n 2 4n Kinetic energy relativistic correction 2mc 2 l = α2 Z 2 2n 4 E n n l Spin-orbit coupling Darwin term Z 4 j j + 1 l l + 1 s(s + 1) E SO = a 3 0 n 3 2l(l + 1/2)(l + 1) Total effect: E Darwin = ħ2 8m 2 c 2 4π Ze2 4πε 0 ψ Ԧr = 0 2 ΔE n = α2 Z 2 n 2 E n n j

58 Fine structure Paul Dirac calculations: Tadeusz Stacewicz

59 Fine structure Paul Dirac (Nobel 1933) calculations: Bohr 3P 3/2 and 3D 3/2 Dirac 2S 1/2 and 2P 1/ Tadeusz Stacewicz

60 Fine structure Willis Lamb (Nobel 1955) calculations: 2S 1/2 and 2P 1/2 QED Quantum ElectroDynamics Lamb shift due to the interaction of the atom with virtual photons emitted and absorbed by it. In quantum electrodynamics the electromagnetic field is quantized and therefore its lowest state cannot be zero (E min = ħω 2 Coulomb potential. ), which perturbs

61 Fine structure

62 10.2 ev Fine structure Lamb shift (Willis Lamb) QED size of the proton! (hydrogen vs muonic hydrogen) [about 1 GHz] for Hydrogen the order of ev (magnetic field of electron B 0.4T) Hyperfine interaction: interaction with nuclear moment [21 cm (1420 MHz) for atomic H]

63 Fine structure The Maxwell wave function of the photon M. G. Raymer and Brian J. Smith, SPIE conf. Optics and Photonics

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