Atomic Physics 3 rd year B1

Transcription

1 Atomic Physics 3 rd year B1 P. Ewart

2 Lecture notes Lecture slides Problem sets All available on Physics web site:

3 Atomic Physics: Astrophysics Plasma Physics Condensed Matter Atmospheric Physics Chemistry Biology Technology Street lamps Lasers Magnetic Resonance Imaging Atomic Clocks Satellite navigation: GPS etc

4 Astrophysics

5 Condensed Matter Zircon mineral crystal C 60 Fullerene

6 Snow crystal

7 Lasers

8 Biology DNA strand

9 Lecture 1 How we study atoms: emission and absorption of light spectral lines Atomic orders of magnitude Basic structure of atoms approximate electric field inside atoms

10 Atomic radiation y 1 y 2 y( t) = y1+ y2 Y( t) Y( t+ t) IY( t) I 2 I Y( t + t) I 2 Oscillating charge cloud: Electric dipole

11 Spectral Line Broadening Homogeneous e.g. Lifetime (Natural) Collisional (Pressure) Inhomogeneous e.g. Doppler (Atomic motion) Crystal Fields

12 Lifetime (natural) broadening Number of excited atoms Electric field amplitude Intensity spectrum N( t) I( w) E(t) Fourier Transform Time, t Exponential decay frequency, w Lorentzian lineshape

13 Lifetime (natural) broadening Number of excited atoms Electric field amplitude N( t) I( w) E(t) `t ~ 10-8 s Intensity spectrum `Dn ~ 10 8 Hz Time, t Exponential decay frequency, w Lorentzian lineshape

14 Collision (pressure) broadening Number of uncollided atoms Intensity spectrum N( t) I( w) Time, t Exponential decay frequency, w Lorentzian lineshape

15 Collision (pressure) broadening Number of uncollided atoms Intensity spectrum N( t) I( w) `t c ~ s `Dn ~ Hz Time, t Exponential decay frequency, w Lorentzian lineshape

16 Doppler (atomic motion) broadening Distribution of atomic speed Doppler broadening N(v) I( w) atomic speed, v frequency, w Maxwell-Boltzmann distribution Gaussian lineshape ` Typical Dn ~ 10 9 Hz

17 Atomic orders of magnitude Atomic energy: Thermal energy: J ~2 ev 1 / 40 ev Ionization energy, H: = Rydberg Constant 13.6 ev 109,737 cm -1 Atomic size, Bohr radius: 5.3 x m Fine structure constant, a = v/c: 1/137 Bohr magneton, m B : 9.27 x JT -1

18 The Central Field 1/r r U(r) ~Z/r Actual Potential

19 Z eff ~Z Important region 1 Radial position, r

20

21 Lecture 2 The Central Field Approximation: physics of wave functions (Hydrogen) Many-electron atoms atomic structures and the Periodic Table Energy levels deviations from hydrogen-like energy levels finding the energy levels; the quantum defect

22 Schrödinger Equation (1-electron atom) Hamiltionian for many-electron atom: Individual electron potential in field of nucleus Electron-electron interaction This prevents separation into Individual electron equations

23 Central potential in Hydrogen: V(r)~1/r, separation of y into radial and angular functions: y = R(r)Y m l(q,f)c(m s ) Therefore we seek a potential for multi-electron atom that allows separation into individual electron wave-functions of this form

24 Electron Electron interaction term: Treat this as composed of two contributions: (a)a centrally directed part (b)a non-central Residual Electrostatic part e - e - +

25 Hamiltonian for Central Field Approximation Central Field Potential ^ H 1 = residual electrostatic interaction Perturbation Theory Approximation: ^ ^ H 1 << H o

26 Zero order Schrödinger Equation: ^ H 0 y = E 0 y ^ H 0 is spherically symmetric so equation is separable - solution for individual electrons: Radial Angular Spin

27 Central Field Approximation: What form does U(r i ) take?

28 Hydrogen atom - Z+ - - Many-electron atom - - Z+ Z+ Z protons+ (Z 1) electrons U(r) ~ 1/r Z protons U(r) ~ Z/r

29 The Central Field 1/r r U(r) ~Z/r Actual Potential

30 Z eff ~Z Important region 1 Radial position, r

31 Finding the Central Field Guess form of U(r) Solve Schrödinger eqn. Approx y. Use approx y to find charge distribution Calculate U c (r) from this charge distribution Compare U c (r) with U(r) Iterate until U c (r) = U(r)

32 Energy eigenvalues for Hydrogen:

33 H Energy level diagram 0 l = s p d n Energy ev Note degeneracy in l 1

34 Revision of Hydrogen solutions: Product wavefunction: Spatial x Angular function Normalization : Eigenfunctions of angular momentum operators Eigenvalues

35 Angular momentum orbitals Y + ( q,f) Y 1 ( q,f)

36 Angular momentum orbitals Spherically symmetric charge cloud with filled shell Y + ( q,f) Y 1 ( q,f)

37 Radial wavefunctions Zr/ a o Zr/ a 10 o Ground state, n = 1, l = 0 1st excited state, n = 2, l = 0 N = 2, l = Zr/ a o 20 2nd excited state, n = 3, l = 0 n = 3, l = 1 n = 3, l = 2

38 Radial wavefunctions l = 0 states do not vanish at r = 0 l 0 states vanish at r = 0, and peak at larger r as l increases Peak probability (size) ~ n 2 l = 0 wavefunction has (n-1) nodes l = 1 has (n-2) nodes etc. Maximum l=(n-1) has no nodes Electrons arranged in shells for each n

39 The Periodic Table Shells specified by n and l quantum numbers Electron configuration

40 The Periodic Table

41 The Periodic Table Rare gases He: 1s 2 Ne: 1s 2 2s 2 2p 6 Ar: 1s 2 2s 2 2p 6 3s 2 3p 6 Kr: ( ) 4s 2 4p 6 Xe: (..)5s 2 5p 6 Rn: ( )6s 2 6p 6

42 The Periodic Table Alkali metals Li: 1s 2 2s Na: 1s 2 2s 2 2p 6 3s Ca: 1s 2 2s 2 2p 6 3s 2 3p 6 4s Rb: ( ) 4s 2 4p 6 5s Cs: (..)5s 2 5p 6 6s etc.

43 H Energy level diagram 0 l = s p d n Energy: Energy E n = - R n 2 Note degeneracy in l ev 1

44 Na Energy level diagram l = s p d 5s 4p Hydrogen n ev 4s 3d Energy: 3p E n = R n* 2 Note no degeneracy in l n* = n - d l Quantum defect 3s 5.14

45 Absorption spectroscopy Atomic Vapour Spectrograph White light source Absorption spectrum

46 Finding the Energy Levels Hydrogen Binding Energy, Term Value T n = R n 2 Many electron atom, T n = R. (n d(l)) 2 d(l) is the Quantum Defect

47 Finding the Quantum Defect 1. Measure wavelength l of absorption lines 2. Calculate: n = 1/ l 3. "Guess" ionization potential, T(n ) i.e. Series Limit o D(l) 4. Calculate T(n ): n = T(n ) - T(n ) i o i 5. Calculate: n* or T(n ) = R /(n - ) i i d(l) d(l) 2 Quantum defect plot T(n ) i

48 Lecture 3 Corrections to the Central Field Spin-Orbit interaction The physics of magnetic interactions Finding the S-O energy Perturbation Theory The problem of degeneracy The Vector Model (DPT made easy) Calculating the Spin-Orbit energy Spin-Orbit splitting in Sodium as example

49 The Central Field 1/r r U(r) ~Z/r Actual Potential

50 Corrections to the Central Field Residual electrostatic interaction: Magnetic spin-orbit interaction: H ^ 2 = -m.b orbit

51 Magnetic spin-orbit interaction Electron moves in Electric field of nucleus, so sees a Magnetic field B orbit Electron spin precesses in B orbit with energy: -m.b which is proportional to s.l Different orientations of s and l give different total angular momentum j = l + s. Different values ^ of j give different s.l so have different energy: The energy level is split for l + 1/2

52 Larmor Precession Magnetic field B exerts a torque on magnetic moment m causing precession of m and the associated angular momentum vector l The additional angular velocity w changes the angular velocity and hence energy of the orbiting/spinning charge DE = - m.b

53 Spin-Orbit interaction: Summary B parallel to l m parallel to s

54 Perturbation energy Radial integral Angular momentum operator? ^ ^ How to find < s. l > using perturbation theory?

55 Perturbation theory with degenerate states Perturbation Energy: Change in wavefunction: So won t work if E i = E j i.e. degenerate states. We need a diagonal perturbation matrix, i.e. off-diagonal elements are zero New wavefunctions: New eignvalues:

56 The Vector Model Angular momenta represented by vectors: l 2, s 2 and j 2, and l, s j and with magnitudes: l(l+1), s(s+1) and j(j+1). and l(l+1), s(s+1) and j(j+1). z Projections of vectors: l, s and j on z-axis are m l, m s and m j lh m l h Constants of the Motion Good quantum numbers

57 Summary of Lecture 3: Spin-Orbit coupling Spin-Orbit energy Radial integral sets size of the effect. Angular integral < s. l > needs Degenerate Perturbation Theory New basis eigenfunctions: j and j z are constants of the motion Vector Model represents angular momenta as vectors These vectors can help identify constants of the motion These constants of the motion - represented by good quantum numbers j s l

58 (a) No spin-orbit coupling (b) Spin orbit coupling gives precession around j (c) Projection of l on z is not constant (d) Projection of s on z is not constant Fixed in space m l and m s are not good quantum numbers Replace by j and m j Oxford Physics: 3rd Year, Atomic Physics

59 Vector model defines: j l s Vector triangle Magnitudes

60 ~ b n,l x ½ { j 2 l 2 s 2 } Using basis states: n, l, s, j, m j to find expectation value: The spin-orbit energy is: DE = b n,l x (1/2){j(j+1) l(l+1) s(s+1)}

61 DE = b n,l x ( 1 /2){j(j+1) l(l+1) s(s+1)} Sodium 3s: n = 3, l = 0, no effect 3p: n = 3, l = 1, s = ½, -½, j = ½ or 3 /2 DE( 1 /2) = b 3p x ( - 1); DE( 3 /2) = b 3p x ( 1 /2) 3p (no spin-orbit) j = 3/2 j = 1/2 2j + 1 = 4 2j + 1 = 2 1/2-1

62 Lecture 4 Two-electron atoms: the residual electrostatic interaction Adding angular momenta: LS-coupling Symmetry and indistinguishability Orbital effects on electrostatic interaction Spin-orbit effects

63 Coupling of l i and s to form L and S: Electrostatic interaction dominates l 2 s 2 L l 1 is 1 S L = l1 + l2 S = s + s 1 2

64 Coupling of L and S to form J L = 1 S = 1 S = 1 L = 1 L = 1 S = 1 J = 2 J = 1 J = 0

65 Magnesium: typical 2-electron atom Mg Configuration: 1s 2 2s 2 2p 6 3s 2 Na Configuration: 1s 2 2s 2 2p 6 3s Spectator electron in Mg Mg energy level structure is like Na but levels are more strongly bound

66 Residual electrostatic interaction 3s4s state in Mg: Zero-order wave functions Perturbation energy:? Degenerate states

67 Linear combination of zero-order wave-functions Off-diagonal matrix elements:

68 Off-diagonal matrix elements: Therefore as required!

69 Effect of Direct and Exchange integrals +K Singlet -K J Triplet Energy level with no electrostatic interaction

70 Orbital orientation effect on electrostatic interaction l 2 l 1 L l 1 l 2 L = l + l 1 2 Overlap of electron wavefunctions depends on orientation of orbital angular momentum: so electrostatic interaction depends on L

71 Residual Electrostatic and Spin-Orbit effects in LS-coupling

72 Term diagram of Magnesium Singlet terms So P1 D2 S P D ns Triplet terms 3pnl 3p 2 5s 4s 3s3p 1 P1 3s3d 1 D2 resonance line (strong) 3s3p 3 P2,1,0 21 3s S0 intercombination line (weak)

73 The story so far: Hierarchy of interactions H O H 1 H 2 H 3 : Nuclear Effects on atomic energy H 3 << H 2 << H 1 << H O

74 Lecture 5 Nuclear effects on energy levels Nuclear spin addition of nuclear and electron angular momenta How to find the nuclear spin Isotope effects: effects of finite nuclear mass effects of nuclear charge distribution Selection Rules

75 Nuclear effects in atoms Nucleus: stationary infinite mass point Corrections Nuclear spin magnetic dipole interacts with electrons orbits centre of mass with electrons charge spread over nuclear volume

76 Nuclear Spin interaction Magnetic dipole ~ angular momentum m = - glħ m l = - g l m B l m s = - g s m B s m I = - g I m N I g I ~ 1 m N = m B x m e /m P ~ m B / 2000 Perturbation energy: H^ 3 = - m I. B el

77 Magnetic field of electrons: Orbital and Spin Closed shells: zero contribution s orbitals: largest contribution short range ~1/r 3 l > 0, smaller contribution - neglect B el

78 B el = (scalar quantity) x J Usually dominated by spin contribution in s-states: Fermi contact interaction. Calculable only for Hydrogen in ground state, 1s

79 Coupling of I and J Depends on I Depends on J Nuclear spin interaction energy: empirical Expectation value

80 Vector model of nuclear interaction F = I + J I and J precess around F F I I I J F J F J

81 Hyperfine structure Hfs interaction energy: Vector model result: Hfs energy shift: Hfs interval rule:

82 Finding the nuclear spin, I Interval rule finds F, then for known J I Number of spectral lines (2I + 1) for J > I, (2J + 1) for I > J Intensity Depends on statistical weight (2F + 1) finds F, then for known J I

83 Isotope effects reduced mass Orbiting about Fixed nucleus, infinite mass + Orbiting about centre of mass

84 Lecture 6 Selection Rules Atoms in magnetic fields basic physics; atoms with no spin atoms with spin: anomalous Zeeman Effect polarization of the radiation

85 r Parity selection rule -r -e f * nl (x,y,z) [ix ˆ + jy ˆ + kz] ˆ f n l (x,y,z) Parity (-1) l must change Dl = + 1

86 Configuration Only one electron jumps

87 Selection Rules: Conservation of angular momentum h J 2 = J 1 h J 2 = J 1 J 1 J 1 DL = 0, + 1 DS = 0 DM J = 0, + 1

88 Atoms in magnetic fields

89 Effect of B-field on an atom with no spin Interaction energy - Precession energy:

90 Normal Zeeman Effect Level is split into equally Spaced sub-levels (states) Selection rules on M L give a spectrum of the normal Lorentz Triplet Spectrum

91 Effect of B-field on an atom with spin-orbit coupling Precession of L and S around the resultant J leads to variation of projections of L and S on the field direction

92

93 Total magnetic moment does not lie along axis of J. Effective magnetic moment does lie along axis of J, hence has constant projection on B ext axis

94 Perturbation Calculation of B ext effect on spin-orbit level Interaction energy Effective magnetic moment Perturbation Theory: expectation value of energy Energy shift of M J level

95 Vector Model Calculation of B ext effect on spin-orbit level Projections of L and S on J are given by

96 Vector Model Calculation of B ext effect on spin-orbit level Perturbation Theory result

97 Anomalous Zeeman Effect: 3s 2 S 1/2 3p 2 P 1/2 in Na

98 Polarization of Anomalous Zeeman components associated with Dm selection rules

99 Lecture 7 Magnetic effects on fine structure - Weak field - Strong field Magnetic field effects on hyperfine structure: - Weak field - Strong field

100 Summary of magnetic field effects on atom with spin-orbit interaction

101 Total magnetic moment does not lie along axis of J. Effective magnetic moment does lie along axis of J, hence has constant projection on B ext axis

102 Perturbation Calculation of B ext effect on spin-orbit level Interaction energy Effective magnetic moment Perturbation Theory: expectation value of energy Energy shift of M J level What is g J?

103 Vector Model Calculation of B ext effect on spin-orbit level Projections of L and S on J are given by

104 Vector Model Calculation of B ext effect on spin-orbit level Perturbation Theory result

105 Landé g-factor Anomalous Zeeman Effect: 3s 2 S 1/2 3p 2 P 1/2 in Na g J ( 2 P 1/2 ) = 2/3 g J ( 2 S 1/2 ) = 2

106 Polarization of Anomalous Zeeman components associated with Dm selection rules

107 Strong field effects on atoms with spin-orbit coupling Spin and Orbit magnetic moments couple more strongly to B ext than to each other.

108 Strong field effect on L and S. m L and m S are good quantum numbers L and S precess independently around B ext Spin-orbit coupling is relatively insignificant

109 Splitting of level in strong field: Paschen-Back Effect N.B. Splitting like Normal Zeeman Effect Spin splitting = 2 x Orbital g S = 2 x g L

110 g S = 2g L

111 Magnetic field effects on hyperfine structure

112 Hyperfine structure in Magnetic Fields Hyperfine interaction Electron/Field interaction Nuclear spin/field interaction

113 Weak field effect on hyperfine structure I and J precess rapidly around F. F precesses slowly around B ext I, J, F and M F are good quantum numbers m F

114 Only contribution to m F is component of m J along F m F = -g J m B J.F x F F F magnitude direction g F = g J x J.F F 2 Find this using Vector Model

115 g F = g J x J.F F 2 F I F = I + J I 2 = F 2 + J 2 2J.F J.F = ½{F(F+1) + J(J+1) I(I+1)} J

116 DE = Each hyperfine level is split by g F term Ground level of Na: J = 1/2 ; I = 3/2 ; F = 1 or 2 F = 2: g F = ½ ; F = 1: g F = -½

117 Sign inversion of g F for F = 1 and F = 2 J = 1/2 F = 2 J = -1/2 I = 3/2 F = 1 I = 3/2 J.F positive J.F negative

118 Strong field effect on hfs. DE = J precesses rapidly around B ext (z-axis) I tries to precess around J but can follow only the time averaged component along z-axis i.e. J z So A J I.J term A J M I M J

119 Strong field effect on hfs. Dominant term Energy Na ground state

120 Strong field effect on hfs. Energy: DE = J precesses around field B ext I tries to precess around J I precesses around what it can see of J: The z-component of J: J Z

121 Magnetic field effects on hfs Weak field: F, M F are good quantum nos. Resolve m J along F to get effective magnetic moment and g F DE(F,M F ) = g F m B M F B ext Zeeman splitting of hfs levels Strong field: M I and M J are good quantum nos. J precesses rapidly around B ext ; I precesses around z-component of J i.e. what it can see of J DE(M J,M I ) = g J m B M J B ext + A J M I M J hfs of Zeeman split levels

122 Lecture 8 X-rays: excitation of inner-shell electrons High resolution laser spectroscopy - The Doppler effect - Laser spectroscopy - Doppler-free spectroscopy

123 X Ray Spectra

124 Characteristic X-rays Wavelengths fit a simple series formula All lines of a series appear together when excitation exceeds threshold value Threshold energy just exceeds energy of shortest wavelength X-rays Above a certain energy no new series appear.

125 Generation of characteristic X-rays Ejected electron X-ray e - e -

126 X-ray series

127 X-ray Intensity X-ray spectra for increasing electron impact energy K-threshold L-threshold E 3 > E 2 > E 1 Max voltage Wavelength

128 Binding energy for electron in hydrogen = R/n 2 Binding energy for hydrogen-like system = RZ 2 /n 2 Screening by other electrons in inner shells: Z (Z s) Binding energy of inner-shell electron: E n = R(Z s) 2 / n 2 Transitions between inner-shells: E i - E j = n = R{(Z s i ) 2 / n i 2 - (Z s j ) 2 / n j2 }

129 Fine structure of X-rays

130 X-ray absorption spectra

131 Auger effect

132 High resolution laser spectroscopy

133 Doppler broadening Doppler Shift: Maxwell-Boltzmann distribution of Atomic speeds Distribution of Intensity Doppler width Notes error

134 Crossed beam Spectroscopy

135 Saturation effect on absorption Absorption profile for weak probe Absorption profile for weak probe with strong pump at w o Strong pump at w L reduces population of ground state for atoms Doppler shifted by (w L w o ). Hence reduced absorption for this group of atoms.

136 Saturation effect on absorption Absorption of weak probe w L Absorption of strong pump w L Probe and pump laser at same frequency w L But propagating in opposite directions Probe Doppler shifted down = Pump Doppler shifted up. Hence probe and pump see different atoms.

137 Saturation of zero velocity group at w O Counter-propagating pump and probe see same atoms at w L = w O i.e. atoms moving with zero velocity relative to light Probe transmission increases at w O

138 Saturation spectroscopy

139 Saturation spectrum of Sodium D 2 line Doppler broadening of hyperfine lines

140 First saturation spectroscopy of atomic Hydrogen T W Hansch et al 1971

141 Principle of Doppler-free two-photon absorption { { Photon Doppler shifted up + Photon Doppler shifted down

142 Two-photon absorption spectroscopy

143

144 Doppler-free Two-photon spectrum of Hydrogen 1S 2S transition Tests QED calculation of electron interaction with proton

145

146 Doppler-free spectrum of OH molecule in a flame

147 THE END