introduction lorentz model of the atom photon physics: atom

Photon physics - Photon Physics, Februari 2005Photon physics: ATOM
Peter van der Straten (Atom Optics) Photon physics February 18, 2010 1 / 55
Introduction
Lorentz model of the atom
Introduction
2 One-electron atoms
3 Angular momentum
4 Radial equation
6 Many-electron atoms
7 Selection rules
Bohr model of the atom
Outline
3 Angular momentum
4 Radial equation
7 Selection rules
Spectroscopy of hydrogen
Table of transitions in hydrogen
Balmer series Hα 656.279 nm Hβ 486.133 nm Hγ 434.047 nm Hδ 410.174 nm
Balmer (1885):
νab = R
c
νab
Bohr’s postulate
Postulate I: That an atomic system can, and can only, exist permanently in a certain series of states corresponding to a discontinuous series of values for its energy, and that consequently any change of the energy of the system, including emission and absorption of electromagnetic radiation, must take place a complete transition between two such states. These states will be denoted as the stationary states of the system. Postulate II: That the radiation absorbed or emitted during a transition between two stationary states is ‘unifrequentic’ and possesses a frequency ν, given by the relation
E ′ − E ′′ = hν,
where h is Planck’s constant and where E ′ and E ′′ are the values of the energy in the two states under consideration.
Bohr’s hydrogen model (1913)
r +
- Additional approximation: the motion of the electron around the nucleus is circular!
Balance:
2.
Correspondence principle (1913)
νorbit = vn
νrad = 1
a0 =
m = 0.529× 10−10m.
Balmer revisited (1913)
rn = n2a0 with a0 = 0.529× 10−10m
Velocity of the electron:
Fine structure constant α and Rydberg constant:
α = 1
n n2a0 = n~
Spectrum of atomic hydrogen (H)
How quantum-mechanical is the atom?
Atomic units (in the remainder of this presentation!):
a0 ≡ e ≡ ~ ≡ 1/4πε0 ≡ 1.
Heisenberg’s uncertainty relation: p ·r ≥ ~ = 1 in a.u. (atomic units)
r +
⟩)1/2 = n2
n = n ≥ 1
The Bohr atom just obeys (for n ≈ 1) to the Heisenberg’s uncertainty rel.
Conclusion: qm is very important for the ATOM!
One-electron atoms
3 Angular momentum
4 Radial equation
7 Selection rules
One-electron atoms
The one-electron problem is spherical symmetric: V (~r) = V (r) !
We have to solve the differential equation in spherical coordinates (r , θ, ) instead of Cartesian coordinates (x , y , z): ψ (~r) → ψ (r , θ, ) ”Central Forces” (see BJ §2.6)
ψE ,`,m(r , θ, ) = RE`(r) radial
functions
Y`m(θ, ) spherical harmonics
One-electron atoms
r
x
z
y
θ
φ
z = r cos θ
∂2
∂φ2
One-electron atoms
r
x
z
y
θ
φ
H = T + V = −~2
2m ∇2 + V (r) =
} + V (r)
Angular momentum
3 Angular momentum
4 Radial equation
7 Selection rules
Angular momentum
r
x
z
y
θ
φ
y = r sin θ sinφ θ ∈ [0, π]
z = r cos θ φ ∈ [0, 2π]
Lz and L2 are the operators for quantizing the eigenfunctions
x , y , z rep. r , θ, φ rep.
Lz −i~ (
∂2
∂φ2
] Peter van der Straten (Atom Optics) Photon physics February 18, 2010 17 / 55
Angular momentum
Y`m(θ, φ) = (−)m [ (2`+ 1)(`−m)
4π(`+ m)!
` (≥ 0): angular momentum quantum number
m (−`,−`+ 1, . . . , `): magnetic quantum number
Summary: LzY`m(θ, φ) = ~mY`m(θ, φ)
L2Y`m(θ, φ) = ~2l(l + 1)Y`m(θ, φ)
Y`m(θ, φ)|Y`′m′(θ, φ) = δ``′δmm′
Angular momentum
Y00 = q
1 4π
Y20 = q
5 16π
Y1 ±1 = q
sin2 θ e±2iφ
Angular momentum
Y30 = q
7 16π
Y3 ±2 = q
Y3 ±1 = q
Y3 ±3 = q
pz = q
3 4π
cos θ
Radial equation
3 Angular momentum
4 Radial equation
7 Selection rules
Radial equation
} + V (r)
Since L2 (and Lz ) only depend on (θ, φ), we look for solutions:
ψE`m(r , θ, ) = RE`(r)Y`m(θ, )

] RE`(r) = ERE`(r)
i.e. now only a differential equation in ”r”! Note, that E is independent of m (degenerate).
Radial equation
Radial equation
Veff(r) ≡ −1
r + `(`+ 1)
E > 0; d2uE`(r)
√ 2mE
and ∫ Ψ∗ΨdV ∝
0 | uE`(r) |2 dr →∞
Radial equation
E < 0; d2uE`(r)
√ −2mE
~2 uE`(r) ∝ e±kr
Choose the solution with (−) sign
Radial equation
We have d2uE`(ρ)
] uE`(ρ) = 0
Boundary conditions: limρ→0 uE`(ρ) = 0 and limρ→∞ uE`(ρ) = e−ρ/2
Step 1: uE`(ρ) = e−ρ/2f (ρ), du(ρ)/dρ = . . . d2u(ρ)/dρ2 = . . .[ d2
dρ2 − d
dρ − `(`+ 1)
ρ2 + λ
d2
d dρ
+ (λ− `− 1)
] g(ρ) = 0
Radial equation
Radial equation (IV)
Boundary conditions: limρ→0 uE`(ρ) = 0 and limρ→∞ uE`(ρ) = e−ρ/2
Step 3: g(ρ) = ∑∞
k k 1
⇒ Divergent !
Radial equation
λ ≡ n = nr + `+ 1 n = 1, 2, 3, . . . n =
( −1
2E
)1/2
Hydrogen 1H
-13.6
-3.4
-1.5
Continuum
Radial equation
0 1 2 3 4 Radius (a0)
0.0
0.5
1.0
1.5
2.0
0.0 0.1
-0.2
0.0
0.2
0.4
0.6
0.8
0.00
0.05
0.10
0.15
0.20
0.00
0.05
0.10
0.15
0.20
0.00
0.05
0.10
0.15
0.20
P ro
ba bi
lit y
Radial equation
-0.1
0.0
0.1
0.2
0.3
0.4
0.00 0.02
-0.04 -0.02
0.00 0.02
0.00
0.01
0.02
0.03
0.04
0.05
0.00 0.02
P ro
ba bi
lit y
Radial equation
The eigenfunctions of the angular dependent differential equation The ”usual” spherical harmonics Y`m(θ, ) with
` 0 1 2 3
notation s=sharp p=principal d=diffuse f=fundamental and |m| ≤ `. The number of angular nodes: ` Degeneracy of the total wavefunctions: ψn`m(r , θ, ) = Rn`(r)Y`m(θ, )
E = − 1
2n2 depends only on n!
ψn`m has n − 1 degeneracy due to ` and (2`+ 1) degeneracy due to m
total degeneracy ∑n−1
`=0 (2`+ 1) = n2
degeneracy 4s 4p 4d 4f 1 3 5 7
total 16 = 42 = n2
Radial equation
Orbitals of electrons in the hydrogen atom
Spin
Outline
3 Angular momentum
4 Radial equation
7 Selection rules
Spin
qm theory predicts that 2`+ 1 must be an integer.
` = 0, 1/2, 1, 3/2, . . .
As discovered in the Stern-Gerlach experiment, the electron has an internal degree of freedom (spin):
` = 1/2 or s = 1/2
[Si ,Sj ] = i~Sk i , j , k cycl. α ≡ χ1/2,1/2 spin− up ↑ S2χs,ms = s(s + 1)~2χs,ms = 3/4~2χs,ms β ≡ χ1/2,−1/2 spin− down ↓ Szχs,ms = ms~χs,ms = ±1/2~χs,ms
Fermion/Boson: All spin-1/2 particles are fermions, or only one particle in each state (Pauli principle). All particles with integer spin are bosons.
Spin
Usual examples: electrons, protons, neutron (fermions), photon (boson) Example: 3He (I=1/2) and 4He (I=0), or 6Li (F=1/2,3/2) and 7Li (F=1,2)
Spin
Pauli exclusion principle
The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. It states that no two identical fermions may occupy the same quantum state simultaneously.
A more rigorous statement of this principle is that, for two identical fermions, the total wave function is anti-symmetric. For electrons in a single atom, it states that no two electrons can have the same four quantum numbers, that is, if n, `, and m` are the same, ms must be different such that the electrons have opposite spins. In relativistic quantum field theory, the Pauli principle follows from applying a rotation operator in imaginary time to particles of half-integer spin. It does not follow from any spin relation in nonrelativistic quantum mechanics.
Spin
Application of Pauli principle
Many-electron atoms
3 Angular momentum
4 Radial equation
7 Selection rules
Many-electron atoms
The “Aufbau” principle and the periodic system
In the central field approximation E only depends on n and `: E = En` with
En` < En′` for n < n′
En` < En′`′ for n + ` < n′ + `′
En` < En′`′ for n + ` = n′ + `′ and n < n′
Aufbau principle, or
E1s < E2s < E2p < E3s < E3p < E4s < E3d < E4p < E5s < . . . n + ` 1 2 3 3 4 4 5 5 5
Degeneracy: 2 spin
× (2`+ 1) m
= 4`+ 2
s : ` = 0 4`+ 2 = 2; p : ` = 1 4`+ 2 = 6 d : ` = 2 4`+ 2 = 10; f : ` = 3 4`+ 2 = 14
Many-electron atoms
1s22s22p63s23p64s23d104p65s24d105p6 Xe
`
Many-electron atoms
Lr lawrencium
Many-electron atoms
Z Element El. conf. Term M IP (eV) r (A)
1 H hydrogen 1s 2S1/2 1.008 13.598 0.53
2 He helium 1s2 1S0 4.003 24.587 0.93
3 Li lithium [He]2s 2S1/2 6.939 5.392 1.52
4 Be beryllium 2s2 1S0 9.012 9.322 1.12
5 B boron 2s22p 2P1/2 10.811 8.299 0.80
6 C carbon 2s22p2 3P0 12.011 11.260 0.77
7 N nitrogen 2s22p3 4S3/2 14.007 14.534 0.74
8 O oxygen 2s22p4 3P2 15.999 13.618 0.74
9 F fluorine 2s22p5 2P3/2 18.998 17.422 0.72
10 Ne neon 2s22p6 1S0 20.183 21.564 1.12
11 Na sodium [Ne]3s 2S1/2 22.990 5.139 1.86
12 Mg magnesium 3s2 1S0 24.312 7.646 1.60
13 Al aluminium 3s23p 2P1/2 26.982 5.986 1.43
14 Si silicon 3s23p2 3P0 28.086 8.151 1.17
15 P phosphorus 3s23p3 4S3/2 30.974 10.486 1.10
16 S sulphur 3s23p4 3P2 32.604 10.360 1.06
17 Cl chlorine 3s23p5 2P3/2 35.453 12.967 0.97
18 Ar argon 3s23p6 1S0 39.948 15.759 1.54
19 K potassium [Ar]4s 2S1/2 39.102 4.341 2.31
20 Ca calcium 4s2 1S0 40.08 6.113 1.97
Many-electron atoms
Chemical binding
Many-electron atoms
0
5
10
15
20
25
0
1
2
3
4
5
6
Atomic radii
Many-electron atoms
Russell-Saunders coupling
+1
+2
+5
+3
−1
−2
−3
−4
0
+4
L
Z
X
Y
Russell-Saunders coupling or L− S coupling: First couple all ~ of the individual electron to ~L and all ~s to ~S , and then couple ~J = ~L + ~S .
Term symbol: 2S+1LJ σ, multiplicity 2S + 1, total orbital angular
momentum L, total angular momentum J and parity σ. Convention: L = 0 (S), 1 (P), 2 (D), 3 (F), 4 (G)
S S
L
Many-electron atoms
since ∑
The “valence shell” contains the electrons “outside” the filled orbitals.
One value of L and S will be split up in a “multiplet” with different values of J:
J = |L− S |, · · · , L + S
Term symbol: 2S+1LJ σ, multiplicity 2S + 1, orbital angular momentum L
and total angular momentum J
Many-electron atoms
L = 1,S = 1 3P
J = 3 3D3
J = 2 3D2
J = 1 3D1
Many-electron atoms
Hund’s rules
In order to determine the energies of the terms, we need Hund’s rules (empirically determined)
1 The term with the largest multiplicity (S) for a given value of L has the lowest energy, and the energy increases with decreasing S .
2 For a given value of S , the term with the highest L has the lowest energy.
np n′p
These rules does not determine, that E (3P) < E (1D)
But this is normally the case:
np n′p
Hc Hc + H1
Selection rules
3 Angular momentum
4 Radial equation
7 Selection rules
Selection rules
Selection rules (in the dipole approx.)
If the wavelength of the radiation is much larger than the distance of the electron(s) from the nucleus,
|~ri | λ,
the interaction between the electro-magnetic field and the atom is the electric dipole interaction:
Hint = −~µ · ~E ,
with ~µ = −e~r the dipole moment of the atom (electrons vs. nucleus). Rewrite ε and ~r from Cartesian (x , y , z) repr. to spherical (r , θ, ) repr.
r0 = z = r
√ 4π
3 Y1±1(θ, )
Selection rules
The polarization of the light is determined by ε (~E = E0ε):
ε = ε0 (ε perp. to z-axis) lin. pol. light ε = ε± (~k perp. to z-axis) cir. pol. light
ψb|ε ·~r |ψa =
b (θ, φ)Ya(θ, φ)Y1q(θ, φ) selection rules
Selection rules
Selection rules derive from the angular part of the overlap integral. Linear polarized light:
ε = ε0
no n selection rule
s = 0 ` = ±1 m = ±1
no n selection rule Example of linear polarized light: 1s ↔ 2p0 but 1s /↔2p±1
Selection rules
ψa = 1s, ψb = 2p0
1 (θ)P0 1 (θ).
Selection rules
Polar part:∫ π
0 ∗(θ)P0
29
33 .
3 ×
√ 3
35 a.u. ' 0.74
Selection rules
+
ωba = (Eb − Ea)/~
3 and W s ba ∝ |ψb|r |ψa|2
2p → 1s
λ3 ab
λab in nm (=10−9m)
Ea
Eb
Selection rules
|ψb|r |ψa|2 = (0.74)2a.u. λab = 121.67nm
W s ba(2p → 1s) = 2.026× 1015 × (121.67)−3 × (0.74)2 = 6.15× 10+8s−1
Lifetime: τ2p = 1/W s
ba = 1.63ns
state 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f
τ(ns) ∞ “∞” 1.6 16 5.4 15.6 230 12.4 36.5 73 Metastable (τ ' 0.15 s)
Introduction
One-electron atoms
Angular momentum
Radial equation

introduction lorentz model of the atom photon physics: atom

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