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Atomic and Molecular Spectroscopy
Spectroscopy
Dr Stuart Mackenzie
Dr Stuart Mackenzie
thermodynamics
Atomic Structure
kinetics
Rate Rate
processes
Statistical Statistical
mechanics
Quantum theory
atoms / molecules
Atomic & Molecular Atomic
& Molecular
Spectroscopy
Quantum Mechanics
Valence
NMR
Photochemistry
Reaction ynamics
Dynamics
Lasers Solids &
surfaces Resources
ƒ Handouts (colour online)
ƒ Tutorials
uto a s
ƒ Books:
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Modern Spectroscopy (4th ed. 2004) JM Hollas
High Resolution Spectroscopy (2nd ed., 1998) JM Hollas
Molecular Spectroscopy (OUP Primer) JM Brown
Spectra of Atoms and Molecules (2nd ed. 2005) Bernath
Spectra of Atoms and Molecules
ed 2005) Bernath
Fundamentals of Molecular Spectroscopy (4th ed. 1994) Banwell & McCash
Atomic Spectra (OUP Primer) TP Softley
Molecular Quantum Mechanics (4th ed.) Atkins and Friedman
Electronic and Photoelectron Spectroscopy, Ellis, Feher and Wright
ƒ Practicals:
P
i l
ƒ
ƒ
ƒ
ƒ
II‐03 HCl, DCl spectra
II‐04 Fluorescence and quenching
p
II‐08 Flame atomic absorption
p
II‐05 I2 visible spectrum
II‐10 Na/Na+ atomic spec II‐17 Computational Raman
II‐18 N2+ spectrum
Lecture 1: General Aspects of Spectroscopy
1.1 Electromagnetic radiation
ƒ Transverse
Transverse wave of perpendicular, sinusoidally
wave of perpendicular sinusoidally
oscillating electric and magnetic fields E = E 0 sin
i ( kkx − ωt + φ )
ƒ
ƒ
ƒ
ƒ
with wavevector, k = 2π/λ
and angular frequency ω = 2πν
and angular frequency, 2πν
Characterised by:
wavelength, λ (in m) or frequency, ν (in Hz) Speed in vacuo defined as cvac = 299 792 458 ms‐1
c = νλ
= νλ = ω/k
cvac is related to the permittivity (electric constant) and permeability (magnetic constant) of free space:
(proof comes from Maxwell’s Equations)
B
A plane electromagnetic wave propagating in the z‐direction c2 =
1
μ 0ε 0
1.2 Quantised Light: Photons
It will usually be convenient to consider light as a stream of zero rest mass It
ill
ll b
i tt
id li ht
t
f
t
particles or packages of radiation called photons with the following properties:
Max Planck (1855‐1947)
ƒ Energy, E= hν
in which h is Planck’s constant, h = 6.626 x 10‐34 Js
ƒ Linear momentum,
Linear momentum p =
p = E/c = h
E/c = hν/c = h/λ
/c = h/λ (de Broglie)
(de Broglie)
Louis de Broglie
(1892‐1987)
(1892
1987)
ƒ (spin) Angular momentum equivalent to a quantum number of 1:
jph = 1 i .e.,, j ph
= 2=
ph
n.b., 1) photons are Bosons (i.e., obey Bose‐Einstein statistics)
2) photons have helicity
2)
photons have helicity (projection of angular momentum on the (projection of angular momentum on the
direction of travel) of ±1 only (i.e., not 0)
1.3 Quantities and Units ‐10 m]
Wavelength, λ: SI unit = m
[or μm, nm or Angström, 1 Å = 10
Å
λ is dependent on the (refractive index of the) medium in which the wave travels
Frequency, ν: SI unit = Hz (i.e., cycles s‐1) [or MHz = 106 Hz , GHz = 109 Hz]
frequency is independent of the medium
Energy, E: SI unit = J,
BUT : It is hard to measure energy directly. Spectra are recorded as line intensities as a function of frequency
f ti
ff
or wavelength.
l th
The conversion to energy appears simple: E = hν = hc/λ
But h is only known to 8 significant figures. Hence, it is convenient to introduce
Wavenumber, a property defined as reciprocal of the vacuum wavelength: and whose units are universally quoted as cm‐1 (n.b. not m‐1)
ν =
1
λvac
Wavenumber is directly proportional to energy, E = hcν and thus we commonly quote “energies” in units of cm‐1.
1.4 Energy levels: The Born Oppenheimer Approximation
tot
=
e
+
n
=
e
+
ee
+
ne
+
n
+
as trivial)
nn
See “Valencee” notes HT year 2
The total energy of a molecular system comprises:
ƒ The translation of the whole molecule, Ttrans (n.b. we’ll neglect this
ƒ Kinetic energy ,Te and Tn of electrons and nuclei, respectively
ƒ Potential energy, Vee and Vnn of electrons and nuclei, respectively
ƒ Potential energy between nuclei and electrons, Vne
The Born Oppenheimer Approximation (Annal. Phys., 84, 457 (1927))
The
orn Oppenheimer Approximation (Annal. Phys., 84, 457 ( 9 7))
Due to the difference in mass between the electron and nuclei, the motion of the two may be separated and the total molecular wavefunction, Ψtot, may, to a good approximation be written
approximation, be written
ψ tot = ψ el q ,Q ψ n Q
(
electron coordinates
) ( )
nuclear coordinates
and the resulting total energy is a simple sum Etot = Eel + Enuc
It will be convenient, though less rigorous, to further factorise ψn further into vibrational and rotational parts so ψtot = ψelψvibψrot and Etot= Eel + Evib+ Erot
Molecular Energy Levels
l
i.e., typically ΔEel >> ΔEvib >> ΔErot
Different electronic states (electronic arrangements
(electronic arrangements,
configurations or terms)
ΔE ≈ 2 x 104 – 105 cm‐1 102 – 5 x 103 cm‐1
Transitions at λ ≈
500 – 100 nm
100 μm –
00 μ
2 μm
μ
Vis – UV infrared
3 – 300 GHz (0.1 – 10 cm‐1)
10 cm – 1 mm
microwave
1.5 The Population of Energy levels
ni
Ei
ΔE
n0
E=0
The Boltzmann Law
The Boltzmann Law
At thermal equilibrium, the population of the i th
energy level is given by:
⎛ Ei ⎞
N
ni = g i exp ⎜⎜ −
⎟⎟
q
⎝ kT ⎠
⎛ Ei
q=
g i exp ⎜⎜ −
levels ,i
⎝ kT
∑
Where:
q is the molecular partition function (see HT Stat. Mech. notes)
gi is the degeneracy of the i th level (the no. states with same energy)
Ei is the energy
is the energy of the i
of the i th level
k is the Boltzmann constant ( = R/NA= 1.381 x 10‐23 J K‐1) T is the Kelvin temperature
Hence relative to n0:
Hence, relative to n
⎞
⎟⎟
⎠
ni g i
⎛ −ΔE ⎞
= exp ⎜
⎟
n0 g 0
kT
⎝
⎠
Ludwig Boltzmann 1844‐1906
1.6 The Interaction of Light and Matter I: A simple classical picture
Consider the ways in which a single photon might interact with a system of two Consider
the ways in which a single photon might interact with a system of two
energy levels E1 and E2, with populations n1 and n2, respectively:
A. Stimulated absorption, M + hν→
i l d b
i
h
M**
ƒ The photon is lost
ƒ The system absorbs energy E = hν = E2‐E1
E2
n2
E1
dn1
dn1
n1 rate of absorption rate of absorption = ∝ ρ E 21 n1 ⇒ = −B 12 ρ E 21 n1
dt
dt
In which B12 is the Einstein Coefficient of Absorption and ρ(E21) is the radiation energy density (energy of radiation field m‐3) at energy E21, which, for a black‐body at temperature T, is given by Planck’s Law y
p
, g
y
( )
radiation density, ρ E = 8π hν 3
c3
1
⎛ E ⎞
⎟ −1
⎝ kT ⎠
exp ⎜
B. Stimulated emission M* + hν → M + 2hν
E2
n2
E1
n1
ƒ Additional
Additional photon created with same frequency, photon created with same frequency
polarization, direction and phase as the original
ƒ The system relaxes, i.e., emits energy
rate of stimulated emission = dn2
dn
∝ ρ E 21 n2 ⇒ 2 = −B 21 ρ E 21 n2
d
dt
dt
d
in which B21 is the Einstein coefficient of stimulated emission.
Einstein showed that for a system to reach equilibrium a 3rd process must occur:
C. Spontaneous emission M* → M + hν
E2
n2
E1
n1
ƒ A photon is created with E = E2 – E1 = hν
ƒ The system relaxes, i.e.,
Th
t
l
i emits energy
it
dn2
dn2
rate of spontaneous emission = rate of spontaneous emission ∝ n2 ⇒ = − An2
dt
dt
and A is the Einstein coefficient of spontaneous emission (or “Einstein A coefficient”)
1.7 The Einstein Coefficients [A. Einstein, Z. Phys.,18, 121 (1917)]
spont.
emission
absn
stim.
stim
emission
dn1
At equilibrium: = 0, i .e., B 12 ρ ( E 21 ) n1 = A21n2 + B 21 ρ ( E 21 ) n2
dt
A21n2
=
Rearranging, ρ ( E 21 ) =
B 12n1 − B 21n2 B
12
c.f. Planck’s Law
f
Yielding:
( )
ρ E 21 = 8π hν 3
c3
{
A21
g1
g2
exp
( )} − B
E 21
kT
21
1
⎛ E 21 ⎞
⎟⎟ − 1
⎝ kT ⎠
exp ⎜⎜
g 1B 12 = g 2B 21 and A 21 =
8π hν 3
c
3
B 21
There is only one independent Einstein coefficient
What are the implications of the fact that the A‐coefficient, A ∝ ν3?
1.8 Interactions of Light and Matter II: A time‐dependent treatment
E2
n2
We will often use pictures like
to consider transitions.
E1
n1
Indeed our approach will be
i) to determine the eigenstates (stationary states) of a system and then ii) consider allowed transitions between these states
i.e., the photon doesn’t expicitly figure
The total wavefunction, Ψtot , satisfies the time‐dependent Schrödinger equation:
∂ψ
where Hˆ = Hˆ 0 +V t and V t = − μE 0 cos ωt
Hˆ ψ = i =
∂t
Eigenstates are the solutions of the t‐independent Schrödinger eqn: Ĥ 0φn = E n0φn
and the full (t‐dep) wavefunction is
φn exp {−iE not / =}
Ψ tot is a linear combination of stationary states ψ =
∑c
n
n
t φn exp {−iE not / =}
Time-dependent coefficients
After some manipulation (see MQM, Ch 6), we arrive at the rate of transition to state Aft
i l ti (
MQM Ch 6)
i
t th
t ft
iti t t t
m from a well‐defined, i.e., pure, initial state, j, to be:
(
)
(
)
0
0
0
0
⎧
i
E
−
E
+
=
ω
t
i
E
−
E
− =ω t ⎫⎪ ∗
E0 ⎪
dc m t
m
j
m
j
ˆ j dτ
=
+ exp
⎨exp
⎬ φm μφ
2i = ⎪
=
=
dt
⎩
⎭⎪
1
2
∫
3
Thus, for non‐zero transition probability (i.e., allowed transitions): 1 E 0 ≠ 0 there must be non‐zero radiation intensity, 1. 0 there must be non zero radiation intensity aand
nd
2. E m0 − E j0 = ± =ω i .e. , energy must be conserved, and ∫
ˆ j d τ ≠ 0 The 3
3. φm∗ μφ
0 The "transition dip
transition dipole mo
ole moment" must be non‐z
must be non‐zero o
1.9 The Transition Dipole Moment, R21
∫
The transition dipole moment, TDM, is defined as R 21 = ψ 2* μψ
ˆ 1d τ = 〈ψ 2 μˆ ψ 1 〉
ˆ =
where the dipole moment operator, μ
∑
qi rˆ i
P
Position
vector off ithh particle
l
i
Charge on i th particle
μ̂ operates upon our initial wavefunction ψ1 producing a new state ψ = μˆ ψ 1 〉
TDM, R21, thus represents the transition amplitude of ending up in our particular state,, ψ2 ,, determined by the overlap integral of y
p
g
ψ2 with ψ :
ˆ ψ1〉
〈ψ 2 ψ 〉 = 〈ψ 2 μ
The rate of transition (or intensity) is the square of this amplitude:
i .e., transition intensity t
iti i t it ∝ R =
21
2
(∫
ˆ 1d τ
ψ 2 μψ
*
)
2
ˆ ψ 1 〉2
= 〈ψ 2 μ
The TDM is, unsurprisingly, closely related to the Einstein B coefficient (after all they both describe the same thing):
3
B 21 =
8π
( 4πε )3h
2
R
=
21
2
0
1
2
R
6ε 0 =2 21
1.10 The Transition Dipole Moment and spectroscopic selection rules
∝
2
21
=
(∫
ψ 2* μψ 1 τ
)
2
= 〈ψ 2 μ ψ 1 〉 2
The TDM is thus the ultimate source of spectroscopic selection rules for “dipole
allowed transitions
transitions”.
i.e., of all the conceivable energetically allowed transitions it determines which
actually occur and encompasses symmetry and angular momentum constraints.
Forbidden transitions have R21 = 0
Allowed transitions have R
transitions have R21 ≠ 0
≠0