I. E-M Radiation and Interaction with Matter (Particles)

I-4

I. E-M Radiation and interaction with matter (particles)

McHale, Ch 3 / Struve, Ch1 / Bernath, Ch1 / Flygare, Ch11

For molecular spectra we take “hybrid” approach →

treat molecules with QM

E-M fields classically – (as waves)

Maxwell’s Eqn describe Classical E-M

E, B fields in phase

mutual perpendicular to k

n(Hz) → freq = 1/l

SI unit: or e = mp = 1.0 vacuum

e ~ 1 – 80 (H2O) material dependent

(no magnetic monopoles) B → magnetic induction

H → magnetic field

(m0e0)-1 = c2 speed of light - related to e0 electric permitivity

Alternate: (add current) current

- charge density x velocity

m0 - magnetic permeability

1st equation - relates to electro-statics – how fixed fields work

How do these relate to E-M? - they couple B, E (3rd, 4th equations)

by substitution can derive wave equations:

wave equations, 2nd order differential in space, time

Solve with general form – B, E will vary with this functionality

Can rewrite using a scalar and vector potential: f, A

if no charge Df = 0

satisfy

clearly E and B are in phase and orthogonal

Wave equation becomes: with solution of: A = A0ei(k.r – wt)

Wave Equations see Flygare: Chap 1

Maxwell eqn: (SI) (e0m0)-1 = c2 vacuum

1. mp = e = 1

2. m0, e0 cont

3. J = rv - current density

4. recall: - gradient operator

then: - divergence, - curl

and - LaPlacian operator

Meaning of equations:

1 → (Faraday) time dep → induce (since cancels )

2 → (Ampere-Oersted) time dependent electric (displacement) field produces

3 → (Coulomb Law) field relate to change

4 → no magnetic monopoles

Define scalar and vector potentials

Scalar Potential:

f: static (time independent) 1.

i.e. fits Max. eqn 1: – due to cross product, no B field)

note:

Poisson’s eqn - scalar potential from charge

(vacuum: LaPlace eqn: )

Vector Potential:

A: time varying consideration 2.

Subst. into Max. Eqn. #1

Constrain variables (f=0)

Scalar vector: (eqn. 5)

(but if use the Lorentz convention)

Wave equation for the scalar potential

Use the definitions with Maxwell #3:

from above take (eqn.5):

plug in and rearrange (no current):

Wave equation for the scalar potential -- Now some arithmetic:

Next take eqn (2):

substitute:

If J = 0 then

Solution:

let A

-k2A

wave vector: k~1/l: k

note: mp = 1 (non-magnetic medium) (refractive index- light)

propagation vector speed of light:

Poynting vector in vacuum n = 1.0

Now use equations

B

E E ІІ A

E, B are time oscillating (w) and spatial varying (k) fields E-M radiation

Interaction of Radiation and Matter

Turn Back to Molecules

In 542 you learned many problems need approximation → many particle systems

ex: Consider benzene 6 – C’s

6 – H’s

36 + 6 – electrons

Huge dimensionality – relatively small molecules.

Here we will discuss variants of two methods

Perturbation Theory – extend to time dependent

Variation Theory – modify for Hartree-Fock (elect structure calculation)

Time dependent Perturbation Theory – Struve, Ch 1

Levine, Ch 9.9 - 10

Electrostatic fields McHale, Ch 3 - moments

Molecule has changes qe and dipole m

en – change of nth particles or electron nucleus

rn – position of nth particles or electron nucleus

Energy of interaction with electrostatic field

where is the scalar potential at the origin

is the electric field

is the quadrupole tensor element ij

Magnetic fields since no magnetic monopoles

where is the magnetic field

[recall is actual field (magnetic indulation), is applied field]

Time Independent Perturbation Theory

Recall if both , let

Connections to energy depend on ,

higher connections → more powers or more terms from use higher order f:

Effect of perturbation is to mix the states with it on, initial state now has some fraction of other states so can say there is some probability

here:

of

having the characteristics of

aside: If time dependant, can view state as evolving in time can change its nature

Polarizability – above we have

but this only addresses “permanent” dipole moment of molecule – applies force to changes, separates them and induces a dipole

modify

classical E:

in general, a is a tensor – molecular response more complex

Compare this to 2nd order perturbation theory see terms to power E2 are

a note eigen slip

Molecules and atoms with biggest electron systems or most loosely bound elections → big a

ex: H = 0.667 Å3, Li = 24.3 Å3, Cs = 59.6

Now light radiation is an electro-magnetic field

interaction will parallel this E = -m – E

but here E = E(t) and B = B(t)

used to modify the approach

Time dependent fields

Maxwell equations lead to description of E-M field

and are in phase, but oriented 90º apart expressed ??? a vector potentialand

the Coulomb Gauge:

f = 0 (free space)

then:

(from) ( II )

(from, and ) ()

We can show [Struve, p.11] that effect on H is:

conservative potential

i.e. change interaction inside the molecule

expand

H0 remember

0 Coulob operator

time independent

2nd term in A2 ~ 0 since fields (pert.) small

time variation let

To use Time dependent Perturbation Theory McHale, Ch 4 consider

time independent w/f

when form complete set

when wave functions must change but – recall expand in complete set

now if is turned off, molecule will be in a time independent state

or cn = 1

cn = 0 nk

Time dependent Schrödinger Equation goes:

multiply left by and use to get orthogonal normality condition

(mn)

rearrange to:

now recall initially ck = 1

cn≠k = 0

so can approximate (i.e. for “short” time)

time variation of wave function is in coefficient: (on “weak” perturb)

where

operative equation:

integrate to give:

can do higher orders but they are not normally needed unless very big

perturbations – ex. laser-intense fields

For linear spectroscopy:

so substitute

expand

1st term:

assume (i.e. E = Ex, B = By, k = k2)

from Struve:

so

Electric dipole transition

dipole moment

expression correct

continuing

Now recall

so probability that at a time = t

system will be in a state

conditions

integral if

Draw get 1, otherwise get 0

now if , delta function not exactly connect get a very sharp peaked

function center at w = -wkm

Also should do this for real part of and result is sine function

(no integral)

Now one can go beyond level to include terms from

prep in y, pull on x

these give rise to

M1 (Lz) E2

magnetic dipole: M1

electric quadrupole: E2

and others could follow

M1 – responsible for EPR, NMR transitions can be important in trans metal spectra

and central to optical activity

E2 – rare but can occur (Electric quadrupole)

Relative sizes: recall expansion

visible light l ~ 5000 Å

infrared light ~ 100,000 Å

r ~ size molecule – medium benzene ~ 5 Å

visible/benzene

uv a little bigger (factor 5)

in order magnitude smaller

Then recall probability bigger reduction yet!

So M1 and E2 effects can be neglected for most molecules except nmr, epr, CD

Selection rules – since for E1

E1 need to have is odd must be even

a) must be opposite parity (odd, even)

b) polarization will affect transitions of oriented molecules

[if gas on liquid average one ]

rotation must have permanent dipole / vibration elect dipole must change

E2, M1 – similar but

a) same parity (even, even) (odd, odd)

b) Orientation can affect

Operate E1 – m-wave, IR, uv-vis absorption (electronic)

M1 – ESR, NMR, CD, weak electronic

E2 – same select rules as Raman, 2 phota but not mechanical

Error in last lecture:

McHale, Chap 4

agree:

Assume ck = 1, cm = 0, mk start at t = 0

Probability:

Note: integral from 0t because assume that cm = 0 at t = 0 (and before)

small correction (after expansion and electric dipole approximation):

Real part eiwt

Substitution:

Now consider probability term, time integral:

so there are 2 terms, one dominates in absorbance

where w = wmk , emission: w = - wmk

Return to probability – square the integral, absorbance: choose w = wmk

on ( – ) term dominant

Dw = wmk – w

Plot: : (wmk – w)

Long time: Note:

Transition Rate:

absorption stimulate

emission

Formal Golden Rule: d(Dv) = 2p d(Dv)

Probability linear in time → longer expose sample to light

the higher probability of a transition

Rate is what we measure experimentally – flex of light

stimulate an absorbance (loss of flex rate abs)

Uncertainty – lifetime

f(Dw, t) has a width:

dt → lifetime of state or duration on pulse (especially f-sec)

Ch 4.3 Book does nice relationship of density of photon states

and the rate of transition. Development not central

Ch 4.4 Then a detailed discussion of polarizability. We will put

this off until we address scattering. Now focus on dipole

Ch. 3 Frequency dependent polarizability – note complex due to relax

here t is a relaxation of state, is rate of decay

express quantum mechanically

this picture misses live widths → relaxation → complex function

can insert into denomination.

Allows quantum mechanic definition of oscillation strength

convenient method of categorizing transition:

How is this evidenced in matter?

Aside: refraction: speed light in vacuum – C (const)

speed light in material –

refractive index =

most non-magnetic m ~ m0 (4p x 10-7 ) ?

relative permitivity

actually complex – real → dispersion (refraction)

imaging → absorption

since index normally > 1, = 1 vacuum, refraction

– will cause denaturation from path on charge n

– will be greater at an absorbance

Absorbance – attenuation intensity:

b = x

g = 2,3 e c

at w ~ wkm

absorbance relates to probability of charge state

McHale,Ch. 3 Polarizability is response of material to electric field

induced dipole moment:

if model e response to force as Hook Law → harmonic oscillation →

multiplication:

see text – time dependent:

Kuernes Kronig:

QM:

see similar resonance big a

Now see oscillation strength:

relative permetivity response of medium to field

E – apparent field / E0 – applied

dielectric constant – factor reduction Coulombic force

Apply field induces polarization (P) in medium to oppose it

Elective susceptibility

l = er - 1

isotropic medium

Parallel polarizability:

if using frequency

same with refraction

since: n2(w) = er(w)

plus into

Now:

so absorb coefficient:

for solution:

relate to dipole expression (Einstein?)

McHale, Ch 6; Struve, Ch 8; Bernath, Ch 1

Einstein relationships are phenormalized expressions of rates of ???

up r12 = N1 B12 r(n) N1 – population lower state

B12 – stimulated rate constant at n

down r21 = N2 (B21 r(n) + A21) r(n) – energy density

A21 – spontaneous rate

note: only interested in n = n12 → resonant frequency

simple kinetics – no light

radiative lifetime: 1st order decay

if light on a long time system comes to equilibrium

N1 B12 r = N2(B21 r + A21) and

solve for DE = hn

(relating r(n) to kinetics)

if let r(n) be a black body light source (also equilibrium)

(gives denomination term)

see that A21 depends strongly on n3 → probability of spontaneous emission increase

as go to the uv

Two emission processes

Compare rates high n → uv – spontaneous dominance

low n → IR – stimulated dominance

Important → spontaneous (fluorescence) – incoherent

→ stimulated (e.g. laser) – same properties as

incident photoreduction and polarization

So how do lasers work in vis-uv (note kT ~ 200 cm-1 – for IR)

non-equilibrium devices → population inversion

must make N2 > N1 (non ???)

Recall Lumbert Law: dI = -g I dx I = I0 e-gx

positive absorption loss of intensity

negative emission (stimulated) amplification

relate power/volume to energy density/time:

assume B12 = B21

relate to einstein: (correct c for n) non degenerate B12 r(n)

r(n) → nI(n)g(n)/c

Lumbert Law:

now back to macroscopic: complex part

of induced polarization

~ 2n(w) k(w)

Relate back to Golden Rule:

rethink Coulent Law:

assume electronic/no stimulated emission

I in increment dx (cross section: 1-unit) (N2 ~ 0)

Beens Law: -dI = 2.303 a(n)CIdx I = I0e-acx a(n) = e(n)

(C-cone M, a(n) – malar absorption (per cm) molec abs

Combine to get rate:

( u(n) energy density )

To account for bandwidth

but normally r(n) constant over bandwidth – take out and compare to relationship for B12

so if meamic spectra, integrate, connect for path and core

can determine B12 experimentally also

in D2

(devices for homework!)

McHale 6.6? Line shapes

Homogenous → “all affect same way” → typical lifetime

time and freq complementary variable (inverse)

Fourier Transform relate then: (on correlation function)

Lomentzian: FWHM

(very long tails)

This concept for single state transition →

for electronic –vibration (mix) or (rotation-vibration) the

distribution of states (if unresolved) shape

Bernath, p31 Inhomogenous broadening → collection of molecules has a distribution

Fig 1-22 of resonant freq n12

Gaussian dist