select/special topics in atomic physics · 2017-08-04 · p. c. deshmukh . department of physics....

P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036

[email protected] Unit 7(i) Lecture 30

Select/Special Topics in Atomic Physics

1

Atomic Photoionization cross-sections, angular distribution of photoelectrons - 1

PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

Ammeter __

+2

from PCD STiAP Unit 6 Probing the Atom: C&S

(1887) PEE experiment of Hertz and Lenard

Bound Continuum

Photoelectrons

PES: ‘complete’ experiment

2

3

| |⟨ ⟩f iTψ ψ

HertzLenard

1887

?from STiAP Unit 6 Probing the Atom: C&S

Einstein1905

2 1| (2 1) ( cos )2

liikrikz

f ll

e ee l Pr ik

δ

ψ θ−− ⎛ ⎞−

⟩ → − + − ⎜ ⎟⎝ ⎠

∑

• Hertz Lennard (1887)

• Einstein (1905)

• Powerful tool to probe

Atom

Ions (positive, negative, Photo-ionization,Photo-detachment )

Molecules

Surface Analysis

……

4PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

5

0


X

YZ

hν

e−

θkφ

Goal: Obtain quantum expressions for AtomicPhotoionization Cross Sections and AngularDistributions so that one may compute these propertiesusing rigorous theoretical models


• C.S.C.O.

Cross-sections (Probabilities)

Angular distribution of photoelectrons

Spin-polarization parameters of the photoelectrons

f iTψ ψ Transition Matrix Element

T : Coupling operatorsDipoleQuadrupole

Other ?7PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

f iTψ ψContinuum function with ‘in-going’ wave boundary condition

Initial state

I.P.A.

H.F.

D.F.

M.C.H.F. / M.C.D.F.

Non-Relativisticor Relativistic?“correlations”

Configuration interaction in the continuum- Inter-channel couplingMg: 2 2 2 4 2

1/2 3/21s 2s 2p 2p 3s1e

3/2 1/23s p , p⎯⎯→ε ε2e

5/2 3/23s d , d⎯⎯→ε εOperator T is specific to the nature of ‘coupling’


Cross sec tion :(transition probability)

− σ

DISCRETE EXCITATION: Photo-excitation

Photo-Ionization / PhotodetachmentFinal state: continuum

Angular distribution of the photoelectrons : , ,β γ ζ


Along with measurements of Spin polarization parameters

‘complete’ experiment

Development of the tools we require…..

Development of techniques

Classical model ‘Old’ quantum theory

Goal: Fully quantum mechanical model.

( )(Mb)

Photoionization cross-section x-secσ


Interaction of EM

radiation with an

atom

Oscillations of

electrons about mean

position.

• Atomic polarizability

• Polarization

• Susceptibility


Induced dipole

moment

“Oscillator Strength” / Transition Probability /Cross-Section

Our Aim: Develop research methodology from first principles

Develop all the necessary building blocks

Quantum Mechanics

Relativistic Effects

Many Body Effects ( correlations)

MULTIPOLE TRANSITIONS12PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

Each electron in an atom is bound to its site by a restoring force

Simplest level: atomic electrons ‘driven’ by Electric Intensity Vector of EM field:

( )i t0ˆE e− ω +θε

( )tkr−

( )..

m r t rk= −

e : electron’s charge

U.Fano & J.W.Cooper Review of Modern Physics 40:3 441 (1968)


Damped oscillator

( ) ( ).. .

m r t rk 2 m rγ= − −


Energy dissipation due to unspecified degrees of freedom

atomic electrons ‘driven’ by Electric Intensity Vector of EM field: ( )i t

0ˆE e− ω +θεDamped + Driven oscillator

( ) ( ) ( ).. .

i t0ˆm r t kr 2 m r eE e− ω +θ= − − γ + ε

e : electron’s charge


Driving force

( ) ( ) ( ) ( ).. .

i td

20,s 0ˆm r t r m r eEm e− ω +θ= − − Γ + εω

( ) ( ) ( )

( ) ( )

.. .i t2

d 0,s 0

.. .i t2

d 0,s 0

ˆm r t m r m r eE e

eˆr t r r E em

− ω +θ

− ω +θ

+ Γ + ω = ε

+ Γ +ω = ε

( ) ( ) ( )i t2 2d 0,s 0

eˆi r t E em

− ω +θ−ω − Γ ω+ω = ε

( ) ( )( ) ( )2 2

Check :

r t i r(t)

r t i r(t) r(t)

= − ω

= − ω = −ω

( ) ( )i t02 20,s d

Eeˆr t e m i

is a solution

− ω +θ= εω −ω − Γ ω


Displacement of the electron from the

equilibrium point is on account of the driving

force and results in

Induced Oscillating Dipole Moment

( ) ( ) ( )i t2 2d 0,s 0

eˆi r t E em

− ω +θ−ω − Γ ω+ω = ε

( ) 0r t 0 as E 0→ →

Classical Damped OscillatorLater, we connect Oscillator Strength to PI cross-section


Connections:

Oscillator Strength

Atomic polarizabilities

Photoionization cross-section


Atomic polarizability

induced dipole moment

per unit electric field

d

Eα =d Eα=

d er=( )

( )

02 20,

0

ˆ

ˆ

i t

s d

i t

Eee em ier

E E e

ω θ

ω θ

εω ω ω

αε

− +

− +

− − Γ= =

2

2 20,

1

s d

em i

αω ω ω

=− − Γ

Induced Dipole Moment


: atomic polarizabilityα

Atomic polarizabilities

induced dipole moment

per unit electric field

d

Eα =

d Eα=2

2 20,

1( )s d

em i

α ωω ω ω

=− − Γ

( ) ( ) ( )

2

0, 0,0,

1( )d

s ss

em

i

α ωωω ω ω ω

ω ω

=⎡ ⎤Γ

+ − −⎢ ⎥+⎢ ⎥⎣ ⎦

( )

2

0,

0, 0,0,

1; ( )2

2

s

ds s

s

em

iω ω α ω

ωω ω ωω

≈ ≈⎡ ⎤Γ

− −⎢ ⎥⎣ ⎦


( ) ( )i t02 20,s d

Eeˆnow, r t em i

− ω +θ= εω −ω − Γ ω


2 2 20,

1( )s d

me i

α ωω ω ω

× =− − Γ

( ) ( )i t02

e mˆr t ( ) E em e

− ω +θ⎡ ⎤= ε α ω⎢ ⎥⎣ ⎦

2

2 20,

1( )s d

em i

α ωω ω ω

=− − Γ

∴

∴

( ) ( ) ( )i t0ˆd t er t E ( )e− ω +θ= = ε α ω

Dipole moment

( ) ( )i t0E ( )ˆr t ee

− ω +θα ω= ε

Average power pumped into the

atomic system by the EM field is the

real part of:

( )

T T

0 0

T

0

1 dW 1 F drQ dt dtT dt T dt

1 F er dtT e

•= =

⎡ ⎤⎛ ⎞= •⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

∫ ∫

∫Concept of complex field is a mathematical tool.

We are interested in its real part………….

Fano & Cooper Revs. Mod. Phys. July 1968 Section 2.1


Average power pumped into the atomic

system by the EM field is the

real part of:

( )

T T

0 0

T

0

1 dW 1 F drQ dt dtT dt T dt

1 F er dtT e

•= =

⎡ ⎤⎛ ⎞= •⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

∫ ∫

∫

( )T

Re0

1 FQ ereT e

R te dR⎡ ⎤⎛ ⎞

= •⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

∫Physical

interest:

Real part23PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

( )T

Re0

1 FQ Re Re er dtT e

⎡ ⎤⎛ ⎞= •⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫

( ) ( )i t0ˆer t ( i )E ( )e− ω +θ= ε − ω α ω

( )i t0

F ˆE ee

− ω +θ= ε

i( t ) i( t )0EF ˆRe e ee 2

− Ω +θ + Ω +θ⎛ ⎞⎡ ⎤= ε +⎜ ⎟ ⎣ ⎦

⎝ ⎠

( )( ) ( )i t i t*0

Re er t

Eˆ ( i) ( )e ( i) ( )e2

− ω +θ + ω +θ

⎡ ⎤ =⎣ ⎦ω ⎡ ⎤= ε − α ω + + α ω⎣ ⎦


( )T

Re0

1 FQ Re Re er dtT e

•⎡ ⎤⎛ ⎞

= ⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

∫( ) ( )

( ) ( ) ( ) ( )

i t i t0

i t i t*0

Eˆ e e2

Eˆ i ( )e i ( )e

2

− ω +θ + ω +θ

− ω +θ + ω +θ

⎧ ⎡ ⎤ε +⎪ ⎣ ⎦⎪⎨⎪ ω ⎡ ⎤⎪ε − α ω + + α ω⎣ ⎦⎩

•

The ‘dot’ product in the integrand:

( ) ( ) ( )( ) ( ) ( )

i2 t

i2

*20

* t

i ( ) i ( )E4 i (

e

) i ( )e

− ω +θ

+ ω +θ

⎡ ⎤− α ω + + α ω +ω⎢ ⎥=⎢ ⎥− α ω + + α ω⎣ ⎦

What shall we get from the oscillatory functions over large time intervals?


( )T

Re0

FRe Re ere

1Q dtT

•⎡ ⎤

= ⎢ ⎥⎣ ⎦

⎛ ⎞⎜ ⎟⎝ ⎠

∫The ‘dot’ product in the integrand:

( )t

i2 t0

t

e dt ? ; For large t (E =0 for t 0

so the range of integration 0 included)

→∞ω +θ

→−∞

= ⟨

−∞→

∫ ∓

( ) ( ) ( )( ) ( ) ( )

i2 t

i2

*20

* t

i ( ) i ( )E4 i (

e

) i ( )e

− ω +θ

+ ω +θ

⎡ ⎤− α ω + + α ω +ω⎢ ⎥=⎢ ⎥− α ω + + α ω⎣ ⎦

26

( )t

i2 t

t

e dt ( )2

→∞ω +θ

→−∞

π= ± δ ω∫ ∓


0 no oscillations

ω = →

( )T

Re0

1 FQ Re Re er dtT e

•⎡ ⎤⎛ ⎞

= ⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

∫

The integrand: ( ) ( )2

*0E i ( ) i ( )4⎡ ⎤= ω + α ω + − α ω⎣ ⎦

( ) ( )*i ( ) i ( )+ α ω + − α ω

( ) ( ) i (a ib) i (a ib) b b 2b 2Im ( )

= + − + − +

= + + = + = α ω

a ibα = +


( )T

Re0

1 FQ Re Re er dtT e

⎡ ⎤⎛ ⎞= •⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫

Related to “oscillator strengths”

and to

atomic photoionization cross-section

20E Im ( )

2ω

= α ω

Average power pumped into the atomic system by

the EM field is given by:


Questions? Write to [email protected]


[email protected] Unit 7(ii) Lecture 31


29



Average power pumped into the atomic system by the

EM field is the real part of:

( )T T T

0 0 0

1 dW 1 F dr 1 FQ dt dt er dtT dt T dt T e

⎡ ⎤⎛ ⎞•= = = •⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫ ∫ ∫


( )T

Re0

1 FQ Re Re er dtT e

⎡ ⎤⎛ ⎞= •⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫

20E Im ( )

2ω

= α ω

Atomic polarizabilitysusceptibility

Absorption coefficientPhoto-absorption cross-section

( )

( )

02 20,

0

ˆ

ˆ

i t

s d

i t

Eee em ier

E E e

ω θ

ω θ

εω ω ω

αε

− +

− +

− − Γ= =

Number of atoms per unit volume in a

dilute atomic gas.

P(r, t) E(r, t)= χsusceptibility : ( ) ( ) χ ω = ηα ω

:η

2

2 20,

1( )s d

em i

α ωω ω ω

=− − Γ


Atomic polarizability

Polarization

d Eα=

2

22 20,

Im ( ) d

s d

em i

ωα ωω ω ω

Γ=

− + Γ

( ) ( )

( )( )( ) ( )

2

2 2

2

2 2

2 20,

0, 0,

Im ( ) d

d

d

s

ds s

em

em

ωα ωω

ω

ω ω

ω ω ω ωω

Γ=

+ Γ

Γ=

+ Γ

−

− +

2 220,

2 2 2 20, 0,

1( ) s d

s d s d

iem i i

ω ω ωα ω

ω ω ω ω ω ω− + Γ

= ×− − Γ − + Γ


2 220 ,

22 20 ,

( ) s d

s d

iem i

ω ω ωα ω

ω ω ω

− + Γ=

− + Γ

( ) ( ) ( )( )

2

22 2

0, 0, 2

0,

Im ( ) d

ds s

s

em

ωα ωω

ω ω ω ωω ω

Γ=

⎧ ⎫Γ⎪ ⎪+ − +⎨ ⎬+⎪ ⎪⎩ ⎭

( ) ( ) ( )( )

2

22 2

0, 0, 2

0,

Im ( )

2 2

d

ds s

s

em

ωα ωω

ω ω ωω

Γ≈

⎧ ⎫Γ⎪ ⎪− +⎨ ⎬⎪ ⎪⎩ ⎭

( )

2

220,

0,

2Im ( )2

2

d

s ds

em

α ωω

ω ω

Γ

≈⎧ ⎫Γ⎪ ⎪⎛ ⎞− +⎨ ⎬⎜ ⎟

⎝ ⎠⎪ ⎪⎩ ⎭

( ) ( ) ( )

2

2 2 20, 0,

Im ( ) d

s s d

em

ωα ωω ω ω ω ω

Γ=

− + + Γ

0,sω ω≈


( )

2

220,

0,

2Im ( )2

2

d

s ds

em

α ωω

ω ω

Γ

≈⎧ ⎫Γ⎪ ⎪⎛ ⎞− +⎨ ⎬⎜ ⎟

⎝ ⎠⎪ ⎪⎩ ⎭

20

Re

EQ Im ( )2

ω= α ω

Eq.2.13/ Fano & Rau / TACS34PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

( )

20

Re

2

220,

0,

222

2

d

s ds

eEQmω

ω ω

ωΓ

⎧ ⎫Γ⎪ ⎪⎛ ⎞− +⎨ ⎬⎜ ⎟⎝ ⎠⎪⎩

=

⎪⎭

( )2

2

0,

2 20

Re 42

2

d

ds

E eQm

ω ω

Γ

⎧ ⎫Γ⎪ ⎪⎛ ⎞− +⎨ ⎬⎜ ⎟⎝ ⎠⎪⎩

=

⎪⎭

( )

d2 20

Re 22 d

0,s

E e 2Q4 m

2

Γ

=⎧ ⎫Γ⎪ ⎪⎛ ⎞ω−ω +⎨ ⎬⎜ ⎟

⎝ ⎠⎪ ⎪⎩ ⎭

( )( )

d

d

0,s 20 2 d0,s

1 2lim2

2

Γ →

Γ

δ ω−ω =π ⎧ ⎫Γ⎪ ⎪⎛ ⎞ω−ω +⎨ ⎬⎜ ⎟

⎝ ⎠⎪ ⎪⎩ ⎭

( )

d

22 d

0,s

definitiondf 1 2d

2→

Γ

ω π ⎧ ⎫Γ⎪ ⎪⎛ ⎞ω−ω +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

=

Definition of

Oscillator Strength

Average power

pumped into the

atomic system by

the EM field.


( )

d2 20

Re 22 d

0,s

E e 2Q4 m

2

Γ

=⎧ ⎫Γ⎪ ⎪⎛ ⎞ω−ω +⎨ ⎬⎜ ⎟

⎝ ⎠⎪ ⎪⎩ ⎭

( )

d

22 d

0

definition

,s

df 1 2d

2→

Γ

=ω π ⎧ ⎫Γ⎪ ⎪⎛ ⎞ω−ω +⎨ ⎬⎜ ⎟

⎝ ⎠⎪ ⎪⎩ ⎭

Definition of

Oscillator

Strength

Average power

pumped into the

atomic system by

the EM field.

2 20e E dfQ

4 m dπ

=ω


Oscillator Strength gives the SPECTRAL

DISTRIBUTION OF THE OSCILLATOR

2 20e E dfQ

4 m dπ

=ω

i fQ WI I

Time dependent perturbation theory

→

⟨ ⟩ ωσ = =

−

Average power pumped

into the atomic system by

the EM field.

Photoionization

Cross-Section


[ ] dimensions?σ →

( )

d

22 d

0

definition

,s

df 1 2d

2→

Γ

=ω π ⎧ ⎫Γ⎪ ⎪⎛ ⎞ω−ω +⎨ ⎬⎜ ⎟

⎝ ⎠⎪ ⎪⎩ ⎭

( )( )

d

d

0,s 20 2 d0,s

1 2lim2

2

Γ →

Γ

δ ω−ω =π ⎧ ⎫Γ⎪ ⎪⎛ ⎞ω−ω +⎨ ⎬⎜ ⎟

⎝ ⎠⎪ ⎪⎩ ⎭( )0,s0 0

df d 2 dd

∞ ∞

ω = δ ω−ω ωω∫ ∫ 1 ( )

( ) ( ) ( )

x a dx

f a f x x a dx

δ

δ

+∞

−∞

+∞

−∞

= −

= −

∫

∫( )0,s

0

df 1d 2 dd 2

∞ ∞

−∞

ω = δ ω−ω ωω∫ ∫


( )0,s0 0

df d 2 dd

∞ ∞


‘OSCILLATOR STRENGTH’

is so defined as to yield UNIT INTEGRAL

( )0,s0

df 1d 2 dd 2

∞ ∞

−∞


( )0,s0

df d d 1d

∞ ∞

−∞

ω = δ ω−ω ω =ω∫ ∫


( )

d

22 d

0

definition

,s

df 1 2d

2→

Γ

=ω π ⎧ ⎫Γ⎪ ⎪⎛ ⎞ω−ω +⎨ ⎬⎜ ⎟

⎝ ⎠⎪ ⎪⎩ ⎭

Atomic Photoabsorption:

Quantum Mechanical TreatmentNon-Relativistic Schrodinger Equation

I.P.A. Independent Particle Approximation

Later!

Relativistic effects,

Correlation / C.I. / Many-Body effects40PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

‘F’ used instead of ‘E’, since

E is used for quantum energy eigenvalues

X

YZ

hν

e−

θkφ

xˆ eε =

( ) ( )i k r t i k r t0

Electric Driving Field of the EM radiation

ˆF F e e− −ω + −ω⎡ ⎤= ε +⎢ ⎥⎣ ⎦i i

Electric component is considered

to be polarized along the x-axis.41PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

0 0E ;ψ

(s) ss s sE E i ;

2Γ

= − ψ

PHOTOABSORPTIONInitial Level: SHARP

42

(Energy, Time) uncertainty :

NO operator for ‘time’


2

1 q qp A(r, t) p A(r, t)2m c cH

Zeq (r, t)r

⎡ ⎤⎛ ⎞ ⎛ ⎞− • −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎢ ⎥=⎢ ⎥+ φ −⎢ ⎥⎣ ⎦

{ }

( )

2 222

2

1 qp A(r, t)2m c

i qH A(r, t) A(r, t)c

Zeq (r, t)r

⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎢ ⎥

= − ∇• + •∇⎢ ⎥⎢ ⎥⎢ ⎥+ φ −⎢ ⎥⎢ ⎥⎣ ⎦


Minimal couplingbetween field and charge- but not with higher multipole moments of the charge distribution

PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 44

lineariqH H' A(r, t)mc

= = − •∇

{ }2 2

quadratic 2

qH A(r, t)2mc

=

2quadratic

2

2linear

H q A mc qAH 2mc qpA 2cp

× =∼

generalizedmomentum

qp A(r, t)c

π = −

[ ] [ ]

1

11 1

qA MLTc

A MLT LT charge

energycharge

−

−− −

⎡ ⎤ =⎢ ⎥⎣ ⎦

= × ×

⎡ ⎤= ⎢ ⎥⎣ ⎦

Schiff’s QM/Problem 4, page 423

Magnitude of A for visible part of the EM waves for radiation in a cavity at several thousand degrees Kelvin

p: mechanical momentum in Bohr orbit

quadraticH : ignorable

Perturbation Hamiltonian

{ }2 2

2

ignore (2 photons) quadratic term

q A(r, t) weak field 2mc

−

−

( )A(r, t) f A(r, t) f A(r, t) f= ∇• + •∇ + •∇

iqH ' A(r, t)mc

= − •∇

( )

( )

221 Zep

2m rHi q A(r, t) A(r, t)2mc

⎡ ⎤−⎢ ⎥

= ⎢ ⎥⎢ ⎥− ∇• + •∇⎢ ⎥⎣ ⎦

( ) ( )A(r, t) A(r, t) f A(r, t)f A(r, t) f∇• + •∇ = ∇• + •∇

( )A(r, t) A(r, t) 2A(r, t)∇• + •∇ ≡ •∇

Coulomb gauge:

=0 &

A 0

φ

∇• =


2A(r, t) f= •∇

0H H H'= + λ

0Ei t

0 (r)e−

ψ

ki tk k

k(t(r, t) c (r)e) − ωΨ = ψ∑

Initial State

Perturbed

State

[ ]k ki t i tk k 0 k k

k k

i c (r)e H(t) (t)H ' c (r)et

− ω − ω∂ ⎧ ⎫ ⎛ ⎞ψ = + λ ψ⎨ ⎬ ⎜ ⎟∂ ⎩ ⎭ ⎝ ⎠∑ ∑

iqH ' A(r, t) mc

qH ' A(r, t) pmc

= − •∇

= •

i (r, t) H (r, t)t∂Ψ = Ψ

∂


[ ]k ki tk k 0 k k

k k

i ti c (r) H H '( (t)c (r)et) et

− ω − ω⎧ ⎫ ⎛ ⎞ψ = + λ ψ⎨ ⎬ ⎜ ⎟⎩ ⎭ ⎝ ⎠

∂∂ ∑ ∑

( )

k k

k k

i t i tk k 0 k k

k k

i t i tk k k k k

k k

i c (r)e H c (r)e

i i

(t) (t)

c (r)e H ' c (r)e(t) (t)

− ω − ω

− ω − ω

⎡ ⎤ ⎡ ⎤⎧ ⎫ ⎛ ⎞ψ + ψ +⎨ ⎬⎢ ⎥ ⎢ ⎥⎜ ⎟⎩ ⎭ ⎝ ⎠⎢ ⎥ ⎢ ⎥=

⎢ ⎥ ⎢ ⎥⎧ ⎫ ⎛ ⎞− ω ψ λ ψ⎢ ⎥ ⎢ ⎥⎨ ⎬ ⎜ ⎟⎩ ⎭ ⎝ ⎠⎣ ⎦ ⎣ ⎦

∑ ∑

∑ ∑

k ki t i tk k k k

k ki c (t) (r)e H ' c (t) (r)e− ω − ω⎧ ⎫ ⎛ ⎞ψ = λ ψ⎨ ⎬ ⎜ ⎟⎩ ⎭ ⎝ ⎠∑ ∑

*fdV (r)ψ ×∫

{ }f ki t i tf k

ki c (t)e c (t) f | H ' | k e− ω − ω⎛ ⎞= λ⎜ ⎟

⎝ ⎠∑

Space integral


{ } ( )f ki tfk

k

(t) (dc 1 c f | H 't)idt

| k e+ ω −ω= λ∑

{ } fki tfk

k

(t) (t)dc 1 c f | H 'd it

| k e+ ω= λ∑

n (n)f f

n 0

c (t) c (t)∞

=

= λ∑ n (n)f f

n 0

c (t) c (t)∞

=

= λ∑

fkin (n) m (m)f k

n 0 m 0

t

kc (t 1 f | H ' | k e

i) c (t)

∞ω

=

+∞

=

λ λ⎧ ⎫

= λ⎨ ⎬⎩ ⎭

∑ ∑∑


fkin (n) m (m)f k

n 0 m 0

t

kc (t 1 f | H ' | k e

i) c (t)

∞ω

=

+∞

=

λ λ⎧ ⎫

= λ⎨ ⎬⎩ ⎭

∑ ∑∑

(0)f

(0)f c do not c (t) 0 depe n nd o t= →

fki t(1) (0)f k

k

1c (t) c f | H ' | k ei

+ ω= ∑fki t(s 1) (s)

f kk

1c (t) c f | H ' | k ei

+ ω+ = ∑


{ } fii t(1)f

d 1c (t) f | H ' | i edt i

+ ω=

fi

ti t '(1)

f0

ic (t) dt ' f | H '(r, t ') | i e+ ω−= ∫

( )2(1)fc (t) transition probability I order→

( ) ( ){ }i k r t i k r t0

iq ˆH ' A ( ) e emc

• −Ω − • −Ω⎛ ⎞= − ε Ω + •∇⎜ ⎟⎝ ⎠


fi

ti t '(1)

f0

ic (t) dt ' f | H '(r, t ') | i e+ ω−= ∫

( ) ( ){ }i k r t ' i k r t '0

iq ˆH ' A ( ) e emc

• −Ω − • −Ω⎛ ⎞= − ε ω + •∇⎜ ⎟⎝ ⎠

( )

( )

fi

fi

(1)f

ti t 'i k r t '

00

ti t 'i k '

00

r t

ic (t)

dt ' f | | i e

iqmc

ˆA ( )e

ˆdt ' f | | i eA ( )e

+ ω

+

• −

• −Ω ω

Ω

−

⎛ ⎞−⎜ ⎟⎝ ⎠

ε ω •∇

ε ω

−⎛ ⎞= ×⎜ ⎟⎝ ⎠

⎡ ⎤+⎢ ⎥

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

∇⎣

•⎦

∫

∫52PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

( )

( )

fi

fi

(1)f

ti t '

0

ti t '

0

0

ik r

ik r

c (t)

f | | i dt 'e

f | |

qA

i d

( )mc

ˆe

ˆe t 'e

− ω−ω

+

•

− • ω+ω

= ×

⎡ ⎤+

ω⎛ ⎞−⎜ ⎟⎝ ⎠

ε •∇

ε•∇

⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

∫

∫

( )( )

( )( )

( )

fifi

fi

tt i t 'i t '

fi0 0

i t

fi

edt 'ei

e 1i

± ω±ω± ω±ω

± ω±ω

=± ω±ω

−=± ω±ω

∫i

f

fi 0; 0ω ⟩ ω⟩53PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution


Google page, on 7th October 2012Commemoration of Niels Bohr’s 127th birthday

( )fi

0

ik

(1)

t

0

r

f

ti '

c (t)

f | | i dt 'e

qA ( )mc

ˆe • − ω−ω

ω⎛ ⎞= ×−⎜ ⎟⎝ ⎠

ε •∇ ∫( )

( )( )fi

iki t

(1)f

i

r0

f

e 1c (t) f | | ii

qA ( ) ˆemc

•− ω−ωω⎛ ⎞− ε •∇⎜ ⎟

⎝ ⎠

−=

− ω−ω


( )

( )( )fi

iki t

(1)f

i

r0

f

e 1c (t) f | | ii

qA ( ) ˆemc

•− ω−ωω⎛ ⎞− ε •∇⎜ ⎟

⎝ ⎠

−=

− ω−ω

2 22(1) ik rif f

0qA ( ) ˆc (t) f | 2F(t, )e | imc

•σ⎛ ⎞= ε •∇ ω−⎜ ⎟⎝ ⎠

ω

( )

( )( )( )

( )( )fi fi

*i t i t

fifi fi

e 1 e 12F(t, )i i

− ω−ω − ω−ω⎡ ⎤ ⎡ ⎤− −ω−ω = ⎢ ⎥ ⎢ ⎥− ω−ω − ω−ω⎣ ⎦ ⎣ ⎦

56

Transition Probability: modulus square


( )

( )( )( )

( )( )fi fi

22(1)

f

2i

*i t i t

fi fi

k r0qA ( ) ˆemc

c (t) f | | i

e 1i

e 1i

− ω−ω − ω−

•

ω

=

⎡ ⎤ ⎡ ⎤− −⎢ ⎥ ⎢ ⎥− ω−ω − ω−ω

ω⎛ ⎞ ε •∇ ×⎜ ⎟

⎣ ⎦ ⎣ ⎦

⎝ ⎠


( )

( )( )( )

( )( )fi fi

*i t i t

fifi fi

e 1 e 12F(t, )i i

− ω−ω − ω−ω⎡ ⎤ ⎡ ⎤− −ω−ω = ⎢ ⎥ ⎢ ⎥− ω−ω − ω−ω⎣ ⎦ ⎣ ⎦

( ) ( )

*i t i te 1 e 12F(t, )i i

− ω − ω⎡ ⎤ ⎡ ⎤− −ω = ⎢ ⎥ ⎢ ⎥− ω − ω⎣ ⎦ ⎣ ⎦

[ ]2

2 2

t2 2sin2 1 cos t 22F(t, )

⎡ ⎤ω⎛ ⎞⎜ ⎟⎢ ⎥− ω ⎝ ⎠⎣ ⎦ω = =

ω ω

t largelim F(t, ) t ( )→

ω = π δ ω

58

fi ω = ω−ω


2

2

t2sin2F(t, )

ω⎛ ⎞⎜ ⎟⎝ ⎠ω =

ω

2 2ik r02(1)

fc (t) fqA ( ) ˆe 2F| | (t, )mc

i•⎛ ⎞ ε •∇ × ω⎜ ⎟⎝ ⎠

ω=

t largelim F(t, ) t ( )→

ω = π δ ω

2 22(1) ik r0f

qA ( ) ˆc (t) f | e | i 2 t ( )mc

•⎛ ⎞= ε •∇ × π δ ω⎜ ⎟⎝ ⎠

ω

2 2ik r0

fiqA ( ) ˆW f | e | i 2 ( )

mc•ω⎛ ⎞= ε •∇ × πδ ω⎜ ⎟

⎝ ⎠

Transition RATE: Transition probability per unit time


X

Y

Z

ke−

hν

dΩ

d : solid angle: circular frequency

of the EM radiation

Ωω


2 2ik r0

fiqA ( ) ˆW f | e | i 2 ( )

mc•ω⎛ ⎞= ε•∇ × πδ ω⎜ ⎟

⎝ ⎠

X

Y

Z

ke−

hν

dΩd : solid angle

: circular frequency of the EM radiation

Ωω


Electric component :

polarized along the x-axis.xˆ eε =

[ ]f

2 2ˆ ik r0ˆfi k

qA ( ) ˆW f | e | i 2 ( )mc

ε •ω⎛ ⎞= ε •∇ × πδ ω⎜ ⎟⎝ ⎠

Energy absorbed per unit time in i fEnergy of the EM radiaflux tion

→

per uEnerg

nit ay absorbed

rea per unper unit time in i f

Energy of the EM rit time adiation→

[ ]f

f

ˆˆˆfi k

k

Wdd I( )

εε ω×σ⎡ ⎤ =⎢ ⎥Ω ω⎣ ⎦62PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

?

i(k r t ) i(k r t )0

0

ˆA(r, t) A ( ) e e

ˆ2A ( ) cos(k r t)

• −ω − • −ω⎡ ⎤= ω ε +⎣ ⎦

= ω ε • −ω

1 AE(r, t)c t

H(r, t) A

∂= −∇φ−

∂= ∇×

Coulomb gauge:

=0 and A 0 φ ∇• =


I( ) S

c E H4

ω =

= ×π

20

22 20 0

I( ) S 2 A ( )

A ( ) A ( )c 2 c

νω = = π ω

λων ω

= ω = ωπ

[ ]f

f

ˆˆˆfi k

k

Wdd I( )

εε ω×σ⎡ ⎤ =⎢ ⎥Ω ω⎣ ⎦

[ ]f

f

ˆˆˆfi k

2ˆ 2k

0

Wdd A ( )

2 c

εε ω×σ⎡ ⎤ =⎢ ⎥ ωΩ⎣ ⎦ ωπ


220I( ) S A ( )

2 cω

ω = = ωπ



[email protected] Unit 7(iii) Lecture 32


65



[ ]f

2 2ˆ ik r0ˆfi k

qA ( ) ˆW f | e | i 2 ( )mc

ε •ω⎛ ⎞= ε •∇ × πδ ω⎜ ⎟⎝ ⎠

f

2 2ik r

ˆ0

2ˆ 2k

0

qA ( ) ˆf |dd A ( )

2 c

e | i 2 ( )mcε

•ω⎛ ⎞ ε •ω×σ⎡ ⎤ =⎢ ⎥ ω

∇ × πδ ω⎜ ⎟

Ω⎣ ω

⎠⎦

π

⎝


[ ] [ ]f

f

f

ˆˆˆfi k

k

ˆˆfi k

220

Wdd I

W

A ( )c

(2

)

εεε ω×σ⎡ ⎤ =⎢ ⎥Ω ω⎣ ⎦

ω×=

ωω

π

f

2 2ik r

ˆ

2k

q ˆf | e | i 2 ( )d mcd

2 c

•ε

⎛ ⎞ω× ε•∇ × πδ ω⎜ ⎟σ⎡ ⎤ ⎝ ⎠=⎢ ⎥ ωΩ⎣ ⎦π

f

ˆ 2 2 22 ik r

2k

d q ˆ4 f | e | i )cd

(m

ε•⎛ ⎞⎡ ⎤ = π × ε•∇ ×δ ω⎜ ⎟⎢ ⎥ ω⎣ ⎦ ⎝ ⎠

σΩ

f

ˆ 2 2 2 2ik r

2k

d 4 e ˆf | e | i ( )d m c

ε•⎛ ⎞σ π⎡ ⎤ = × ε •∇ ×δ ω⎜ ⎟⎢ ⎥Ω ω⎣ ⎦ ⎝ ⎠

f

ˆ 2 2 2ik r

2k

d 4 ˆf | e | i ( )d m

ε•σ π α⎡ ⎤ = × ε•∇ ×δ ω⎢ ⎥Ω ω⎣ ⎦


f

ˆ 2 2 2ik r

2k

d 4 ˆf | e | i ( )d m

ε•σ π α⎡ ⎤ = × ε•∇ ×δ ω⎢ ⎥Ω ω⎣ ⎦

f

ˆ 2 2 2ik r

fi2k

d 4 ˆf | e | i ( )d m

ε•σ π α⎡ ⎤ = × ε•∇ ×δ ω−ω⎢ ⎥Ω ω⎣ ⎦

f

ˆ 2 3 2ik r

fi2k

d 4 ˆf | e | i (E E )d m

ε•σ π α⎡ ⎤ = × ε•∇ ×δ −⎢ ⎥Ω ω⎣ ⎦

Sakurai / Modern Quantum Mechanics / Eq.5.7.14 / page 337

2 18 2L : 1 Mb=10 cm−⎡ ⎤⎣ ⎦68PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

φ

rθ

dr

dθ

dϕ

X

Y

Z


fk

e−

Transitions to a number of degenerate states at energy E is possible scaling

Each of these

states contributes

to photoionization

i fik r ˆf | e | i• ε •∇

X

ZConstruct a grid of points labeled

by integers spaced at uniform

distances along the X, Y, Z axes.

Y

ke−

hνxˆ eε = dΩ

f

ˆ 2 3 2ik r

fi2k

d 4 ˆf | e | i (E E )d m

ε•σ π α⎡ ⎤ = × ε•∇ ×δ −⎢ ⎥Ω ω⎣ ⎦


X

YZ

ke−

hν

xˆ eε = dΩf

ˆ 2 3 2ik r

fi2k

d 4 ˆf | e | i (E E )d m

ε•σ π α⎡ ⎤ = × ε •∇ ×δ −⎢ ⎥Ω ω⎣ ⎦

71

3-dimensional orthogonal

space of independent

integers , , .x y zn n nPCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

Each of these

states contributes

to photoionization

i f

φ

rθ

dr

dθ

dϕ

X

Y

ZVolume

spanned by the

three

displacements

through

sinr dθ ϕrdθ

dr

2

2

( )( )( sin ) sin

dV dr rd r dr dr d dr drd

θ θ ϕ

θ θ ϕ

=

=

= Ω

Number of

states in the

volume element 2n dndΩ


fk

ik r ˆf | e | i• ε •∇

e−

73

Box normalizationwith Born von Karmann boundary conditions

How many wavelengths fit in the box?

2

2

x x

xx

xx

n L

n Lk

nkL

λπ

π

=

=

=

( )

( )

2 2 22 2 2

22 2 22 2 2 2

2

2 22 2

2

x y z

x y z

kE k k km m

E n n n nm L mL

π π

= = + +

⎛ ⎞= + + =⎜ ⎟⎝ ⎠


74

Number of states in

the volume element

2

2

n dnddnn dEddE

= Ω

= Ω

( )

( )

2 22

2

2 2

2

2 2

2

2

2 2

4

E nmL

dE ndnmL

ndnmL

π

π

π

=

=

=

2

2 2

2

2 2

14

4

dn mLdE n

m L kk n

π

π

=

=

12

2

12

2 2

2 2

2

1 2 221 1 22

mEk

dk mE mdEdk m mdE k k

−

⎛ ⎞= ⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠

= =

2

24dn dk LdE dE

knπ

=

( )

2 2 2 2

22 2 2

22

2

2

x y z

x y z

n n n n

L k k k

L k

π

π

= + +

⎛ ⎞= + +⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠

2kn L

π⇒ =


2

2

24 2L

dk L dk LdE dE

ππ π

==

75

Number of states in

the volume element

2

2

2

2

n dnd

n dEd

dk LdE

dndE

Edn dπ

= Ω

= Ω

= Ω 12

2

2 2

2

1 1 22

mEk

dk m mdE k k

⎛ ⎞= ⎜ ⎟⎝ ⎠

= =

22 2

2Ln kπ

⎛ ⎞= ⎜ ⎟⎝ ⎠

2kn L

π=


22

2 2dk L dEdd

LE

kπ π

⎛⎝

= ⎞⎜ ⎟

⎠Ω

32

22L mk

kdEd

π⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= Ω

32

2dk dEddE

L kπ

⎛ ⎞= Ω⎜ ⎟⎝

⎟⎠

⎛ ⎞⎜⎝ ⎠

3

22L mk dEdπ

⎛ ⎞ ⎛ ⎞= Ω⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

f

ˆ 2 3 2ik r

fi2k

d 4 ˆf | e | i (E E )d m

ε•σ π α⎡ ⎤ = × ε•∇ ×δ −⎢ ⎥Ω ω⎣ ⎦

76

Number of states in

the volume element

32

22L mkn dnd dEdπ

⎛ ⎞ ⎛ ⎞Ω = Ω⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

f

ˆ

k

2 3 2ik r

fi2

3

2

L mk

dd

4 ˆf | e | i d(E E )m

E2

ε

•

σ⎡ ⎤ =⎢ ⎥Ω⎣ ⎦

π α× ε ⎛ ⎞ ⎛ ⎞

⎜ ⎟•∇ δ − ⎜ ⎟⎝ ⎠ ⎠ω π ⎝∫


77

f

ˆ 32 3 2ik r

2 2k fi

d 4 L mkˆf | e | id m 2

ε•σ π α⎡ ⎤ ⎛ ⎞ ⎛ ⎞= × ε•∇ ⎜ ⎟ ⎜ ⎟⎢ ⎥Ω ω π⎣ ⎦ ⎝ ⎠ ⎝ ⎠

f

ˆ 32 32

2 2k fi

d 4 L mkMd m 2

εσ π α⎡ ⎤ ⎛ ⎞ ⎛ ⎞= × ⎜ ⎟ ⎜ ⎟⎢ ⎥Ω ω π⎣ ⎦ ⎝ ⎠ ⎝ ⎠

Dirac delta function integration:


ik r ˆM f | e | i•= ε •∇

78

f

ik r

ik r ik ri3

ˆM f | e | i

1 ˆdV e e (r)L

•

− • •

= ε •∇

⎛ ⎞ ⎡ ⎤= ε •∇ ψ⎜ ⎟ ⎣ ⎦⎝ ⎠∫

f

ˆ 32 32

2 2k fi

d 4 L mkMd m 2

εσ π α⎡ ⎤ ⎛ ⎞ ⎛ ⎞= × ⎜ ⎟ ⎜ ⎟⎢ ⎥Ω ω π⎣ ⎦ ⎝ ⎠ ⎝ ⎠

( )

( )( ) ( )

f

f

i k k ri3

i k k ri3

1 ˆM dV e (r)L1 1 ˆdV e i (r)

iL

− •

− •

⎡ ⎤= ε •∇ ψ⎢ ⎥⎣ ⎦

⎡ ⎤= ε • − ∇ ψ⎢ ⎥⎣ ⎦−

∫

∫PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution

79

( )( ) ( )fi k k r

i3

1 1 ˆM dV e i (r)iL

− •⎡ ⎤= ε • − ∇ ψ⎢ ⎥⎣ ⎦− ∫

( )fi k k ri f3

i ˆM dV (r) k eL

− •− ⎡ ⎤= ψ ε•⎢ ⎥⎣ ⎦∫

( ) ( ) ( )

( ){ } ( )

f

f

i k k ri3

i k k ri f3

1 1 ˆM dV (r) i eiL

1 ˆdV (r) i k k eL

− •

− •

⎡ ⎤= ψ ε• − ∇⎢ ⎥⎣ ⎦−

⎡ ⎤= ψ ε• −⎢ ⎥⎣ ⎦

∫

∫ˆ k 0ε • =


80

( )fi k k rf i3

i ˆM dV k (r)eL

− •− ⎡ ⎤⎡ ⎤= ε • ψ⎣ ⎦ ⎢ ⎥⎣ ⎦∫

( )fi k k rf i3

i ˆM k dV (r)eL

− •− ⎡ ⎤⎡ ⎤= ε • ψ⎣ ⎦ ⎢ ⎥⎣ ⎦∫

ˆˆ cos ?fkε γ• = =


X

Y

Zk

e−

hν

dΩ

81

θφ γ

ˆˆcosˆ ˆ

f

x r

k

e e

γ ε= •

= •

ˆ ˆ ˆ ˆsin cos sin sin cosr x y ze e e eθ φ θ φ θ= + +

cos sin cosγ θ φ=


82

( )

( )

[ ] ( )

f

f

f

i k k rf i3

i k k rff i3

i k k rfi3

i ˆM k dV (r)eLik ˆˆ k dV (r)eLik dV (r)eL

cos

− •

− •

− •

− ⎡ ⎤⎡ ⎤= ε • ψ⎣ ⎦ ⎢ ⎥⎣ ⎦

− ⎡ ⎤

γ

⎡ ⎤= ε • ψ⎣ ⎦ ⎢ ⎥⎣ ⎦

− ⎡ ⎤= ψ⎢ ⎥⎣ ⎦

∫

∫

∫

( )f2 2

2 f3

i k k ri

2 dV (co rkL

s eM ) − •⎡ ⎤⎡ ⎤= ⎣ ψ⎢ ⎥γ⎦ ⎣ ⎦∫The integral we are interested in is proportional

to the Fourier Transform of the initial state

cossin cosγθ φ=


83

( )

( )

( )

f

f

f0

i k k ri

i k k rnlm

3Zr2 i k k ra

0

I dV (r)e

dV (r)e

1 ZdV e ea

− •

− •

− − •

= ψ

= ψ

⎡ ⎤⎛ ⎞⎢ ⎥= ⎜ ⎟⎢ ⎥π ⎝ ⎠⎢ ⎥⎣ ⎦

∫∫

∫

{ }

32

02

20 2 20 f 2

0

Z8a1 ZI

a 1Z a k ka

⎛ ⎞π⎜ ⎟⎛ ⎞ ⎝ ⎠= ⎜ ⎟π ⎡ ⎤⎝ ⎠ ⎧ ⎫

+ − ⎨ ⎬⎢ ⎥⎩ ⎭⎣ ⎦

1s wavefunction



f

ˆ 32 32

2 2k fi

d 4 L mkMd m 2

εσ π α⎡ ⎤ ⎛ ⎞ ⎛ ⎞= × ⎜ ⎟ ⎜ ⎟⎢ ⎥Ω ω π⎣ ⎦ ⎝ ⎠ ⎝ ⎠

( )f2 2

i k k r2 fi3

2kM dV (rL

cos )e − •⎡ ⎤⎡ ⎤= ψ⎣ ⎦ ⎢ ⎥⎦γ⎣∫

( )

( )

f

f0

i k k ri

3Zr2 i k k ra

0

I dV (r)e

1 ZdV e ea

− •

− − •

= ψ

⎡ ⎤⎛ ⎞⎢ ⎥= ⎜ ⎟⎢ ⎥π ⎝ ⎠⎢ ⎥⎣ ⎦

∫

∫

{ }

32

02

20 2 20 f 2

0

Z8a1 ZI

a 1Z a k ka

⎛ ⎞π⎜ ⎟⎛ ⎞ ⎝ ⎠= ⎜ ⎟π ⎡ ⎤⎝ ⎠ ⎧ ⎫

+ − ⎨ ⎬⎢ ⎥⎩ ⎭⎣ ⎦

22 2 2f

3

kM cos IL

⎡ ⎤= γ⎣ ⎦

85

( )

{ }

f

ˆ

k

5 3 2 322 30f

42 3 222 2fi0 f

dd

64 Z a cosk4 L mkm L 2

Z a k k

εσ⎡ ⎤ =⎢ ⎥Ω⎣ ⎦

π γπ α ⎛ ⎞ ⎛ ⎞× ⎜ ⎟ ⎜ ⎟ω π⎝ ⎠ ⎝ ⎠⎡ ⎤+ −⎢ ⎥⎣ ⎦

( )

{ }f

5 3 2ˆ 30f

42ˆ 2 2k fi0 f

Z a cos32 kdd m

Z a k k

ε γασ⎡ ⎤ =⎢ ⎥Ω ω⎣ ⎦ ⎡ ⎤+ −⎢ ⎥⎣ ⎦Bransden & Joachain ‘Physics of Atoms & Molecules’ Eq.4.157 / page 190 // Eq.4.208; page 227


86

( )

{ }f

5 3 2ˆ 30f

42ˆ 2 2k fi0 f

Z a cos32 kdd m

Z a k k

ε γασ⎡ ⎤ =⎢ ⎥Ω ω⎣ ⎦ ⎡ ⎤+ −⎢ ⎥⎣ ⎦Born approximation

2 2 2 2f f

High Energy Approximation

k kI.P. 2m 2m

ω = + ≈ → ≈2fk

2mω≈

2f

f f f

f

kkc2m

k p vk 1k 2cm 2cm 2c

= = ⟨⟨⟨ f

k 1k

⟨⟨⟨


2 2 c kcω πν πλ

= = =

X

YZ

fke−

hν

dΩ

87

θφ γ

( ) ( )2 2 2 cos

f f

f f

k k k k

k k kk θ

− • −

= + −


88

( ) ( ) 2 2

22

2 2

2 cos

1 2 cos

f f f f

ff

f f

k k k k k k kk

kkkkk k

θ

θ

− • − = + −

⎛ ⎞⎟⎜ ⎟⎜= + − ⎟⎜ ⎟⎟⎜⎝ ⎠

22 v

1 cosff fk k k

cθ

⎛ ⎞⎟⎜ ⎟− −⎜ ⎟⎜ ⎟⎜⎝ ⎠

f

f

vkk 2c


22

2

1 2 cos

v1 2 cos

2

f ff

ff

kk k kk

kc

θ

θ

⎛ ⎞⎟⎜ ⎟⎜− − ⎟⎜ ⎟⎟⎜⎝ ⎠⎛ ⎞⎟⎜ ⎟= −⎜ ⎟⎜ ⎟⎜⎝ ⎠

89

( )

{ }f

5 3 2ˆ 30f

42ˆ 2 2k fi0 f

Z a cos32 kdd m

Z a k k

ε γασ⎡ ⎤ =⎢ ⎥Ω ω⎣ ⎦ ⎡ ⎤+ −⎢ ⎥⎣ ⎦

22 v

1 cosff fk k k

cθ

⎛ ⎞⎟⎜ ⎟− −⎜ ⎟⎜ ⎟⎜⎝ ⎠

22 2 2 2 2 f0 f 0 f

2 2 2 2 2 f0 f 0 f

vZ a k k Z a k 1 cosc

vZ a k a k cosc

⎛ ⎞+ − ≈ + − θ⎜ ⎟⎝ ⎠

≈ + − θ


90

22 2 2 2 2 f0 f 0 f

2 2 2 2 2 f0 f 0 f


vZ a k a k cosc

⎛ ⎞+ − ≈ + − θ⎜ ⎟⎝ ⎠

≈ + − θ

22 2 2 2 2 f0 f 0 f

2 2 2 2 f0 f 0 f


va k a k cosc

⎛ ⎞+ − ≈ + − θ⎜ ⎟⎝ ⎠

≈ − θ


2 2 20high energy approximation: fa k Z⟩⟩⟩

91

( )

{ }f

5 3 2ˆ 30f

42ˆ 2 2k fi0 f

Z a cos32 kdd m

Z a k k

ε γασ⎡ ⎤ =⎢ ⎥Ω ω⎣ ⎦ ⎡ ⎤+ −⎢ ⎥⎣ ⎦22 2 2 2 2 2 f

0 f 0 f 0 f

2 2 f0 f

vZ a k k a k a k cosc

va k 1 cosc

+ − ≈ − θ

⎛ ⎞= − θ⎜ ⎟⎝ ⎠

( )( )f

5 3 2ˆ 30f

44k fi 2 2 f

0 f

Z a cos32 kdd m va k 1 cos

c

ε γασ⎡ ⎤ =⎢ ⎥Ω ω⎣ ⎦ ⎛ ⎞− θ⎜ ⎟⎝ ⎠


92

( )( )f

5 3 2ˆ 30f

44k fi 2 2 f

0 f

Z a cos32 kdd m va k 1 cos

c

ε γασ⎡ ⎤ =⎢ ⎥Ω ω⎣ ⎦ ⎛ ⎞− θ⎜ ⎟⎝ ⎠

( )f

5 2 2ˆ

4k fi 0 f f

sin cosd 32 Zd m a k v1 cos

c

ε θ φ⎛ ⎞σ α⎡ ⎤ = ⎜ ⎟⎢ ⎥Ω ω⎣ ⎦ ⎛ ⎞⎝ ⎠ − θ⎜ ⎟⎝ ⎠

2cos φ Distribution with respect to the direction of polarization of the electric field


93

( )f

5 2 2ˆ

4k fi 0 f f

sin cosd 32 Zd m a k v1 cos

c

ε θ φ⎛ ⎞σ α⎡ ⎤ = ⎜ ⎟⎢ ⎥Ω ω⎣ ⎦ ⎛ ⎞⎝ ⎠ − θ⎜ ⎟⎝ ⎠

For unpolarized light:

22 2

0

1 1cos cos d2 2

π

φ = φ φ =π ∫

( )f

5 4unpolarized2 f

k fi 0 f

vd 16 Z sin 1 cosd m a k c

−⎛ ⎞σ α ⎛ ⎞⎡ ⎤ = θ − θ⎜ ⎟ ⎜ ⎟⎢ ⎥Ω ω⎣ ⎦ ⎝ ⎠⎝ ⎠


( )f

5unpolarized2 f

k fi 0 f

vd 16 Z sin 1 4 cosd m a k c

⎛ ⎞σ α ⎛ ⎞⎡ ⎤ ≈ θ + θ⎜ ⎟ ⎜ ⎟⎢ ⎥Ω ω⎣ ⎦ ⎝ ⎠⎝ ⎠

94

( )f

5unpolarized2 f

k fi 0 f

vd 16 Z sin 1 4 cosd m a k c

⎛ ⎞σ α ⎛ ⎞⎡ ⎤ ≈ θ + θ⎜ ⎟ ⎜ ⎟⎢ ⎥Ω ω⎣ ⎦ ⎝ ⎠⎝ ⎠

Total

f

unpolarized2unpolarized

k0 0

5

50 f

d sin d dd

128 Z 13m a k

π π

θ= φ=

σ⎡ ⎤σ = θ θ φ⎢ ⎥Ω⎣ ⎦

⎛ ⎞π α= ⎜ ⎟ω ⎝ ⎠

∫ ∫


( )

2 2f

5/25 2f f

5/25/2

k2m

k k

2m

ω ≈

⇒

=

⎛ ⎞= ω⎜ ⎟⎝ ⎠

Total

5unpolarized

5/2 7/20

128 Z 13m a2m

⎛ ⎞π ασ = ⎜ ⎟ ω⎛ ⎞ ⎝ ⎠

⎜ ⎟⎝ ⎠

752

3

E , Z , n

−

−

σ→

BUT! Breakdown of the Independent Particle Approximation in High-Energy PhotoionizationPhysical Review Letters 78:24 p.4553-4556 (1997)

95

f

ˆ 2 2 2ik r

2k

d 4 ˆf | e | i ( )d m

ε•σ π α⎡ ⎤ = × ε•∇ ×δ ω⎢ ⎥Ω ω⎣ ⎦

f

ˆ2

k

d

Physical Dime

L

i

d

ns onsε⎡ ⎤σ⎡ ⎤ =⎢ ⎥⎢ ⎥Ω⎣ ⎦⎢ ⎥⎣ ⎦


ik r

we shall now study the

ˆmatrix element f | e | i• ε •∇



f

ˆ 2 2 2ik r

2k

d 4 ˆf | e | i ( )d m

ε•σ π α⎡ ⎤ = × ε•∇ ×δ ω⎢ ⎥Ω ω⎣ ⎦

f

ˆ2

k

d

Physical Dime

L

i

d

ns onsε⎡ ⎤σ⎡ ⎤ =⎢ ⎥⎢ ⎥Ω⎣ ⎦⎢ ⎥⎣ ⎦

Total

5unpolarized

5/2 7/20

128 Z 13m a2m

⎛ ⎞π ασ = ⎜ ⎟ ω⎛ ⎞ ⎝ ⎠

⎜ ⎟⎝ ⎠

75 32E , Z , n

− −σ→


[email protected] Unit 7(iv) Lecture 33


97



X

YZ

hν

e−

θkφ


f

ˆ 2 2 2ik r

2k

d 4 ˆf | e | i ( )d m

ε•σ π α⎡ ⎤ = × ε•∇ ×δ ω⎢ ⎥Ω ω⎣ ⎦


f

ˆ 2 2 2ik r

2k

d 4 ˆf | e | i ( )d m

ε•σ π α⎡ ⎤ = × ε•∇ ×δ ω⎢ ⎥Ω ω⎣ ⎦

f

ˆ2

k

d

Physical Dime

L

i

d

ns onsε⎡ ⎤σ⎡ ⎤ =⎢ ⎥⎢ ⎥Ω⎣ ⎦⎢ ⎥⎣ ⎦

Total

5unpolarized

5/2 7/20

128 Z 13m a2m

⎛ ⎞π ασ = ⎜ ⎟ ω⎛ ⎞ ⎝ ⎠

⎜ ⎟⎝ ⎠

75 32E , Z , n

− −σ→

ik r

we shall now study the

ˆmatrix element f | e | i• ε •∇

100

( )i k r re 1 expansion in powers of

"dipole" approximation

+≈

λi

( ) ( )ik r ik r1ˆ ˆf | e | i f | e i | ii

• •ε •∇ = ε• − ∇−

kx

i r iˆf | e | i f | |p i• ε •∇ =


ik r ik riˆ ˆf | e | i f | e p | i • •ε •∇ = ε•

Good for large wavelength, low energyUp to ~6keV above ionization threshold

However, beyond the dipole approximation….

101

[ ]ik r0

iˆf | e | i f | m x,Hi

| i• ε •∇ =

[ ] [ ]

k

2k k k k k k k

k k k k k k k k k k k k

k k k k k k

k

r ,p r p p p p r

r p p p r p p r p p p r

r ,p p p r ,p2i p

⎡ ⎤ = −⎣ ⎦= − + −

= +

=

ik rdipoleap

xprox.

piˆf | e | i f | | i• ⎤ε •∇ =⎦

2

k 0 k kp i[r ,H ] [r , ] p2m m

= =


Matrix element of momentum / position operator

102

[ ]r0

ik0

m xH H x1ˆf | e | i f | | i• ε •∇ = −

[ ]ik r0

iˆf | e | i f | m x,Hi

| i• ε •∇ =

( )ik ri f2

mˆf | e | i E | ixE f |• ε •∇ = −


ik rdipoleap

xprox.

piˆf | e | i f | | i• ⎤ε •∇ =⎦

103

( )f i

f

x

i

imf | | i E E f | | i

im

p x

f | | ix

= −

= ω

Momentum & Length forms of the matrix elements

Astrophysical Journal, (1945)vol. 102, p.223 S.Chandrasekhar

fi fi fip im r = ω


- from - PCD STiAP Unit 4 HF SCF 104

*2 2

11 1 2 1 1,12 2

,( ) ( )

( ) ( ) ( ) ( )⎡ ⎤

= ⎢ ⎥⎢ ⎥⎣ ⎦∫ j iex

j i js sj i sim m m

u r u rV q u q dV u r

rδ χ ζ

( )1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) = + −∑ ∑direct exi j i j i i i

j jf r u q V q u q V q u q u qε

( )1 1 1 1 1 ( ) ( ) ( ) ( ) = + −d exi i i i if r u q V u q V u q u qε

( )1 1 1 1 ( ) ( ) ( ) = +i HF i i if r u q V u q u qε “deceptively simple”- Bransden & Joachain

*2 2

1 212

( ) ( )( )

⎡ ⎤= ⎢ ⎥⎢ ⎥⎣ ⎦∫

directj jcoulomb

j

u q u qV q dq

r


( ), , 0r V r r′ ≠⎡ ⎤⎣ ⎦

For non-local potential:

2

k 0 k kp i[r ,H ] [r , ] p2m m

= =

fi fi fi p im r = ω

Equivalence of momentum/velocity and the length forms of the matrix element

requires

Outer subshell radial wave functions and d waves for ε=0 for Ne, and Ar

Cooper minimum:J. W. Cooper, Physical Review,128, 2, 681 (1962)

fi fi fip im r = ω

Matrix elements for p→d transitions in Ne, Ar and Kr


Tanima Banerjee ….Physical Review A75, 042701 (2007)


fi fi fip im r= ω

ik r ik riˆ ˆ M f | e | i f | e p | i • •= ε •∇ = ε•

x fif | | i im | ip f | x= ω

Bransden & Joachchain: Physics of Atoms & Molecules Section 4.3 ‘Dipole approximation’

( )2

ik r r re 1 i 2 cos k, r 1 (dipole approximation)• ⎛ ⎞= + π +Ο ≈⎜ ⎟λ λ⎝ ⎠

( ) ( )D fifi fi fi

mi iˆ ˆ ˆM f | p | i im r r− ω= ε• = ω ε• = ε •


E2 Cooper minimum

Hg 6s RRPA

610−

510−

100 150 200 250 300 350 400 450 500 550 6000.0

0.5

1.0

1.5

2.0

2.5

3.0

Hg 4f Dipole cross-section in RRPA couplingall the E1 channels from 6s,5d,5p,4f,4d

4d3/24d5/2

4f5/2

4f7/25s

4f5/2(6s,5d,5p,5s,4f,4d)

4f7/2(6s,5d,5p,)5s,4f,4d)

σ E1 M

b

hν eV

Near threshold photoionization

Hg 4f

150 200 250 300 350 400 450 500 550 600

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Hg 4f Dipole Matrix Elements in RRPA, coupling all E1 channels from 6s,5d,5p,4f,4d,4p,4s4f5/2

4f7/2

|4f5/2->&d3/2||4f5/2->&d5/2|

|4f7/2->&d5/2| |4f5/2->&g7/2|

|4f7/2->&g7/2|

|4f7/2->&g9/2|

hν (eV)

7/2,5/24 f9/2,7/2

5/2,3/2

gd

ε

ε

11,0

lj

δδ

= ±= ±

( 0)

lR r rshape resonance

→ →Centrifugal barrier effect


112

xfi

1f | | i f | | iim

x p=ω

fE

iEAbsorption : O.S. 0Emission : O.S. 0

⟩⟨

QM 2fifi fidefinition

Oscillator Strength: 2mf r

3

→ ω=


( )

( )

classical dmodel

22 d

0,s

0,s0

definitiondf 1 2d

2

df d d 1d

∞ ∞

−∞

→

Γ

ω π ⎧ ⎫Γ⎪ ⎪⎛ ⎞ω−ω +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

ω = δ ω−ω ω =ω

=

∫ ∫

fif

f 1=∑

QM

Thomas – Reiche – Kuhn sum rule

113

definition

2 2fi fifi fi fi2

Oscillator Strength:2m 2mEf r r= =

ω

definition

2x fifi fi

2mf x3ω

=

2y fifi fi

2z fifi f

definition

definitio in

2mf y3

2mf z3

ω=

ω=


114

2x fi fifi fi

2m 2mf x i | x | f f | x | i3 3ω ω

= =

xx fifi

if

i | p | f2mf f | x | i3 imω

=ω x

fi

f | x | i1 f | p | i

im

=

=ω

{ } xxfi if

fi

f | p | i2 1f im i | x | f3 ( i) im

⎧ ⎫= ω ⎨ ⎬− ω⎩ ⎭ ( )

xi | x | f f | p | i23 i

=+

( )xx

fi

i | p | f f | x | i2f3 i

=−

if fiω = −ω


fE

iE

( )xx

fi xfi

1i | p | f2f3

f | p | iimi

⎧ ⎫⎨ ω− ⎭

×⎩

= ⎬

115

( )xx

fi

i | x | f f | p | i2f3 i

=+

xfi x

2if i | p | f f | x | i3

=

xfi x

2if i | x | f f | p | i3

= −

xfi x x

if i | p | f f | x | i | x | f f ii3

| p |⎡ ⎤= −⎣ ⎦

x xxfi

f f

if i | p | f f | i | x | f f | pi3

| ix | −⎡ ⎤= ⎣ ⎦∑ ∑


( )xx

fi

i | p | f f | x | i2f3 i

=−

116

x xxfi

f f

if i | p | f f | i | x | f f | pi3

| ix | −⎡ ⎤= ⎣ ⎦∑ ∑

xfi x x

f

if i | p x xp | i3

⎡ ⎤= −⎣ ⎦∑

[ ]xx,p i= ( )xfi

f

i 1f i3 3

= − =∑

fif

f 1=∑ Thomas – Reiche – Kuhn sum rule

Heisenberg arrived at the law in reverse!

[ ]xx,p i=

ff f 1=∑


117

one electronatom

definiti

2fifi fn io

2mf r−

ω=

N electronato

2N

(N) ( jm

definition

)fifi fi

j 1

2mf r=

−

ω= ∑

N electronatom

d

2N

(N) ( j)fifi fi

f f jefiniti 1on

2mf Nr=

−

= =ω∑ ∑ ∑


118

2fi fi

f

r2m

ω =∑

one electronatom

definiti

2fifi fn io

2mf r−

ω=

fif

f 1=∑

fi

21fi

fr

2mω =∑

fi

2fi

f

n :

sum rules for various mom ts

r

en

ω∑


PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution119

[ ]f

2 2 2ˆ ik rˆfi 2 2 k

4 e I( ) ˆW f | e | i ( )m c

ε •π ω= ε•∇ ×δ ω

ω

( ) ( )

ik r ik r

D fifi fi fi

iˆ ˆM f | e | i f | e p | i

mi iˆ ˆ ˆM f | p | i im r r

• •= ε •∇ = ε•

− ω= ε• = ω ε• = ε•


[ ] ( )f

2 22fi

fi2

2 2ˆ ˆfi 2 2 k

4 e I m ˆ r( )W ( )m c

ε ⎡ ⎤ωε•⎢ ⎥

⎣ ⎦

π ω= ×δ ω

ω

[ ]f

2 2ˆ 2ˆfi fi2 k

4W I( )cos D ( ); D=erc

ε π= ω γ ×δ ω

2 1cos3

γ = [ ]2 2D

fi fi2

4W I( ) D ( )3 cπ

= ω ×δ ω

Dipole approx.

[ ] ( )f

2 2 222

f

2ˆ 2ˆfi f k 2 2i i

4 eW I( ) ( 4 e I( )ˆ r cos r ( )c

)c

ε π= ω ×δ

πε ω = ω γ ×δ ω•

“Dipole approx.”


[ ]2 2D

fi fi2

4W I( ) D ( )3 cπ

= ω ×δ ω

[ ]D 2 2fifi fi2

W 4 D ( ) B3π

= ×δ ω =ρ

Einstein Bfi coefficient for (stimulated) absorption i f

f

i

( )f i

: energy density of the radiationE E at frequency

ρ ω

−ω =

[ ]( )

2 22 2D

fi fi2 2fi

4 I( ) 4 I( )D ( ) D ( )W 3 c 3 cI / c

π ω π ω×δ ω ×δ ω

= =ρ ρ ω

Bransden & Joachchain: Physics of Atoms & Molecules Section 4.4 ‘Einstein Coefficients’


( )iif

f

if f f

i

if

f

f

ifdN + dt

N : total number of atoms in state f: Eins

A N sponteneous

A sponteneno

B N stimulate

tein coefficient for emission: Einstein coefficien

d

B stimulaus

tet for emisd sion

ρ= ω+ →

Number of atoms per unit time making the emission transition f i

( )fifi i if

i

dN B Ndt

N : total number of atoms in state i

= ρ ω

Number of atoms per unit time making the absorption transition i f

f

i

( )f i

: energy density of the radiationE E at frequency

ρ ω

−ω =


122

( )( )

( )( )

if f if f if

fi i if

if if if i

fi if f

kT

A N B N1

B N

A B NB N

eω

+ ρ ω=

ρ ω

+ ρ ω=

ρ ω

=

At equilibrium, the two rates are equal

( )fi

iffi

kTfi if

A

B e Bω⇒ ρ ω =−

fi if ifEinstein coefficients absorption : i f : B emission : f i : B ,A→ →

( )ifif f if f if

dN A N B Ndt

= + ρ ω ( )fifi i if

dN B Ndt

= ρ ω

(thermal equilibrium)

( )fi

3fi

fi 2 3kT

1Planck's law: c e 1

ω

ωρ ω =

π−

if fi3fi

if if2 3

B B

A Bc

=

ω=π

Relationship between the three Einstein coefficients



[email protected] Unit 7(v) Lecture 34


123

Atomic Photoionization cross-sections, Angular distribution of photoelectronsCOOPER – ZARE formula


PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution124

[ ]f

2 2 2ˆ ik rˆfi 2 2 k

4 e I( ) ˆW f | e | i ( )m c

ε •π ω= ε•∇ ×δ ω

ω

( ) ( )

ik r

D

dipoleapprox.

fifi fi fi

i iˆ ˆM f | e p | i f | p | i

mi ˆ ˆM im r r

•= ε • ε •

− ω= ω

≈

ε• = ε•


[ ]f

2 2ˆ 2ˆfi fi2 k

4W I( )cos D ( ); D=erc

ε π= ω γ ×δ ω

[ ] ( )f

2 2 222

f

2ˆ 2ˆfi f k 2 2i i

4 eW I( ) ( 4 e I( )ˆ r cos r ( )c

)c

ε π= ω ×δ

πε ω = ω γ ×δ ω•

“Dipole approximation” @ low energy, long wavelength

( )2

ik r r re 1 i 2 cos k, r 1 (dipole approximation)• ⎛ ⎞= + π +Ο ≈⎜ ⎟λ λ⎝ ⎠

ik re 1 (dipole approximation)

• ≈


QM 2fifi fidefinition

fif

Oscillator Strength: 2mf r

3f 1

→ ω=

=∑

( )

( )

classical dmodel

22 d

0,s

0,s0

definitiondf 1 2d

2

df d d 1d

∞ ∞

−∞

→

Γ

ω π ⎧ ⎫Γ⎪ ⎪⎛ ⎞ω−ω +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

ω = δ ω−ω ω =ω

=

∫ ∫

( )

quantum dmechanics fi

2definition 2 d0,s

fdf 1 2=d

2

Γ

ω π ⎧ ⎫Γ⎪ ⎪⎛ ⎞ω−ω +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

Quantum mechanical frequency distribution of the oscillator strengthOscillator strength per unit frequency range at a given frequency


QM 2fifi fi fidefinition f

Oscillator Strength: 2mf r ; f 1

3

→ ω= =∑

( )

quantum dmechanics fi

2definition 2 d0,s

fdf 1 2=d

2

Γ

ω π ⎧ ⎫Γ⎪ ⎪⎛ ⎞ω−ω +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

Quantum mechanical frequency distribution of the oscillator strength. Oscillator strength per unit frequency range at a given frequency.

( )

quantum s dmechanics

2definition 2s d0,s

fdf 1 2=d

2

Γ

ε π ⎧ ⎫Γ⎪ ⎪⎛ ⎞ε − ε +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

∑Sum over all discrete states.Fano & Rau, Eq. 2.26


n ,

2

E1

(n )many electron atoms

=

= −− μ

Fano & Cooper Revs. Mod. Phys. 40:3 (1968) p441

Fano & Rau, page 34

Eigen-amplitudes of the transition matrix elements are slowly varying functions of energy across the ionization threshold.Quantum defect theory: many electron atomsSeaton / Fano

: almost independent of energyμ

H:SO(4)

( ) 2 ( )

0

1 2 1 1 ( cos )2

.4.6 / 64 / &

likzTot

ikri k

ll

e l e Pr ik

Eq page Fano Ra

e

ur

δ θψ− ∞

−

=

− ⎧ ⎫⎡ ⎤+ − −⎨ ⎬⎣− ⎦⎩ ⎭−

→∞ ∑→

2 1| (2 1) ( cos )2

liikrikz

f ll

e ee l Pr ik

δ

ψ θ−− ⎛ ⎞−

⟩ → − + − ⎜ ⎟⎝ ⎠

∑

| |f iTψ ψ⟨ ⟩

[ ] =Θ128from PCD STiAP Unit 6 Probing the Atom: C&S

( )

( )( )

( ) 2 ( )

0

1 2 1 1

,

(cos )2

l

Tot

i kzi t

i

l

tkr

kl

r t

e e l e Pr ik

rω

ω δ θ

ψ

∞

−

+

=

+ ++ ⎧ ⎫⎡ ⎤+ −

⎤

⎨ ⎬⎣ ⎦⎩ ⎭

⎦

+

→∞

∑

→

θθ π+

( )( )

cos( ) cos

cos cos( )l lP P

θ π θ

θ θ π

+ = −

− = +

129PCD STiAP Unit 6 Probing the Atom: C&S

Photoelectron escape/exit direction: unique

[ ] =Θ( )

( )ˆ( , ) ( )

i

i kr ti kz t

k rer t A e f

r

ωωψ

−−+

→∞⎡ ⎤

→ + Ω⎢ ⎥⎣ ⎦

collision Time Reversal symmetry

| | ?f iTψ ψ⟨ ⟩ =

130

sin( )2( ) (2 1) (cos )

ll

Total l lrl

lkrr c i l P

kr

π δψ θ→∞

− +⎯⎯⎯→ +∑

( ) ( )*4 ˆ ˆ( ) (2 1) 2 1

sin( )2

l

Total

Li l m m

l f lrl m l

l

r e i l Y k Y rl

lkr

kr

δ πψ

π δ

−−→∞

=−

⎡ ⎤⎯⎯⎯→ + ⎢ ⎥+⎣ ⎦

− +×

∑ ∑

Photoelectron Angular Distributions – J.Cooper and R.N.Zare

Presented at the Theoretical Physics Institute, University of Colorado, Summer, 1968


INGOING wave boundary conditions (Unit 6)


Spherical harmonics addition theorem

| | ?f iTψ ψ⟨ ⟩ =

131

( ) ( )*4 ˆ( ) (2 1) 2 1

sin( )2

ˆ

l

Total

Li l m m

l lrl m l

l

fr e i l Y Yl

lr

k r

k

kr

δ πψ

π δ

−−→∞

=−

⎡ ⎤⎯⎯⎯→ + ⎢ ⎥+⎣ ⎦

− +×

∑ ∑

( ),

ˆ( ) ( , ) ( )Total

ml kl

l mr a l m Y r R rψ − =∑

*, ˆ( ) 4 ( )lil ml f

where

a l m e Y ki δπ −=



: polar angle measured with respect to the polarization directionˆ ˆxeε

Θ

=

X

YZ

fke−dΩ

132

θφ

01

4c

ˆ| | | |

os ( , )

| |

| | | |3

f i f i f x i

f i f im

T x r

r Y

e

r

ψ ψ ψ ψ ψ ψ

ψ ψ ψ πγ ψ=

⟨ ⟩ = ⟨ ⟩ = ⟨ • ⟩

= ⟨ Θ Φ⟩ = ⟨ ⟩


01

1 3( , ) cos2 4

mYπ

= Θ Φ = Θ

γ =Θ

( )ˆˆ , fkγ εΘ = =

: polar angle w.r.t. polarization direction ˆ ˆxeε

Θ

=

X

YZ

fke−dΩ

133

θφ

01

4cos| ( , )3

| | | | |f i f i f imT r Yrψ ψ ψ ψ πγ ψ ψ=⟨ ⟩ = ⟨ ⟩ = Θ Φ⟨ ⟩


01

1 3( , ) cos2 4

mYπ

= Θ Φ = Θ

γ =Θ

( )ˆˆ , fkγ εΘ = =

θθ π+

Time Reversal symmetry

01| | | ( , ) |f i f iT rCψ ψ ψ ψ⟨ ⟩ = ⟨ Θ Φ ⟩

4( , ) ( , ) 2 1

renormalized spherical harmonics

q qk kC Y

kπ

Θ Φ = Θ Φ+

( ) ( )

[ ]

0 '1 '

,

0

2

1,

' '0

'

ˆ ˆ( , ) | ( , ) |

(

( )

|

, ) | ( , ) |

|

'

(

'

)kl n l

f

m ml l

l m

l mll

i

a l m Y

T

r C Y r

a l m lm C

r drR r R r

d m

r

l

ψ ψ∞

⟨ Θ Φ ⟩

= ⟨ Θ

⎡ ⎤⎢ ⎥⎣

⟨

Φ

⟩

⎦⟩

=

∑

∫∑

134

01| | | ( |, )f i ifT rCψψ ψ ψ⟨ ⟩ = Θ Φ⟨ ⟩

( ),

( ) ˆ( , ) ( )Total

ml kl

l m

a l m Y r R rrψ − =∑( )' ' ' ' ' ˆ| ( ) | ' 'i

n l m n lr R r r l mψ⟨ ⟩ = ⟨ ⟩

( ) ( )

01

01 ' ' '

,

'ˆ( , )

| | | ( , ) |

| ( , ) ˆ( )( ) |f

ml k

m

i

ll m

f

l n l

i

a l m Y r R

T

r

rC

Rr Y rC r

ψ ψψ ψ⟨ ⟩ = ⟨ Θ Φ ⟩ =

= ⟨ Θ Φ ⟩∑

Cooper-Zare / Eq.II.7


Radial and Angular integration: space integral

Slide 131

135

[ ] 0' 1

,| | ( , ) | ( , ) | ' 'f i ll

l mT a l m d lm C l mψ ψ⟨ ⟩ = ⟨ Θ Φ ⟩∑

( )01 1

1 '| ( , ) | ' ' 1 | | '

0 'l m l l

lm C l m l C lm m

− ⎛ ⎞⟨ Θ Φ ⟩ = − ⟨ ⟩ ⎜ ⎟−⎝ ⎠

[ ] ( ) ( )',

| |

1 '( , ) 1 1

' 0 '

f i

l m gll

l m

T

l la l m d l

m m

ψ ψ

−>

⟨ ⟩ =

⎧ ⎫⎛ ⎞− −⎨ ⎬⎜ ⎟−⎝ ⎠⎩ ⎭

∑

Wigner-Eckart Theorem


( ) ( )' 1

21 NOTE! | | ' 1 1

' 1 is the greater of & ' and

'

2

l l gl C l l ll lwher

m

e l l g

m

l

− +

> >

>

⟨ ⟩ = − = −

− +=

=

Radial integral

136

f

ˆ 2 2 2ik r

2k

d 4 ˆf | e | i ( )d m

ε•σ π α⎡ ⎤ = × ε•∇ ×δ ω⎢ ⎥Ω ω⎣ ⎦

( )f

ˆ 22 2

2k

i f2ˆ

m E E f | x | id 4 ( )d m

εσ π α⎡ ⎤ = × ×δ ω⎢ ⎥Ω ω⎣−

⎦

( )f

ˆ 22 2

2 f f i2k

id 4 ( )d m

|E T |m Eεσ π α⎡ ⎤ = × ×δ ω⎢ ⎥Ω ω⎣ ⎦

⟩− ⟨ψ ψ

[ ] ( ) ( )',

| |

1 '( , ) 1 1

' 0 '

f i

l m gll

l m

T

l la l m d l

m m

ψ ψ

−>

⟨ ⟩ =

⎧ ⎫⎛ ⎞− −⎨ ⎬⎜ ⎟−⎝ ⎠⎩ ⎭

∑


cos| | | | | |f i f i f ix r Tγψ ψ ψ ψ ψ ψ⟨ ⟩ = ⟨ ⟩ = ⟨ ⟩

137

( )f

ˆ 22 2

2 f f i2k

id 4 ( )d m

|E T |m Eεσ π α⎡ ⎤ = × ×δ ω⎢ ⎥Ω ω⎣ ⎦

⟩− ⟨ψ ψ

f

ˆ

k

dd

εσ⎡ ⎤ α⎢ ⎥Ω⎣ ⎦

[ ] ( ) ( )',

| |

1 '( , ) 1 1

' 0 '

f i

l m gll

l m

T

l la l m d l

m m

ψ ψ

−>

⟨ ⟩ =

⎧ ⎫⎛ ⎞− −⎨ ⎬⎜ ⎟−⎝ ⎠⎩ ⎭

∑

( ) ( )

( ) ( )

1

1

1

2

2

2

' 11 '

'

' 2*2 '

1 '1 ( , ') 1 1' 0 '2 ' 1

1 ' ( , ') 1 1

' 0 '

l m gl l

m l

l m gl l

l

l la l m d l

m ml

l la l m d l

m m

−>

−>

⎧ ⎫⎛ ⎞⎡ ⎤ − − ×⎨ ⎬⎜ ⎟⎣ ⎦ −+ ⎝ ⎠⎩ ⎭⎧ ⎫⎛ ⎞⎡ ⎤ − −⎨ ⎬⎜ ⎟⎣ ⎦ −⎝ ⎠⎩ ⎭

∑ ∑

∑initialstates

f i| T |⟨ψ ψ ⟩ *f i| T |⟨ψ ψ ⟩


138

f

ˆ

k

dd

εσ⎡ ⎤ α⎢ ⎥Ω⎣ ⎦( ) ( )

( ) ( )

1

1

1

2

2

2

' 11 '

'

' 2*2 '

1 '1 ( , ') 1 1' 0 '2 ' 1

1 ' ( , ') 1 1

' 0 '

l m gl l

m l

l m gl l

l

l la l m d l

m ml

l la l m d l

m m

−>

−>

⎧ ⎫⎛ ⎞⎡ ⎤ − − ×⎨ ⎬⎜ ⎟⎣ ⎦ −+ ⎝ ⎠⎩ ⎭⎧ ⎫⎛ ⎞⎡ ⎤ − −⎨ ⎬⎜ ⎟⎣ ⎦ −⎝ ⎠⎩ ⎭

∑ ∑

∑

f

ˆ

k

dd

εσ⎡ ⎤ α⎢ ⎥Ω⎣ ⎦

( ) ( )

( ) ( )

1

1

1

2

2

2

11

2*2

11 ( , ) 1 102 1

1 ( , ) 1 1

0

l m gl l

m l

l m gl l

l

l la l m d l

m ml

l la l m d l

m m

−>

−>

⎧ ⎫⎛ ⎞⎡ ⎤ − − ×⎨ ⎬⎜ ⎟⎣ ⎦ −+ ⎝ ⎠⎩ ⎭⎧ ⎫⎛ ⎞⎡ ⎤ − −⎨ ⎬⎜ ⎟⎣ ⎦ −⎝ ⎠⎩ ⎭

∑ ∑

∑

Notation simplification: ‘prime’ no longer required


139

f

ˆ

k

dd

εσ⎡ ⎤ α⎢ ⎥Ω⎣ ⎦( ) ( )

( ) ( )

1

1

1

2

2

2

11

2*2

11 ( , ) 1 102 1

1 ( , ) 1 1

0

l m gl l

m l

l m gl l

l

l la l m d l

m ml

l la l m d l

m m

−>

−>

⎧ ⎫⎛ ⎞⎡ ⎤ − − ×⎨ ⎬⎜ ⎟⎣ ⎦ −+ ⎝ ⎠⎩ ⎭⎧ ⎫⎛ ⎞⎡ ⎤ − −⎨ ⎬⎜ ⎟⎣ ⎦ −⎝ ⎠⎩ ⎭

∑ ∑

∑

: max 4 terms,

since = 1l

lδ ±

∑

( ) ( )( )( )( )

( ) ( )( )( )

1/22 21

1/22 2

1 1 11

0 2 1 2 3 1

1 11

0 2 1 2 1

l m

l m

l l l mm m l l l

l l l mm m l l l

− −

−

⎡ ⎤+ + −⎛ ⎞= − ⎢ ⎥⎜ ⎟− + + +⎝ ⎠ ⎢ ⎥⎣ ⎦

⎡ ⎤−⎛ ⎞ −= − ⎢ ⎥⎜ ⎟− + −⎝ ⎠ ⎣ ⎦

* ˆ( , ) 4 ( )lil ml fa l m i e Y kδπ −=


From slide 131

140

( ){ } ( ){ }( )

2 2 11

1 2 1 2

1 2

1 2 1 1

2*

'

1 21 2

16 ˆ ˆ( ) ( ) 2 1

1 1 1

0 0

l lill m ml f l f l l l l

m l l

l l g g

i i e Y k Y k d dl

l l l ll l

m m m m

δ δπ − −

+ + +> >

⎡ ⎤ ⎡ ⎤− ×⎣ ⎦ ⎣ ⎦+

⎧ ⎫ ⎧ ⎫⎛ ⎞ ⎛ ⎞× − ×⎨ ⎬ ⎨ ⎬⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠⎩ ⎭ ⎩ ⎭

∑ ∑ ∑

( ) ( ) ( )( ) ( )

2 21 1 1 1 1 1

2 21 1

( 1) 1 ( 2) 6 1 cos2 1 1

l l l l l l

l l

l l l l l ll l l

σ σ σ σ δ δβ

σ σ− + + − + −

− +

− + + + − + −=

⎡ ⎤+ + +⎣ ⎦

[ ]

( )f

ˆTotal

2k

22

d 1 P cosd 4

1P cos 3cos 12

ε σσ⎡ ⎤ = +β Θ⎢ ⎥Ω π⎣ ⎦

Θ = Θ−

2' ' '

0

( ) ( )ll kl n ld r drR r rR r∞

= ∫

Cooper-Zare formula

f

ˆ

k

dd

εσ⎡ ⎤ α⎢ ⎥Ω⎣ ⎦


Radial integrals

])1()[12()cos()1(6)2)(1()1(

21,

21,

111,1,2

1,2

1,

+−

−++−+−

+++−+−+++−

=ll

llllll

RllRlRRllRllRll

ωω

ωωωω δδβ

J. Cooper and R. N. Zare, Lectures in Theoretical Physics: Atomic Collision Processes, 1969

141

[ ]f

ˆTotal

2k

d 1 P cosd 4

ε σσ⎡ ⎤ = +β Θ⎢ ⎥Ω π⎣ ⎦


Bransden & Joachain –Physics of Atoms & Molecules / Eq.13.282

( )21 3cos 1 02β

+ Θ− ≥

( )23cos 1 12β

Θ− ≥ −

( )22

1P cos 3cos 12

Θ = Θ−

2

mincos 0 2⎡ ⎤Θ = ⇒β ≤⎣ ⎦

2

maxcos 1 1⎡ ⎤Θ = ⇒β ≥ −⎣ ⎦

1 2⇒ − ≤ β ≤

2

00 cos 1≤ Θ ≤ π

≤ Θ ≤


( ) ( ) ( )( ) ( )

2 21 1 1 1 1 1

2 21 1

( 1) 1 ( 2) 6 1 cos2 1 1

l l l l l l

l l

l l l l l ll l l

σ σ σ σ δ δβ

σ σ− + + − + −

− +

− + + + − + −=

⎡ ⎤+ + +⎣ ⎦

21

21

0 photoionization of ns subshell2 2 independent of ( , )l

l

for l

n Eσβσ

+

+

=

= = →⎡ ⎤⎣ ⎦

[ ] ( )

( )f

ˆ2Total Total

2k

2 2Total Total

d 11 P cos 1 2 3cos 1d 4 4 2

1 3cos 1 3cos4 4

ε σ σσ⎡ ⎤ ⎡ ⎤= +β Θ = + × Θ−⎢ ⎥ ⎢ ⎥Ω π π⎣ ⎦ ⎣ ⎦

σ σ⎡ ⎤= + Θ− = × Θ⎣ ⎦π πWalker-Waber- relativisticJohnson-Lin

- relativistic, with correlations

Further modifications:

Cooper-Zare formula

1st order corrections : Additional terms

. 1 .ik re ik r≈ +2fi f izxω ψ ψ

fi f y iLω ψ ψ

( ) ( ) ⎥⎦⎤

⎢⎣⎡ ++−+=

Ωφθθγδθβ

πσσ cossincos1cos3

21

422nl

dd

: E2 Electric Quadrupole

:M1 Magnetic Dipole.

E2 Electric QuadrupoleAngular Distribution Asymmetry Parameters

,δ γ

“Beyond the dipole….”

Journal of Physics B 30, L727 (1997)

( ) ( ) ⎥⎦⎤

⎢⎣⎡ ++−+=

Ωφθθγδθβ

πσσ cossincos1cos3

21

422nl

dd

3 analyzers in TOF spectrometer:θ=00; φ=900,

θ=54.70; φ=900 (E1 magic angle analyzer),

θ=54.70; φ=00 (E2 analyzer).

[ ]βπ

σσ+=

Ω1

4nl

dd

0 1cos54.7 ;43

nldd

σσπ

= =Ω

( )[ ]816.0314

γδπ

σσ++=

Ωnl

dd

Journal of Physics B 30, L727 (1997)

145

Cooper-Zare

Walker-Waber

Johnson-Lin

Photoelectron Angular Distributions– J.Cooper and R.N.ZarePresented at the Theoretical Physics Institute, University of Colorado, Summer, 1968The relativistic theory of the angular distribution of photoelectrons in jj coupling T E H Walker and J T Waber 1973 J. Phys. B: At. Mol. Phys. 6 1165

Relativistic Random Phase ApproximationW. R. Johnson and C. D. Lin, Phys. Rev. A, 20, 966-977, 1979.

Walter R Johnson


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