Select/Special Topics in Atomic Physics · 2017-08-04 · P. C. Deshmukh . Department of Physics....
Transcript of 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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
6PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
• 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’
8PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
Cross sec tion :(transition probability)
− σ
DISCRETE EXCITATION: Photo-excitation
Photo-Ionization / PhotodetachmentFinal state: continuum
Angular distribution of the photoelectrons : , ,β γ ζ
9PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
10PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
Interaction of EM
radiation with an
atom
Oscillations of
electrons about mean
position.
• Atomic polarizability
• Polarization
• Susceptibility
11PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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)
13PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
Damped oscillator
( ) ( ).. .
m r t rk 2 m rγ= − −
14PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
15PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
16PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
17PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
Connections:
Oscillator Strength
Atomic polarizabilities
Photoionization cross-section
18PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
19PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
: 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
− −⎢ ⎥⎣ ⎦
20PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( ) ( )i t02 20,s d
Eeˆnow, r t em i
− ω +θ= εω −ω − Γ ω
21PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
22PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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⎣ ⎦
24PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )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?
25PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )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∫ ∓
26PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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 = +
27PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )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:
28PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
Questions? Write to [email protected]
P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036
[email protected] Unit 7(ii) Lecture 31
Select/Special Topics in Atomic Physics
29
Atomic Photoionization cross-sections, angular distribution of photoelectrons - 2
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
⎡ ⎤⎛ ⎞•= = = •⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦∫ ∫ ∫
30PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )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
31PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
32PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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≈
33PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )
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.
35PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )
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
36PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
37PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
[ ] 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∫ ∫
38PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )0,s0 0
df d 2 dd
∞ ∞
ω = δ ω−ω ωω∫ ∫
‘OSCILLATOR STRENGTH’
is so defined as to yield UNIT INTEGRAL
( )0,s0
df 1d 2 dd 2
∞ ∞
−∞
ω = δ ω−ω ωω∫ ∫
( )0,s0
df d d 1d
∞ ∞
−∞
ω = δ ω−ω ω =ω∫ ∫
39PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )
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’
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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 −⎢ ⎥⎢ ⎥⎣ ⎦
43PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
∇• =
45PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
∂
46PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
[ ]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
47PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
{ } ( )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⎨ ⎬⎩ ⎭
∑ ∑∑
48PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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+ = ∑
49PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
fki t(1) (0)f k
k
1c (t) c f | H ' | k ei
+ ω= ∑Perturbation is switched on at
some reference time 0t 0=(0)k kic = δ when the system is in a
known initial state | i⟩
fii t(1)f
1c (t) f | H ' | i ei
+ ω=
50PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
{ } 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 + •∇⎜ ⎟⎝ ⎠
51PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 54
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
55PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )
( )( )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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )
( )( )( )
( )( )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 •∇ ×⎜ ⎟
⎣ ⎦ ⎣ ⎦
⎝ ⎠
57PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )
( )( )( )
( )( )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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
59PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
X
Y
Z
ke−
hν
dΩ
d : solid angle: circular frequency
of the EM radiation
Ωω
60PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
61PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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 ∇• =
63PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
64PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
220I( ) S A ( )
2 cω
ω = = ωπ
Questions? Write to [email protected]
P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036
[email protected] Unit 7(iii) Lecture 32
Select/Special Topics in Atomic Physics
65
Atomic Photoionization cross-sections, angular distribution of photoelectrons - 3
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
[ ]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
⎝
66PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
[ ] [ ]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⎣ ⎦
67PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
69PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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⎣ ⎦
70PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
72PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
= = + +
⎛ ⎞= + + =⎜ ⎟⎝ ⎠
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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⇒ =
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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=
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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 ⎝∫
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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:
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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 • =
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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• = =
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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=
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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=
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 84
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
⟨⟨⟨
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
− • −
= + −
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
90
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
22 2 2 2 2 f0 f 0 f
2 2 2 2 f0 f 0 f
vZ a k k Z a k 1 cosc
va k a k cosc
⎛ ⎞+ − ≈ + − θ⎜ ⎟⎝ ⎠
≈ − θ
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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⎜ ⎟⎝ ⎠
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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⎣ ⎦ ⎝ ⎠⎝ ⎠
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )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 ⎝ ⎠
∫ ∫
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )
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⎣ ⎦⎢ ⎥⎣ ⎦
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
ik r
we shall now study the
ˆmatrix element f | e | i• ε •∇
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 96
Questions? Write to [email protected]
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→
P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036
[email protected] Unit 7(iv) Lecture 33
Select/Special Topics in Atomic Physics
97
Atomic Photoionization cross-sections, angular distribution of photoelectrons - 4
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
X
YZ
hν
e−
θkφ
98PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
f
ˆ 2 2 2ik r
2k
d 4 ˆf | e | i ( )d m
ε•σ π α⎡ ⎤ = × ε•∇ ×δ ω⎢ ⎥Ω ω⎣ ⎦
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 99
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 •∇ =
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
= =
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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 •∇ = −
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
- 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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 105
( ), , 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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 108
Tanima Banerjee ….Physical Review A75, 042701 (2007)
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 109
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 •
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 110
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 111
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=
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )
( )
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=
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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 | −⎡ ⎤= ⎣ ⎦∑ ∑
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( )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=∑
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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∑ ∑ ∑
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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 distribution
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•
Bransden & Joachchain: Physics of Atoms & Molecules Section 4.3 ‘Dipole approximation’
[ ] ( )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.”
120PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
[ ]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’
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 121
( )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 =
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
Questions? Write to [email protected]
P. C. Deshmukh Department of PhysicsIndian Institute of Technology MadrasChennai 600036
[email protected] Unit 7(v) Lecture 34
Select/Special Topics in Atomic Physics
123
Atomic Photoionization cross-sections, Angular distribution of photoelectronsCOOPER – ZARE formula
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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•
Bransden & Joachchain: Physics of Atoms & Molecules Section 4.3 ‘Dipole approximation’
[ ]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)
• ≈
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 125
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 126
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
127PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
INGOING wave boundary conditions (Unit 6)
Photoelectron escape/exit direction: unique
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 −=
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
Photoelectron escape/exit direction: unique
: 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⟩ = ⟨ ⟩
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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⟨ ⟩
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
( ) ( )' 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
−>
⟨ ⟩ =
⎧ ⎫⎛ ⎞− −⎨ ⎬⎜ ⎟−⎝ ⎠⎩ ⎭
∑
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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 ⟩
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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 −=
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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⎣ ⎦
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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⎣ ⎦
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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 ≤
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution 142
( ) ( ) ( )( ) ( )
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
PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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146PCD STiAP Unit 7 Photoionization cross-section, pe angular distribution
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