Download Atomic and Molecular Physics for Physicists Ben-Gurion University of the Negev

Ben-Gurion University of the Negev
Atomic
Atomic and
and Molecular
Molecular Physics
Physics
for
for Physicists
Physicists
Ron Folman
Chapter 1: The Physics of radiation:
Classical Oscillating electric dipole and its radiation
Main References: Corney A. Atomic and Laser
Spectroscopy, Oxford UP, 1987; QC 688.C67
1987; Many good refs like:
J. D. Jackson, Classical Electrodynamics (Wiley:
New York, 1998).
Exercises:
Dudi Moravchik.
www.bgu.ac.il/atomchip
www.bgu.ac.il/nanofabrication
www.bgu.ac.il/nanocenter
Our course is all about how matter
interacts with the wave-length
or frequency spectrum of the ElectroMagnetic radiation.
ν=C/λ
∆ν = ∆λ C / λ2
Some important numbers near
the visible:
10.6µm CO2 laser
1.5µm communication standard
1.064µm Nd:Yag laser
0.75µm high limit of visible
0.4µm low limit of visible
0.2µm low limit of normal optics
The reason of why we see the stars (SHU):
The atmosphere absorption has a tiny
window in the visible!!!
But as you see below, unfortunately,
most of us can no longer see the stars!
Our course is about the interaction of the EM radiation with the INTERNAL
degrees of freedom of Atoms and Molecules and not external degrees of freedom.
(as the two are always connected, perhaps a better definition would be that in this
Course we are interested with radiation interacting with a single free atom)
An example of an external degree of freedom interacting with EM radiation:
Radiation from oscillating atoms in solids - Black Body radiation
(we will touch upon this topic briefly when we discuss the emergence of QM)
Another nice example of light-matter interaction we wont discuss in this course is the
Interaction with bulk. For example the fascinating subject of mirrors (see Feynman’s
Book on light and matter), or phonon-light e.g. Brillouin scattering.
Third example of what we wont discuss: Mossbauer effect (radiation-nucleus)
How do we analyze the radiation from matter (this is called spectroscopy):
For example, here is one of the early methods.
Interpretation of the pictures:
How does EM radiation look like (SHU):
What are the parameters of the radiation:
Speed in vacuum v (=c)
(Group velocity ∂ω/∂k, phase velocity ω/k
w(k) is the dispersion relation)
Speed in material v=c/n
Wave length λ
Frequency f=v/ λ
Energy density ED=1/8π (E·E + B·B)
Energy flux (Poynting’s formula) c/4π (ExB)= c ED
Polarization (circular, linear)
In this course we will be interested with the interaction of an electron with the radiation.
(the electron together with the positive nucleus creates a dipole)
The electrons responds or generates the radiation:
(SHU)
First experimental observation
Heinrich Rudolf Hertz (1988)
"“I do not think that the
wireless waves
I have discovered
will have any
practical application."
Maxwell equations (1864):
First of all, I expect you to be fluent in scalar and vector mathematics:
AxB=(AyBz-AzBy , AzBx-AxBz , AxBy-AyBz)
2
∆
∆
∆
∆ = (∂/∂x,∂/∂y,∂/∂z) , Laplacian = ∆
= (∂2/∂x2+∂2/∂y2+∂2/∂z2)
∆
A*B=(AxBx+AyBy+AzBz)
∆
Operations: Curl V= ∆ x V , Div V= ∆ * V, Grad S= ∆ S
(V=Vector, S=Scalar)
Identities: Div Curl A = 0 , Curl Grad A = 0 , Curl Curl = Grad Div – Laplacian
More identities:
Young Maxwell at the University
Homework:
Ohm’s law
Static continuity equation
Static Maxwell equation
1. Prove that:
2. If a current j is in the xz plane, and σ1 changes into σ2 along the z=0 plane,
Then the relationship between j(1) (above the z=0 plane) and j(2) (below the plane) is:
Gauss Law
Gauss law for magnetism
(absence of magnetic monopoles
Faraday’s law of induction
Ampere’s law (with Maxwell’s addition)
E – electric field (Volt per meter)
H – magnetic field (Ampere per meter)
D – electric displacement (Coulomb per sq. meter)
B – magnetic induction (Tesla, Gauss, Weber per sq. meter)
Maxwell’s addition: he notices that Amprere’s law Curl H = J gives Div J=0
(as Div Curl =0) and this contradicts the continuity eq. Div J + ∂ρ/∂t = 0.
He thus added ∂D/∂t recognizing that changing electric fields act like currents,
likewise producing magnetic fields.
In Linear materials
Where
the polarization density P (in coulombs per square meter)
and magnetization density M (in amperes per meter)
and where:
where:
χe is the electrical susceptibility of the material,
χm is the magnetic susceptibility of the material,
ε is the electrical permittivity of the material, and
µ is the magnetic permeability of the material
In non-dispersive, isotropic media, ε and µ are time-independent scalars,
and Maxwell's equations reduce to
In vacuum without charges and currents:
And Maxwell’s eq. receive the form:
Furthermore, Maxwell showed that waves of oscillating electric and magnetic fields travel
through empty space at a speed that could be predicted from simple electrical experiments
—using the data available at the time, Maxwell obtained a velocity of 310,740,000 m/s.
Maxwell (1865) wrote:
This velocity is so nearly that of light, that it seems we have strong reason to conclude that
light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance
in the form of waves propagated through the electromagnetic field according to electromagnetic
laws.
How did he do that?
Home work: Use the equations in vacuum and the identity Curl Curl=Grad Div – Laplacian
and arrive at:
Now note that these are wave equations
where the wave velocity v in vacuum is
V
=C
The currently accepted values for the speed of light, the permittivity,
and the permeability are summarized in the following table
Symbol
Name Numerical Value SI Unit of Measure
Generally
V=C/n
where
n=(µε)−1/2
Type
Speed of light
meters per second defined
Permittivity
per meter farads derived
Permeability
per meter henries defined
Plane wave solutions
If
is a generic solution of the wave equation, i.e.
then we choose as the general solution for the electric field
where E0 gives an amplitude and a direction.
We then observe:
Namely
and
Namely
Electric Dipole
Greek :di(s) = double and polos = pivot
We are interested in an oscillating dipole
which in the case of the atoms actually means
an oscillating electron (as the nucleus is 2000
times heavier)
We now introduce for the first time fields generated by time varying distributions
of current and charge, and try and solve Maxwell’s equations again:
For this purpose we invent something called the Vector and Scalar potentials
Since Div Curl = 0 this definition automatically satisfies Maxwell’s Div B = 0.
We now introduce the new vector potential into
and find Curl (E + ∂A/∂t) = 0
and consequently (because Curl Grad = 0) we get: E = - GradΦ - ∂A/∂t
The other two Maxwell eq. give:
Lap.Φ + ∂(DivA)/∂t = - ρ/ε
and
Lap.A – 1/v2 ∂2A/∂t2 – Grad (DivA+ 1/v2 ∂Φ/∂t)= - µ J
With the Lorenz gauge DivA + 1/v2 ∂Φ/∂t = 0
We find two solvable eq. for A and Φ:
Lap. Φ - 1/v2 ∂2Φ/∂t2 = - ρ/ε
Lap.A - 1/v2 ∂2A/∂t2 = - µ J
Side note:
We invented A as a mathematical assistant, but it appears it has physical presence!
Lets look at the Aharonov-Bohm experiment:
Although the particle always sees zero
Magnetic field
So in classical terms there is no effect
on the particle, but in QM
Where the phase difference one gets
Between the two paths is
Dependent on the magnetic flux.
Note: The vector potential admitted by a solenoidal field is not unique.
If A is a vector potential then so is
Solving the two inhomogeneous wave equations for A and Φ:
Lap. Φ - 1/v2 ∂2Φ/∂t2 = - ρ/ε
Lap. A - 1/v2 ∂2A/∂t2 = - µ J
 ρ ( r ', t ' )  dV '
Φ ( r,t) =
4πεε 0 ∫
r −r'
1
µµ0  J ( r ', t ' )  dV '
A( r, t ) =
4π ∫
r −r'
Taking t’=t-|r-r’|/v and taking |r-r’| to be approximately
r − r ' ≅ r − rˆ ⋅ r '
And taking the time dependence of the currents and the charges to be sinusoidal
ρ ( r , t ) = ρ ( r ) exp ( −iωt )
J ( r , t ) = J ( r ) exp ( −iωt )
We find the solution of A to be
J ( r ' ) exp ( −ikrˆ ⋅ r ' ) dV '
µ0
A( r , t ) =
exp{i ( kr − ωt )}∫
4π
( r − rˆ ⋅ r ' )
We are allowed to expand the exponent because r’ is about 10^-10m and in the
visible region λ = 5000A so that k = 5 10^-7 m^-1 and kr’=10^-3.
k2
2
exp ( −ikrˆ ⋅ r ' ) = {1 − ikrˆ ⋅ r '+ i
( rˆ ⋅ r ' ) − .....}
2!
2
Taking only the first term and also taking the denominator in the previous eq. for A
as r, we get the electric dipole approximation!
µ0 exp{i ( kr − ωt )}
A( r, t) =
J ( r ' ) dV '
∫
4π
r
And after some manipulation we get:
A( r, t) = −
µ0 exp{i ( kr − ωt )}
iω
p
4π
r
Where p is the electric dipole moment which is defined as:
p = ∫ r 'ρ ( r ' ) dV '
Solutions of the Dipole Approx.
H=
E=
i curlH
ωε 0
ωk  1 i 
 +
 ( rˆ ∧ p ) exp{i ( kr − ωt )}
4π  r kr 2 
1 k2
 1 ik  
ˆ
ˆ
ˆ
ˆ
=
r
∧
p
∧
r
+
{3
r
r
⋅
p
−
p
}
( )
( )  r 3 − r 2  × exp{i ( kr − ωt )}
4πε 0  r

The magnetic fields are always perpendicular and the electric fields
Have both radial and perpendicular components.
3 spatial regions
Near Field (d<r<λ) – similar to a static dipole (van der Waals forces)
1  3 rˆ ( rˆ ⋅ p ) − p 

 exp( −iωt )
E=
3


4πε 0
r


Intermediate (d<r ~λ)
Far field (d<λ<r) - Spherical waves expanding from the origin!
exp{i ( kr − ωt )} 1
k2
H=
rˆ ∧ p )
(
4πε 0
r
Z0
Z0=1/ε0c
exp{i ( kr − ωt )}
k2
E=
rˆ ∧ p ) ∧ rˆ
(
4πε 0
r
Some features:
• E and H are perpendicular to each other and to k
• the ratio of amplitudes of E and H is Z0
• the observed polarization is dependent on the shape of the dipole oscillations
and the position of the observer
Power Angular Dependence
We start from the Poynting vector
dP 1
= Re{r 2 rˆ ⋅ ( E ∧ H * )}
dΩ 2
And introduce E and H from the previous slide
For a linear oscillator we get
2
2
dP
1  k 
2
p
=
sin
θ


d Ω 2 Z 0  4πε 0 
2
and for circular oscillator
2
2
dP
1  k 
2
=
+
p
1
cos
θ)
(


d Ω 4 Z 0  4πε 0 
2
k 4c
pEl =
p
The total being
12πε 0
2
2
2
dP
1  k 
=

 rˆ ∧ ( rˆ ∧ p )
d Ω 2 Z 0  4πε 0 
2