Atomic Physics
Atoms with dipoles – surface effects
Any sort
of
collision
…
1. Ion Velocity:
E/mc2 = E(MeV)/0.5(MeV) = ½ MN v2 / mc2
Therefore v/c = √ [ 4E(MeV) (m/MN)]
= ( 2/√1840 ) √[(E (MeV)/A(amu)]
Giving v = 13.9 mm/ns √[(E (MeV)/A(amu)]
Note:
2.
c = 300 mm/ns
Atomic velocity (from Bohr theory):
Quantum number n, nuclear charge Z, Vn = (2πe2/ħ) (Z/n)
= v0 (Z/n)
Where v0 = velocity of 1st Bohr orbit = c/137
Van de Graaff accelerators
The GSI –
Darmstadt
accelerators
Fluorescence in an Electric field
1st fast-beam experiment - a little complicated…
Fixed E-field
Variable magnetic field
Fluorescence in an Electric field
Incident protons pick up electrons in the
target into excited states – consider just the
n=2 states..
Apply an electric or magnetic field after the
target and measure Lyman-α emission:
What happens?
Stark effect in atoms (in brief)
For an electric field E(z) in the z-direction, the first order stark energy correction
is ΔWE(1) = eE <ψ| z |ψ>
However, if the state |ψ> has a definite parity (+ or -), this integral is zero.
Hence, we need to use the second order expression:
1st case The Quadratic Stark effect:
ΔWE(2) = e2E2 ∑(γ′J′) |<γJMJ | z |γ′J′MJ>|2 /[WγJ - Wγ′J′ ]
Notes: (1) the coupled states must have opposite parity and have the same MJ.
(2) We have the selection rule J′ = J ± 1 (or L′ = L ± 1 and S=S′ in LS-coupling.
(3) The energies are proportional to (MJ)2
2nd case: the “linear” Stark effect: when states of opposite parity are
degenerate or closely degenerate – (almost only hydrogen), we can set up a
secular equation – the diagonal elements will be zero [see case (1)] , while the
off-diagonal terms will be the Stark coupling elements.
Stark effect in atoms (2)
Consider 2 states a and b – eg the 2s1/2 and 2p1/2 states in hydrogen which are almost
degenerate except for the Lamb shift L. In an electric field E ......
The perturbed energies W(E)a and W(E)b will be the solutions of the determinant:
|
W(E)aa – L -W W(E)ab
| =0
|
W(E)ba
W(E)bb – W
|
Where the 2 diagonal elements W(E)aa and W(E)bb are zero
And the 2 off-diagonal elements are equal: W(E)ab = W(E)ba = -3ea0Z E
Ignoring L, we get the solutions W = ± 3ea0Z E
=> a linear variation for the 2 ML = 0 states, and NO variation for the 2p, ML= ±1 states
For very small electric fields where 3ea0Z E is comparable or smaller than L, we will
see a quadratic dependence
[Homework: solve above equation for these two states
when the field splitting 3ea0Z E = L
Note: the 2 ML=0 wavefunctions are also mixtures of the field-free 2s and 2p states:
Show that:
ψ(E)1 = α ψ(0)1 + βψ(0)2
ψ(E)2 = α ψ(0)2 - βψ(0)1
Where α and β mixings are dependent on the field E.]
Electric fields - dipoles
Incident protons pick up electrons in
the target into excited states – consider
just the n=2 states. Apply an electric
or magnetic field after the target.
Try the Stark effect ideas..
Observed decay of Lyman-α
in external electric and magnetic fields
Decay of Lyman-α
in an external electric field,
showing the higher frequency
J=1/2-3/2 quantum beats.
The lower curve is the difference
of measurements in the 2 E-field
directions, the top curve shows
the summed decay.
Mixing of opposite parity
Example of the n=2 s-states and p-states of hydrogen
Notes:
1. Time dependent wavefunctions.
2. Non-definite parity states!
3. Electric field mixing.
4. Decay modes.
The Eck effect
The effect of reversing the
direction of the electric field seen
by the hydrogen atom as it leaves
the target…
Explain the second graph!...
The Eck theory of atoms having an
electric dipole - PRL 31, 270 (73)
2s-state
2p state
The quantum beat amplitudes depend on
the phase angle α between the s and p
excitation amplitudes fs and fp0, and the
cross-sections. With no coherence, the
amplitude and phase of the beats are the
same (only the V2 term is nonzero).
Eck’s result is given below:
From the data one can determine that the <center> of the electron cloud is not at
the proton, but can be in front or behind the proton. Further measurements show
that the electron cloud tends to be in front of the proton as it exits the surface, but
there is a significant energy-dependence of this feature – not yet explained.
Deuterium Balmer-alpha decay – note the electric field plates
Density matrix formalism for quantum beats
(useful for describing photon decays of
coherent states…)
Define the wavefunction as:
And the density matrix parameters ρ
where the energies are complex
(including the decay constant γ) as
defined above
Note:
Diagonal terms
In general
Off-diagonal terms
Removing cylindrical symmetry
Since the hydrogen experiment
indicates that electron pick-up at
the final surface is important, what
are the important symmetries in the
atomic wavefunctions?
What happens if we tilt the surface,
and remove “cylindrical symmetry”
in the excitation system?
What happens in the decay
processes for such wavefunctions?
We need to understand the links between such wavefunctions (e.g. their
angular momentum properties) and the optical polarization properties of
the light emitted.