Download • Cross sections • Atomic units • Atomic and molecular beams

AAMOP WS 2011/2012
Experimental methods
José R. Crespo López-Urrutia
12.10.2011
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Cross sections
Atomic units
Atomic and molecular beams
Supersonic jets
Crossed beam experiments
Nanodroplets
Laser cooling and trapping of atoms
José R. Crespo López-Urrutia
[email protected]
E0/T0
10.10.2011
E1
12.10.2011
T1
17.10.2011
E2
19.10.2011
T2
24.10.2011
T3
26.10.2011
-
31.10.2011
E3
02.11.2011
T4
07.11.2011
E4
09.11.2011
T5
14.11.2011
E5
16.11.2011
T6
21.11.2011
E6
23.11.2011
T7
28.11.2011
E7
30.11.2011
Motivation and introduction. Organizational issues.
Atomic units. Cross sections. Coincidence measurements. Time-of-flight
methods. Counting statistics. Atomic beams.
Spin and relativity, from Schrödinger to Dirac equation. Solutions with
negative energy, Dirac sea, antiparticles.
Sources of singly and highly charged ions. Electron and ion detection and
energy analysis.
Bound-state solutions of Dirac equation, spectroscopy, fine-structure
effects
Higher-order corrections to Dirac equation: QED, hyperfine-structure and
isotope shift effects
No lecture
Lasers, synchrotrons, free-electron lasers. Photon detection. solid-state
detectors, microcalorimeters.
Continuum-state solutions of Dirac equation, plane and distorted waves,
multipole decompositions
Classical optical spectroscopy. Laser spectroscopy. Ultrashort pulse
lasers. Frequency combs.
Angular momentum, coupling of momenta, angular momentum theory,
Clebsch-Gordan coefficients
Spectroscopy outside the visible range in electron beam ion traps, and
storage rings. EUV, VUV, X-ray spectroscopy,
Independent particle model, central field approximation, spectroscopy of
few-electron atoms/ions
Hydrogen-like ions: Quantum electrodynamics, hyperfine structure, gfactor. Few-electron ions.
Spectra of many-electron ions, jj and LS coupling, advanced manyelectron approaches
Photoionization and photorecombination. Quantum interference.
T8
05.12.2011
Photoionization and recombination, atomic collisions, Coulomb
ionization and excitation, dielectronic recombination
T9
07.12.2011
Simple molecules: H2. Molecular ions. Born-Oppenheimer approximation.
Rovibrational spectra: Raman, Stokes effects.
T10
12.12.2011
Quasimolecules: Ultracold atoms and ions, optical lattices, Geonium,
coupling of mechanical and electronic dynamics
E8
14.12.2011
Electronic correlations, many-body effects and Auger decay. Bound
electrons in strong fields. Collisional excitation and ionization.
E9
19.12.2011
Penning and Paul ion traps. Ultra-precision mass spectrometry.
E10
21.12.2011
Atom and ion traps: Laser and evaporative cooling methods.
T11
09.01.2012
Stark and Zeeman effects. Symmetry and mixing of electronic states.
Induced transitions.
T12
11.01.2012
Radiative decay and absorption, evaluation of matrix elements,
symmetry and selection rules
E11
16.01.2012
Ions and atoms in strong laser fields.
E12
18.01.2012
Atomic momentum spectroscopy: COLTRIMS, reaction microscopes.
T13
23.01.2012
Non-dipole effects, two and multi-photon processes, second-order
perturbation theory, Green's function approach, two-photon
spectroscopy
E13
25.01.2012
Attophysics: Dynamic investigations of molecular vibrations
and reactions
T14
30.01.2012
Interaction of charged particles with matter: Statistical approach
T15
01.02.2012
Basics of the density matrix theory. Mixed quantum states.
Experimental methods
• Collision dynamics
• Cross sections
Why collision physics?
• Practical importance: many applications in technology and
medicine, environmental physics
• Basic research: fundamental understanding of the observed
phenomena. Difficult challenge: few (many) body systems
Only two-particle systems can be solved analytically!
Structure: H atom
Dynamic problems with analytical solution:
γ+ H
+
e- + I
Potential scattering
~~~~~>
Photoabsorption in H
Collision kinematics
Conservation laws of:
• Scalars: energy, number of particles, electrical charge,...
• Vectors: momentum, angular momentum,...
before
mb
v =0
ma
after
r
r
r
p a = pb + p B
MA
MB
p a|| = pb|| + p B||
r
r
0 = p b⊥ + p B ⊥
Q value:
Difference of total kinetic energy before and after collision
Q = (EB + Eb) - Ea
endotherm:
exotherm :
Q<0
Q>0
What is a (collision/reaction) cross section?
incoming
particle beam
Detector
Probability
for a reaction:
J
N scattering centres
with effective area σ
Area A
N ⋅σ
P=
A
Total cross section:
σ=
Number of reactions per scattering centre and time interval
Current density j of incoming particles
Differential cross section:
⎛ dσ ⎞ =
⎜
⎟
⎝ dΩ ⎠ϑ
Flux of particles scattered in solid angle interval dΩ
Current density j of incoming particles
Total cross section: σ = ∫ (dσ / dΩ) ⋅ dΩ
Typical units of area and cross section:
• Atomic physics: cm2 ; 1 a.u.2 = 0.88.10-16 cm2
• Nuclear physics: 100 fm2 = 10-24 cm2 = 1 barn
Target
Particle beam
Target density n (cm-3)
n
Length L
L
Number of reactions
time
=
Projectiles
time
Number of scattering
centres per area unit
.
n.L
.
σ
Mean free path length: < λ > = 1/(n.σ)
Differential cross section
1
dσ
⎛ dσ ⎞ =
⎜
⎟
⎝ dΩ ⎠ 2π sin(ϑ ) dϑ
with
dΩ = 2π sin(ϑ )dϑ
Projectile (incident energy E)
ϑ
b
Impact parameter
Ring area:
dΩ
2π.b.db
with dσ = 2π.b.db follows:
2πb
db
b
db
⎛ dσ ⎞
=
⋅
=
⋅
⎜
⎟
Ω
d
⎝
⎠ 2π sin(ϑ ) dϑ sin(ϑ ) dϑ
Dependence of scattering angle from impact parameter
ϑ = ϑ(b, E)
Franck-Hertz experiment
Hg vapour
C
Filament
(e- Quelle)
2 inelastic collisions
1 inelastic collision
elastic collisions
Current A-B
Hg excitation energy: 4.9 eV
Voltage C-A / V
Experimental methods
• Atomic units
• Atomic beams
Atomic units
Atomic units (a. u.) are a convenient "natural" measure of sizes
and magnitudes. The hydrogen atom serves as a standard.
Basic constants:
Angular momentum:
h
Electron mass:
m0
Elementary charge:
e
Dielectric constant:
4πε 0
Atomic units
Length: Bohr radius
h2
(4πε 0 )
a0 =
2
m0 e
= 0,529·10-10 m
Energy: potential energy of two elementary charges with distance
m0 e 4
e02 1
1
E1 = 2
=
2
a0 4πε 0
h (4πε 0 )
a0
= 4.3·10-18 J = 27.2 eV
Velocity: classical velocity on 1st Bohr orbit
v0 =
e2
(4πε 0 )h
Momentum:
Time:
=α c
Fine structure constant:
α=
e2
(4πε 0 )hc
m0 e02
p0 = m0 v0 =
= 2 ·10-24 kg m/s
(4πε 0 )h
a0
h3
2
(
4
πε
)
=
0
v0 m0e04
= 2.4 ·10-17 s = 24 as
⇒ c = 137a.u.
Atomic units
Examples:
Momentum of an electron with 100 eV kinetic energy
Ee = pe2/(2m)
Ee = 100 / 27.2 a.u.
⇔ pe = 2me Ee = 2 ⋅100 / 27.2 = 2.7 a.u.
Energy of a He atom with the same momentum pi =2.7 a.u.
Ei = pi2/(2MHe)
MHe = 4*1836
Ei = 2.72/(2.4.1836) = 0.00049 a.u. = 13 meV
or, more directly:
Ei = Ee (me/MHe) = 13 meV )
Atomic and molecular beams
An ideal target requires:
• negligible interaction between the particles of the
target gas with the environment
• high (but not too high) density
Target gas
• well defined velocity (cold gas)
pressure P0
• low divergence, spatial confinement
Capillary,
Air
nozzle etc.
Vacuum
chamber
(experiment)
Emission
characteristics
Important quantities:
• capillary or nozzle diameter d and length L
• mean path length
Λ at pressure P
Relation of particle density and pressure (at room temperature)
n [cm-3] = 2.7.1016 . P [mbar]
Emission characteristics
Aperture (diameter d)
Nozzle (diameter d, length L)
Λ
Λ>d
Λ > d, L
Λ>d
Λ<L
I(ϑ) = cosϑ
low density
high density
Effusive atomic and molecular beams
Effusive flow: low density
⇒ collisions between the particles can be neglected
Λ: mean free path length
d: diameter of the source aperture
The collision rate R depends on the particle velocity v,
target density n and collision cross section
σ
Example:
at T = 800 K and P = 1 mbar
→
Λ = 8 mm
The cross section σ is of the order of one atomic unit:
Most probable velocity in a
Maxwell-Boltzmann
distribution vw
Mean velocity in
the gas reservoir
Emission
characteristis of a
thin aperture and of
a nozzle with L = d
Typical flux per solid
angle:
5×1016 atoms /(s·sr)
Lithium atomic (effusive) beam source with oven
atomic Li beam
Li
a)
b)
c)
d)
Li oven with heated nozzle
Heating wires
Skimmer
Heat shielding
Stern-Gerlach’s experiment
What Wikipedia
says:
The real result:
without magnetic field
with magnetic field
Rabi and Ramsey molecular beam methods
Norman Ramsey
Rabi became interested in the molecular beam method while visiting Stern in Hamburg
(Stern-Gerlach experiment) because of the possibility to influence a single atom or
molecule. Rabi won the Nobel Prize in 1944 for the resonance method to record the
magnetic properties of nuclei.
One of his students, Norman Ramsey, built a molecular beam to measure nuclear magnetic
moments, and developed the method of successive oscillatory fields. He won the 1989 Nobel
Prize for this technique and the development of atomic clocks.
Supersonic gas jets
•
•
•
•
P0
very low divergence
high density
well defined velocity (cold beam: a few K)
rather complex apparatus (vacuum system)
Λ
Gas expands from high pressure (P0 1-100 bar)
through a small nozzle into high vacuum (P1)
What happens ?
• acceleration to supersonic speed
• formation of a shock front
(finite pressure P1 in the chamber)
• internal energy (temperature) is
transformed into kinetic energy
Λ<d
Λ<L
P1
Shock front
P0 , T
Nozzle
P1
zone of
silence
Supersonic gas jets
During the adiabatic expansion of the gas from initial pressure P0 to P
through the nozzle a part of the disordered thermal motion of the particles
(given by P0, T0) is transformed into directed translational energy.
Energy conservation gives for the enthalpy H (total energy):
H = Etherm + PV = 3/2 kT + kT = 5/2 kT
vjet
after expansion:
Ekin = 5/2 kT
→Temperature decreases
(adiabatic expansion)
→ Kinetic energy (directed)
increases
dn/dv / arb. units
vjet = (5kT/M)1/2
2
1.5
Maxwell (300 K)
1
0.5
0
0
500
1000
1500
2000
2500
3000
v / m/s
Ekin ≅ 70 meV for expansion at T = 300 K
vjet ≅ 1700 m/s for Helium
Supersonic expansion
reduces target temperature
Supersonic gas jets
Shock front
Atomic hydrogen: radiofrequency discharge source
H2
H
Dissociation
Skimmer
Nozzle
Supersonic expansion condition: Product of pressure and nozzle diameter
P0.d ≅ 10 ... 1000 mbar·mm (for T=10 ... 300 K)
Hydrogen discharge (30 K):
He gas jet (300 K):
P0 ≅ 10 mbar, d ≅ 3 mm
P0 ≅ 30 bar, d ≅ 30 µm
Crossed molecular beams
(2) velocity selector
10-4
(4) D2 or H2 supersonic beam source
10-6
(7) skimmer
(1) heated effusive
F atom beam source
10-7
(11) ultrahigh vacuum,
triply differentially pumped
mass spectrometer,
10-9
10-10
10-11
Experiment for reactive scattering
F + D, DF + D and F + H2 + HF + H
detector chamber.
Pressures are indicated: (3) liquid nitrogen cooled cold trap, (5) heater, (6) liquid nitrogen feed line,, (8) tuning
fork chopper, (9) synchronous motor, (10) cross correlation chopper for time-of-flight velocity analysis
Y. T. Lee, Nobel lecture 1986, Molecular beam studies of elementary chemical processes
Crossed molecular beams
Liquid helium droplet sources
• Embedding molecules, clusters and weakly bound complexes in
helium nanodroplets allows to study them by means of absorption and
emission spectroscopy with vibrational/rotational resolution.
• In the superfluid helium droplet the guest (“dopant”) molecules can
rotate at very low temperatures with little interactions with the
medium. The dopant immersed in the centre of the droplet does not
interact with any surfaces.
Some types of dopants do not penetrate the helium droplet but orbit
around it.
Helium nanodroplets (clusters)
To study molecular spectra, large helium clusters (droplets) are formed in the nozzle.
In the scattering chamber they can capture a “guest” molecule.
The clusters are irradiated with a tunable laser. When the radiation is in resonance with the
guest molecule, the absorbed energy evaporates helium atoms and the mass-spectrometer
signal decreases.
[M. Hartmann et al., Science 272, 1631 (1996)]
Helium nanodroplets apparatus
Doping helium nanodroplets with molecules
Absorption spectrum of complex molecules
Absorption spectrum of the monomer of PTCDA molecule in helium
nanodroplets (black). Blue curve: convolution with a gaussian distribution
(FWHM=600cm-1). Orange curve: spectrum measured in a solution. PTCDA:
perylene-3,4,9,10-tetracarboxylic-3,4,9,10-dianhydride)
Frank Stienkemeier group, Freiburg University
Slowing down atoms with light
Light force due to photon scattering
• Interaction with a laser beam transfers a momentum hk for each
absorbed photon
• Spontaneous emission is isotropic → zero net momentum transfer
• Spontanteous force is limited by saturation (maximum rate due to
lifetime)
laser beam
momentum
transfer
n·hk
isotropic emission
of n photons
Optical molasses
red detuning
velocity of atom
Atoms moving toward the laser experience a force
roughly proportional to their velocity: friction
Several laser beams generate a dense, viscous atom cloud
Magneto-optical trap (MOT)
v
F
+
F
-
2 counterpropagating beams, tuned
below resonance (δ<0).
Doppler shift δv=kv brings 1 beam
closer to resonance:
Capture in velocity space.
Lithium: vc=4m/s; 13.3mK
1D-toy model of a (J=0→J‘=1)- MOT
Zeeman-shift δ by B-field gradient adds
position dependent detuning: Restoring force
Cooling and trapping in position space
Net force:
(
)
(
F + δ + ( v , z ) + F − δ − ( v, z )
→ F ≈ −ηv − κz
)
Cooling limit: photon momentum hk
futher cooling towards BEC: evaporation
“Single collision“ experiments: since the 1960s
Requirements:
Crossed beams (projectile and target)
Detectors for low-energy electrons and ions
Spectrometer
with detector
Coincident (e,2e) measurement
e+H→H++e+e
Projectile beam
∼∼
∼
∼
∼
-
Gas target
∼∼∼
+
-
H atom
H. Ehrhardt, Freiburg 1969
A modern “Franck-Hertz” experiment
• Free metal atoms are
excited by electron impact
•Incident electron energy
and spin are controlled
• Angle and energy
dependence of the scattered
or ejected electrons
• Polarization and intensity
of decay photons are
determined
Quantum scattering amplitudes and
phases describing the interaction are
determined to test theoretical models
Limitations of conventional spectrometers
Ion impact
pa
p0
ϑ
Atom
Δp
prec
p = 2mE
a) 5 MeV/u
p+
b) 1 GeV/u U92+
(relativistic)
v=
p
m
p = 26 000 a.u., v = 14 a.u.
p = 4.5·108 a.u., v = 110 a.u.
Δp
= 10 −5
p
Δp
= 2 ⋅10 −9
p
→ changes in projectile trajectory are not measurable!
Statistical limitations
Double ionization
e − + He → He 2+ + 3e −
pa
Atom
p0
pc
pb
Toroidal spectrometer
(Université Paris XI)
Main problem:
small solid angle
Electron gun
Count rate:
5
d
σ
N& D =
j0 NT ΔΩ1 ΔΩ 2 ΔΩ3ΔE eff ε1 ε 2 ε 3
dΩ1dΩ 2 dΩ 2 dE1dE2
N& D = 10 −22
cm 2
nA
1
11
−6
2
eV
⋅
⋅
⋅
⋅
=
100
10
10
3
0
.
002
s
eV 2
mm 2
one hit every 10 min!
Imaging spectrometers
• Detection of ions and electrons
• Developed (ca. 1985) for target
ion spectroscopy
Gas-Jet
•Recoil-Ion Momentum
Spectroscopy (RIMS)
Electrons
Ions
EFie
•Cold Target Recoil-ion
Momentum Spectroscopy
(COLTRIMS)
ld
Projectile
Time-of-flight and position:
full momentum information
Large acceptance (up to 4π):
multicoinicdence
Ion trajectory
EFie
ld
Position-sensitive
detector
Ende 13.10.2010
Spin states preparation
• Accurate measurement of the neutron magnetic moment:
µN= (- 1.91304275 ± 0.00000045) nuclear magnetons.
• Test of time reversal symmetry:
upper limit for the neutron electric dipole moment: (-3 ± 5)X10-26 e·cm
• Hydrogen hyperfine splitting agrees with present quantum
electromagnetic theory to within the accuracy of the theoretical
calculation and can be used to obtain information on the proton
structure.