• Cross sections• Atomic units• Atomic and molecular beams• Supersonic jets• Crossed beam experiments• Nanodroplets• Laser cooling and trapping of atoms

AAMOP WS 2011/2012Experimental methods

José R. Crespo López-Urrutia

12.10.2011

José R. Crespo López-Urrutia crespojr@mpi-hd.mpg.de

Photoionization and photorecombination. Quantum interference.30.11.2011E7

Spectra of many-electron ions, jj and LS coupling, advanced many-electron approaches

28.11.2011T7

Hydrogen-like ions: Quantum electrodynamics, hyperfine structure, g-factor. Few-electron ions.

23.11.2011E6

Independent particle model, central field approximation, spectroscopy of few-electron atoms/ions

21.11.2011T6

Spectroscopy outside the visible range in electron beam ion traps, and storage rings. EUV, VUV, X-ray spectroscopy,16.11.2011E5

Angular momentum, coupling of momenta, angular momentum theory, Clebsch-Gordan coefficients

14.11.2011T5

Classical optical spectroscopy. Laser spectroscopy. Ultrashort pulse lasers. Frequency combs.

09.11.2011E4

Continuum-state solutions of Dirac equation, plane and distorted waves, multipole decompositions

07.11.2011T4

Lasers, synchrotrons, free-electron lasers. Photon detection. solid-state detectors, microcalorimeters.

02.11.2011E3

No lecture 31.10.2011-

Higher-order corrections to Dirac equation: QED, hyperfine-structure and isotope shift effects

26.10.2011T3

Bound-state solutions of Dirac equation, spectroscopy, fine-structure effects

24.10.2011T2

Sources of singly and highly charged ions. Electron and ion detection and energy analysis.19.10.2011E2

Spin and relativity, from Schrödinger to Dirac equation. Solutions with negative energy, Dirac sea, antiparticles.

17.10.2011T1

Atomic units. Cross sections. Coincidence measurements. Time-of-flight methods. Counting statistics. Atomic beams.

12.10.2011E1

Motivation and introduction. Organizational issues.10.10.2011E0/T0

Basics of the density matrix theory. Mixed quantum states.01.02.2012T15

Interaction of charged particles with matter: Statistical approach30.01.2012T14

Attophysics: Dynamic investigations of molecular vibrations and reactions

25.01.2012E13

Non-dipole effects, two and multi-photon processes, second-order perturbation theory, Green's function approach, two-photon spectroscopy

23.01.2012T13

Atomic momentum spectroscopy: COLTRIMS, reaction microscopes.18.01.2012E12

Ions and atoms in strong laser fields.16.01.2012E11

Radiative decay and absorption, evaluation of matrix elements, symmetry and selection rules

11.01.2012T12

Stark and Zeeman effects. Symmetry and mixing of electronic states. Induced transitions.

09.01.2012T11

Atom and ion traps: Laser and evaporative cooling methods.21.12.2011E10

Penning and Paul ion traps. Ultra-precision mass spectrometry.19.12.2011E9

Electronic correlations, many-body effects and Auger decay. Bound electrons in strong fields. Collisional excitation and ionization.14.12.2011E8

Quasimolecules: Ultracold atoms and ions, optical lattices, Geonium, coupling of mechanical and electronic dynamics

12.12.2011T10

Simple molecules: H2. Molecular ions. Born-Oppenheimer approximation. Rovibrational spectra: Raman, Stokes effects.

07.12.2011T9

Photoionization and recombination, atomic collisions, Coulomb ionization and excitation, dielectronic recombination

05.12.2011T8

• Collision dynamics• Cross sections

Experimental methods

Why collision physics?

Structure: H atom Dynamic problems with analytical solution:

+ e- + I

Potential scattering

~~~~~>~~~~~>γ + H

Photoabsorption in H

• Practical importance: many applications in technology and medicine, environmental physics

• Basic research: fundamental understanding of the observedphenomena. Difficult challenge: few (many) body systems

Only two-particle systems can be solved analytically!

Collision kinematics

Conservation laws of:• Scalars: energy, number of particles, electrical charge,... • Vectors: momentum, angular momentum,...

before after

ma MA

mb

MB

v =0

Bba ppp rrr +=|||||| Bba ppp +=

⊥⊥ += Bb pp rr0

Q value:Difference of total kinetic energy before and after collision

Q = (EB + Eb) - Ea

endotherm: Q < 0exotherm : Q > 0

What is a (collision/reaction) cross section?

N scattering centreswith effective area σ

incomingparticle beam

Detector

J

Area A

Probabilityfor a reaction:

ANP σ⋅=

σ = Number of reactions per scattering centre and time interval

Current density j of incoming particles

=⎟⎠⎞

⎜⎝⎛

Ω ϑ

σdd Flux of particles scattered in solid angle interval dΩ

Current density j of incoming particles

Total cross section:

Differential cross section:

∫ Ω⋅Ω= ddd )/( σσTotal cross section:

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

Length L

Target density n (cm-3)Particle beam

Number of reactions

time=

Projectiles

time. n . L . σ

Number of scatteringcentres per area unit

Mean free path length: < λ > = 1/(n.σ)

L

n

Differential cross section

ϑσ

ϑπσ

dd

dd

)sin(21=⎟

⎠⎞

⎜⎝⎛

Ωwith ϑϑπ dd )sin(2=Ω

with dσ = 2π.b.db follows:

ϑϑϑϑππσ

ddbb

ddbb

dd ⋅=⋅=⎟

⎠⎞

⎜⎝⎛

Ω )sin()sin(22

ϑ = ϑ(b, E)

b

Impact parameter

Ring area: 2π.b.db

ϑ

dΩ

Projectile (incident energy E)

Dependence of scattering angle from impact parameter

C

Franck-Hertz experiment

Hg vapour

Filament(e- Quelle)

2 in

elas

tic

colli

sion

s

1 in

elas

tic

colli

sion

elas

tic

colli

sion

s

Cur

rent

A-B

Hg excitation energy: 4.9 eV

Voltage C-A / V

• Atomic units• Atomic beams

Experimental methods

Atomic units

Atomic units (a. u.) are a convenient "natural" measure of sizesand magnitudes. The hydrogen atom serves as a standard.

e

Angular momentum: h

0mElementary charge:

04πε

Electron mass:

Dielectric constant:

Basic constants:

Length: Bohr radius = 0,529·10-10 m)4( 020

2

0 πεem

a h=

Energy: potential energy of two elementary charges with distance 0a

00

20

20

2

40

1 41

)4(1

πεπε aeemE ==

h= 4.3·10-18 J = 27.2 eV

Velocity: classical velocity on 1st Bohr orbit

cev απε

==h)4( 0

2

0 ..137 uac =⇒c

eh)4( 0

2

πεα =Fine structure constant:

Momentum:h)4( 0

200

000 πεemvmp == = 2 ·10-24 kg m/s

Time: 204

00

3

0

0 )4( πεemv

a h= = 2.4 ·10-17 s = 24 as

Atomic units

Examples:

Momentum of an electron with 100 eV kinetic energy

Ee = pe2/(2m)

Ee = 100 / 27.2 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 )

..7.22.27/10022 uaEmp eee =⋅==⇔

Atomic units

Atomic and molecular beamsAn ideal target requires: • negligible interaction between the particles of the target gas with the environment• high (but not too high) density• well defined velocity (cold gas)• low divergence, spatial confinement

Vacuumchamber (experiment)

Air

Target gaspressure P0

Capillary,nozzle etc.

Emissioncharacteristics

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]

Nozzle (diameter d, length L)

I(ϑ) = cosϑ

Λ > d Λ > d, L Λ > dΛ < L

Λ

low density

Emission characteristics

high density

Aperture (diameter d)

Effusive atomic and molecular beams

The cross section σ is of the order of one atomic unit:

Λ: mean free path lengthd: 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

Effusive flow: low density⇒ collisions between the particles can be neglected

Most probable velocity in a Maxwell-Boltzmanndistribution vw

Emission characteristis of a thin aperture and of a nozzle with L = d

Typical flux per solid angle: 5×1016 atoms /(s·sr)

Mean velocity in the gas reservoir

Li

atomic Li beam

Lithium atomic (effusive) beam source with oven

a) Li oven with heated nozzleb) Heating wiresc) Skimmerd) Heat shielding

The real result:without magnetic field with magnetic field

Stern-Gerlach’s experiment

What Wikipedia says:

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.

Rabi and Ramsey molecular beam methods

Norman Ramsey

Supersonic gas jets

• 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

P0 , T

Nozzle zone of silence

Shock frontP1

Λ < dΛ < L

P1

P0

During the adiabatic expansion of the gas from initial pressure P0 to Pthrough 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):

→Temperature decreases(adiabatic expansion)

→ Kinetic energy (directed) increases

H = Etherm + PV = 3/2 kT + kT = 5/2 kT

after expansion:

Ekin = 5/2 kT

vjet = (5kT/M)1/2

Ekin ≅ 70 meV for expansion at T = 300 Kvjet ≅ 1700 m/s for Helium

0 500 1000 1500 2000 2500 3000

0

0.5

1

1.5

2

vjet

Maxwell (300 K)

v / m/s

dn/dv

/ arb

. uni

ts

Supersonic gas jets

Supersonic expansion reduces target temperature

Shock front

Supersonic gas jets

Atomic hydrogen: radiofrequency discharge source

Nozzle

Skimmer

H2 HDissociation

Supersonic expansion condition: Product of pressure and nozzle diameter

P0.d ≅ 10 ... 1000 mbar·mm (for T=10 ... 300 K)

Hydrogen discharge (30 K): P0 ≅ 10 mbar, d ≅ 3 mmHe gas jet (300 K): P0 ≅ 30 bar, d ≅ 30 µm

Crossed molecular beams

Y. T. Lee, Nobel lecture 1986, Molecular beam studies of elementary chemical processes

Experiment for reactive scattering F + D, DF + D and F + H2 + HF + H

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

(1) heated effusive F atom beam source

(2) velocity selector

detector chamber.

(4) D2 or H2 supersonic beam source

(11) ultrahigh vacuum, triply differentially pumped mass spectrometer,

(7) skimmer

10-4

10-6

10-7

10-9

10-10

10-11

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.

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.

Helium nanodroplets (clusters)

[M. Hartmann et al., Science 272, 1631 (1996)]

Helium nanodroplets apparatus

Doping helium nanodroplets with 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

Absorption spectrum of complex molecules

Slowing down atoms with light

Light force due to photon scattering

• Interaction with a laser beam transfers a momentum hk for eachabsorbed photon

• Spontaneous emission is isotropic → zero net momentum transfer

• Spontanteous force is limited by saturation (maximum rate due to lifetime)

isotropic emissionof n photons

laser beam momentumtransfern·hk

Optical molasses

red detuning velocity of atom

Atoms moving toward the laser experience a forceroughly proportional to their velocity: friction

Several laser beams generate a dense, viscous atom cloud

v

F+ F-

2 counterpropagating beams, tunedbelow resonance (δ<0).

Doppler shift δv=kv brings 1 beamcloser to resonance:

Capture in velocity space.

Lithium: vc=4m/s; 13.3mK

Magneto-optical trap (MOT)

1D-toy model of a (J=0→J‘=1)- MOT

Zeeman-shift δ by B-field gradient addsposition dependent detuning: Restoring force

Cooling and trapping in position space

Net force:

( ) ( )zvF

zvFzvFκη

δδ−−≈→+ −−++ ),(),(

Cooling limit: photon momentum hkfuther cooling towards BEC: evaporation

“Single collision“ experiments: since the 1960sRequirements:

Crossed beams (projectile and target) Detectors for low-energy electrons and ions

Gas target

Projectile beam

Spectrometerwith detector

∼∼∼ ∼∼

--

+ ∼∼∼ -H atom

H. Ehrhardt, Freiburg 1969

Coincident (e,2e) measuremente+H→H++e+e

• Free metal atoms areexcited by electron impact

•Incident electron energyand spin are controlled

• Angle and energydependence of the scatteredor ejected electrons

• Polarization and intensityof decay photons aredetermined

A modern “Franck-Hertz” experiment

Quantum scattering amplitudes and phases describing the interaction are determined to test theoretical models

Limitations of conventional spectrometers

Ion impact

b) 1 GeV/u U92+ p = 4.5·108 a.u., v = 110 a.u. (relativistic)

a) 5 MeV/u p+ p = 26 000 a.u., v = 14 a.u.

mEp 2=mpv =

510−=Δpp

9102 −⋅=Δpp

→ changes in projectile trajectory are not measurable!

Atom

p0pa

prec

ϑΔp

Double ionization

Atomp0pa

pbpc

321321021221

5εεεσ eff

TD ENjdEdEddd

dN ΔΔΩΔΩΔΩΩΩΩ

=&

Toroidal spectrometer(Université Paris XI)

seV

mmnA

eVcmND

1002.03101010010 261122

222 =⋅⋅⋅⋅= −−&

−+− +→+ eHeHee 32

Electron gun

Count rate:

one hit every 10 min!

Main problem: small solid angle

Statistical limitations

Time-of-flight and position:full momentum informationLarge acceptance (up to 4π):multicoinicdence

Imaging spectrometers

Gas-Jet

Ions

Electrons

Projectile

E-Field

E-Field

Ion trajectory Position-sensitivedetector

• Detection of ions and electrons• Developed (ca. 1985) for target ion spectroscopy

•Recoil-Ion Momentum Spectroscopy (RIMS)

•Cold Target Recoil-ion Momentum Spectroscopy (COLTRIMS)

Ende 13.10.2010

• 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 theoreticalcalculation and can be used to obtain information on the proton structure.

Spin states preparation