construction of a slotted disk velocity selector for ... · 2.1 supersonic molecular beams 2.1.1...

Construction of a slotted disk velocity selector for supersonic molecular beams

Masterarbeit am Fachbereich Physik

August 2014

Betreuer: Prof. Dr. Reinhard Dörner PD Dr. Till Jahnke

Jasper Becht

Contents

Contents 3

1 Motivation 5

2 Physical and technical principles 9

2.1 Supersonic molecular beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 The free jet expansion into the vacuum . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Relaxation of the internal energy . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Speed ratio and velocity distribution in gas jets . . . . . . . . . . . . . . . . . 11

2.2 The problem of velocity selection of uncharged particles . . . . . . . . . . . . . . . . 14

2.2.1 Velocity selection by gravitation . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Mechanical velocity selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Working principles of slotted disk velocity selectors . . . . . . . . . . . . . . . . . . . 17

2.3.1 Particle trajectories and transmission function . . . . . . . . . . . . . . . . . 17

2.3.2 Technical implementation of slotted disk velocity selectors . . . . . . . . . . . 20

2.3.2.1 Static and dynamic unbalance . . . . . . . . . . . . . . . . . . . . . 20

2.3.2.2 Integration into a vacuum system . . . . . . . . . . . . . . . . . . . 20

3 Applications and prospects of velocity selectors 23

3.1 Optimizing the resolution in COLTRIMS-experiments . . . . . . . . . . . . . . . . . 23

3.1.1 COLTRIMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2 E�ects of velocity distribution and jet characteristics on the resolution . . . . 24

3.1.3 Optimizing the resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Applications in crossed molecular beam experiments . . . . . . . . . . . . . . . . . . 25

4 Construction of the Apparatus 27

4.1 Determination of the geometric parameters . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Limitations of the geometric parameters due to mechanical constraints . . . . 27

4.1.2 Calculation of the construction geometry by the usage of a numerical program 28

4.2 Transmission function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 The apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.1 Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.2 Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.3 Axis and bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.3.1 Rotor and shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.3.2 Eigenfrequencies of the rotor . . . . . . . . . . . . . . . . . . . . . . 32

4.3.3.3 Bearings and suspension . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.4 Mounting and lifting mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.5 Integration in a COLTRIMS-system . . . . . . . . . . . . . . . . . . . . . . . 35

3

4 CONTENTS

5 Tests and conclusion 37

5.1 Measurement of the running stability on atmosphere . . . . . . . . . . . . . . . . . . 37

5.1.1 Measurement of frequency signal from the motor drive unit . . . . . . . . . . 37

5.1.2 Measurement with a photoelectric barrier . . . . . . . . . . . . . . . . . . . . 38

5.1.3 Comparison of light barrier- and driver encoder measured frequency . . . . . 39

5.1.4 Running behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 E�ect on the transmission behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Conclusion and outlook 43

7 Acknowledgements 45

Bibliography 47

8 Appendix 51

8.1 Frequently used acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

8.2 Software being used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

8.3 Program code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.3.1 Velocity distribution of a supersonic gas jet . . . . . . . . . . . . . . . . . . . 52

8.3.2 Plotting program for an double-elipsoidal Maxwell distribution . . . . . . . . 52

8.3.3 Output parameters of a slotted disk velocity selector . . . . . . . . . . . . . . 53

8.3.4 Plotting program for the 2-dimensional rendering of a selector geometry . . . 54

8.3.5 Transmission function of a slotted disk velocity selector . . . . . . . . . . . . 55

Chapter 1

Motivation

Since its development in the early decades of the 20th century, atomic and molecular physics aims

at the description of our world on the interatomic and intraatomic scale. Theoretical research in

this branch of physics describes the intraatomic interactions between the protons and electrons in

an atom and the interatomic interactions between the atoms in a molecule using the theories of

quantum mechanics and quantum electrodynamics. The calculation of the properties of a quantum

mechanical system is performed by solving the Schrödinger equation for the wave function ψ(~x, t):

i~∂

∂tψ (~x, t) = Hψ(~x, t)

i~∂

∂tψ (~x, t) =

(− ~2

2m4+ V (~x, t)

)ψ(~x, t)

which is a partial di�erential equation containing the partial time derivative of the wave function

and the hamiltonian H. [Sha94] The hamiltonian contains the second derivative in space and

the potenial V (~x, t). The only atomic system for which the Schrödinger equation can be solved

analytically is the hydrogen atom. For all elements with more than one electron the Schrödinger

equation can only be solved numerically or by means of approximation methods. When quantum-

electrodynamical e�ects are considered, the calculations become more complicate, even for the

hydrogen atom. [Jen06] The experimental branch of the �eld of atomic and molecular physics

revises the predictions made with the theoretical models, and also gives new impetus by sometimes

discovering unexplained physical e�ects.

Due to the fact that a quantum mechanical system cannot be looked at in the traditional way, atoms

and molecules are investigated in fragmentation processes and spectroscopic methods. [Whi03]

Fragmentation, for example ionization of atoms and molecules and dissociation of molecules, can

be achieved by using particle beams or highly intense laser �elds. By measuring the momenta of

the fragments, ions and electrons, the quantum mechanical properties of the system are investi-

gated. In an atom, this can be binding energies, transition probabilities between states and electron

angular distributions. In molecules, for example, the shape of the molecule and its bond lengths

are investigated in dissociation experiments. An usual technique of an atomic or molecular physics

experiment is crossing a gas jet containing the investigated atoms or molecules with a laser or

particle beam in an ultra-high vacuum. [Tit11][Sco88]

In the atomic and molecular physics research group from the Goethe University in Frankfurt,

we watch atomic and molecular processes using the COLdTargetIonMomentumRecoilSpectroscopy

(COLTRIMS) imaging technology. In this technique a gas jet is overlapped with a particle or laser

beam and the momenta of the charged fragments are measured. The fragments are imaged on two

detectors by guiding them with electric and magnetic �elds. The momenta are then calculated

5

6 CHAPTER 1. MOTIVATION

back using the electric- and magnetic �eld strength and the positions on the detectors. [Sch06]

Due to the initial velocity of a particle in the gas jet, the momentum measured in gas jet direction

equals the sum of the particle weight times the gas jet velocity and the momentum caused by the

fragmentation process pf . The latter one is the actual objective of the measurement, and can be

obtained by subtracting the momentum shift caused by the initial velocity v∞ of a particle in the

gas jet from the measured momentum pm:

pf = pm −m · v∞

To measure the momentum data pf with high resolution, it is crucial to determine the initial

velocity in the gas jet v∞ as precisely as possible, because a velocity spread 4v in the gas jet

has also a blurring e�ect on the momentum data pd. In order to achieve a small thermal velocity

spread in the gas jet up to now two techniques were used. A MagnetoOpticalTrap (MOT) cools

gas atoms trapped in a small volume via tuned laser beams using the e�ect of Doppler-shifting.

A COLTRIMS system using a MOT, a so-called MOTRIMS, was built in Heidelberg. [GHAW13]

The other method of achieving a small velocity spread 4v is the adiabatic cooling occurring in a

supersonic jet expansion which is the method used in our group:

A gas which is expanded adiabatically through a small nozzle with a source pressure p0 and temper-

ature T0 into an ultra-high vacuum, transfers its thermal energy into kinetic energy of the forward

motion. During cooling down, the particles in the gas jet become faster, a process which is limited

by cluster-released heating, an e�ect caused by binding energy released when the atoms or molecules

in the gas jet form clusters at low temperatures. [HKE03] The gas jet reaches a �nal temperature

T∞, a�ected by the interatomic or -molecular cross-section, and in thermal equilibrium between

adabatic cooling and cluster-released heating, and a �nal velocity

v∞ =

√5kBT0m

which depends on the particle weight m and source temperature T0. The �nal temperature T∞in the gas jet depends on source pressure p0, source temperature T0, nozzle diameter d and the

gas species in a complicated way. [Pau00a] The velocity distribution in the gas jet has a Gaussian

shape whose spread depends on the �nal temperature T∞:

4v =

√2kBT∞m

The velocity spread is the crucial value on which the momentum resolution in gas jet direction

depends. The resolution is therefore strongly dependent on the speed ratio

S =v∞4v

which is a parameter widely used for describing the experimental quality of a gas jet. In order to

decrease the velocity spread 4v and therefore increase the speed ratio, the nozzle can be cooled.

For some gases, the lowering of the �nal temperature T∞, and therefore of the velocity spread 4v,is su�cient for resolving the desired physical e�ects in the momentum data. These are especially

very light gases with a low clustering temperature, for example helium. In helium gas jets, �nal

temperatures down to mK can be achieved, corresponding to speed ratios higher than S=100. On

the other hand, for some heavier gases with a higher cluster temperature, like Argon, the minimal

achievable velocity spread is still too high to resolve all physical e�ects which are the subject of

investigation, for example AboveThreshholdIonisation (ATI) -structures. [Hen13]

7

Figure 1.1: Velocity selection using a slotted disk velocity selector: A gas jet with an initial velocitydistribution f(v) passes the slits of the rotating disks of the velocity selector with the transmission functionT(v). The transmitted velocity distribution F(v) contains only velocities between vmin and vmax.

In order to overcome this experimental limit, a slotted disk velocity selector was constructed and

built. This device installed between the nozzle and the experiment �lters the particles dependent

on their velocities, transmitting only those traveling with a velocity between the well de�ned limits

vmin and vmax. A slotted disk velocity selector consists of two or several disks attached to a

rotating axis which are slotted at the margin. [Pau00b] The device is aligned parallel to the gas jet

with the gas jet traversing the slits of the rotating disks, and implements the green wave principle

mechanically: Particles, after having passed a slit in the �rst disk, can only pass through slits in all

remaining disks if they have a velocity v ≈ v0 between vmin and vmax, which depend on the disk

geometry and rotation frequency. The principle is shown if Fig. 1.1.

Besides improving the resolution in COLTRIMS-experiments using supersonic gas jets, a slotted

disk velocity selector could also be used for carrying out measurements with gases from thermal

sources, which have a broad Maxwell-Boltzmann distribution, or in crossed beam experiments which

investigate chemical reactions between two species in crossed gas jets.

8 CHAPTER 1. MOTIVATION

Chapter 2

Physical and technical principles

2.1 Supersonic molecular beams

2.1.1 The free jet expansion into the vacuum

In a supersonic jet expansion, gas is expanded through a nozzle (diameter d) adiabatically into a

region of lower pressure pa. Let T0 and p0 be source temperature and -pressure, respectively, and

κ be the heat capacity ratio1. If the source pressure exceeds the ambient pressure pa by a factor

p0pa≥(κ+ 1

2

) κκ−1

(2.1)

which is less than 2,1 for all gases, the expansion is supersonic [Pau00a]. This leads to a pressure

at the nozzle which exceeds the ambient pressure pa, therefore the �ow is said to be underexpanded.

The Mach number Ma , which describes the ratio of local beam velocity to the speed of sound, is

1 at the nozzle exit and increases with increasing distance. Though, the gas �ow is insensitive to

boundary conditions downstream and overexpands to pressures lower than pa. This leads to the

barrel-shock, which re-compresses the divergent free jet boundary. The recompression of the �ow

decreases the Mach number and re-increases density and pressure, till the Mach disk appears at a

distance

xMd

= 0, 67

√p0pa

(2.2)

The area between nozzle and Mach disk is called zone of silence because it is insensitive to any

boundary conditions. At the Mach disk, the �ow re-expands, which could theoretically entail

a second Mach disc. Practically, this is never the case under the conditions of molecular beam

experiments because of the very low ambient pressure pa (normally less than 10−4 mbar), but

multiple Mach disks can be seen for example behind the engines of jet �ghters. Fig. 2.1 shows

schematically a free jet expansion compared to measured data, which shows clearly the di�erent

regions in the the expansion zone.

During the expansion, the gas cools down to T∞. Via collisions, internal energy is transfered

into kinetic energy of the total �ux, which results in a narrowing of the velocity distribution and

therefore a lowering of the temperature.

1κ, the heat capacity ratio or isentropic exponent is the ratio between cP (constant pressure heat capacity) andcV (constant volume heat capacity): κ = cP

CV

9

10 CHAPTER 2. PHYSICAL AND TECHNICAL PRINCIPLES

Figure 2.1: a) Schematic representation of a supersonic jet expansion [Pau00a]; b) Shadowgraph of afree jet expansion (top), isopycnics (middle) and pressure distribution along the �ow axis (bottom). Datameasured by Irie et al. [IYKS03]

2.1.2 Relaxation of the internal energy

The transformation of internal energy into kinetic energy of forward motion of the gas jet during the

expansion obeys the conservation of enthalpy h. [Sco88] This allows the calculation of the average

terminal velocity v∞:

1

2v2∞ = h− h0 =

ˆ T0

T

cpdT′

(2.3)

Using cp = κκ−1

kBm with the heat capacity ratio κ and the assumption T∞ � T0, which is a good

approximation for most experiments, we obtain for the terminal velocity

v∞ =

√κ

κ− 1

2kBT0m

(2.4)

For a monoatomic gas like argon or helium, this becomes

v∞ =

√5 · kBT0m

(2.5)

In �rst-order approximation, one can assume an expansion to be spherical, which means that the

density of the gas jet decreases by ρ ∝ r−2. The rapid decrease in density involves also fast decreasein the number of two-body and three-body collisions a molecule undergoes after leaving the nozzle

(see Fig. 2.2). This can be identi�ed by a transition from the continuous-regime directly behind

the nozzle to the free-molecular regime far out in the expansion, where almost no collisions occur.

This process is di�cult to describe, because neither continuum nor free molecular �ow mathematics

apply. A simpli�cation of the process is made by the quitting surface model, which introduces a

�ctive border, at which all collision processes stop, between the two regimes. [Sco88]

The collision rates and -cross-sections determine the relaxation of internal degrees of freedom and

the cooling process as well as clustering, which is dependent on the three body collision rate. Due

to the decrease of collision rates during the expansion, kinetic processes stop, they freeze out. The

number of two body collisions experienced by a single molecule in a gas jet is normally of the

2.1. SUPERSONIC MOLECULAR BEAMS 11

Figure 2.2: Number of two-body and three-body collisions remaining in the expansion as a function ofthe distance in nozzle diameters, calculated with di�erent models. [Pau00a]

order 102 to 104 [Pau00a]. Vibrational relaxation cross sections are usually smaller than rotational

relaxation cross sections. [PD12] Therefore, the relaxation of rotational degrees of freedom, which

needs up to 102 binary collisions, is usually realized in a free jet expansion, but vibrational modes

need more than 104 collisions to adapt and therefore remain almost unchanged during the expansion.

When atoms or molecules in a gas jet cluster, they release the binding energy in the form of heat

which increases the beam temperature, an e�ect that opposes the adiabatic cooling. This clustering

limitation of the cooling [HKE03] is normally an unwanted process because it decreases the speed

ratio and therefore the resolution of the experiment. Exceptions are experiments in which clusters

are the object of research, for example the very exceptional quantum systems of the helium dimer

and the helium trimer. [VZB+14]

The clustering limitation of the cooling and therefore the speed ratio is an intrinsic constraint

dependent on the gas to certain molecular beam experiments, a problem that can be solved by the

usage of a slotted disc velocity selector.

2.1.3 Speed ratio and velocity distribution in gas jets

Classic equilibrium-thermodynamics apply only to the very �rst zone behind the nozzle. Due to the

decrease in the collision rates with proceeding expansion, collision and relaxation processes parallel

and perpendicular to the jet axis decouple. This means that it is not possible to de�ne a thermal

equilibrium temperature, in the sense of Maxwell-Boltzmann statistics, anymore 2.

The distribution function f (~x,~v, t) of the gas particles during the expansion obeys the Boltzmann

equation for the case of no external forces and no explicit time dependence:

~v · ∇f (~x,~v, t) =

(δf

δt

)coll

(2.6)

where the collision integral(δfδt

)coll

includes the intermolecular potential and -cross-section, both

depending on the gas species. A rigorous solution to the Boltzmann equation for given boundary

conditions is not possible, but several approximative methods have been used up to now. [Pau00a]

For example, Toennies and Winkelmann used an ellipsoidal drifting Maxwell model for the veloc-

ity distribution function and a simple Lennard-Jones potential for the intermolecular interaction.

[TW77]

2Still, it is possible to de�ne an e�ective temperature Teff that lies between T‖ and T⊥ [Sco88]


f (v) dv = n ·√

m

2 · π · kB · T‖·(

m

2 · π · kB · T⊥

)· e−

m2·kB ·T‖

(v‖−w)2

· e−m

2·kBT⊥v2⊥ (2.7)

The ellipsoidal drifting Maxwell model describes the decoupling of parallel and perpendicular re-

laxation by introducing a parallel and a perpendicular temperature T‖ and T⊥, respectively. The

calculation of the �ow-�eld parameters T‖, T⊥, v‖, v⊥, n was thus realized by Toennies and Winkel-

mann by using the method of characteristics.

This model includes implicitly the assumption that it is possible to de�ne a physical reasonable

perpendicular temperature T⊥ . Actually, the velocity distribution in perpendicular direction does

not really look like a Gauss distribution but more like the sum of two Gaussians, so that a def-

inition of a temperature is not well justi�ed. [Sco88] Therefore, the velocity distribution can be

described using a double-ellipsoidal Maxwell function which contains a second Maxwell distribution

in perpendicular direction :

f (v) dv = n·√

m

2 · π · kB · T‖·(

m

2 · π · kB · T⊥

)·e−

m2·kB ·T‖

(v‖−w)2

·(p1e− m

2·kB ·T⊥1v2⊥ + p2 · e−

m2·kB ·T⊥2

v2⊥)

(2.8)

The second Maxwell distribution in perpendicular direction has a higher temperature T⊥2, which

represents the broad tails in the perpendicular velocity distribution and beam pro�le observed for

example by Cattolica et al. [CRTW74] Both Maxwell distributions are weighted by the factors p1and p2, which can be empirically determined. Fig. 2.3 shows an example for a velocity distribution

in an argon gasjet. The program code of the used function is displayed in appendix 8.3.2.

Figure 2.3: Distribution of v‖ (vpar) and v⊥ (vperp) in an Argon gasjet, based on a double-ellipsoidalMaxwell function.The parameters used are: S = 30, T0 = 300K, T‖ = 0.83K, T⊥1 = 0.6K, T⊥2 = 4K, p1 = 0.8,p2 = 0.2

In typical gas target experiments, the gas target is required to have a small diameter in the target

zone. To achieve this, one or multiple skimmers are placed between the nozzle and the target zone,

which cut the gas jet down to a small diameter. The �cutting� of the gas jet by the skimmer cools

2.1. SUPERSONIC MOLECULAR BEAMS 13

the perpendicular velocity distribution in a complicated way, and gas molecules scattered back

at the walls of the skimmer may blur the velocity distribution in parallel direction and therefore

increase T‖.

The speed ratio S, a parameter very important for experiments, is given by the ratio between the

average terminal velocity v∞ and the thermal velocity breadth 4vtherm =√

2 · kBT‖∞/m :

S =v∞

4vtherm(2.9)

It is related to the FWHM of the velocity distribution by

S = 2 ·√ln (2) · v∞

4vFWHM= 1, 66 · v∞

4vFWHM(2.10)

In the time- and momentum resolving COLTRIMS3 imaging technique, momentum data parallel

to the jet axis is always blurred by the velocity spread 4vFWHM . Also Time-Of-Flight (TOF)

data and Velocity-Map-Imaging (VMI) 4 data from other experiments are a�ected by the velocity

spread of the jet. [Whi03]

This makes it obvious why the speed ratio is so important for experimentalists using gas targets.

The speed ratio is strongly correlated to the product of source diameter d and stagnation pressure

p0, and several theoretical and empirical models for the prediction of S exist. [Pau00a, Sco88] Fig.

2.4 shows data for di�erent noble gases in dependence on the stagnation pressure p0, measured by

Hillenkamp et al. 2003.

Figure 2.4: Speed ratio S and parallel temperature T∞‖ versus source stagnation pressure p0 for di�erentnoble gases, with constant source conditions (T0 ≈ 295K, d = 100µm) [HKE03]

The gas with the highest achievable speed ratio is helium. Its probability to cluster is very low

compared to other rare gases, and because of the very low binding energy of helium clusters (95 neV

for the dimer and 11 µeV for the trimer), the heating e�ect of the clustering is very low. [Bec12]

Moreover, due to quantum mechanical e�ects, the interatomic cross section is very high allowing

cooling, which is collision-dependent, down to very low temperatures in the range of mK. This and

the phenomenon that the speed ratio decreases signi�cantly to higher atomic weights make helium

well suitable as a carrier gas for seeded beams5

3COLdTargetIonMomentumRecoilSpectroscopy (COLTRIMS) is a method to image atomic and molecular frag-mentation processes. A further description is in Section 3.1.1

4Velocity Map Imaging is a technique to image molecular and atomic dissociation and scattering processes. Theions are focused on a detector by an electrostatic lens in a way that each spot on the detector represents a certainvelocity. A brief description of this principle is given in [EP97]

5The seeded beams technique uses a carrier gas that can achieve a high speed ratio, usually helium, but sometimes


2.2 The problem of velocity selection of uncharged particles

In experimental atomic and particle physics, it is very common that the knowledge or determination

of the velocity of the regarded particles, respectively their energy, is essential for the accomplishment

of the experiment. To determine the velocity of an arbitrary particle beam its velocity distribution

can either be cooled or �cut�.

Figure 2.5: Cooling (above) and cutting (below) of velocity distributions

Cooling results in a sharpening of its velocity distribution which equals reducing the FWHM. An

example of this is the seeding of an arbitrary gas in a helium gas jet, which leads to very good speed

ratios because of the low temperature helium can reach during an adiabatic expansion. [HKE03]

(See footnote 5) The other possibility is �cutting� the velocity distribution function f(v), which can

be expressed as a multiplication with a velocity-dependent transmission function T (v). [Pau00b]

F (v) = T (v) · f(v) (2.11)

Very often, the transmission function T (v) and therefore also the �nal velocity distribution F (v)

have well de�ned minimum and maximum velocities vmin and vmax.

In the early days of atomic and molecular physics, velocity selection was the only way of measuring

the velocity of particles. For example, the speed of light and the Maxwell-Boltzmann distribution of

a gas were measured using slotted disc velocity selectors. [Eld27] Velocity selection is also essential

in particle physics, where the beam energy in particle accelerators needs to be well de�ned. This is

often accomplished by applying the velocity-dependent Lorentz force ~FL = q · (~v× ~B) of a magnetic

�eld to the particles in combination with an electric �eld ~E perpendicular to ~B and ~v. This leads

to velocity-dependent trajectories where the velocity v = EB is the only one not de�ected. The

velocity spread transmitted is dependent on the size of the hole (see Fig. 2.6). This principle is an

example for the �cutting� of a velocity distribution.

For uncharged particles , velocity selection (cutting of the velocity distribution) is much more chal-

lenging: The only forces that can usually be applied to them are mechanical forces and gravitation.

also other gases like argon. The gas which is the actual object of research (large molecules for example) is mixedinto the carrier gas. This allows to reach lower temperatures than in an unseeded gas jet. [PD12]

2.2. THE PROBLEM OF VELOCITY SELECTION OF UNCHARGED PARTICLES 15

Figure 2.6: Velocity-selection of charged particles using electromagnetic �elds [hyp14]

Methods for cooling neutral particles are the Magneto Optical Trap (MOT) for atoms (see chapter

1 and [KL+10] for a good description) and the Stark Decelerator (see section 3.2) that slows down

polar molecules.

2.2.1 Velocity selection by gravitation

Consider particles streaming through a skimmer with the width d, in the direction of another hole

with the diameter d in a distance l which is mounted at a drop h below the skimmer. Then it is

obvious that the particles would reach the hole only if the distance they fall down while traveling

equals h±d. Let v0 be the desired velocity, and vmin and vmax the pursued minimum and maximum

velocities, respectively. The situation is shown in Fig. 2.7. By equating the time a particle travels

with the time it needs to fall down by a height he, we obtain

ttravel = tfall (2.12)

l

v=

√g

2 · he(2.13)

By inserting v0 and h, vmin and h+d and vmax and h−d into this relation, respectively, we obtainafter some algebra

vmax − vminv0

=

√h

h− d−√

h

h+ d(2.14)

and

l = v ·

√2 · hg

(2.15)

where the left side of equation 2.14 is the inverse of a speed ratio S:

v0vmax − vmin

=1

S(2.16)

The parameters d, S and v0 are normally de�ned by the experiment: Argon has an almost pressure-

independent speed ratio of S = 30 [HKE03], and we want to increase it to S=70. Its average velocity

at room temperature is about v0 = 560ms , and a standard skimmer-diameter is d = 0, 3mm. [Hen13]

Solving equations 2.14 and 2.15 for these requirements leads us to

h = 2, 1 cm

l = 35m

Obviously a distance of 35 m is much to large for being integrated into a typical atomic or molecular

physics experiment. Moreover, it was assumed implicitly that the beam exits the skimmer perfectly


parallel, which is not the case. This entails a decrease of the intended speed ratio and a decrease

of the intensity. Finally, we can conclude that velocity selection by gravitation is not a practicable

method for most atomic and molecular physics experiments requiring high speed ratios and involving

fast-travelling particles. However, a gravitational velocity selector was implemented by S. Gerlich

from the group of Markus Arndt in Vienna for quantum interference experiments using large, and

therefore relatively slow (see formula 2.5), organic molecules. [GE11]

Figure 2.7: Geometrical model of gravitational velocity selection

2.2.2 Mechanical velocity selection

The basic principle of mechanical velocity selection can be shown with a simple analogy: First, a

�door� is opened at a time t0 shortly for letting a particle pass, and then after a certain distance

l at a time t1, a second �door� is opened shortly. Consequentially, a particle can only pass both

doors if its velocity is v = l/(t1 − t0), otherwise, it would be scattered at the second barrier. One

way to implement this principle is the slotted disk velocity selector, which consists of at least two

disks rotating on the same axis with at least one hole per disc, where the holes are rotated by an

angle α against each other. This principle is shown in Fig. 2.8.

Figure 2.8: The basic principle of a slotted disc velocity selector shown at a selector with just two discs:The holes are rotated by an angle α against each other, so that a particle can only pass both holes if itsvelocity suits the rotational frequency.

Just one hole per disc leads to a very low transmission, therefore the disks can contain several holes.

The presence of these additional holes allows more (velocity-dependent) trajectories through the

selector. These are called sidebands and can be eliminated by the addition of more discs. [HB60]

An increasing of the number of discs has the ancillary e�ect that the assembling and alignment of

the apparatus get more intricate. Nevertheless, selectors with up to 8 discs were built. [TRR62]

A principle that completely avoids the problem of sidebands and alignment is the slotted helical

cylinder selector. [Pau00b] The disadvantage is the high weight and and the big e�ort that has to

be made to mill the helical paths into the massive body. Furthermore, also slotted ring selectors

with their rotation axis perpendicular to the beam axis have been constructed. These two principles

are illustrated in Fig.2.9.

2.3. WORKING PRINCIPLES OF SLOTTED DISK VELOCITY SELECTORS 17

Figure 2.9: A helical slotted cylinder selector (left) completely avoids the problem of sidebands by gaplesspredetermining the trajectory of the particle. A slotted ring selector has the same �open door� principleas a slotted disk velocity selector, except for that its rotational axis is perpendicular to the beam axis.Picture taken from [Pau00b]

2.3 Working principles of slotted disk velocity selectors

2.3.1 Particle trajectories and transmission function

Consider a selector composed of two identical disks with a radius r, a thickness d and a number

of N slits in periodical arrangement in a distance l (from front side to front side) mounted on a

rotatable axis (see Fig. 2.10). There is no angular shift between the slits of the disks which makes

the calculation and the alignment easier. Let a1 be the width of the slits and a0 be the distance

from slit to slit on the circular arc. The velocity selector rotates with a frequency f .

A particle traveling with a velocity v0 starting at the �rst disk can reach a slit with an o�set of

noff slits away on the second disk if the time it needs to reach the second disk ttravel equals the

time tn the disk needs to turn noff slits further:

ttravel = tn (2.17)

Using the relation that time equals distance over velocity we obtain

l

v0=noff · (a0 + a1)

2 · π · r · f(2.18)

Solved to v0, we obtain the formula for the velocity v0 we want to select.

v0 =2 · π · r · l

noff · (a0 + a1)· f (2.19)

As we see, the velocity selected by a slotted disk velocity selector is directly proportional to its

speed of rotation. This formula is the principal relation important for the work with slotted disk

velocity selectors. All signi�cant parameters can be calculated using this formula or modi�ed forms

of it.

If a particle starts at the turning up edge of the slit in the �rst disk and reaches the slit in the

second disk near the vanishing edge it has a bit more time to travel and may therefore be a bit

slower than v0. Let its velocity be vmin. On the other hand, if the particle starts on the vanishing

edge of the slit in the �rst disk and meets the slit in the second disk on the upcoming one, it must

have a velocity vmax faster than v0.

Carrying out the preceding calculations for both cases, we obtain

vmax =2 · π · r · (l − d)

noffset · (a0 + a1)− a1f = v0 ·

1− dl

1− a1noffset·(a0+a1)

(2.20)

vmin =2 · π · r · (l + d)

noffset · (a0 + a1) + a1f = v0 ·

1 + dl

1 + a1noffset·(a0+a1)

(2.21)


Figure 2.10: Two-dimensional schematic representation of particle trajectories in a slotted disk velocityselector. v

(n−1)s is the faster and v

(n+1)s the slower sideband velocity. The position of an optional

intermediate disk to block the sidebands is indicated by the dotted grey lines.

Of course, a particle passing the slotted disk velocity selector is not principally forced to hit the

slit in the second disk which is noff positions away. If it is slower, it can hit for example the slit at

the position noff + 1, and, if it is faster, the one at noff − 1. These additional velocities possibly

transmitted are the sidebands mentioned in section 2.2.2. They are described by the same formulas

as v0, only that noff has to be replaced by the order of the particular sideband. Depending on the

width of the velocity distribution in the gas jet, sideband velocities can be a problem, and they are

suppressed by intermediate disks, as mentioned in 2.2.2.

The transmission Tr of the velocity selector is given by the ratio of the width a1 of the slits versus

the size of one period:

Tr =a1

a1 + a0(2.22)

A good resolution (and therefore a high speed ratio), meaning small di�erences between vmin/maxand v0, can be achieved by a small slit width a1 � a0 or by a high o�set number. [vSV72] This

follows from formula 2.20 and 2.21 and can be illustrated by Fig 2.10. A slit width much smaller

than the material between two slits leads to a low transmission Tr and therefore to a low rate in the

experiment, which is especially unfavorable if the experiment is run with a pulsed laser: The overlap

between the laser pulses and the gas pulses could be very low in unfortunate cases. Increasing the

o�set number noff and therefore the o�set angle between two corresponding slits requires a higher

rotational speed, so that high frequencies become inevitable to achieve both a good resolution and

a high transmission, which was also shown by van Steyn and Verster by numerical calculations.

[vSV72]

The number of intermediate disks necessary to suppress sidebands is dependent on the desired

speed ratio

SSD =v0

vmax − vmin(2.23)

the slit geometry described by a0 and a1 and the width of the velocity distribution in the gas


jet. Devices with up to 8 disks were built in order to completely eliminate sidebands, [TRR62]

The speed ratio SSD selected by a slotted disk velocity is de�ned by the selected velocity over the

velocity spread, which di�ers from the de�nition for gas jets which uses the FWHM. Therefore, the

speed ratio SSD is �better� than the normal speed ratio S because of using the absolute velocity

spread instead of the FWHM.

There is no algorithm for determining the ideal geometry of a slotted disk velocity selector, which

results in the necessity of the usage of iterative methods. This can be carried out by either using

a geometrical model like in Fig. 2.10, implementing an iterative computer program with a lot of

if-statements or performing a Monte-Carlo-simulation. In the early years of the usage of slotted

disk velocity selectors, when computers did not exist, the two-dimensional geometrical model was

very common and the only method to easily determine the transmission properties of a particular

alignment. [HB60] In 1971, van Steyn and Verster from the Eindhoven University of Technology in

the Netherlands performed a light-weight iterative Monte-Carlo-like simulation written in ALGOL

on one of the �rst mainframe computers. They published their results in the form of dimensionless

parameters that lead to a favorable, side-band-free construction geometry. [vSV72]

When practically planning a slotted disk velocity selector, however, some parameters are already

predetermined by the technical properties of the particular experiment.

The transmission function T (v) is given by

T (v) =noff · (a0 + a1)

2 · π · r

[(1 +

a1noff · (a0 + a1)

)−(

1 +d

l

)v0v

]vmin ≤ v ≤ v0

T (v) =noff · (a0 + a1)

2 · π · r

[−(

1− a1noff · (a0 + a1)

)+

(1− d

l

)v0v

]v0 ≤ v ≤ vmax (2.24)

T (v) = 0 v ≤ vmin and v ≥ vmax

The transmission function with its shape of a triangle with slightly hyperbolic curvatured sides, is

outlined in Fig. 2.11. [Pau00b] Due to the only light curvature of the sides, a triangular function

can be used as a reasonable approximation.

The �nal velocity distribution F (v) achieved with a slotted disk velocity selector is given by the

product function of the initial velocity distribution f(v) and the transmission function T (v):

F (v) = T (v) · f(v) (2.25)

Figure 2.11: Transmission curve of a slotted disk velocity selector. The sides of the triangle are slightlycurved. [Pau00b]


2.3.2 Technical implementation of slotted disk velocity selectors

The two basic elements a slotted disk velocity selector consists of is the axis on which the slotted

disks are mounted and the motor. The axis is supported by two roller bearings or ball bearings at

its ends. The motor can be mounted directly around the axis, with the rotor being attached to the

shaft and the stator surrounding rotor and shaft. ([PCV+04], see Fig. 2.13) The other possibility is

mounting the motor in a separate case and connecting it to the axis by a �exible coupling which can

be provided by a short wire [TRR62] or by a special shaft coupling, which was used in the present

case. A construction with the motor directly mounted on the axis has the advantage that it can

be realized very compact. The construction and implementation of such an assembly, on the other

hand, is intricate because shaft, rotor and stator have to be adapted to each other regarding their

speci�c dimensions. Mounting the motor separately, however, allows the usage of a commercially

available, vacuum compatible motor.

The major demands on the motor are a high frequency stability and vacuum compatibility. Brush-

less motors with a permanent magnet rotor whose bearings are lubricated with a vacuum compatible

grease ful�ll these requirements. [Pau00b] Additionally, a slotless stator can be used, which produces

a smoother rotational motion. [Mub08] Reading out the rotational frequency can be implemented

by a LED and a photodiode [Pau00b], and some motor types supply a frequency read-out at the

drive unit. [LLC13]

2.3.2.1 Static and dynamic unbalance

An important issue to face in the construction of a slotted disk velocity selector is the unbalance

of the rotor. The centrifugal force produced by a so-called static unbalance is

Fsu = m · er · ω2 (2.26)

with the rotor weight m, the angular speed ω and the eccentricity er which is the shift between

principal inertia axis and rotational axis. [GNP05] If the inertia axis and the axis of rotation are

tilted, for example, by two weights at opposite sides of the rotor and di�erent positions relative to

the axis, one speaks about a dynamic unbalance (see Fig. 2.12).

Figure 2.12: Static and dynamic unbalance [Mah14]

Even if the rotating assembly consisting of shaft and disks is built very light-weight and machined

with the highest precision, the high rotational frequencies of several 100 Hz which are reached

by slotted disk velocity selectors can result in serious unbalance forces. High unbalances cause

vibrations which can be very strong depending on the eigenfrequencies of the di�erent parts of the

assembly. In the worst case an unbalance can lead to the destruction of the equipment. To avoid

such issues, the rotor of the velocity selector can be balanced or mounted in damped bearings.

Rubber O-rings around the bearings leave some clearance for the rotor to �nd its principal inertia

axis and they absorb the energy of unwanted oscillations. [HB60]

2.3.2.2 Integration into a vacuum system

The place of application of a slotted disk velocity selector in an atomic- or molecular physics

experiment is between the gas nozzle and the target chamber. Normally, very low pressures in the


target chamber are required, and therefore it is expedient to mount the expansion chamber, the

chamber with the velocity selector and the target chamber di�erentially pumped. One skimmer each

before and behind the velocity selector can help improving the gas jet characteristics. Dependent

on jet density and velocity distribution in the particular experiment, the amounts of gas scattered

at the velocity selector could be large. This requires the installation of a Turbo Molecular Pump

(TMP) with high suction strength at the chamber with the velocity selector.

Figure 2.13: Cross-sectional view of a slotted disk velocity selector with directly integrated motor: (1)rotor axis, (2) spacers, (3) laminated rotor (motor), (4) stator windings (motor), (5) vacuum casing,(6) aluminium block, (7) aluminium support plates, (8) ball bearings, (9) emergency bearings, (10)rubber O-rings for damped bearing support, (11) slotted disks, (12) LED and photodiode for frequencymeasurement. Picture taken from [Pau00b].

Chapter 3

Applications and prospects of

velocity selectors

3.1 Optimizing the resolution in COLTRIMS-experiments

3.1.1 COLTRIMS

COLdTargetRecoilIonMomentumSpectroscopy (COLTRIMS) is a method for imaging atomic and

molecular ionization and dissociation processes in gas targets. The atoms or molecules are frag-

mented by a well-focused ion- or photon beam, and the fragments (ions and electrons) are directed

at two detectors mounted opposite each other by a combination of an electric and a magnetic �eld.

The latter one forces the electrons on helical trajectories due to their large charge/mass-ratios. (see

Fig. 3.1)

The positions of the particles on the detectors, which are de�ned by the jet- and by the beam axis,

give insight into the momenta of the ionisation- or dissociation fragments along these directions. If

the fragmenting beam is a pulsed laser or synchrotron with well de�ned pulse timing, the Time-

OfFlight along the spectrometer axis gives additional momentum information perpendicular to gas

jet and laser- or ion beam. [Tit11][Bec12]

Figure 3.1: Principle of the COLTRIMS imaging method. A gas jet and a photon- or ion beam collidein a small reaction volume, and the dissociated fragments are detected by two MultiChannelPlate (MCP)detectors which are aligned parallel to the reaction layer. Image taken from [San09]

23

24 CHAPTER 3. APPLICATIONS AND PROSPECTS OF VELOCITY SELECTORS

The ions and electrons can be matched by momentum conservation, and the sum of all particle en-

ergies from the same process gives the KineticEnergyRelease (KER) of a fragmentation process. By

imaging particle energies and momenta in ionisation- and dissociation processes, COLTRIMS gives

insight into the internal energy levels of atoms and molecules, molecular shapes and intramolecular

distance distributions. [HJ+10],[VZB+14]

3.1.2 E�ects of velocity distribution and jet characteristics on the reso-

lution

With the knowledge about the strength of the electric- and magnetic �eld and the equations of

motion in the spectrometer, the momenta of the charged fragments are calculated back from the

positions on the detector and the time of �ight. The resolution along the beam axis is dependent

on the localisation of the reaction volume, which is determined by the length of the laser focus and

the diameter of the gas jet, and also on the gas jet velocity distribution in perpendicular direction.

In the direction of the spectrometer axis, the resolution is only dependent on the diameter of the

laser focus, which can be very small down to some µm. [Bec12] The resolution of the data on the

ion detector in gas jet direction e�ectively only depends on the velocity distribution in the gas jet

(the diameter of the laser focus is negligible) which is the crucial matter for many experiments,

especially with heavy gases or gases that can not be cooled due to their clustering properties.

[HKE03] A localisation of the reaction volume with a simultaneous limiting of the perpendicular

velocity distribution is usually carried out using a cross of adjustable collimators. (See Fig. 3.2)

Figure 3.2: In�uence of a collimator on the gas jet properties: The jet is strongly localized and the velocitydistribution in perpendicular direction (below) is narrowed while the velocity distribution in transversaldirection is not a�ected.

3.1.3 Optimizing the resolution

Some ideas for possible COLTRIMS-experiments would require a high momentum resolution in all

directions, which can be realized by the usage of slotted disk velocity selector. Currently, our group

is working on the implementation of a highly resolving COLTRIMS-experiment, which involves the

usage of nano-positioned collimators for a more precise gas jet localisation and the present slotted

disk velocity selector for velocity selection. (See Fig. 3.3) One of �rst objectives is re-performing

the measurement of the energy quantization in the double-ionization of argon, which was carried

out by Kevin Henrichs in 2012. [Hen13] In this experiment, 2 electrons and a doubly charged ion

are created. By momentum conservation, the sum of the momenta of all three particles is equal to

the initial momentum of the neutral argon atom in the gas jet. One of the electrons is not observed,

but its energy is calibrated using momentum conservation. Therefore, this measurement is severely

limited by the momentum spread of the neutral atom. Due to the low speed ratio of argon (S=30),

the AboveTresholdIonization (ATI) -peaks that were observed appeared relatively blurred. For a

3.2. APPLICATIONS IN CROSSED MOLECULAR BEAM EXPERIMENTS 25

�ner resolution in the ATI-structures, it was decided that the momentum error in gas jet direction

should be less than 0.25 a.u 1 momentum. In SI-units, this is

4p ≤ 5 · 10−25kg ·ms

This leads to a maximum velocity spread of

4v ≤ 4pmAr

= 7.5m

s

The necessary speed ratio of the slotted disk velocity selector is then obtained using formula 2.23

and the terminal velocity of an argon gasjet at room temperature:

SSD ≥v∞4v

=554ms7.5ms

= 74

This value was used as lower limit for the calculation of possible disk geometries carried out in

section 4.1.

Figure 3.3: Principle of translational and perpendicular beam cooling: The perpendicular velocity distribu-tion (below) is narrowed by nano-positioned collimators, and the velocity distribution in parallel directionis narrowed by a slotted disk velocity selector.

Furthermore, the usage of a slotted disk velocity selector could help improving the momentum-

and therefore also the energy resolution also in experiments with gases that have a speed ratio too

low for measurements without velocity selection or with thermal sources. Perhaps, new unknown

structures in the energy distribution of ionization processes could be resolved.

3.2 Applications in crossed molecular beam experiments

The crossed beam technique is a widely used method of investigation in physical chemistry and

molecular physics. In crossed beam experiments, two atomic or molecular beams are crossed at

an angle of typically 90° and undergo chemical reactions or scattering in the collision volume.

The products of the reaction are subsequently detected and can be analyzed considering their

angular distribution or occupation of internal states. [LZMM13, GGZ07] Crossed molecular beam

experiments give insight into molecular cross-sections, potential-energy-surfaces, reaction energies

and the distribution of quantum states in reactants and reaction products.

The span of analysis methods is quite large: The simplest experiments use quadrupole mass spec-

trometers, which often are mounted rotatably in the layer of both beams [GGZ07], a method

1a.u. (atomic units) are a system of units that simpli�es calculations with atomic- and nuclear physics quantities.1 a.u. momentum =1.993·10−24 kg·m·s−1. A description of the system and the conversions can be found in [NIS14]

26 CHAPTER 3. APPLICATIONS AND PROSPECTS OF VELOCITY SELECTORS

that gives insight about reaction products and -energies. An experiment using Laser-Induced-

Fluorescence (LIF) was carried out by Lahankar et Al. from the Montana State University inves-

tigating the reaction O(3P ) +D2 −→ OD(X2Π) +D which displayed the OD open-shell electronic

structure. [LZMM13]

The team of Sebastiaan Y.T. van de Meerakker from the Fritz-Haber-Institut of the Max-Planck-

Gesellschaft in Berlin performed an experiment in which they scattered a Stark-decelerated NO

beam with a rare gas jet and imaged the di�raction oscillations in NO, after pumping and probing

it, in a VMI-System. [vZOV14]

The reaction between the species in the beams depends on the collision energy of the molecules in

the center-of-mass system. Due to this reason, in most applications the knowledge of the velocity

distribution is crucially for the data analysis and an intrinsic constraint to the resolution of the

experiment. For example, the studies of Guo et al. on the formation of hexenediynyl radicals out

of tricarbon molecules and Allene were impaired by the large velocity spread in their tricarbon gas

jet, which gave the exoergicities they investigated a relatively large error. [GGZ07]

A stark decelerator like the one which the team of Sebastiaan Y.T. van de Meerakker used in their

collision experiment of NO with a rare gas jet is relatively large and expensive compared to a slotted

disk velocity selector, and furthermore, it can only be applied to polar molecules. A slotted disk

velocity selector has the advantage of being applicable to almost all kinds of gases, except very

corrosive ones. Moreover, it can also be used on laser-ablation sources which vaporize solid state

matter or on thermal sources with a Maxwellian velocity distribution.

The major downside concerning the usage of a slotted disk velocity selector is its �nite transmission

T smaller than one. The application of a slotted disk velocity selector on only one of both beams

would raise no problems. The usage of slotted disk velocity selectors on both beams, however,

would reduce the collision rate by a factor of T 2 or T1 · T2 if two di�erent types are used. The

synchronization of both velocity selectors would be very di�cult, because normally, both beams

don't have the same average velocity because of di�erent molecular masses. This problem could

theoretically be solved by changing the source temperatures. But the phase synchronization of two

slotless brushless motors by a precision of less than 0.1° or less seems almost impossible to achieve.

Chapter 4

Construction of the Apparatus

4.1 Determination of the geometric parameters

The device was constructed under the premise that it should be easy to exchange disks and inter-

mediate spacers to allow the usage of di�erent geometries. Firstly, two di�erent geometries were

computed. One geometry has two disks and has therefore sideband velocities, but on the other

hand, it is easy to align because of its insensitivity to angular errors. [Pau00b] The other geometry

is a high-precision and high-transmission �ve-disk geometry with no sidebands in the vicinity of v0.

It was designed at the limits of the precision that was possible to machine, and is therefore very

di�cult to align.

Both designs were planned for the usage with an uncooled argon gas jet, meaning the gas nozzle is

at room temperature of approximately T0 = 295K. This leads to a jet velocity of (see 2.1.2):

v0 =

√5 · kB · T0mAr

= 554m

s

Of course, the velocity selector can also be used in other gas jets by adjusting its rotational fre-

quency to the particular jet velocity according to formula 2.19. In the following spreadsheet, both

geometries and their properties are summarized:

number of disks slit width rotational frequency alignment

2 coarse high easy, insensitive to angular errors

5 very �ne low di�cult, very sensitive to angular errors

4.1.1 Limitations of the geometric parameters due to mechanical con-

straints

As mentioned in 3.1, a momentum resolution of less than 0.25 a.u for argon in the experiment was

required, which corresponds to a speed ratio of S=74 at room temperature. (See section 3.1.3) Most

slotted disk velocity selectors were built for achieving speed ratios less than 30. [HB60, PCV+04]

As shown in section 2.3.1, the velocity spread can be decreased by increasing the o�set number noffand therefore the rotational frequency, or by decreasing the slit width a1. For the design of the �ve

disk geometry, we decided to keep the rotational frequency as low as possible due to safety reasons.

This meant that the slit width had to be kept as small as possible. The fabrication of thin metal

sheets with high precision is usually carried out by etching, electro-forming or laser-cutting. For

example, Pirani et Al. had the disks of their two disk velocity selector machined by laser-cutting

and electroforming. [PCV+04] We decided to let our disks be machined by laser cutting regarding

the following reasons: The precision that can be achieved is very high with tolerances less than

50µm. [Inc14] Moreover, laser cutting is a fully atomized Computer Numerical Control (CNC)

27

28 CHAPTER 4. CONSTRUCTION OF THE APPARATUS

guided process, which involves a fast and easy programing of the machines using Computer Aided

Design (CAD) data formats, and therefore lower costs.

RJ Lasertechnik informed us that they would be able to cut slits with a width of 0.1 mm. We

decided to realize the �ve-disk geometry with the mentioned minimum slit width of a1 = 0.1mm

and a wall width of a0 = 0.2mm between 2 slits, using a disk thickness of d = 0.1mm. Due to the

�nite length of the laser focus, it is not possible to cut shapes that are considerable smaller than the

thickness of the material. The two-disk geometry was designed with a slit width of a1 = 0.2mm, a

wall width of a0 = 1.5294mm and a sheet thickness of d = 0.2mm.

Theoretically, the frequency of rotation could be held low by giving the disks a large radius r and

distance l, but in fact, these parameters are limited by the size of the vacuum chamber in which

the selector is to be installed. An additional constraint for the length l is also the fact that the

density of a gas jet decreases with increasing distance from the gas nozzle.

The length of the selector unit was set to l = 150mm and the disk radius to r = 60mm to allow

the integration into a vacuum chamber with an inner diameter of 250mm.

4.1.2 Calculation of the construction geometry by the usage of a numer-

ical program

Due to the lack of a deterministic method for �nding a suitable disk geometry mentioned in Section

2.3.1, the inevitable try-and-error-method was used. A small function written in Matlab/Octave

was used to calculate the output parameters of a two disk geometry and its necessary rotational

frequency in dependence on the o�set number noff , the desired selected velocity v0 and the slit

parameters a1 and a0. Disk radius r, length l and disk thickness d were �xed to the values mentioned

in Section 4.1.1.

A black-box diagram of the function is shown below, and the code is displayed in Appendix 8.3.

Input Output

rotational frequency f

o�setnumber noff Minimum transmitted velocity vminvelocity v0 =⇒ Maximum transmitted velocity vmax

slit width a1 Fixed: inferior sideband velocity v(n+1)max

wall width a0 Disk radius r upper sideband velocity v(n−1)min

Disk thickness d Speed Ratio S

Length l

Di�erent values for the input parameters were given to the function and modi�ed within the bounds

of what is technical reasonable until the speed ratio met the requirements and the frequency was

on a value not to high which was decided to be less than 400 Hz.

For the design of the �ve-disk high-precission geometry, additionally the positions of the interme-

diate disks had to be determined. An other program was written for this purpose, which displayed

a two dimensional �rolled out� diagram of the selector in the style of the one in Fig. 2.10. The

transmission behaviour of possible sidebands could then be studied by the usage of vector graphics

software. This also entailed the possibility to review if a sideband is blocked by several or just by

one disk, and also if it is blocked near the edge of a slit, which could lead to problems concerning

the �nite precision of the alignment. The following diagram shows the working principle of the

program:

4.2. TRANSMISSION FUNCTION 29

Input Output

o�setnumber noffvelocity v0 Positions of intermediate disks lislit width a1 =⇒ 2D graphical output

wall width a0 Fixed Transmission properties vmin/max, S

intermediate disks o�sets nioff Disk radius r

Disk thickness d

Length l

By successively modifying the o�set positions nioff of the intermediate disks, an almost sideband-

free con�guration was found. The next 5 sidebands in each direction were completely blocked,

which corresponds to a range between 500ms and 610ms . In these regions, the tails of the velocity

distribution of argon are e�ectively zero. The sideband-free region is also large enough for the usage

of the �ve-disk geometry with most other supersonic sources.

The slits of both geometries have a height h = 2mm in radial direction (see Fig. 4.3).

Parameter 2 disk design 5 disk design

v0 554 ms 554 m

s

f 288Hz 147Hzr 60mm 60mml 150mm 150mma1 0.2mm 0.1mma0 1.5294mm 0.2mmh 2mm 2mmTr 0.12 0.33noff 17 50n2 - 21l2 - 63mmn3 - 25l3 - 75mmn4 - 29l4 - 87mm

Table 4.1: Geometric parameters of the two- and the �ve disk design. ni and li are the o�set numbersand disk positions of the intermediate disks, respectively.

4.2 Transmission function

The transmission functions T (v) of both geometries were computed according to formula 2.24. The

initial velocity distribution f(v) of the argon gasjet at room temperature was computed using a

Gauss distribution and an estimated speed ratio of S=30. This value depends only very weakly on

source conditions and was for example measured by Hillencamp et al. ([HKE03], See Fig. 2.4) and

a theoretical calculation of this value is displayed in the �rst chapter of [Sco88]. The functions used

for the computation of f(v) and T (v) are displayed in the appendix, and the results are shown in

Fig. 4.1 and Fig. 4.2.

It is remarkable that the transmission function T (v) is much narrower than the gas jet's velocity

distribution f(v) which is almost constant over the breadth of the central transmission peak, which

is the reason that the transmitted velocity distribution F (v) is almost identical to the transmission

function. The next sideband peaks are negligible because they have much less than 10% of the

height of the central peak.


Figure 4.1: Transmission function T(v), initial velocity distribution f(v) and �nal velocity distributionF(v)=f(v)·T(v) for the two-disk geometry

The �ve-disk geometry has no transmitted sidebands at all in the range of the velocity distribution.

E�ectively, the transmission function is blurred by the deviations in the rotation frequency caused

by the �nite running stability of the motor. The e�ect of the frequency deviations of the motor on

the transmission behavior is considered in section 5.2, using the data obtained by the photoelectric-

barrier measurements of the running behaviour.

Figure 4.2: Transmission function T(v), initial velocity distribution f(v) and �nal velocity distributionF(v)=f(v)·T(v) for the �ve-disk geometry

4.3. THE APPARATUS 31

4.3 The apparatus

The device was designed using the CAD software Autodesk Inventor. CNC-compatible data �les

of the di�erent parts were generated out of the CAD data and sent to the respective producers.

4.3.1 Discs

The disks are made of hardened 1.4031 Mo-steel, a molybdenum and chrome based steel alloy. This

material has very good mechanical properties like high tensile strength and low internal stress.

[Gmb07] The disks were machined by RJ Lasertechnik GmbH by CNC-guided lasercutting. Due to

reasons of mechanical stability, the slits were not cut from the outside surrounding into the material

but cut out of the surface leaving a margin around the slits. Four periodically arranged rectangular

incisions were cut into the margin to facilitate the alignment. They can also be used for velocity

measurement using a photodiode (see Fig. 2.13).

Figure 4.3: Total View (left) and close-up view (right) of a disk of the two-disk geometry. The disks ofthe �ve-disk geometry di�er only by the parameters a1 and a0.

4.3.2 Motor

The motor is a vacuum-compatible 42BS1156V brushless 2 pole DC motor supplied by Koford

Engineering LLC. The slotless stator design eliminates any cogging torque which is crucial for the

accurate operation of a slotted disk velocity selector. The motor, which is very compact with its

diameter of only 42 mm, is rated up to a power of 1000 Watts and reaches a no-load-speed of 925

Hz. [LLC13] The 3 Phases of up to 24 V are supplied by a Koford S24V30A sensorless motor driver.

The motor driver is connected to a TDK Lambda power supply, which provides a DC Voltage of

21 V and up to 165 A.

The rotational frequency is regulated by a potentiometer, but also the option for using a pulse-

width-modulated square wave exists. Reading out the frequency is realized with an encoder at the

motor driver that gives out a 5 V square wave with 3 times the rotational frequency (due to the

three phases of the motor).

The motor is �t into a massive block of aluminium which provides a high thermal conductivity

and therefore reduces the risk of overheating. The temperature of the motor is monitored by a

thermosensor which is �t into a small whole drilled into the aluminium block and it is bond to the

motor by a vacuum compatible thermal grease.


4.3.3 Axis and bearings

4.3.3.1 Rotor and shaft

The rotor consists of the axis, the exchangeable selector unit which is composed of the disks and

intermediate spacers, and the clamping mechanism (See Fig. 4.4). The clamping mechanism

consists of two pairs of half-rings which �t into assigned grooves in the axis at both ends of the

selector unit. The pairs of half rings are hold on the axis by two stepped rings, which they conversely

held in their position on the axis. The selector unit is clamped against one of the stepped rings by

the tightening torque of four screws which are fastened through the other stepped ring.

The shaft has a diameter of 10 mm within the selector unit (disks and spacers) and of 8 mm at the

ends where it is born. It is connected to the motor at one end by a �exible metal-bellow-coupling

supplied by RW Kupplungen.

The whole assembly was fabricated with the highest grades of precision to reduce unbalances to a

minimum. Shaft and disks were fabricated of premium steel, and most other parts like the spacers

are made of aluminium to reduce weight. The total weight of the rotor is 0.26 kg for both the two-

and �ve-disk geometry.

Figure 4.4: Cutaway view of the rotor: (1) shaft, (2) disks, (3) spacers, (4) ball bearings, (5) bearingcasing, (6) rubber suspension rings, (7) half-ring pairs, (8a) stepped holder ring, (8b) stepped holder ringwith screw threads, (9) bellow coupling to motor

4.3.3.2 Eigenfrequencies of the rotor

A theoretical analysis of the running behaviour of a rotor during the construction process can avoid

nasty surprises when putting the device into operation. The modal analysis allows to compute

the oscillation modes or eigenforms and eigenfrequencies of a rotor. The mechanical problem is

translated into an algebraic one, where the oscillation modes correspond to the eigenvectors and

the eigenfrequencies to the eigenvalues. [SIl04] Often, modal analysises are also performed on the

object rotating at several chosen frequencies, considering that a rotational motion increases the

sti�ness and therefore the eigenfrequencies of an object. [BHMT]

The modal analysis was performed in the simulation environment of Autodesk Inventor. The rotors

of the velocity selector in the �ve- and two-disk-con�guration were regarded separately in stagnation

and in rotational motion, both in their purposed frequencies. The input parameters were the CAD-

�le of the regarded object, the materials and the rotational frequency, the number of modes to

be computed and the window of frequencies to be considered. The output parameters were the

computed eigenfrequencies and the eigenmodes which were represented by de�ection and inking of

the material in arbitrary units.


Figure 4.5: Graphical representation of the �rst oscillation mode of the two-disk con�guration at arotational frequency of f=288 Hz. The corresponding modal frequency is fmod=1715,43 Hz. The unitsof the de�ection are arbitrary.

Results

For both con�gurations, no possibly dangerous eigenfrequencies were found. The �rst four modal

frequencies at zero rotational speed were below 100 Hz for both con�gurations. For the operation

at the designated speed, only one modal frequency was found for each design. The corresponding

eigenforms are displayed in Fig. 4.5. and Fig. 4.6.

con�guration rotational frequency f computed modal frequency fmod

2 disk con�guration 146 Hz 1715,43 Hz

5 disk con�guration 288 Hz 266,03 Hz

Figure 4.6: Graphical representation of the �rst oscillation mode of the �ve-disk con�guration at arotational frequency of f=146 Hz. The corresponding modal frequency is fmod=266,03 Hz.

Due to their small thickness, the disks are the elements most likely to oscillate, so a separate analysis

only regarding the disks was performed. The results are displayed in Fig. 4.7.


Figure 4.7: Eigenforms and eigenfrequencies of the disks for zero speed, 146 Hz and 288 Hz whichcorresponds to the rotational frequencies of the �ve-disk and two-disk con�guration, respectively.

A modal frequency could be dangerous for the device if it is close to the designated rotational

frequency. Because this was not the case here, there was no necessity for any changes in the

assembly of the rotor, so the components could be produced in the originally planned way.

4.3.3.3 Bearings and suspension

The axis was �t at both ends into two 8X22X71 ultra-precision high speed ball-bearings supplied

by GMN, which were greased using DuPont Kryptox LVP vacuum lubricant. The bearings are

located in an aluminium housing which is mounted damped into two nitrile rubber rings. The

damping leaves some clearance for the rotor to �nd its physical axis of rotation in the case of a

small unbalance as described in 2.3.2.1. Furthermore, it absorbs the energy of unwanted vibrations

which especially occur during the acceleration process when the rotor passes some eigenfrequencies.

[GNP05]

Figure 4.8: View of the selector in the vacuum chamber: (1) translation stage, (2) guiding shafts, (3)linear plan bearings, (4) steel rod connection to manipulator ,(5) motor, (6) motor casing, (7) metalbellow coupling, (8) gas jet

1The dimensioning of ball bearings is read in the following form: 1. Number: Inner diameter, 2. Number: Outerdiameter 3. number: Height


4.3.4 Mounting and lifting mechanism

The slotted disk velocity selector is mounted into a CF-250 vacuum chamber (inner diameter 250

mm and length 450 mm) on a translation stage to be movable into and out of the beam path which

is depicted in Fig. 4.8. The translation stage is guided by four shafts via linear plain bearings

and its position can be adjusted using a translational manipulator. Due to the size of the velocity

selector, the beam has an o�set of 96 mm to the central axis of the tube which is achieved by the

usage of o�set �anges.

4.3.5 Integration in a COLTRIMS-system

At the time of the �nalization of this thesis, the slotted disk velocity selector was in the state

of installation in a COLTRIMS-system. The selector and its chamber are mounted between the

expansion stage which houses the gas nozzle and the reaction chamber with the COLTRIMS-system.

Due to a large TMP mounted in the lower section of the reaction chamber, the slotted disk velocity

selector and its chamber are mounted upside-down, which means that the velocity selector hangs

from the top of the chamber into the beam path, held by a steel rod connected to the manipulator

that allows to lift the selector out of the beam path. The gas jet passes two skimmers before

entering the selector chamber. One is placed in the zone of silence, directly behind the nozzle, and

one separates the expansion chamber from the selector chamber for di�erential pumping. A metal

sheet with a small whole of 300 µm separates the selector chamber from the target chamber. Due

to the low pressures of less than 10−11 mbar in the target chamber, the gas scattered at the velocity

selector must be pumped di�erentially, and the hole in the metal sheet provides also a sharper

localisation of the gas jet. The selector chamber is pumped by a TMP with a diameter of 100 mm.

The integration of the velocity selector in the COLTRIMS system is depicted in Fig. 4.9.

Figure 4.9: Installation of the slotted disk velocity selector in the COLTRIMS system (large chamber onthe left). The selector chamber is rendered transparent for showing the mounting of the velocity selector.The path of the gas jet is marked blue and the path of the laser beam red. The whole setup is supportedby an aluminium frame which is not displayed in this �gure.

Chapter 5

Tests and conclusion

5.1 Measurement of the running stability on atmosphere

Before the installation of the device in the vacuum system was carried out, measurements of the

running behaviour and frequency stability had been performed. The rectangular frequency signal

from the drive unit (see section 4.3.2) was read out, and furthermore a light-barrier-measurement

using a HeNe laser �ring through the slits on a photodiode was implemented.

5.1.1 Measurement of frequency signal from the motor drive unit

As mentioned in 4.3.2, the motor drive unit gives out a 5 V square wave with 3 times the rotational

frequency. The supplier assured that this frequency coincides with the physical rotational frequency

of the permanent magnet rotor, which we could con�rm. (see measurement 5.1.3) We connected

the encoder output of the motor driver to a multimeter with a frequency counter. The display of

the multimeter was recorded with a camera every second, due to its rate of one update per second.

To obtain the corrected rotational frequency, the value from the multimeter had do be divided by

3. The measurement was started when the motor leveled o� at continuous operation, indicated

by a relatively stable value of 378 Hz displayed by the multimeter, a low current provided by the

power supply and by a smooth and steady sound. The results are displayed in Fig. 5.1.

Figure 5.1: Measurement with the frequency encoder from the drive unitLeft: Corrected rotational frequency. Right: Corresponding density distribution of the frequency with aGaussian �t function

Regarding the development of the frequency in dependance on time, one sees the frequency staying

relatively stable over the measurement, except for some small runaways. The density distribution

37

38 CHAPTER 5. TESTS AND CONCLUSION

shows a standard deviation of ±0.864 Hz, which involves a deviation in the rotational frequency of

less than 0.7 %.

5.1.2 Measurement with a photoelectric barrier

For resolving the running behaviour within one revolution, the signal from a photoelectric barrier

spanning the slits of the selector was measured. This was implemented by the usage of a small

HeNe laser as collimated light source and a photodiode as light detector. The laser was aligned in

such a way that it hit the photodiode after passing through a slit in one of the disks. The disks

with the 0.2 mm-slits (two-disk geometry) were used, because the slits of the �ne geometry were

too small and too close to each other to di�er between the signals induced by two passing slits.

The signal from the photodiode was in the magnitude of some mV, and could directly be displayed on

an oscilloscope. For the proper measurement it was ampli�ed, �ltered and converted to a rectangular

pulse signal by a threshold-triggered 5V pulse generator before being sent to the data-accusation

computer. There, the square pulse signal was recorded by a Time to Digital Converter-Card (TDC)

and processed by the data acquisition software COBOLD.

The results were displayed as a density distribution of the time tss between the signals of two slits

and as a three-dimensional density distribution of the number of counts of the time di�erences tssversus the total time. The total density distribution was used to calculate the distribution curve

of the corresponding rotational frequencies by using that a signal comes N (number of slits in the

disk) times per revolution:

frot =1

tss ·N(5.1)

The graph of the rotational frequency distribution and of the development of the tss-distribution

are shown in Fig. 5.2 and Fig. 5.3.

Figure 5.2: Measurement with photoelectric barrier: Frequency distribution

The frequency distribution (Fig 5.2) is in good agreement with the the Gaussian �t function that

corresponds to a standard deviation of 0.764 Hz. Its development over time (see Fig. 5.3) is on

average relatively constant with a white-noise-like modulation.

5.1. MEASUREMENT OF THE RUNNING STABILITY ON ATMOSPHERE 39

Figure 5.3: Measurement with photoelectric barrier: Development of the time di�erence between topassing slits in total time

5.1.3 Comparison of light barrier- and driver encoder measured fre-

quency

The last measurement of the running behaviour aimed at examining possible di�erences between the

frequency given out by the encoder and the physical rotational frequency. For this measurement,

the signal of the photodiode was connected to the oscilloscope, which was adjusted in a such way

that it showed as many peaks as possible. The multimeter displaying the signal from the driver

encoder and the oscilloscope displaying the signal from the photodiode were both aimed at with

the camera and pictured simultaneously. Using the time between to slits passing by tss obtained

by the oscilloscope, the physical rotational frequency was calculated using formula 5.1. The results

for di�erent rotational speeds are displayed in table 5.1.

Driver encoder Photoelectric barrier Deviation

126.33 125.68 +0.36%173.67 174.42 -0.43%182.67 183.49 -0.45%182.33 183.49 -0.63%206.33 206.42 -0.04%

Table 5.1: Values for the rotation frequency measured with the driver encoder and photoelectric barrier

Both values are in good agreement with each other, and the deviation is less than half a percent

except in one case. The values displayed by the multimeter connected to the driver unit seem to be

a bit higher here which could also be due to the fact that the multimeter does not display �oating

point numbers and to the error of the measurement with the oscilloscope. However, it is conspicuous

that the physical rotational frequency does not di�er signi�cantly from the frequency given out by

the driver encoder. This allows the usage of the driver encoder for frequency monitoring which has

the advantage that a photoelectric barrier (see Fig. 2.13) is not necessary.

5.1.4 Running behavior

Due to safety reasons, the slotted disk velocity selector was surrounded by massive lead blocks when

turned up for the �rst time. This turned out not to be necessary, because no signi�cant vibrations


could be observed when the selector was run in continuous operation at frequencies higher than

100 Hz. When being turned up, some vibrations can be observed for short moments, obviously

when the rotor is passing some eigenfrequencies. The velocity selector was also run in a vacuum

test chamber in a relatively bad vacuum of ∼10 mbar without any abnormalities observable.

When being turned up, the motor driver pulls a current of up to 12 A, but in continuous operation

at frequencies between 100 Hz and 200 Hz, the current always stays below 1.5 A. The motor shows

no signi�cant heating, and even during operational times of more than two hours, its temperature

always keeps below 40 °C. This is considerably below its rated maximum temperature of 150 °C.

[LLC13]

When the device is run in atmosphere, air swirls are produced by the slits in the disk. They are

strong especially for the �ve-disk geometry, which has 1256 slits per disk (compared to 218 on the

two-disk geometry).

5.2 E�ect on the transmission behaviour

The e�ect of the frequency spread on the transmission behaviour was estimated. The velocity

spread σfv corresponding to the frequency spread σf = 0.764Hz was calculated for each geometry,

by using formula 2.19. This value was used to obtain the velocity FWHM corresponding to the

standart deviation σfv , caused by the frequency spread :

4vfFWHM = 2.3548 · σfv

The velocity FWHM caused by the natural spread of the transmission function was calculated using

4vvFWHM =1

2· (vmax − vmin)

assuming the transmission function being triangular which is a reasonable approximation. (See

section 2.3.1). The corrected velocity FWHM4vCorFWHM was then obtained using

FWHMf =√FWHM2

1 + FWHM22

=⇒

4vCorFWHM =

√(4vfFWHM

)2+ (4vvFWHM )

2

due to the fact that both spread functions, the velocity distribution transmitted by the velocity

selector and the velocity spread caused by the motor, are convoluted. [AK71] The �nal speed ratio

in the gas jet is calculated using the regular de�nition

S = 1.66 · v04vCorFWHM

and not the �slotted disk velocity selector de�nition� of the speed ratio, because there are no vminand vmax anymore due to the convolution with a Gaussian function. One must be aware that

this de�nition is the �weaker� de�nition of a speed ratio, becauses it uses the FWHM and not an

absolute (constrained by vmin and vmax) velocity spread.

5.3. CONCLUSIONS 41

2 disks 5 disks

vmin[m/s] 550.99 550.70vmax[m/s] 557.05 557.35

vFWHMmin [m/s] 552.50 552.35vFWHMmax [m/s] 555.52 555.674vvFWHM [m/s] 3.02 3.32S = v0

4vvFWHM304.52 258.64

σfv [m/s] 1.47 2.88

4vfFWHM [m/s] = 2.355 · σfv 3.36 6.784vCorFWHM [m/s] 4.52 7.55Scor = v0

4vCorFWHM

203.47 121.83

Table 5.2: Corrected transmission properties of the slotted disk velocity selector concerning the e�ect ofthe frequency deviation

The e�ect of the frequency deviations on the �ve-disk geometry is much larger than the e�ect on the

two-disk geometry. The blurring of the transmitted velocity distribution also spreads the sideband

peaks of the two disk geometry (See Fig. 4.1) towards the main transmission peak. The spreading

factor is just 1.35 and therefore the sidebands may still not have a serious disturbing e�ect on the

experiment.

Nevertheless, these calculations are just estimations, because the frequency stability was measured

only at one rotation speed, and it is not known if it changes signi�cantly dependent on the frequency.

Additionally, the measurements were performed on atmosphere, which caused air swirls on the disks

that possibly amplify the deviations in the rotation frequency.

5.3 Conclusions

The running behavior of the velocity selector was measured using a photoelectric barrier and with

the encoder unit.

The measurement with the encoder unit shows the frequency stability over longer timescales, mean-

ing several periods of rotation. The frequency stays relatively stable, except for some small run-

aways. (See Fig. 5.1) These distortions in the rotational speed do not last very long (less than 2

seconds), obviously they are directly corrected by the driver unit. During our tests, the average

frequency was observed to stay stable over running periods of more than two hours.

The photoelectric barrier and laser based measurement, on the other hand, gave insight into the

running stability within a single revolution by measuring the turning time between two slits. This

measurement yielded a standard deviation in the frequency distribution of 0.764 Hz, which is less

than 0.5%. (see Fig. 5.2). The running behavior of the device is displayed in Fig. 5.3, which

shows the density distribution of the time di�erence tss between two slits. The small deviations in

the frequency seem to be completely noisy and not following any systematical behaviour. The lack

of any periodical modulations in the running behaviour, which was also con�rmed by observing a

large number of photodiode-peaks on the oscilloscope, indicates that the motor has no cogging at

all, which is due to its slotless design.

The standard deviation in the rotational frequency measured by the photoelectric barrier (σ=0.764

Hz) is a bit smaller than the value measured with the driver encoder (σ=0.864) This may be due

to the higher precision (the multimeter can only display integers) of the measurement with the

photodiode and due to its higher number of samples.

The e�ect of the frequency deviations on the transmission behaviour was estimated. The e�ect on

the two-disk geometry is small, in contrast to the e�ect on the �ve-disk geometry which is rather

large. These estimations, however, are too fraught with uncertainties to jump to conclusions and

the actual transmission behaviour can only be investigated by tests with a gas jet in the vacuum

chamber.

Chapter 6

Conclusion and outlook

The resolution in atomic and molecular beam experiments strongly depends on the the width of

the velocity distribution in the gas jet. In order to improve the resolution in COLTRIMS imaging

experiments, a slotted disk velocity selector was constructed and built. The device, using a mechan-

ical implementation of the �green wave�-principle, only transmits velocities between two boundary

velocities vmin and vmax. The slotted disk velocity selector consists of an axis carrying the slotted

disks and a motor connected to the axis by a �exible coupling. Disks and intermediate spacers are

exchangeable. The axis is mounted damped, in order to absorb the energy of unbalance-caused

oscillations and to give the axis some clearance. The device, which is designed for rotational speeds

up to 400 Hz, is run by a vacuum-compatible two-pole permanent magnet slotless motor. Motor

and axis are attached to an aluminium frame. Two disk geometries with di�erent transmission

properties were designed.

The device was designed in the CAD-Software Autodesk Inventor, and before the components were

given into production, a modal analysis was performed to review if the eigenfrequencies of the rotor

were critical for operation, which turned out not to be the case.

Several running tests were performed before the installation of the device in a COLTRIMS-system.

Using an encoder function provided by the motor driver unit, the frequency was read out and

its development was monitored over time. This data was also used to calculate the frequency

distribution. Moreover, a photoelectric-barrier measurement using a HeNe laser and a photodiode

was performed measuring the time the disks need to turn one slit further. The time distribution

was recorded and also used to calculate the frequency distribution. Both methods indicate a

high frequency stability of the selector over time which is crucial for successful performance in

experiments.

The e�ect of the deviations in the frequency on the transmission behavior was calculated, but the

reasonability of these calculations is hard to estimate and will become apparent when the �rst

experimental tests with a gas jet will be performed.

However, due to the fact that the frequency instability has probably a signi�cant worsening e�ect

on the transmission behavior of the �ve-disk geometry and that this geometry turned out to be

really hard to align, this geometry is likely not to be used. The two-disk geometry is not so sensitive

to the frequency instabilities due to its higher rotation frequency (slotless motors run more stably

in the range of high frequencies [LLC13]) and insensitivity to errors in the angular alignment. It is

currently mounted on the selector and will be used in the �rst tests.

The slotted disk velocity selector is mounted in a vacuum chamber which is designed to be placed

between gas jet expansion stage and experimental chamber. All three stages are di�erentially

pumped.

Currently, the selector and the chamber are installed in a COLTRIMS-system which is in the state

of going into operation. As soon as the gas jet and the laser are aligned and all electronics are put

into operation, �rst tests using an argon gasjet will be performed. The next objective will be to

43

44 CHAPTER 6. CONCLUSION AND OUTLOOK

re-perform the double-ionization experiment with argon which was originally performed by Kevin

Henrichs in 2011 (see section 3.1.3), in order to display the discovered ATI-structures with higher

resolution.

With the knowledge of the results of the �rst tests, the slotted disk velocity selector could be

optimized. As mentioned in section 5.2, the actual e�ect of frequency deviations on the transmission

behaviour is hard to estimate in advance, and test results with a gas jet could help to design new

disk geometries, in case that the present two disk geometry turns out to be insu�cient. As soon as

the slotted disk velocity selector has proven the ability to operate stably and reliably over longer

periods of time, new experiments, for example with sources whose velocities are Maxwell-Boltzmann

distributed, could be planned.

Chapter 7

Acknowledgements

First, I would like to thank Professor Reinhard Dörner for giving me the opportunity to work on

this project. It was inciting and motivating, and I learned a lot. Thank you!

Lothar Schmidt, Till Jahnke and Markus Schö�er must be mentioned for their generous help in

the investigation of technical problems.

I would like to acknowledge Martin Pitzer for the good cooperation during the installation of the

velocity selector in his experimental apparture.

I also thank...

...the whole atomic physics group for the motivating and relaxed atmosphere. Working here was

an unforgettable experience. I wish you all luck and succes for your future research and careers!

...and last but not least my family for supporting me and giving me the opportunity to study

physics!

45

46 CHAPTER 7. ACKNOWLEDGEMENTS

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50 BIBLIOGRAPHY

Chapter 8

Appendix

8.1 Frequently used acronyms

ATI - Above Treshold Ionization

Multiphoton-ionization of an atom by more than the minimum number of photons required

CAD - Computer Aided Design

Constructional drawing of 3-dimensional models using vector-based software

CNC - Computer Numerical Control

Automized guidance of machine-tools like milling- or lasercutting machines

COLTRIMS - COLd Target Ion Momementum Recoil Spectroscopy

Imaging method for dissociation processes

FWHM - Full width half maximuum

The distance between two points with half the maximum height on a Gaussian or similar function

MOT - Magneto Optical Trap

A trap using lasers and magentic �elds for cooling atom clouds down to temperatures of several

µK.

TMP - Turbo Molecular Pump

TOF - Time of Flight

Time an atom or molecule fragment needs to hit the detector after the fragmentation process in an

atomic or molecular physics experiment. The time of �ight contains momentum information.

VMI - Velocity Map Imaging

Imaging method for dissociation processes

8.2 Software being used

Computer Aided Design - Software:

-Autodesk Inventor Professional 2013

Computations:

-GNU Octave 3.6.4

51

52 CHAPTER 8. APPENDIX

Plotting:

-GNU Octave 3.6.4

-QTI Plot

Data Acquisition

-COBOLD

A data acquisation software developed in the atomic and molecular physics research group of the

Goethe Universität in Frankfurt.

8.3 Program code

In this section the program code of several programs and functions used for computations of gas

jet properties and slotted disk velocity selectors is displayed. The language is Matlab/Octave.

8.3.1 Velocity distribution of a supersonic gas jet

The function which is written in Matlab/Octave accepts the velocity in the form of a vector v and

returns the vector p containing the values of the velocity distribution function. Source temperature

T, speed ratio S and the atomic/molecular weight in atomic mass units are de�ned in the function

body.

%V e l o c i t y d i s t r i b u t i o n o f a Ga s j e t

funct ion p=v e l o d i s ( v )

T=295; %Source Temperature

S=30; %Speed Rat i o

m=39.948; %Weight i n atomic mass u n i t s ( he r e : Argon )

ckb=1.38065E−23;cu=1.66054E−27;p=exp (−((S^2)*(m*cu )/(5* ckb*T) )* ( v−sqr t (5* ckb*T/(m*cu ) ) ) . ^ 2 ) ;

end

8.3.2 Plotting program for an double-elipsoidal Maxwell distribution

This program displays a coloured surface plot of the parallel and perpendicular velocity distribution

in a supersonic gas jet. Physical constants and gas jet properties are de�ned in the header.

ckb=1.38065E−23;cu=1.66054E−27;cme=9.10938E−31;cmp=1.67262E−27;cmn=1.67493E−27;ce =1.602177E−19;ch=6.62607E−34;m=39.948* cu ; %Atomic we ight i n amu

T0=300; %Source Temperature

w=sqr t (5* ckb*T0/m) ;

S=30; %Achieved Speed Rat i o

Tpar=(m/(2* ckb ) ) * ( (w/S )^2 ) ;

Tperp1 =0.6 ;

8.3. PROGRAM CODE 53

Tperp2=4;

Vpar=l i n space ( 480 , 620 , 101 ) ;%Range o f the p a r a l l e l v e l o c i t y , s t e p s

Vperp=l i n space (−60 ,60 ,101) ;%Range i f the p e r p e n d i c u l a r v e l o c i t y , s t e p s

[ vpar , vpe rp ]=meshgrid ( Vpar , Vperp ) ;

p1=0.8 ; %Weight ing f a c t o r s

p2=0.2; f

v=exp(−(m/(2* ckb*Tpar ) ) * ( ( vpar−w) . ^ 2 ) ) . *

( p1*exp(−(m/(2* ckb*Tperp1 ) )* ( vpe rp .^2))+

p2*exp(−(m/(2* ckb*Tperp2 ) )* ( vpe rp . ^ 2 ) ) ) ;

mesh( vpar , vperp , f v )

x l abe l ( " vpar [m/ s ] " ) ;

y l abe l ( " vperp [m/ s ] " ) ;

z l abe l ( " f ( vpar , vpe rp ) " ) ;

8.3.3 Output parameters of a slotted disk velocity selector

The output parameters of a slotted disk velocity selector are calculated according to the formulas

in chapter 2. The input parameters are the o�set number n, the desired selected velocity v0 and the

slit width a1 and wall width a0. Length l, radius r and disk thickness d are de�ned in the function

body.

funct ion twod i skgeomet ry (n , v0 , a1 , a0 )

%v0 i n m/s , a1 , a0 und d i n mm; they a r e conve r t ed to m i n t e r n a l l y

d=0.2;

a1=a1 /1000 ;

a0=a0 /1000 ;

d=d/1000 ;

r =0.06;

l =0.15;

f=v0*n*( a0+a1 )/(2* p i * r * l ) ;

vmin=2*p i * f * r *( l+d )/ ( n*( a0+a1)+a1 ) ;

vmax=2*p i * f * r *( l−d )/ ( n*( a0+a1)−a1 ) ;v_down_max=2*p i * r *( l−d )* f / ( ( n+1)*( a0+a1)−a1 ) ;v_down=2*p i * r * l * f / ( ( n+1)*( a0+a1 ) ) ;

v_down_min=2*p i * r *( l+d )* f / ( ( n+1)*( a0+a1)+a1 ) ;

v_up_max=2*p i * r *( l−d )* f / ( ( n−1)*( a0+a1)−a1 ) ;v_up=2*p i * r * l * f / ( ( n−1)*( a0+a1 ) ) ; v

_up_min=2*p i * r *( l+d )* f / ( ( n−1)*( a0+a1)+a1 ) ;

S=v0 /( vmax−vmin ) ;

Frequenz=f

v0=v0

vmin=vmin

vmax=vmax

v_up

v_up_max v

_up_min

v_down

v_down_max

v_down_min

Sp e ed r a t i o=S

end


8.3.4 Plotting program for the 2-dimensional rendering of a selector ge-

ometry

This program plots a two-dimensional surface of the �rolled out�-model of a slotted disk velocity

selector in the method of Fig. 2.10. Due to the grid-stepwidth necessary to resolve a �ne disk

geometry, the programm needs running times up to one hour on an extra-cooled Intel i5-3210M

Quadcore processor with 2.50 GHz.

funct ion g r i d 2 ( s c h r i t t w e i t e , abstand , s t e g b r e i t e 1 , s t e g t i e f e ,

l aenge , r a d i u s , v0 , o f f s e t z a h l , n2 , n3 , n4 , n5 )

%A l l e D i s tanzangaben i n mm, Ge s chw i nd i g k e i t e n i n m/ s

ypos6=l a enge ; anzah l=o f f s e t z a h l +16;

x range=(anzah l +1)* abstand ;

y range=ypos6+4* s t e g t i e f e ;

y s t a r t=−5* s t e g t i e f e ;

i n v s t e i=abstand * o f f s e t z a h l / ypos6 ;

ypos1=0; ypos2=n2* abstand / i n v s t e i ;

ypos3=n3* abstand / i n v s t e i ;



ypos6=l a enge ;

i f ( n2==0)

B2=0;

e l s e

B2=1;

end i f

i f ( n3==0)

B3=0;

e l s e

B3=1;

end i f

i f ( n4==0)

B4=0;

e l s e

B4=1;

end i f

i f ( n5==0)

B5=0;

e l s e

B5=1;

end i f

x=[0: s c h r i t t w e i t e : x range ] ;

y=[ y s t a r t : s c h r i t t w e i t e : y range ] ;

[X ,Y]=meshgrid ( x , y ) ;

f o r i =[1 : anzah l ] ;

z ( i , : )= h e a v i s i d e ( x−(0+( i −1)* abstand ) , 0 ) . *

h e a v i s i d e ((0+( i −1)* abstand)+ s t e g b r e i t e 1−x , 0 ) ;end

Z=sum( z , 1 ) ;

G1=Z.* h e a v i s i d e (Y−ypos1 , 0 ) . * h e a v i s i d e ( ypos1+s t e g t i e f e −Y , 0 ) ;

G2=B2*Z .* h e a v i s i d e (Y−ypos2 , 0 ) . * h e a v i s i d e ( ypos2+s t e g t i e f e −Y , 0 ) ;


8.3. PROGRAM CODE 55



G6=Z.* h e a v i s i d e (Y−ypos6 , 0 ) . * h e a v i s i d e ( ypos6+s t e g t i e f e −Y , 0 ) ;

%R=h e a v i s i d e (X− i n v s t e i *Y−s t e g b r e i t e 1 ) . *

h e a v i s i d e (−(X− i n v s t e i *Y−s t e g b r e i t e 1 )+( abstand−s t e g b r e i t e 1−s t e g t i e f e * i n v s t e i ) ) ;G=G1+B2*G2+B3*G3+B4*G4+B5*G5+G6 ;%+R;

%Die T r a j e k t o r i e i s t auskomment ie r t !

c on t ou r f (X,Y,G)

s a v ea s (1 , "/home/ j a s p e r /Studium/Ma s t e r a r b e i t /

Lochsche iben−Pu l s e r /Berechnungen / E r g e bn i s s e / a k t u e l l . png" )

s a v ea s (1 , "/home/ j a s p e r /Studium/Ma s t e r a r b e i t /

Lochsche iben−Pu l s e r /Berechnungen / E r g e bn i s s e / a k t u e l l . svg " )

n=o f f s e t z a h l ;

a=abstand /1000 ;

r=r a d i u s /1000 ;

l=l a enge /1000 ;

f r e qu en c y=v0*n*a /(2* p i * r * l )

v0=v0

vmin=v0*(1+ s t e g t i e f e / l a enge )/

(1+( abstand−s t e g b r e i t e 1 )/

( o f f s e t z a h l * abstand ) ) vmax=v0*(1− s t e g t i e f e / l a enge )/

(1−( abstand−s t e g b r e i t e 1 )/ ( o f f s e t z a h l * abstand ) )

D i s k z ah l=2+B2+B3+B4+B5

O f f s e t z a h l=o f f s e t z a h l

O f f s e tw i n k e l=o f f s e t z a h l * abstand / r a d i u s

ypos1=ypos1

ypos2=ypos2

ypos3=ypos3

ypos4=ypos4

ypos5=ypos5

ypos6=ypos6

Abstand=abstand

S c h l i t z w e i t e=abstand−s t e g b r e i t e 1S t e g b r e i t e=s t e g b r e i t e 1

S ch e i b end i c k e=s t e g t i e f e

Laenge=l a enge

Rad ius=r a d i u s

%" a l l e Laengenangaben i n mm"

end

8.3.5 Transmission function of a slotted disk velocity selector

This program returns a vector containing the values of the transmission function as a function of the

velocity, according to formula 2.24. The selector geometry is de�ned in the header of the function.

All distances are in mm.

%Transm i s s i on o f a SDVS ac co r d i n g to the Formula I n Pau l i : Beams I I

funct ion Tr=t r a n sm i s s i o n ( v ) %r e t u r n s a v e c t o r Tr i n dependence o f v

Frequency = 288.03 %Frequency

v0 = 554 %s e l e c t e d v e l c o i t y

vmin = 550.99 %minimum v e l o c i t y


vmax = 557.05 %maximum v e l o c i t y

v_up = 588.62 %upper s i d eband v e l o c i t y

v_up_max = 592.12 %upper s i d eband maximum v e l o c i t y

v_up_min = 585.18 %upper s i d eband minimum v e l o c i t y

v_down = 523.22 %lowe r s i d eband v e l o c i t y

v_down_max = 525.90 %lowe r s i d eband maximum v e l o c i t y

v_down_min = 520.58 %lowe r s i d eband minimum v e l o c i t y

a1=0.2 ; %S l i t w idth [ a l l d i s t a n c e s i n mm]

a0=1.5294; %Wall w idth

d=0.2; %d i s k t h i c k n e s s

l =150; %s e l e c t o r l e n g t h

r =60; %d i s k r a d i u s

n=17; %o f f s e t number

s c a l e =2500/1.066; %a r b i t r a r y s c a l i n g f a c t o r

gamma=a1 /(n*( a1+a0 ) ) ;

beta=d/ l ;

e t a=a1 /(2* p i * r ) ;

Tr0=(( h e a v i s i d e ( v−vmin ) . * h e a v i s i d e ( v0−v )* ( e ta /gamma) . *

((1+gamma)−(1+beta )* ( v0 . / v )))+( h e a v i s i d e ( vmax−v ) . *h e a v i s i d e ( v−v0 )* ( e ta /gamma).*(−(1−gamma)+(1−beta )* ( v0 . / v ) ) ) ) * s c a l e ;

%Transm i s s i on f u n c t i o n o f the main peak

Trdown=(( h e a v i s i d e ( v−v_down_min ) . * h e a v i s i d e (v_down−v )* ( e ta /gamma) . *

((1+gamma)−(1+beta )* ( v_down . / v )))+( h e a v i s i d e (v_down_max−v ) . *h e a v i s i d e ( v−v_down )* ( e ta /gamma).*(−(1−gamma)+(1−beta )* ( v_down . / v ) ) ) ) * s c a l e ;

%Transm i s s i on f u n c t i o n o f the l owe r s i d eband

Trup=(( h e a v i s i d e ( v−v_up_min ) . * h e a v i s i d e (v_up−v )* ( e ta /gamma) . *

((1+gamma)−(1+beta )* ( v_up . / v )))+( h e a v i s i d e (v_up_max−v ) . *h e a v i s i d e ( v−v_up )* ( e ta /gamma).*(−(1−gamma)+(1−beta )* ( v_up . / v ) ) ) ) * s c a l e ;

%Transm i s s i on f u n c t i o n o f the upper s i d eband

Tr=Tr0+Trdown+Trup ; %Sum of main peak and next two s i d eband peaks

end

Erklärung nach § 28 (12) Ordnung für den Bachelor- und dem Masterstudiengang Hiermit erkläre ich, dass ich die Arbeit selbstständig und ohne Benutzung anderer als der angegebenen Quellen und Hilfsmittel verfasst habe. Alle Stellen der Arbeit, die wörtlich oder sinngemäß aus Veröffentlichungen oder aus anderen fremden Texten entnommen wurden, sind von mir als solche kenntlich gemacht worden. Ferner erkläre ich, dass die Arbeit nicht - auch nicht auszugsweise - für eine andere Prüfung verwendet wurde. Frankfurt, den

construction of a slotted disk velocity selector for ... · 2.1 supersonic molecular beams 2.1.1...

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