F47 — Cyclotron frequency in a Penningtrap

Blaum group

June 25, 2015

Contents1 Introduction 5

2 Theory 72.1 Cyclotron motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Guiding center motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Penning trap motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Penning trap electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Penning trap frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Real Penning traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Detection and Manipulation 133.1 Detecting electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Coupling and driving the modes / Measuring the frequencies . . . . . . . . . 143.3 Driving the coupling modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Experiment 174.1 Overview of the experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Mechanical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.1 Measurement electronics . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2.2 Measurement control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.1 Identifying the Components . . . . . . . . . . . . . . . . . . . . . . . . 214.3.2 Finding the Resonance Frequency . . . . . . . . . . . . . . . . . . . . 214.3.3 Preparing for an Antenna Frequency Scan . . . . . . . . . . . . . . . . 224.3.4 Manual Antenna Frequency Scan . . . . . . . . . . . . . . . . . . . . . 234.3.5 Rough Scripted Antenna Frequency Scan . . . . . . . . . . . . . . . . 234.3.6 Fine Scripted Antenna Frequency Scan . . . . . . . . . . . . . . . . . 244.3.7 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2

List of Figures2.1 Motion of charged particle in electromagnetic fields. . . . . . . . . . . . . . . 82.2 Cut-view of the electric field inside a Penning trap. . . . . . . . . . . . . . . . 82.3 Motion of a charged particle in a Penning trap. . . . . . . . . . . . . . . . . . 92.4 Penning trap electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Electrical model of the endcap connection. . . . . . . . . . . . . . . . . . . . . 133.2 Detection scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1 Overview of the experimental setup. . . . . . . . . . . . . . . . . . . . . . . . 174.2 Overview of the measurement electronics. . . . . . . . . . . . . . . . . . . . . 194.3 Timings of the measurement sequence. . . . . . . . . . . . . . . . . . . . . . . 204.4 Typical antenna frequency scan graph. Try to notice and understand the

patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

List of Tables1.1 Applications of mass spectrometry and relative mass uncertainty that is needed. 5

4.1 Typical values and maximum-/minimum-ratings. . . . . . . . . . . . . . . . . 22

3

1 IntroductionMany experiments in modern physics focus on measuring fundamental properties of elementaryparticles (electrons) and compound particles (nucleons, atoms, ions, molecules). Examplesfor fundamental properties of interest are charge, mass, and the magnetic moment.The mass m is closely related to mass-energy E via the famous relationship

E = mc2 .

This relationship can be used to convert mass measurements to energy measurements.For example, the mass difference between the mother nucleus and daughter nucleus in aradioactive decay is equivalent to the decay energy. Measuring such energies is important inmany areas of physics:Astrophysics: One of the “11 Science Questions for the New Century” that were formulatedby the NRC in 2002 was: “How were the elements from iron to uranium made?” Massspectrometry provides key parameters (e.g. neutron separation energies) for building modelsthat explain how nuclei are formed in supernovas and stars.Nuclear physics: The measurement of nuclear masses helps to improve mass models. Themasses of radioactive nuclei are of particular interest. One goal is to find the “island ofstability”, which is predicted by many different models of nuclear masses.Tests of QED: Measuring the binding energy of the 1s-electron in hydrogen-like uranium(U91+ allows to test QED in strong fields1. The ionization energy of U91+ from the groundstate is approximately 200 keV and far out of the reach of laser spectroscopy. High precisionmass measurements allow to measure this energy.Neutrino physics: The determination of the neutrino mass is an active research fieldin particle physics and astrophysics. A precise measurement would yield an importantparameter in theories beyond the Standard Model of physics. The neutrino mass is alsoneeded for estimating the energy that is required for neutrino production during the big bang.Penning traps can help to measure this value by determining β-decay energies. The differencebetween the total energy of a β-decay and the maximum energy that the β-electrons canhave (provided by a separate measurement) is equal to the mass energy of the νe-neutrino.

Penning-traps are useful tools in mass spectrometry, but they can also be used to measureother properties of charged particles, such as magnetic moments or electric dipole moments.

1The electric field near the uranium nucleus is one of the strongest available electrostatic fields (cf. Schwingerlimit).

Table 1.1: Applications of mass spectrometry and relative mass uncertainty that is needed.

Field Rel. mass uncert.Chemistry: identification of molecules 10−5 − 10−6

Nuclear physics: shells, sub-shells, pairing 10−6

Nuclear fine structure: deformation, halos 10−7 − 10−8

Astrophysics: r-process, rp-process, waiting points 10−7

Nuclear mass models and formulas: IMME 10−7 − 10−8

Weak Interaction studies: CVC hypothesis, CKM unitarity 10−8

Atomic physics: binding energies, QED 10−9 − 10−11

Metrology: fundamental constants, CPT < 10−10

5

This FP-experiment is an introduction to Penning-trap mass spectrometry. The theorychapter, Chapter 1, describes how charged particles can be stored in a Penning trap, and howtheir trajectories can be decomposed into independent periodic modes. Chapter 2 focuses onhow to detect and manipulate the charged particles in a Penning trap. Chapter 3 introducesthe experimental setup, the measurement technique, and the experiments that you canperform.

6

2 Theory

2.1 Cyclotron motionWhen a particle with mass m and charge q moves through a magnetic field ~B, it experiencesthe Lorentzian force

~FL = q ~v × ~B .

In the special case ~v ⊥ ~B, the particle’s trajectory will be a circle (Fig. 2.1, left). In the moregeneral case, ~v has a component parallel to the magnetic field, and the trajectory is a spiral.By setting ~FL equal to the centripetal force, you can quickly show that the frequency of

the circular motion is independent of the velocity:

ωc = q

mB . (2.1)

This frequency is called the cyclotron frequency. The basis of Penning-trap mass spec-trometry is to determine the cyclotron frequencies of two different particles in the samemagnetic field ~B. When the ratio of these frequencies is calculated, the magnetic field cancelsout and the charge ratio reduces to an integer fraction. The frequency measurement cantherefore be used to calculate a mass ratio:

ωc1

ωc2

= q1q2

m2m1

. (2.2)

By using one particle with a well-known or defined mass, such as a 12C+ ion, the mass ofthe other particle can be determined.

State-of-the-art superconducting magnets can be carefully tuned to have a magnetic fieldthat changes only 1 part in 108 over a 1 cm3 region in the center. When a charged particleis inserted into such a field, it orbits around the field lines (in other words, it is trappedradially), but it is free to drift axially, along the field lines, out of the homogeneous part ofthe magnetic field. This considerably reduces the useful time in which the particle can bestudied or in which its cyclotron frequency can be determined.

The idea of a Penning trap is to have a weak, electric field that axially pushes the particlestowards a defined point. But before describing this field, let us consider the motion ofparticles in strong ~B-fields with simultaneous, weak ~E-fields.

2.2 Guiding center motionIf in addition to the magnetic field ~B there is a weak1 electric field ~E, the trajectory ofcharged particles can be separated into a fast cyclotron motion, caused by ~B, and a slowdrift of the center of the cyclotron motion, caused by ~E. The center of the cyclotron motionis also called the guiding center.For example: If ~E ‖ ~B, the radial motion is undisturbed, and the particle is accelerated

axially, along the field lines. The trajectory is a spiral that has more and more space betweenits loops (Fig. 2.1, center).When ~E ⊥ ~B, the particle accelerates and decelerates as it moves on its cyclotron orbit.

The radius of the orbit grows as the particle moves in the direction of ~E and shrinks again1Weak meaning that the electric forces are much smaller than the magnetic forces.

7

B B E B E

Figure 2.1: Motion of a charged particle in a magnetic field (left), in magnetic field parallelto an electric field (center), and a magnetic field orthogonal to an electric field(right).

during the other half of the orbit. This causes the center of the cyclotron motion to driftsideways, orthogonal to both the electric and the magnetic field (Fig. 2.1, right). This driftis called the E-cross-B drift.The general case can be described by a superposition of ~E‖ and ~E⊥.

We now have the tools to describe the motion of charged particles in a Penning trap.

2.3 Penning trap motionThis section gives a qualitative description of the motion. For a more rigorous quantitativetreatment, see Sec. 2.5.By convention, the magnetic field points towards the z-direction, so that ~B = B~ez. The

electric field that leads to something called Penning trap can then be written as

~E = c

x/2y/2−z

, (2.3)

with some arbitrary constant c (Fig. 2.2). The important part is happening in the Ez-component: Assuming the signs of c and charge q are the same, the electric field pushesthe particles towards z = 0. Since this force is proportional to the z-displacement, like in aspring, the z-motion is a harmonic oscillation (Fig. 2.3).The field in the xy-plane (the “radial” plane) is a necessary trade-off, as dictated by

Gauss’s law: The field lines that are coming in towards the trap center in the z-direction,

B

z

x

Figure 2.2: Cut-view of the electric field inside a Penning trap.

8

need to come out somehow. In an ideal Penning trap, the radial components of the ~E pointevenly away from the trap center.

The radial components of ~E pull the particles away from the trap center. As they accelerateoutwards, their cyclotron orbit increases. When the cyclotron motion takes the particles backtowards the trap center, their cyclotron orbit decreases. In total the center of the cyclotronmotion (the guiding center) performs a slow E-cross-B-drift around the center of the trap(Fig. 2.3). In Penning trap terms, this slow drift is called the magnetron motion.

The drift of the guiding center slightly reduces the frequency of the fast cyclotron motion,see Sec. 2.5.

It is important to point out that the magnetron motion is unstable2! The particles sit ona potential hill in the radial plane. If there are any damping processes, such as collisionswith background gas, the particles roll down the potential hill and are lost. In cryogenictraps that are cooled to 4K, particles can easily be stored for years and longer, but inthe room-temperature Penning-trap of this experiment, the storage times are on the orderof 100ms.

2.4 Penning trap electrodesThe electric field ~E of a Penning trap corresponds to the potential

Φ = c

2

(−ρ2

2 + z2)

, (2.4)

where ρ2 = x2 + y2. To create this potential inside the Penning trap, there are threecylindrically symmetric electrodes that follow the equipotential lines of Φ: Two electrodesfollow the contours

z(ρ) = ±√z2

0 + ρ2

2 , (2.5)

with z0 as the closest point to the trap center. These electrodes are called the endcaps(Fig. 2.4). Both endcaps are held at the same potential. The third electrode, called the ring

2It can be argued that the term Penning “trap” is a misnomer, but “Penning arbitrarily long storage device”does not have the same ring to it.

z

x

y ω−

ω+

ωz

B

Figure 2.3: Motion of a positively charged particle in a Penning trap. The motion consists ofthe slow magnetron drift around the trap center (blue), the axial motion (red),and the cyclotron motion (yellow). The superposition of all three motions isshown in purple.

9

z

x

B

z0

ρ0

V0

Upper endcap

Ring electrode

Lower endcap

Figure 2.4: Penning trap electrodes.

electrode, follows the contour:

ρ(z) = ±√ρ2

0 + 2z2 , (2.6)

with ρ0 being the distance between trap center and ring electrode. In the trap of theFP-experiment, ρ0 =

√2z0.

Let V0 be the voltage between the ring electrode and the endcaps. Then the potential inthe trap can be written as:

Φ(ρ, z) = V0

z20 + 1

2ρ20

(−1

2ρ2 + z2

)this trap= V0

2z20

(−1

2ρ2 + z2

)(2.7)

2.5 Penning trap frequenciesTo quantify our qualitative understanding of the Penning trap trajectories, we have to solvethe equations of motion

~r = q

m

(~r × ~B + ~E

). (2.8)

With ~B = B~ez and Equation 2.7 for the potential Φ(ρ, z) that defines the electric field~E = −~∇Φ, we can rewrite this asxy

z

= q

mB

y−x

0

+ qV02mz2

0

xy−2z

(2.9)

The z-component of this differential equation is independent of the radial components, and ithas the same structure as the equation of an undamped, harmonic oscillator. It is solved by

z(t) = z0eiωzt with ωz =

√qV0mz2

0. (2.10)

The frequency ωz is called the axial frequency. Using the above definition for ωz andEquation 2.1 for ωc, the radial equations of motion can be written as(

xy

)= ωc

(y−x

)+ 1

2ω2z

(xy

). (2.11)

10

These equations can be solved by introducing the function u(t) = x(t) + iy(t). With the helpof this function, the radial equations of motion can be reformulated as

u = −iωcu+ 12ω

2zu . (2.12)

Guessing a solution of the form u(t) = u0e−iωt leads to two independent solutions with the

frequencies

ω± = 12

(ωc ±

√ω2c − 2ω2

z

). (2.13)

The frequency ω+ is the faster frequency. This frequency is often called the reducedcyclotron frequency, because for typical trap parameters, it is only slightly smaller thanthe free space cyclotron frequency ωc. The other frequency, ω−, is called the magnetronfrequency.In order for the trajectory to be stable, the frequencies must be real-valued. This is only

the case if ω2c > 2ω2

z . This requirement is equivalent to

B >

√2mq

V0z2

0. (2.14)

In other words, if the voltage between the trap electrodes is too big, then the electric forcesbecome too large and the ions are no longer contained radially by the magnetic field. This iscalled the stability limit.

In typical Penning traps, ω− ωz ω+ < ωc. The magnetron frequency ω− may becomelarger than ωz, but this is only true if the trap is operated dangerously close to the stabilitylimit.The fact that the motion of particles in a Penning trap can be broken down into three

independent, harmonic modes is very important from a practical standpoint, because itensures that the frequencies can be measured separately, and that the measurements do notdepend on difficult-to-control initial conditions.

Also, the free space cyclotron frequency ωc can easily be calculated from the trap eigenfre-quencies ω+, ω−, and ωz. There are two ways to do it:

ωc = ω+ + ω− (2.15)

ω2c = ω2

+ + ω2− + ω2

z (2.16)

These relationships can easily be verified using the results of Equation 2.13 and Equa-tion 2.10. In ideal Penning traps, they give identical results.

One further relationship is quite useful when calculating one frequency from two measuredfrequencies:

ω+ω− = 12ω

2z . (2.17)

It follows directly from Equation 2.13.

2.6 Real Penning trapsIn a real Penning trap, the convenient separation of the particle motion into three independent,harmonic modes is not strictly true. Imperfections lead to frequency offsets, to energydependent frequency shifts (each frequency depends on the energies of all modes) and tocoupling (energy exchange) between the modes. Relevant imperfections are:

11

• Particle interactions. If there is more than one charged particle in the Penning trap,particle interactions lead to chaotic frequency shifts and couplings. High precisionPenning traps are designed to work with single ions, or with at most two ions in thetrap. The interactions in the electron cloud of the FP-experiment are the biggestlimitation of this device.

• Misalignment between ~B and ~E. This can happen when the trap electrodes are tiltedwith respect to the magnet. In Penning traps used for mass spectrometry, typicalfrequency shifts are on a 10−7-level, but it can be shown that this shift cancels outwhen using Equation 2.16 to calculate ωc.

• Ellipticity of the ~E field, where the radial components of ~E are stronger in one directionthan the other. This affects measurements on a 10−10-level.

• Cylindrically symmetric ~E-field imperfections. When, for example, Ez is not perfectlyproportional to z but additionally also depends on z3, then the trapped particles’z-oscillation has the characteristics of a “stiffening” (or “weakening”, depending on thesign) spring, and the axial frequency changes as a function of amplitude. Relevant ona 10−8-level

• Cylindrically symmetric ~B-field imperfections. They lead to similar effects as the~E-field imperfections, but are typically relevant on a 10−10-level.

• Relativistic shifts. These are especially important for light particles. For light ions, thereduced cyclotron frequencies are typically shifted on a 10−10-level

• Image charges in the trap electrodes. Image charges always pull the charged particlesto the closest electrode, away from the trap center. This modifies the equations ofmotions and is relevant on a 10−10-level.

Most of these frequency shifts are energy dependent. Current research in Penning trapmass spectrometry pushes for lower motional amplitudes (lower ion energy), smaller trapimperfections, and better characterization of the remaining shifts.

12

3 Detection and Manipulation

3.1 Detecting electronsWhen a charged particle is placed near a conducting surface, an image charge is induced inthe surface. If the charged particle moves, the image charge moves as well. This is called animage current.

The radial motion of particles in a Penning trap causes an image current that mostly flowsin the ring electrode. Since the ring electrode is rotationally symmetric, the image currentcan flow freely, and it does not lead to a measurable voltage differences across the ring.The image current of the axial motion is mainly induced in the endcaps. It flows from

one endcap to the other. Depending on how the endcaps are connected to each other, thiscurrent can be measured.One could, for example, place a large, 1 MΩ resistor between the endcaps. The image

current would cause a voltage drop across this resistor, which could be amplified and measured.But unfortunately, the traps have a self-capacitance Ceff of several pF, which is electricallyparallel to the resistor. At typical axial frequencies of a few MHz, the overall impedance1 isdominated by Ceff, and only a few kΩ.

Instead, in the FP Penning trap, the endcaps are connected to each other via an inductor L(Fig. 4.4). The inductor impedes the flow of the oscillating image current, and gives rise toa voltage that can be measured. Together with the self-capacitance of the trap, Ceff, theinductor forms and LCR-circuit. (R is given by the losses inside the coil.) At the resonancefrequency of this LCR-circuit, ωLCR = 1/

√LC, the impedance is maximal and real-valued.

The z-oscillation of the trapped particles acts as a current source parallel to the trapcapacitance Ceff. If the particles oscillate with ωz near ωLCR, their image current causes ameasurable voltage drop across the coil. As a side effect, the voltage drop also back-acts onthe moving charges and damps their motion, until they are in thermal equilibrium with theRLC-circuit. This is how “hot” electrons can be detected: Assuming that the ring voltage isadjusted correctly, they appear as a peak on top of the frequency spectrum of the detectioncircuit.

1The impedance Z is the resistance to alternating currents, U = ZI. It can be real-valued (resistors),imaginary (inductors, capacitors), or complex.

z

x

B Endcap

Ring

Endcap

exte

rnal

coi

l

CeffLcoil

Rcoil

Figure 3.1: Electrical model of the endcap connection.

13

When the electrons are loaded in the FP-experiment, they start out hot, but they thermalizewithin a few milliseconds, before the peak can be measured. Therefore, we use a slightlymodified detection scheme:

z

x

B

fdet

Ctiny

47 48 49 50 510

0.2

0.4

0.6

0.8

1.0

Det

ecte

d sig

nal i

n a.

u.

fdet in MHz

Amp

Figure 3.2: Detection scheme. The coil is excited through a small capacitance Ctiny. Whenthe ring voltage is carefully adjusted so that fz = fLCR, then the electrons shortout the excitation signal.

The detection circuit is excited with a frequency ωdet, which is chosen to be equal to (or atleast close to) the resonance frequency of the detection circuit, ωLCR. This causes a currentto flow through the coil and the equivalent trap capacitance Ceff. If the ring voltage happensto be at a value where ωz = ωdet, then the electrons are excited and an image current flowsacross the endcaps. This image current is an additional path for the excitation signal to flow,and it typically has a much lower impedance than the LCR-circuit. In other words, whenresonant, the electron cloud shorts out the LCR-circuit.We apply the excitation signal through a tiny capacitor (1 pF), and use an amplifier

followed by a spectrum analyzer to observe the power of the signal, measured across thecoil. Since the signal is applied through such a high impedance, the image current from theelectrons can short it out very effectively. When there are electrons in the trap, and whenthey are resonant, the detected signal power drops to almost zero. The amount by how muchthe detected signal power drops is a rough measure of the number of electrons. A largersignal drop means more electrons.

3.2 Coupling and driving the modes / Measuring thefrequencies

We can measure the frequencies (cyclotron, reduced cyclotron, axial and magnetron) bycoupling the modes and then driving them.We will next explain what coupling and driving mean and how this is done.A short reminder on vector fields:An electric field can be described by:

~E = E0x

100

+ E0y

010

+ E0z

001

+ E1x

x00

+ E1y

0y0

+ E1z

00z

+Eyx

yx0

+ Ezx

z0x

+ Ezy

0zy

14

+ many non linear terms, such as:

E2x

x2

00

, Ezx2 =

z0x2

Where Ewhatever are constants determining how strongly each element contributes.Another way of writing this:

~E =

non coupling terms︷ ︸︸ ︷E0x + E1xx+ E2xx2 + ...

E0y + E1yy + E2yy2 + ...

E0z + E1zz + E2zz2 + ...

+

linear coupling terms︷ ︸︸ ︷Eyxy + EyzzEyxx+ EyzzEyxx+ Eyzy

+

non-linear coupling terms and other scary terms︷ ︸︸ ︷Ey2xy2 + ...+ Ez2xz

2 + ...Eyx2x2 + ...+ Eyz2z2 + ...Eyx2x2 + ...+ Ey2zy

2 + ...

(3.1)

Actually any vector field can be described using an infinite sum of such terms, not justelectric fields.

Homogenous electric fields have only the constant terms E0x,0y,0z. The more inhomogenousan electric field is, the larger the Ewhatever coefficients will be.Driving the modes:The three independent modes of the Penning trap motion can be individually driven (the

energy increased or decreased) with dipolar drives. For instance, the z-motion can be drivenwith a field of the form

~Edrive,z =

001

ERF cos(ωRFt+ φ) . (3.2)

Such a field can be created with a voltage Vdrive(t) across the endcaps2.When ωRF = ωz, then the trapped particles are driven on resonance, and the amplitude

starts to grow linearly3. If there is no damping (by the detection circuit or by backgroundgas), then the particles would be lost quite quickly, would it not be for anharmonicity: Asthe amplitude grows, the unavoidable anharmonic imperfections become more and morerelevant and the particles’ frequencies change, until they are no longer resonant with thedrive. Only when the drive is very strong it can drive the trapped particles effectively enoughto drive them out of the trap.The driving field for the radial modes is similar, only that the field must point into the

radial direction. Such a field can be created by splitting the ring electrode in two halves andapplying the drive between them.

3.3 Driving the coupling modesOther important resonances are coupling resonances. For example, a driving field of the form

~Ecoupl =

z0x

ERFz0

cos(ωRFt+ φ) (3.3)

can be used to couple the axial to the radial modes. (The constant z0 is in the formulafor dimensional purposes.) Imagine that a trapped particle only has energy in the reduced

2A voltage across the endcaps leads to a field that, to first order, can be approximated by the field of acapacitor, where the two capacitor plates have a distance of 2z0.

3It’s better to think of the amplitude as a vector. Depending on the phase between drive and the particlemotion, the magnitude of the amplitude might first decrease.

15

cyclotron mode, so that its x-motion can be described with x(t) = r+ cos(ω+t). Furthersuppose that the coupling field is static (ωRF = 0).The x-motion, together with the static coupling field, causes a periodic force to “wiggle”

the particle in z-direction. This wiggling happens with the frequency ω+ of the reducedcyclotron motion. It is not resonant with the z-motion, and no energy is transferred into thez-motion.

However, if the coupling field is not static, but instead an oscillating field with the frequencyωRF, then the force in the z-direction will be (ignoring phases)

Fz ∝ Ez(t) ∝ r+ cos(ω+t) cos(ωRFt)

= r+2 cos(ω+t+ ωRFt) + r+

2 cos(ω+t− ωRFt) . (3.4)

The last step uses the trigonometric identity cos(a) cos(b) = 12 cos(a+ b) + 1

2 cos(a− b). IfωRF is chosen such that ωRF = ω+ ± ωz, then one of the two terms in the sum is resonantwith ωz, and the z-amplitude will start to grow.

But as the z-amplitude grows, there will now be a “wiggling” in the x-direction. It canbe shown that this force will be either in-phase or out-of-phase with the original motion,depending on the choice of ωRF. For ωRF = ω+ − ωz, the original x-amplitude will bedecreased as the z-amplitude is increased, and vice versa. The amplitudes oscillate (Rabioscillations).The other sideband, at ωRF = ω+ + ωz, causes the x-amplitude to increase as the z-

amplitude increases, and both amplitudes grow exponentially. The trapped particle is quicklylost from the trap.There are many such coupling fields that can couple modes together in many different

ways. What is characteristic about all of them, is that the frequency needed to drive thecoupling is at a linear combination between the frequency of the coupled modes.In high-precision Penning traps, there are different electrodes for creating the coupling

and the driving fields effectively. In the FP-experiment, there is only one antenna. The fieldof this antenna has a very inhomogeneous field with many dipole, quadrupole, and higherorder components. This antenna can be used to drive all of the eigenmodes and most of thecoupling resonances.Alternatively, in the FP-experiment we can apply a drive signal across the endcaps to

drive only the z-mode. This is used for electron detection.

16

4 ExperimentThis chapter describes the experimental setup, the measurement control and the dataanalysis. You will learn how to use professional measurement equipment, how to controlexperimental routines by software and about experimental techniques used in high-precisionmass spectrometry.

4.1 Overview of the experimental setupThe Penning trap electrodes are mounted inside a vacuum tube. A cylindrical coil wrappedaround the outside of the tube generates the magnetic field. Electrical feedthroughs connectthe electrodes inside the vacuum to the measurement electronics on the outside. The timingsequences are controlled by an Arduino single-board microcontroller, and data is read out bya PC.

Turbopump

Vie

wpo

rt

B-field coilTo vacuum gauge

Feedthroughs(encased)

Am

plifi

eran

d co

il bo

x

BNC conn.

Filament Trap

Figure 4.1: Overview of the experimental setup.

4.2 Mechanical setupTo reach a sufficiently low vacuum pressure (10−8 mbar or better), the vacuum chamber ispumped by a turbopump backed by a scroll pump. The turbopump is connected to a CF63cross. Connected to the other three ports of the cross are a viewport, a pressure gauge, andthe vacuum tube that houses the Penning trap. Through the viewport, you can see the trap,the electron gun and the trap electrodes. The cylindrical coil for generating the magneticfield is mounted on the outside of the vacuum tube. The other end of the vacuum tube isclosed off by a flange that has feedthroughs built into it for the electrical connections withthe Penning trap.The Penning trap consists of three hyperbolically shaped electrodes that are insulated

from each other. The central electrode, called ring electrode, is made of copper, and thetwo endcaps are made of wire mesh. Next to the lower endcap1 is the electron gun. The

1The trap is mounted horizontally, but by convention, the endcap closer to the viewport is called “bottomendcap”.

17

electrons are produced by thermal emission: A tungsten filament is heated by an electricalcurrent running through it. When this filament is set to a negative voltage, electrons arerepulsed from the filament through a grounded aperture plate (a plate with a hole in themiddle) towards the trap.

The electrons can enter the trap region through the wire mesh of the lower endcap. Theseprimary electrons cannot be trapped, since their energy is too high. Instead, secondaryelectrons, which are produced by collisions of the primary electrons with the residual gas,are trapped. During a measurement sequence with electrons in the trap, loading of furtherelectrons is suppressed by switching the filament to ground potential.A small wire between the top endcap and the ring is used as an antenna. When AC

signals are sent through the antenna, an electromagnetic field will be produced inside thetrap. This field is highly inhomogeneous and so includes components of many orders – dipole,quadrupole and higher. This creates coupling (practically, energy transfer) between themodes of motion. The effectiveness of the coupling is determined by the frequency of thesignal we introduce to the antenna. Certain frequencies will create efficient energy transferbetween the modes. Assuming that we continuously excite (give energy to) the axial mode,what happens when we efficiently couple it to another mode? What will happen to thetrapped electron? Which frequencies do we expect to create efficient coupling? How can weuse the antenna to measure these frequencies?

4.2.1 Measurement electronicsFig. 4.2 provides an overview over the Penning trap and the measurement electronics.

The magnetic field is generated by running a current of several 100mA through a cylindricalcoil. The current is controlled by a programmable power supply that can apply currents ofup to ≈ 1.3A through the coil’s resistance of ≈ 50 Ω.The electrostatic trapping field is generated by applying a positive voltage to the ring

electrode while the two endcaps are grounded DC-wise (We still send AC signals throughthem to excite the axial motion of the electrons, but the average voltage is 0). The ringelectrode voltage is generated by a function generator, and then amplified with a constant(but adjustable) gain by the HV200 amplifier. (You’ll need to know the gain for yourmeasurements. How can you measure it?) The function generator supplies the ring voltagewith a signal that decreases linearly from a certain (programmable) starting point to 0, in aperiodic fashion. In this way we scan (ramp) the ring voltage. We do this in order to detectthe electrons (Sec. 3.1). (How does scanning the ring voltage help us to detect the electrons?Hint: Which parameter depends on the ring voltage and what happens when it reaches acertain special value?).

The radio-frequency excitation signal for the fant is generated by a HP8657 signal generator.In order to apply the excitation during a well-defined period, the generator is not directlyconnected to the antenna, but through a radio-frequency switch instead. The switch is closedfor 0 and open for 5V, as in, if the voltage at the control port of the switch is 0V, the RFgenerator signal goes into a 50Ω resistor and nothing special happens, and if its 5V, the RFgenerator signal goes to the antenna. Thus, the excitation time and period can be defined bythe length of the trigger pulse applied to the control port of the radio-frequency switch.

The current used to heat the filament is provided by one of the 4 channels of the 4-channelHameg power supply. This channel “floats”, meaning that neither the positive nor negativeside of the channel are connected directly to ground. This allows the channel to be floatedupwards or downwards (it can be “biased”) by another, non-floating channel of the powersupply. This filament bias voltage is applied through a fast HV-switch, which switchesbetween the negative bias voltage and ground.The coil and the low-noise amplifier of the axial-frequency detection system are enclosed

in a metal box that is mounted on the outside of the vacuum system, onto the feedthroughflange. The supply voltages for the amplifier are provided by a 3-channel Hameg powersupply.

18

HPh8657ATriggerhGenerator

HV200Amplifier

VoutVin

Oscilloscope

Antenna

HM70444-channel

powerhsupply

+-

Filament

B-f

ield

coi

l

Pre. Amp.

RF in

RF-switch

RF outtrigger in

DSA815

RF in

RF out (f )

trigger in

Det

Computer

trigger in

FilamenthBiasHVhSwitch

VoutVin

VV

Agilenth33500BFunctionhGenerator

Ramp Out trigger in

RF-signalhgeneratorfRF

+ -+ -

+-V

Figure 4.2: Overview of the measurement electronics.

The detection system, trap and cables form an RLC circuit. It is connected to the DSA815 network analyzer (spectrum analyzer) which excites it (and so the axial frequency of theelectrons) when in Tracking Generator (TG) mode. The resulting voltage signal is amplifiedby a low-noise amplifier and given back to the DSA815 as input. The trigger signals forthe measurement sequence (loading the trap with ions, exciting them with the antenna,and sweeping the ring voltage to detect them) are generated by an Arduino-based triggergenerator. An oscilloscope displays the voltages of the measurement sequence.

4.2.2 Measurement controlThe experimental setup is controlled by a computer. You will receive a folder containingready-made functions for controlling the setup, as well as an example file for setting up abasic measurement, which you will need to modify and expand. The scripts will save datainto text files which can be viewed using Scilab.The trapped electrons only remain trapped for a few 100ms, and are lost after they’re

measured (some are lost by the measurement proccess, which involves exciting them them tothe point where they are lost from the trap, and the rest are then lost due to the scanning ofthe ring voltage. When its negative all electrons are expelled from the trap). Therefore, inorder to perform multiple measurements we need to continuously perform a cycle of loadingnew electrons, exciting them, detecting them and expelling them from the trap. Each cyclelasts 250ms and is composed of the following steps:The measurement scheme thus consists out of following steps (see Fig. 4.3):

1. Loading the trap (tload, typ. 25ms)

2. Waiting for voltages to stabilize (twait1, typ. 25ms)

3. Exciting the electrons with fant (texc, typ. 50ms)

19

0 50 100 150 200 250

-10

-8

-6

-4

-2

0

2

0 50 100 150 200 250

-1.0

-0.5

0.0

0.5

1.0

0 50 100 150 200 250

0

20

40

0 50 100 150 200 2500.0

0.5

1.0

1.5

Time in ms

Ant

enna

RF

inV

Rin

gVo

ltage

inV

Sign

alPo

wer

ina.

u.Fi

lam

ent

bias

inV

tloadtwait1

texc

twait2tdet

tramp

tSWT

Figure 4.3: Timings of the measurement sequence. All times are controlled by the Arduinotrigger generator, except tramp and tSWT. tramp is given by the frequency of thefunction generator, and tSWT is a setting on the spectrum analyzer.

20

4. Additional waiting time (twait2, typ. 25ms)

5. “Counting”/Detection of the electrons (tdet, typ. 100ms)

6. Expelling all electrons. This happens at the end of tdet.

Loading the trap: During the loading time the filament is switched to a negative biasvoltage, as described in Sec. 4.2. The HV-switch used for switching the bias voltage can becontrolled by a TTL (5V) signal, which is generated by the trigger generator.

Excitation of the Electrons: As described in Sec. 4.2.1, fant is not directly applied to theantenna, but is guided through an RF switch. The switch can be controlled with a TTLsignal, which is generated by the trigger generator.

Counting/Detection of the Electrons: As explained in Sec. 3.1, the electrons can bedetected by scanning over different values of the ring voltage. The electrons’ axial frequencydepends on the ring voltage, so by scanning over the ring voltage we scan over the electrons’axial frequency. When the electrons’ axial frequency equals the RLC circuit’s resonancefrequency, the electrons short out the excitation signal, reducing its amplitude and signalingthat we have electrons in the trap. The lower the amplitude, the more electrons we have inthe trap. (We will learn how to do this in the next section) The ring voltage is ramped to anegative value, so that all electrons are expelled from the trap, and is then set back to apositive value, to prepare for the next measurement cycle.

4.3 MeasurementWe will measure the different frequencies (cyclotron, reduced cyclotron, axial and magnetron)as a function of the coil current and the ring voltage and use the resulting graphs to measuredifferent system parameters (go over the equations in the theory section and try to thinkwhich parameters can be measured from such graphs). We will next detail the steps for suchmeasurements. Keep in mind that original ideas are not only permitted, but encouraged.

4.3.1 Identifying the ComponentsBefore we start the measurements, get more familiar with the experimental setup by iden-tifying the different devices shown in Fig. 4.2. Follow the cables and see what connectsto what and try to understand what each device does. Please don’t plug/unplug anycables or change the values of the power supplies manually. This can mess upthe experiment and even destroy certain components. (Its okay to change the ringvoltage and the coil current remotely using the computer) The devices you can manuallyoperate are the Network Analyzer, the oscilloscope and the HP8657A antenna power source.Feel free to play with them and fully explore their functionalities. Push lots of buttons, itsfun.Make yourself familiar with the minimum/maximum ratings given in Tab. 4.1

4.3.2 Finding the Resonance FrequencyWe need to find the resonance frequency of the RLC circuit. If we knew the capacitance andthe inductance we could have calculated it outright (what is the formula for the resonancefrequency of an RLC circuit?), however; we don’t. We will use a device called a (scalar)Network Analyzer to determine the frequency – the DSA815. The DSA815 has both inputand output connections connected to the RLC circuit. To use it, first preset it to its factorysettings. In its default mode of operation, the DSA815 shows a Fourier transform of thereceived signal. (The horizontal axis is the frequency and the vertical axis is the amplitude

21

Table 4.1: Typical values and maximum-/minimum-ratings.

VALUE DEVICE TYP MIN MAX UNIT

Pressure MaxiGauge 1.2 · 10−8 1 · 10−7 mbar

Coil Current 3CH Hameg 1 −1.3 1.3 A

Filament Current 4CH Hameg 0.42 0.45 A

Filament Bias 4CH Hameg −10 −30 0 V

fant power HP8657A 5 15 dBm

fdet power DSA815 −30 −10 dBm

Ring Voltage Agilent 33500B 1 0 3 V

RF Switch DC+ 4CH Hameg 5 5 5 V

RF Switch DC− 4CH Hameg −5 −5 −5 V

Amplifier Supply 3CH Hameg 3.4 3.4 3.4 V

Amplifier Gate 1 3CH Hameg −0.84 −1.5 0 V

Amplifier Gate 2 3CH Hameg −0.34 −1.5 0 V

of that frequency). What graph did you expect to get and which graph did you really get?Why are the two not identical? Try to play with the settings of the device (frequency range,scale...) to find the resonance. It is expected to be somewhere in the [10;150]MHz range.You will notice that the graph is noisy. To fix this we’ll turn on the Tracking Generator(TG). In this mode, the DSA815 plots the same graph as before, but instead of relying on theinput pulse alone in order to measure it, it sends pulses of its own with different frequencies(to the RLC circuit) and measures the amplitudes of the returned pulses. (Note the 30 dBattenuator connected to the output of the DSA815) Turning the TG mode on might notseem to do anything. Try to find a parameter in the TG menu which you can modify to fixthis. Write down the value you found for the resonance frequency of the RLC circuit. (It isdependent on the values of the ring voltage and the coil current, so make sure to note thoseas well)

4.3.3 Preparing for an Antenna Frequency ScanNext we will use another mode of operation for the network analyzer called zero span modein order to detect the electrons that are in the trap. Unlike in the previous modes, where thenetwork analyzer displays amplitude vs frequency, in zero span mode the network analyzerdisplays amplitude vs time of a specific frequency (actually a narrow band of frequenciesaround a specific frequency). It sends pulses to the RLC circuit, as before, only that in zerospan mode they all have the same frequency, and the returned amplitude (scaled by someirrelevant proportionality constant) is shown as a function of time. To use it, set the centerfrequency to the one you found in the previous part and set the DSA815 to zero span mode.(Don’t turn TG mode off)

Next we will synchronize the scan of the DSA815 with the scanning of the ring voltage inthe trapping cycles (see Fig. 4.3). This effectively turns the horizontal axis from time to ringvoltage. (Why?)

To do that, set trigger mode to external and set the Arduino trigger generator to continuousmode by using the following python script:import devices

22

devices.set_trigger_num(0)devices.send_trigger()Now we see the amplitude of the RLC circuit’s resonance frequency as a function of time

(or, as a function of the ring voltage) You will initially not see anything and will need toovercome several problems:1. Modify some of the settings of the DSA815.Hint: What are the units of each axis? What are the ranges of values of each axis? Which

ranges would we like them to have?2. Verify that the trap settings are tuned for electron capture.Hint: Which parameters related to the trap can you control using the python scripts?

What are their current values? What are their typical values?Once you see something, change the center frequency of the zero span mode (by no more

than 0.5MHz in either direction) while observing the graph and find the optimal frequencyfor which you get the graph you would expect to get. You can use the wheel to perform thissmoothly.

Why do we need to change the frequency? Why is using the circuit’s resonance frequencynot good enough? Why do we get this graph? Does it have a dip or a peak? Why the oneand not the other? Why does the dip/peak appear where it does and not elsewhere? (Inother words – what is special about its position / what happens in that position?)How can we change the position of the dip/peak? Try to do so by changing one of the

experimental setup’s parameters by using the python code. Try to find the frequency whichgives the maximal MMD (Maximal Minimal Depth – the vertical difference between thehighest and the lowest points in the graph. This is the height of the peak/dip. In case wherethe peak/dip keeps appearing and disappearing, you’re not using the right frequency. (Evena 0.5MHz deviation can cause this problem)

4.3.4 Manual Antenna Frequency ScanThe HP8657 signal generator is capable of sending an AC signal to the antenna. As mentionedin the theoretical section, this makes the antenna produce an electromagnetic field containinghigher order components (dipole, quadrupole and higher) which couple the three modes ofmotion in the trap (in other words, it creates energy transfer between the modes). We canselect different frequencies for the antenna’s electromagnetic field. Most will not cause anynoticeable effects, because they will not create any noticeable energy transfer, but specificfrequencies will. Which frequencies are these, and what affect do you think they will have onthe measured signal? (Refer to the theoretical section for help) Test your guess by turningthe HP8657 on and manually scanning for different frequencies while observing the networkanalyzer’s signal. Try to find frequencies which strongly affect the network analyzer’s signaland write down a few of those frequencies. Don’t forget to write down the values of otherimportant parameters like ring voltage and coil current.

A common problem is that the signal of the antenna doesn’t seem to affect the system. Ifthat happens double check the settings of the HP8657.

4.3.5 Rough Scripted Antenna Frequency ScanThe HP8657 can be remotely controlled using the computer. Use the example python file toscan over the antenna excitation frequencies in a range of a few dozen MHz which containsome of the special frequencies you found in the previous part, and measure the MMD ofthe detected signal for each of them. Observe the resulting graph and try to understandthe result. Write down the ring voltage and coil current values you used (remember – Thering voltage you give as input in the python script is amplified before it is sent to the trap.What is the real voltage? What is the amplification factor?). Now change the ring voltageor the coil current and repeat the measurement, writing down the new values. What is thedifference between the two graphs?

23

You might want to set the RSA to display the data in linear scale. Some data points havevery low values, close to 0. When plotting in log scale, almost 0 translates to almost -infinity,and so when using log scale you’ll get noisy graphs with lots of false dips.

Note - When the trap is not in use the coil current should be 0.1A (arbitrary low value toprevent it from heating up significantly).

Therefore, before leaving home, if you’re not preforming a measurement, set the coil currentto 0.1A, and if you are, make sure theres a line of code to change the coil current to 0.1A atthe very end of your file, so that the coil can cool down after your script finishes running)

4.3.6 Fine Scripted Antenna Frequency ScanModify the script further to perform two sets of measurements across a larger frequencyrange of [1;350]MHz, one set with constant ring voltage and changing coil current and oneset with changing ring voltage and constant coil current. (Refer to the table for the valuesyou should use, or play with the ring voltage and the coil current [reminder – only by usingthe computer’s scripted commands, not physically]). These measurements will take a fewhours to complete and can be left to run on their own after you leave for the day.Remember to make sure theres a line of code at the very end of the script to change the

coil current to 0.1A.

4.3.7 Data AnalysisAnalyze the data and obtain the different frequencies from it. Plot graphs of the frequenciesas functions of either the ring voltage or the coil current and fit them. What parameters canyou measure using the graphs?

Optional: (Pick one)1. Make a graph of the MMD as a function of the waiting time and use it to determine

the lifetime of the electrons in the trap.2. Find the phase space for trapping (all possible combinations of ring voltage and coil

current that enable trapping)Very Optional:Be original, try to think of new things you can measure or do.Hopefully:Learn a lot and have fun.

24

100 150 200 250 300 3500

2

4

6

8

10

12

14

16

18

0 50fRF in MHz

Am

plitu

de in

a.u

.

Figure 4.4: Typical antenna frequency scan graph. Try to notice and understand the patterns.

25