STUDY OF AN ONLINE ELECTRON ELASTIC SCATTERING SYSTEM FOR
RADIOACTIVE BEAM

By

Ambar C. Rodriguez Alicea

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Physics—Master of Science

2024

ABSTRACT

Information about the absolute charge radius of nuclei is a critical ingredient for theoretical models

of the atomic structure, including the ongoing search for violations to parity conservation and

physics beyond the Standard Model. This observable is best measured experimentally using the

elastic electron scattering technique in which the angular dependence of the nuclear response is

proportional to the charge distribution. Such information is scarce for nuclei beyond bismuth and

almost non-existent for radioactive isotopes. Several worldwide facilities are either in the process

of or have already coupled electron linac systems to ion storage rings or ion traps for the possibility

to extend scattering experiments from stable to exotic nuclei, such as SCRIT - RIKEN in Japan,

and ELISE - FAIR and DERICA - Dubna in Europe. This thesis investigates the development of

an electron gun, acceleration system, and ion trap that could be coupled to a beamline like the ones

at the Facility for Rare Isotope Beams (FRIB) to perform scattering experiments along with laser

spectroscopy to probe rare isotopes. We first benchmark our simulation by modeling the electron

Accelerator Test Facility (ATF) at Brookhaven National Laboratory (BNL) and compare our results

to measurements of the electron transverse profile. We also extracted the parameters of an ion trap

(e.g, trapping potential) that a linac with the desired properties could achieve, and estimated the

achievable luminosity of the system. This preliminary study provides some baseline for a possible

coupling of a new electron acceleration system to FRIB that could enhance the scientific reach of

this facility and expand our knowledge of the nuclear structure.

This thesis is dedicated to my daughter, my mother and my beloved shrimp; thank you for
supporting me with love and always believing in me. Los amo.

iii

TABLE OF CONTENTS

CHAPTER 1

MOTIVATION .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Electron elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Beyond stable isotopes

1
2
5

CHAPTER 2

THEORY .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Electron elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Beam dynamics . .

7
7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

.

.

.

.

CHAPTER 3

. 23
TECHNOLOGIES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 23
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
3.1 Electron Gun .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Acceleration cavities
3.3
. 40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
3.4 Safety considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Ion trap .

.
.
.

.

.

.

.

.

.

.

CHAPTER 4

PRELIMINARY STUDIES . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Electron accelerator measurements and modeling . . . . . . . . . . . . . . . . 47
Ion trap and design strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2

CHAPTER 5

SUMMARY AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . 62

BIBLIOGRAPHY . .

. .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

APPENDIX A: BEAM PROFILE IMAGES . . . . . . . . . . . . . . . . . . . . . . . . . . 71

APPENDIX B: CODES .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

iv

CHAPTER 1

MOTIVATION

The study of nuclear physics sheds light into the fundamental forces and particles that constitute

our universe. The recently built Facility for Rare Isotope Beams (FRIB), located on the campus

of Michigan State University (MSU), started operation on June 10, 2022 [1]. This facility aims

at unravelling the formation and evolution of stars (e.g., how are heavy elements formed from

hydrogen? How many nucleons can we packed into a nucleus? What are the levels occupied by

protons and neutrons inside isotopes? etc.) through the production and study of rare isotopes, their

decay through beta (𝑒±) and gamma ray emissions, and nuclear breakout or pick-up reactions. The

BEam COoler and LAser spectroscopy (BECOLA) [2] facility at FRIB focuses on one of the most

fundamental properties of isotopes: their nuclear charge radii. They obtain the radius by exciting

an electronic transition from two different isotopes with equal atomic number and measuring the

the difference in transition energy between the two. With this information, and sometimes the

help of atomic structure theory, they can infer the nuclear size. This measurement is crucial for

developing accurate theoretical models of the atomic structure, as well as searches for violations

to parity conservation laws and physics beyond the Standard Model of particles [3]. Despite its

importance, data on this observable outside the valley of stability is scarce, with the most significant

contributions dating back to the 1980’s [4, 5]. This gap in knowledge hampers the advancement of

nuclear theory and its applications and yet, it remains wildly unexplored.

In Figure 1.1 we present a compilation of all the existing isotopes, identified by their atomic

number Z and number of neutrons N, and highlight those atoms for which the charge radii has been

measured, either by electron elastic scattering or other methods.

It is important to make the distinction between nuclear charge radii measured using elastic

electron scattering to those using laser spectroscopy. The latter technique, although more precise,

can only measure the difference between two isotopes (sometimes this is all the information the

scientist care about) and thus rely on reference atoms for which the absolute charge radius has been

measured using some other independent method in order to determine the value of the radii for the

1

Figure 1.1 Visual representation of all discovered isotopes (grey) with color
highlight on those for which the charge radius has been measured (green)
and for the isotopes for which the radius was measured using electron elastic
scattering (red) [4, 5].

unstable isotope. One of the few methods to this day that can measure the absolute charge radius

is electron elastic scattering. Notice the significant gaps in data even within the stable isotopes

in Figure 1.1 starting around Z = 35:

this lack of scattering measurements inevitably means

that the absolute charge radius for isotopes in that area, measured using techniques such as laser

spectroscopy, will carry greater uncertainty as they need to rely on theoretical models to extrapolate

for reference atoms. Another way to possibly measure the charge radius of atoms is by means of the

transition X-ray energy of muonic atoms; however, just as electron scattering, these experiments

have been done mainly on stable isotopes [3]. Furthermore, electron (or positron) elastic scattering

allows direct measurement of the charge distribution, while others methods mostly use assumptions

of the finite distribution of the nucleus to extrapolate the radius information.

1.1 Electron elastic scattering

The first scientist to perform electron scattering experiments with the purpose of determining the

nuclear structure was Robert Hofstadter. In his 1953 paper [6], he explained the equipment and

2

experimental procedure employed to try and find evidence for the finite dimension of the atomic

nucleus. Hofstadter and his team deviated the main beam from the Stanford Linear Accelerator

(SLAC), about 1.922 nA average current concentrated in a few millimeter spot size with energies

between 25 MeV and 150 MeV, into a scattering chamber containing thin foils of beryllium, gold,

lead and tantalum. The detection scheme consisted of an analyzing magnet with 40.64 cm curvature

of inhomogeneous field up to 1.3 T, followed by a collimator and a Čerenkov counter attached to a

photomultiplier. With this system, Hofstadter and his team were able to measure the elastic curves

of the materials and founded no clear diffraction minima nor maxima, evidence that the nuclei had

no sharp boundaries. They used the exact calculations for a point charge scattering and multiplied

that curve by the approximate form factor of an exponential charge distribution assumption fitted to

the experimental data to obtain an estimate for the charge radius. Among other elements, the result

of his calculation gave the approximate radius of 𝑅𝐵𝑒 = 2.2 fm and 𝑅𝐴𝑢 = 7.95 fm for beryllium

and gold respectively [6]. Today, the acceptable values for these radii are 𝑅𝐵𝑒 = 2.51 fm and

𝑅𝐴𝑢 = 5.43 fm according to [5]. Based on these preliminary results, he continued to develop the

fundamental experimental and theoretical principles that we use today for nuclear charge radius

measurements.

As indicated previously, the current for the electron beam used in Hofstadter experiment was

considerably small. Yet, even with such a low current, he was able to measure multiple form factor

minima. The low concentration of electrons was compensated by a high number of target particles,

made from solid foils with thicknesses from 0.20 mm and up to 50 mm. When considering the

current in combination with the target densities, these experiments had luminosities on the order

of 9 × 1029 cm−2s−1 to 7 × 1033 cm−2s−1. Figures 1.2 and 1.3 show the reach of these experiments,

allowing to probe up to the third minimum of diffraction:

the q𝑖={1,2,3} in the figures are the

momentum transfers corresponding to the first, second and third minima.

Figures 1.2 and 1.3 assumed a two-parameter Fermi charge density distribution, presented in

(1.1), for which the radius parameter is taken to be 𝑅 = 1.1 (fm) · 𝐴−3 for the atomic mass number

𝐴 of the stable isotopes, and the surface diffuseness parameter is 𝑎 = 0.57 fm for all nuclei. Note

3

Figure 1.2 The required luminosities needed to access the 1st, 2nd and 3rd
minimum of diffraction as a function of the atomic number Z [3].

Figure 1.3 [a] The charged (Coulomb) form factor squared of the two-parameter Fermi distribution
as a function of the momentum transfer q, with the minima positions labeled as q𝑖 for 𝑖 = 1, 2, 3.
[b] The momentum transfer for the first and second minima as a function of the atomic number Z
[3].

that 𝜌0 is just a normalization constant that satisfies (1.2) for 𝑍 number of protons [3].

𝜌(𝑟) =

𝜌0
1 + 𝑒(𝑟−𝑅)/𝑎

∫ ∞

0

𝜌(𝑟)4𝜋𝑟 2𝑑𝑟 = 𝑍

4

(1.1)

(1.2)

For this type of distribution, the square form factor as a function of momentum transfer q would

appear as it is shown in Figure 1.3 [a]. The positions of the three minima shown and the height of

the maxima are directly correlated to the values of the radius and surface diffuseness parameters,

which are then used to determine the gross nuclear charge size and shape. It is then necessary to

cover a momentum transfer range beyond the first maximum, that is, approaching q2, in order to

determine the correct shape of the nuclear charge distribution. It is clear from Figure 1.2 that in

order to achieve this, one needs at least a luminosity of 1027 cm−2s−1 for 𝑍 ≥ 30 nuclei, and few

orders of magnitude higher for lighter nuclei.

In the case of rare isotopes, one cannot create solid targets with particle densities in the order

of 1021 cm−3. For example, FRIB can at most deliver beam bunches with 1013 particles, with

some rare isotopes having as few as 103. Let’s assume that we have 109 ion particles confined in a

cylindrical potential trap of radius 2.5 mm. From (3.4), we can calculate the required electron beam

current for a luminosity around 1027 cm−2s−1 to be on the order of 100 mA. This need for high

current of the electron beam imposes limitations on the technologies that can be used at facilities

like FRIB to conduct elastic electron scattering experiments.

1.2 Beyond stable isotopes

By coupling an electron linac to an ion beamline such as the ones here at FRIB, researchers can

extend electron scattering experiments from stable nuclei to exotic-radioactive ones. Such type of

innovative facilities are planned or already happening in Japan [7], Europe [8] and Russia [9]. This

thesis covers preliminary investigations for the potential of integrating a compact electron linear ac-

celerator with a beamline at FRIB to perform such scattering experiments. The research presented

here focuses on the structure and feasibility of an electron acceleration system that could be coupled

with FRIB’s capabilities. Chapter 2 provides a review of the theory for behind the experiments of

electron scattering, and provides descriptions of the theory of electron beam production, accelera-

tion and steering. Following this, a review of different available technologies that can be used to

produce and accelerate electron beams is presented in Chapter 3. This chapter also gives mention

to some safety concerns that must be considered when building particle accelerators. In Chapter 4

5

we present the experimental measurements taken from Brookhaven National Laboratory (BNL) -

Accelerator Test Facility (ATF) in conjunction with data from a simulation of the same system.

From the measurements of BNL’s system, beam transverse profile images, the radius of the beam

as well as its emittance were calculated and compared to the simulation. Finally, a brief summary

and future steps are given in Chapter 5.

6

CHAPTER 2

THEORY

2.1 Electron elastic scattering

In the intricate world of nuclear and particle physics, the study of electron interactions has remain

a central component for unraveling the mysteries of matter at its most fundamental level. Electron

scattering, a phenomenon where electrons deviate from their path due to the influence of forces from

a target, is particularly crucial for understanding the underlying structures and properties of atoms.

Among various types, electron elastic scattering, where the the total kinetic energy is conserved

post-interaction, provides insights into the atomic and nuclear structure. In this section, we will

explore the mechanisms of electron elastic scattering, specifically its theoretical foundations.

2.1.1 Overview of the scattering processes

In a scattering experiment, the object of study is referred to as the target, and it is probed by an

accelerated beam of particles with well defined properties. In Figure 2.1 we show the different

types of scatterings, where "a" is the incoming probe particle and "b" is the target particle. These

interactions are broadly classified into elastic or inelastic processes, with the primary distinction

between the two lying in the conservation of kinetic energy.

• Elastic scattering is a process wherein the total kinetic energy of the particles involved is

conserved. In this type of scattering, particles collide and deviate from their path, but their

total kinetic energy before and after the collision remains unchanged.

• In contrast, inelastic scattering involves a change in the kinetic energy of the interacting

particles. In these interactions, part of the original kinetic energy is transformed into another

form, such as excited states, heat, mass, vibrations or light, resulting in a loss of kinetic

energy in the system.

• Also depicted in Figure 2.1 are colliding beam reactions. This reaction scheme could be

either elastic or inelastic depending on the conservation of the total kinetic energy of the

7

system, but it involves a special type of configuration where both the target and the probing

particles are accelerated [10].

Figure 2.1 The various type of scattering processes: (a)
elastic scattering; (b) inelastic scattering (production of an
excited state which then decays into two particles); (c) in-
elastic scattering (direct production of new particles); (d)
reaction of colliding beams [10].

2.1.2 Elastic scattering and nuclear charge radii

For the remainder of our discussion, we will focus on the traditional set-up from Figure 2.2, with

a fixed target and an incoming probing electron beam. We want to use electrons because their

DeBroglie wavelength, when accelerated to a momentum of 𝑝 ≥ 100 MeV/c, corresponds to

roughly the size of the nucleus, 𝜆 ≤ 10 fm. Electron elastic scattering studies for the study of

nuclear charge radius typically span the energy window of 100 ≤ 𝑝 ≤ 1000 MeV/c [11].

The target for scattering experiments may be solid, gas or liquid. In any case, we assume that

the molecules or atoms that make up the target have a constant density of 𝑛𝑏, and are distributed in

a volume characterized by a length 𝑙 and an area 𝐴 along and perpendicular to the beam’s motion.

We assume that the area 𝐴 is equal or smaller than the beam size, such that it equals the interaction

area. The incoming beam, with 𝑁𝑎 particles per second at a velocity 𝑣𝑎, interacts with the target via

a potential 𝑉 (𝑟). The potential changes the state of each electron’s wave function from Ψ𝑖 = 𝑒𝑖𝒌𝑖·r

8

Figure 2.2 Illustration of the electron scattering arrange-
ment.

into Ψ 𝑓 = 𝑒𝑖𝒌 𝑓 ·r, by deviating its trajectory by an angle 𝜃. Let us assume that the only force

mediating the interaction is due to the nuclear Coulomb potential of the form

𝑉 (𝑟) = −

𝑍𝑒2
4𝜋𝜖0

∫ 𝜌𝑒 (r’)
|r − r’|

𝑑3𝑟′

(2.1)

where 𝑍 is the proton number of the target, 𝑒 = 1.6021766 × 10−19 C is the magnitude of the

elementary unit charge, 𝜖0 = 8.8541878 × 10−12 F/m is the permittivity of free space, 𝜌𝑒 is the

electric charge distribution of the nucleus, r’ is the location of the charge creating the potential, and

r is the probing location. If we assume that the nuclei have spherical symmetry, the charge density

distribution can be defined as

𝜌𝑒 (𝑟) =

∫

1
2𝜋2

𝐹 (𝑞)

sin(𝑞𝑟′)
𝑞𝑟

𝑞2𝑑𝑞

(2.2)

where 𝒒 = 𝒌𝑖 − 𝒌 𝑓 is the change in the electron’s momentum or momentum transferred to the

nucleus. 𝐹 (𝑞) is the Fourier transform of the charge density distribution, most commonly known

as the form factor, and defined as

∫

𝐹 (q) =

𝑒𝑖𝒒·r𝜌𝑒 (r’)𝑑3r’

(2.3)

Under an ideal elastic scattering, we assume that the kinetic energy of the electron is conserved,

| 𝒌𝑖 | = | 𝒌 𝑓 |, in which case 𝑞 =

2| 𝒌𝑖 |
ℏ

sin(𝜃/2) [11].

In the laboratory, we measure the intensity of deflected electrons as a function of scattering

angles, 𝐼 (𝜃, 𝜙). We then normalize this intensity by the total number of incident electrons from the

9

beam, 𝑁𝑎, and the interaction area, 𝐴, to get the scattering probability, also known as the differential
where Ω = ∫ ∫ sin 𝜃𝑑𝜃𝑑𝜙 is the solid angle. This ratio has units of area per solid

cross section 𝑑𝜎
𝑑Ω

angle [12].

𝐼 (𝜃, 𝜙)
𝐴 · 𝑁𝑎

=

𝑑𝜎
𝑑Ω

We can express the differential cross section in terms of the scattering amplitude,

normalized by the change in momentum

𝑑𝜎
𝑑Ω

=

𝑘𝑖 − 𝑘 𝑓
𝑘𝑖

| 𝑓 (𝜃, 𝜙)|2

Under the first Born approximation, the scattering amplitude is defined as

𝑓𝐵1 = ⟨Ψ 𝑓 |𝑉 (𝑟)|Ψ𝑖⟩ = −

𝑍𝑒2
4𝜋𝜖0

∫ 𝑒𝑖𝒒·R
|R|

𝐹 (𝑞)𝑑3𝑞

(2.4)

𝑓 (𝜃, 𝜙),

(2.5)

(2.6)

for nuclei with radius R. In this way, we can connect the measured scattering probability to the

charge density distribution contained in the form factor. In practice, 𝜌𝑒 is often approximated using

truncated infinite series, such as a Fourier or Bessel expansion, and the measurements are fitted to

obtain approximate values for the first few coefficients in the series [11, 13].

Following Hofstadter approach, we can modulate the Mott scattering formula for a point source

with the form factor such that

𝑑𝜎
𝑑Ω

=

(cid:19)

(cid:18) 𝑑𝜎
𝑑Ω

Mott

|𝐹 (𝑞)|2 =

(cid:32) 𝑍𝑒2
4𝑘 2
𝑖

(cid:33) cos2(𝜃/2)
sin4(𝜃/2)

|𝐹 (𝑞)|2

(2.7)

where 𝑘𝑖 is the energy of the probing electrons [13]. This approximation includes effects due to the

electron spin, but the recoil of the nucleus as well as magnetic interactions between nucleons and

electrons have been neglected.

2.1.3 Higher order corrections to the Born approximation

In the Born approximation, where the energy transferred by the electron to the nucleus is assumed

to be carried by one virtual photon, the probability for an electron to scatter off a point charge, also

10

known as the differential cross section, is given by the Mott scattering formula [14]

(cid:19)

(cid:18) 𝑑𝜎
𝑑Ω

Mott

4𝑍 2𝑒4𝑘 2
𝑞4𝑐4

=

(cid:18)

1 −

(cid:19)

.

𝑞2𝑐2
4𝑘 2

(2.8)

For energies higher than 100 MeV we can neglect the electron mass and the Mott scattering

formula can be expressed as (with 𝑘𝑖 = 𝑘)

(cid:19)

(cid:18) 𝑑𝜎
𝑑Ω

Mott

=

𝑍 2𝑒4
4𝑘 2

cos2(𝜃/2)
sin4(𝜃/2)

(2.9)

as used in the previous section.

The condition for validity of the Born approximation is given by 𝑍𝛼 ≪ 1, where 𝛼 ≃ 1/137 is

the dimensionless fine structure constant. For heavier nuclei, one must multiply the Mott scattering

by the form factor to accurately represent the extended shape of the nucleus.

The Born approximation can be corrected for radiative effects, straggling and dispersive effects

that occur during the scattering interaction process. The first two account for the emission of

real and virtual photons, while the latter accounts for only virtual photons. Note that the photons

emitted through radiation or straggling must be of very low energies, or else the event wouldn’t be

accounted as an elastic event (they will be out of the detector acceptance) [14].

• Radiative corrections: This effect accounts for the emission of real and virtual photons

while the electron gets accelerated or decelerated during its interaction with the nucleus’

electric field.

• Straggling correction: This effect considers the possibility that the electron undergoes

multiple small-angle scatterings within the target nucleus. It is dependent on the electron

beam energy, and the atomic number and thickness of the target.

• Dispersive corrections: This correction accounts for the reduction of the multi-body problem

between the incoming electron with the nucleus into the simplified one-body problem of an

electron interacting with a static field. It involves an expansion of the Hamiltonian of the

system.

11

The reader is referred to references [14, 15, 16, 17] for more information about these corrections

to the Born approximation.

2.2 Beam dynamics

Particle accelerator systems use the electromagnetic force to accelerate, focus and guide the beam.

The force felt by electrically charged particles with a velocity v and charge 𝑞 when moving through

a magnetic field B, and an electric field E is called the Lorentz Force (2.10). The energy gain of

the particle as it crosses the volume with the fields is obtained from (2.11). Although the magnetic

field does not contribute in the acceleration of the particle, it plays a vital role in focusing, steering

and bending particles, as will be explained later in this chapter.

F = 𝑞(v × B + E)

Δ𝐸gain =

∫ r2

r1

F · 𝑑r = 𝑞

∫ r2

r1

E · 𝑑r

(2.10)

(2.11)

In the following sections, we will take a closer look at the theory behind electron beam emission

and manipulation. We will first explore the theory for photoemission and thermal emission of

electrons, which are the two main processes used for electron beam production. Then, we will

explain how magnetic fields are used to bend and focus a beam using a simplified model of

the system. Finally, a brief explanation of how Direct Current (DC) and Radio Frequency (RF)

acceleration works will be given, including the "phasing" (proper alignment of the time at which

the beam enters the RF cavity) and its importance when designing an acceleration system.

2.2.1 Electron beam emission

The two most common processes used to generate an electron beam are photoemission and

thermionic emission.

In the photoemission process, powerful laser sources are shone onto a

cathode surface, exciting the electrons in the material with sufficient energy to leave the surface as

an applied electric field guides them to an accelerating cavity. For the thermionic emission, the

source of energy for the electrons comes from heating the cathode.

12

2.2.1.1 Photo-cathode guns

When an electron absorbs a photon, it can be excited from a bound states in the material all the

way to a free state in vacuum. This one-step model is very crude and unrealistic: it assumes that

the electron’s energy as well as its momentum are conserved. More widely known is the three-step

model, which breaks the emission process into: (1) photon absorption and electron excitation, (2)

transport of the ejected electrons to the material surface, and (3) escape into vacuum [18].

Photon absorption and electron excitation

In the first step, we assume that all the energy levels below the Fermi energy (𝐸𝐹) are filled,

while all the energy states above it are empty: this approach idealizes the photocathode material

as a conductor at 0 K. One also assumes that the excitation probability is exclusively dependent

on the photon energy and the number of available states for electrons to move from and to (e.g.,

number of states below and above 𝐸𝐹). From the reflectivity 𝑅(𝜈) of the material, the probability

of absorption 𝐴(𝜈) into the material (which is the same as the transmission) as a function of the

incident light’s frequency 𝜈 is

𝐴(𝜈) = 1 − 𝑅(𝜈)

(2.12)

One would need to also know the material’s absorption coefficient at ℎ𝜈 to determine the

absorption depth, where ℎ = 6.62607 × 10−34 J/Hz is Planck’s constant. As for the number of

available states, the probability of exciting the electron from an energy state 𝐸0 to 𝐸0 + ℎ𝜈 is given

by

𝑃(𝐸, ℏ𝜈) =

𝑁 (𝐸)𝑁 (𝐸 − ℏ𝜈)
𝑁 (𝐸′)𝑁 (𝐸′ − ℎ𝜈)𝑑𝐸′

∫ 𝐸𝐹 +ℎ𝜈
𝐸𝐹

(2.13)

in which we assumed that the momentum is not conserved and thus the probability of transition

is exclusively dependent on the material’s electronic density of states. 𝑁 (𝐸) is the number of states

for the energy 𝐸 = 𝐸0 + ℎ𝜈, while the integral accounts for the total number of possible transitions.

13

Transport of the ejected electrons

After we account for the probability of excitation, we need to consider the probability of the

electron to effectively reach the surface of the material. This process is significantly different for

metallic and semiconductor materials. For metallic cathodes, electron-electron (e-e) scatterings

are the primordial energy loss processes that inhibit the electron from reaching the surface. On

the other hand, e-e scattering are forbidden in semiconductors when the photon’s energy is less

than double the band gap energy (as is the case for most applications). Thus, electron-phonon

(e-p) scatterings is the mechanism that dominates. It is important to mention that one single e-e

scattering can reduce the energy of the involved electrons sufficiently for these to be lost, while

e-p scatterings mostly affect the electron’s momentum, such that an electron can undergo multiple

scattering events and still retain sufficient energy to escape.

For metals, we can assume that the probability 𝑆(𝐸) of an excited electron with energy 𝐸 > 𝐸𝐹

to interact with another electron of energy 𝐸 < 𝐸𝐹 is purely proportional to the number of electrons

with energy 𝐸0, 𝑁 (𝐸0), and the number of empty states 𝑁 (𝐸0 + Δ𝐸) and 𝑁 (𝐸 − Δ𝐸). 𝑆(𝐸) is

then defined as in (2.14), and we can obtain the scattering length 𝜆𝑒 (𝐸) from (2.15), where 𝜆0 is

an empirical constant of proportionality.

𝑆(𝐸) ∝

∫ 𝐸𝐹

∫ 𝐸−𝐸𝐹

2𝐸𝐹 −𝐸

𝐸𝐹 −𝐸0

𝜆𝑒 (𝐸) =

∫ 𝐸𝐹
2𝐸𝐹 −𝐸

∫ 𝐸−𝐸𝐹
𝐸𝐹 −𝐸0

𝑁 (𝐸0)𝑁 (𝐸 − Δ𝐸)𝑁 (𝐸0 + Δ𝐸)𝑑 (Δ𝐸)

(2.14)

𝜆0√︁𝐸 − 𝐸 𝑓

𝑁 (𝐸0)𝑁 (𝐸 − Δ𝐸)𝑁 (𝐸0 + Δ𝐸)𝑑 (Δ𝐸)

(2.15)

Then, the probability that an electron created at a depth 𝑑 will reach the surface is given by

(2.16). From this probability, and knowing that the absorption length 𝜆 𝑝ℎ is determined by the

imaginary part of the index of refraction 𝑛 = 𝜂 + 𝑖𝑘 and the incident photon’s wavelength 𝜆 as in

(2.17), we can integrate over all the possible depths of absorption 𝑑 and get the fraction of electrons

reaching the surface without scattering from (2.18).

𝑃 = 𝑒− 𝑑

𝜆𝑒

14

(2.16)

𝜆 𝑝ℎ =

𝜆
4𝜋𝑘

𝑇 (𝐸, 𝜈) =

𝜆𝑒 (𝐸)/𝜆 𝑝ℎ (𝜈)
1 + 𝜆𝑒 (𝐸)/𝜆 𝑝ℎ (𝜈)

(2.17)

(2.18)

The multiple scattering events that can happen in the case of a semiconductor make the derivation

of an analytical expression practically impossible. For this reason, computational methods, such as

Monte Carlo techniques, are commonly employed to simulate the movement of electrons towards

the surface in semiconductors.

Escape into vacuum

The final step requires the perpendicular component of the excited electron’s momentum, 𝑘⊥,

to satisfy the inequality depicted in (2.19), where Φ is the work function energy. Note that it is not

only necessary that the electron’s energy surpasses the work function, but it’s direction of motion

must also be close to perpendicular with the surface.

ℏ2𝑘 2
⊥
2𝑚

≥ Φ

(2.19)

During experiments, photocathode electron guns undergo an "activation" process before extrac-

tion. In the process, the laser is directed at the cathode while some gas is allowed to circulate inside

the chamber. The circulating gas particles attach to the cathode and form a negative affinity layer

that reduces the effective work function and allows electrons to escape more easily, thus increasing

the Quantum Efficiency (QE).

More details about the three-step model, and modification to this theoretical framework, can

be found in [19]. In terms of losses, the first step accounts for the loss due to reflection of the

incident light, while the second step reflects on the losses due to scattering processes. The final step

accounts for the loss of electrons for which the direction of motion wasn’t at the right angle with

the surface. In general, semiconductors yield higher QE due to their band gap. This gap doesn’t

allow electrons to be excited into states that lack sufficient energy to escape, and additionally it

prohibits e-e scatterings. As a result, the mean free path of electrons in metals is much smaller than

15

the laser penetration depth, resulting in unproductive absorption that will never leave the material,

while for semiconductors the mean free path can even be larger than the laser penetration, thus

fundamentally all excited electrons could potentially be extracted [18].

2.2.1.2 Thermionic guns

Photoemission is not the only source available to produce electron beams: thermionic emission

has also proven to be capable of delivering high average current beams with relatively small radial

sizes. This type of emission is described by the Richardson-Laue-Dushman equation (2.20) for a

temperature 𝑇. For this equation, 𝑘 𝐵 is the Boltzmann constant, and 𝑚 and 𝑒 are the mass and

elementary charge of the electron, respectively. This equation provides the current density 𝐽𝑅𝐿𝐷

emitted from the cathode. Note that the work function 𝑊 can be calculated from (2.21) and relates

the electric potential 𝑒Φ with the Fermi level 𝐸𝐹 [20].

| 𝑱𝑅𝐿𝐷 | =

𝑒𝑚

2𝜋2ℏ3 (𝑘 𝐵𝑇)2𝑒−𝑊/(𝑘 𝐵𝑇)

𝑊 = 𝑒Φ − 𝐸𝐹

(2.20)

(2.21)

2.2.2 Beam focusing and bending

The nominal trajectory in accelerator science is along the path of an ideal particle, perfectly aligned

with the center of every instrument in the system.

In real life, accelerator systems deal with

bunches of particles, not individual ones, and none of these particles perfectly follows the nominal

trajectory. To prevent particles from continuously steering away and being lost inside the cavities’

wall, focusing elements are strategically placed to correct for angular divergence and place such

particles closer to the nominal path. Although it is common practice to use magnetic fields to

manipulate the trajectory of the beam (specially at high energies), the electric force could and is

also utilized at low energies. In this section, we will look in detail at the physics of the focusing

process with magnetic fields only.

Let us define a Cartesian coordinate system 𝐾 = (𝑥, 𝑦, 𝑠) whose origin moves with the center of

16

the beam, thus, the nominal trajectory. The beam motion is aligned with the 𝑠 axis, while the 𝑥 axis

points in the horizontal direction and 𝑦 in the vertical direction. In the areas where a magnetic field

bends the beam’s direction, the coordinate system 𝐾 must be adjusted for it. We will assume that

this bending occurs exclusively in the horizontal direction, thus rotations are limited to the 𝑦-axis.

Giving a rotation by an angle 𝜑, the axes 𝑥 and 𝑠 are transformed into the axes 𝒙0 and 𝒔0 as

(2.22), or in differential form by (2.23).

𝒙0 = 𝒙 cos 𝜑 + 𝒔 sin 𝜑,

𝒔0 = −𝒙 sin 𝜑 + 𝒔 cos 𝜑

𝑑𝒙0
𝑑𝜑

= 𝒔0

,

𝑑𝒔0
𝑑𝜑

= −𝒙0

(2.22)

(2.23)

An infinitesimal length of the particles’ path is described as 𝑑𝑠 = 𝑅𝑑𝜑 for a radius of curvature

𝑅, and it follows that the curved trajectory is

𝑑𝜑
𝑑𝑡

=

1
𝑅

𝑑𝑠
𝑑𝑡

(2.24)

Since there is a unique correspondence between the path length 𝑠 and time 𝑡, one can replace

time derivatives by the corresponding derivative with respect to the 𝑠 spatial coordinate as

(cid:164)𝑥 =

𝑑𝑥
𝑑𝑠

𝑑𝑠
𝑑𝑡

= 𝑥′ (cid:164)𝑠,

(cid:165)𝑥 =

𝑑 (𝑥′)
𝑑𝑡

(cid:164)𝑠 + 𝑥′ 𝑑 (cid:164)𝑠
𝑑𝑡

= 𝑥′′ (cid:164)𝑠2 + 𝑥′ (cid:165)𝑠.

Then, for some particle with coordinates

𝒓 = 𝑠𝒔0 + 𝑥𝒙0 + 𝑦𝒚0

and mass 𝑚 under the influence of the magnetic force 𝑩 = 𝐵𝑥 ˆ𝑥 + 𝐵𝑦 ˆ𝑦, such that

(cid:165)𝑟 =

𝑒
𝑚

( (cid:164)𝒓 × 𝑩),

17

(2.25)

(2.26)

(2.27)

(2.28)

it follows that

(cid:165)𝑟 =

𝑒
𝑚

− (cid:0)1 + 𝑥
𝑅

(cid:1) (cid:164)𝑠𝐵𝑦

(cid:0)1 + 𝑥
𝑅

(cid:1) (cid:164)𝑠𝐵𝑥

𝑥′ (cid:164)𝑠𝐵𝑦 − 𝑦′ (cid:164)𝑠𝐵𝑥

(2.29)






We can assume that the effects on the longitudinal component, 𝑠, are negligible since the

particles are moving a relativistic velocities, and we can focus our discussion only on the transverse

components. In addition, we can assume that (cid:165)𝑠 ≈ 0 such that the forward velocity of the particles

remains virtually unchanged as it transverse the region with magnetic fields. Let the momentum

be defined as 𝑝 = 𝑚𝑣 where 𝑣 = (cid:164)𝑠(1 + 𝑥/𝑅), it can be shown that the equations of motion for this

particle are

𝑥′′ −

(cid:16)1 +

𝑥
𝑅

(cid:17) 1
𝑅

= −

𝑦′′ =

(cid:16)1 +

𝐵𝑦

(cid:16)1 +

𝐵𝑥

𝑣𝑒
(cid:164)𝑠𝑝
𝑣𝑒
(cid:164)𝑠𝑝

(cid:17)

(cid:17)

𝑥
𝑅
𝑥
𝑅

(2.30)

Since the dimensions of the beam are relatively small compared to the radius of curvature,

𝑥, 𝑦, 𝑠 ≪ 𝑅, we can expand the magnetic field in the vicinity of the nominal trajectory as

𝑩(𝑥, 𝑦) = (cid:2)𝐵𝑦0 ˆ𝑥 + 𝐵𝑥0 ˆ𝑦(cid:3) +

(cid:20) 𝑑𝐵𝑦
𝑑𝑥

𝑥 ˆ𝑥 +

(cid:21)

𝑦 ˆ𝑦

+

𝑑𝐵𝑥
𝑑𝑦

1
2!

(cid:34) 𝑑2𝐵𝑦
𝑑𝑥2

𝑥2 ˆ𝑥 +

(cid:35)

𝑦2 ˆ𝑦

𝑑2𝐵𝑥
𝑑𝑦2

1
3!

(cid:34) 𝑑3𝐵𝑦
𝑑𝑥3

+

𝑥3 ˆ𝑥 +

(cid:35)

𝑦3 ˆ𝑦

𝑑3𝐵𝑥
𝑑𝑦3

+ ...

𝑒
𝑝 𝑩(𝑥, 𝑦) =

(cid:20) 1
𝑅𝑥

ˆ𝑥 +

(cid:21)

ˆ𝑦

1
𝑅𝑦

+ (cid:2)𝑘𝑥𝑥 ˆ𝑥 + 𝑘 𝑦 𝑦 ˆ𝑦(cid:3) +

1
2!

(cid:2)𝑚𝑥𝑥2 ˆ𝑥 + 𝑚𝑦 𝑦2 ˆ𝑦(cid:3) +

1
3!

(cid:2)𝑜𝑥𝑥3 ˆ𝑥 + 𝑜𝑦 𝑦3 ˆ𝑦(cid:3) + ...

(2.31)

Note that the brackets, in order from left to right, represent the dipole field through bending

radius 𝑅, the quadrupole field with quadrupole strength 𝑘, and the sextupole and octupole fields

with strengths 𝑚 and 𝑜 respectively. The two lowest multipoles are the only ones used for linear

beam optics, since they are either constant and used for steering (dipole) or linear and used

for focusing/defocusing (quadrupole). Normally the sextupole, octupole and higher orders of

18

multipoles are either used for field corrections or unwanted field errors, reason for which we will

only use the dipole and quadrupole moments for this example and will assume that only horizontal

dipole deflection is present.

If the momentum spread of the particles is low, as it is for typical accelerators where Δ𝑝/𝑝 < 1%,

we can rewrite the momentum as

1
𝑝

=

1
𝑝0

(cid:18)

1 −

(cid:19)

.

Δ𝑝
𝑝0

(2.32)

If all the previous assumptions are taken we can neglect the squared terms, after which we obtain

the linear equations of motion for a particle traveling through magnetic structures in accelerator:

𝑥′′(𝑥) +

(cid:18)

1
𝑅2(𝑥)

(cid:19)

− 𝑘 (𝑠)

𝑥(𝑠) =

1
𝑅(𝑠)

Δ𝑝
𝑝

𝑦′′(𝑠) + 𝑘 (𝑠)𝑦(𝑠) = 0

(2.33)

These derivations followed the explanation from [21].

2.2.3 Beam acceleration

2.2.3.1 Basic overview

There are two main common methods to accelerate a charged particle: with a static electric

field (E-field) or with alternating RF electric fields, the most intuitive and simple of the two being

the former. The E-field is produced by a high voltage DC supply and the beam travels between two

or more electrodes with increasing voltage. After the beam exits the static field region, it usually

continues its trajectory in a field-free drift tube with constant energy until it hits a target. This

method imposes a limitation on the particle’s energy based upon the maximum achievable voltage,

that is primarily limited by corona discharge. The electrons and ions near the high voltage electrode

are released with sufficient energy to cause an avalanche when they interact with the residual gas,

and thus lead to a breakdown. Up to a few MeV are normally achievable; yet, the technology is still

useful for some applications that do not require high energy.

For nuclear physics and particle accelerator facilities, the customary type of acceleration used

today is based on alternating electric fields produced from an RF source. Via this method, the

19

particles travel along a cavity with high-frequency alternating voltage of the form displayed in

(2.34) and illustrated in Figure 2.3: the particles gain energy when crossing the electric field, then

drift through a field-free space while the field switches polarity, such that they enter the next region

when the electric field is again aligned to the direction of acceleration of the beam leading to further

increase in its energy [21].

𝑈 (𝑡) = 𝑈𝑚𝑎𝑥 sin(𝜔𝑡)

(2.34)

[a]

[b]

Figure 2.3 Schematics for the basic principles of RF accelerating cavities for
proton acceleration. [a] Design with alternating polarity drift tubes [22]. [b]
Parabolic box cavity design [23].

2.2.3.2 Cavity phases

When particles exits the 𝑖th drift tube, their energy had increase following (2.35), where Ψ0 is

20

the average phase of the RF voltage the particles perceive as they cross the applied field sections

with a maximum achievable voltage 𝑈𝑚𝑎𝑥. The length of each subsequent drift tube (𝑖) must

increase in proportion to the energy gain of the particles (𝐸𝑖), that is related to their momentum

(𝑝𝑖) and velocity 𝑣𝑖 through (2.36) and (2.37).

𝐸𝑖 = 𝑖𝑞𝑈𝑚𝑎𝑥 sin Ψ0

√︃

𝐸𝑖 =

𝑚2

0𝑐4 + 𝑝2

𝑖 𝑐2

𝑝𝑖 = 𝛾𝑖𝑚0𝑣𝑖

𝛾𝑖 =

1

√︃

1 − 𝛽2
𝑖

,

𝛽𝑖 = 𝑣𝑖/𝑐

(2.35)

(2.36)

(2.37)

(2.38)

𝛾𝑖 is the Lorentz factor and 𝑚0 the rest mass of the particle. We consider the period time 𝜏𝑅𝐹 as

the time it takes the alternating field to complete a full cycle as

Therefore, the necessary length for a drift tube, such that the particles arrive at the region with

𝜏𝑅𝐹 =

𝜆𝑅𝐹
𝑐

(2.39)

the electric field at its peak, must satisfy (2.40) for the non-relativistic case (𝐸𝑖 = 1

2 𝑚𝑣2

𝑖 ) or (2.41)

for the relativistic case (see (2.36)) [21]. Note that in the case of electron beams (𝑚0 = 0.511 MeV)

their velocity approaches the speed of light with just few MeV and the length of the tubes remains

constant.

𝑙 𝑁 𝑅
𝑖

=

1
𝑣 𝑅𝐹

√︂𝑖𝑞𝑈𝑚𝑎𝑥 𝑠𝑖𝑛Ψ0
2𝑚

𝑙 𝑅
𝑖 =

𝜏𝑅𝐹
2

√︃

𝑖(𝑞𝑈𝑚𝑎𝑥 sin Ψ0)2 − 𝑖𝑚2
𝛾𝑚0

0𝑐4

21

(2.40)

(2.41)

In accelerator systems, it is not uncommon to have multiple accelerating sections. Consequently,

particles are often found to acquire energies off-peak 𝑈𝑚𝑎𝑥, and thus becoming out of phase relative

to the RF voltage. In other words, if the cavities are designed for particles to arrive at exactly the

maximum power of the voltage, the beam would become unstable and be lost. Instead of using

Ψ0 = 𝜋/2 at the peak voltage 𝑈𝑚𝑎𝑥, some positive degree below Ψ0 is used at 𝑈𝑒 𝑓 𝑓 < 𝑈𝑚𝑎𝑥. The

optimal phase should be around 30° or 𝜋/6, with the exact value depending on the real structure.

A visual concept of this optimal phasing is provided in Figure 2.4.

Figure 2.4 Visual representation for the needed phase focusing be-
tween the beam and the RF voltage. Beam stability is achieved for
phases around 𝜋/6 and 3𝜋/2.

For a structure designed to receive an ideal particle at the nominal phase Ψ0 = 𝜋/6 at some time

𝑡, a slightly faster particle that arrives at the time 𝑡 − 1 would perceive a weaker accelerating voltage

𝑈𝑡−1 < 𝑈𝑡. On the other hand, a particle moving slightly slower and arriving at time 𝑡 + 1, would

feel a voltage 𝑈𝑡+1 > 𝑈𝑡 that would then accelerate it. Thus, this phase promotes a longitudinal self

focusing effect. From Figure 2.4, it is clear that a design for maximal voltage amplitude is unstable

as it produces the opposite effect: faster particles are still accelerated while slower particles are

decelerated as they perceive the field in the opposite direction. Choosing even higher values for the

phase, such as 3𝜋/2, cause deceleration of the beam [21].

22

CHAPTER 3

TECHNOLOGIES

3.1 Electron Gun

In this section, we will delve into the intricate details of electron injector systems and their

application in high performance environments, with an emphasis on photonic guns. We will

briefly discuss the thermal emittance of the beam and how it must be considered in the design, the

materials most commonly used in photocathodes, as well as the challenges of operating electron

guns. The chapter also presents the latest advancements in photocathode technology, including the

design and performance of specific high-efficiency photocathode and thermionic guns. This chapter

offers a thorough exploration of electron injectors’ complexities, challenges, and the cutting-edge

technological solutions developed to optimize their performance.

3.1.1 Beam Emittance, Brightness and Luminosity

An essential aspect of the injector’s performance is the beam’s emittance and brightness. The

brightness of the beam describes its angular divergence, and it is dependant on the beam’s emittance,

which can be defined as the area of the ellipse in the phase space of transverse velocity as a function

of position. In concrete terms, the brightness is the ratio of the beam current to the transverse

emittance, as shown in (3.1.1). This important beam parameter, the emittance 𝜖, is determined after

extraction by the balance between betatron oscillations and damping, which ultimately depends on

the magnet structure [21]. Understanding and optimizing both of these parameters is crucial for

maximizing the beam’s performance.

The space charge effect, the Coulomb repulsion of electrons within the beam bunch, can

rapidly increase the emittance, specially so for high current guns. This can be compensated by

using extraction DC voltages above 100 kV. For most high performance applications, somewhere

between 400−600 kV could suffice for emittance compensation. The normalized thermal emittance

of the beam, 𝜖, when it comes out from the photocathode is directly proportional to the laser spot

size 𝜎𝑙𝑎𝑠𝑒𝑟, and to the square root of the electrons’ transverse energy 𝑘 𝐵𝑇. Since the beam’s

23

brightness is dependant on the emittance, it is thus dependant on these parameters as well. This

can be seen in (3.1) and (3.1.1) for the emittance and brightness respectively.

𝜖 = 𝜎𝑙𝑎𝑠𝑒𝑟

√︂ 𝑘 𝐵𝑇
𝑚𝑐2

(3.1)

Here, 𝑚 is the electron’s mass and 𝑐 is the speed of light. The brightness, 𝐵𝑛, on the other hand,

is described as

𝐵𝑛 =

2𝐼
𝜋2𝜖 (𝑛, 𝑥)𝜖 (𝑛, 𝑦)

(3.2)

where 𝐼 is the beam current, and 𝜖 (𝑛, 𝑥) and 𝜖 (𝑛, 𝑦) are the normalized emittances in the

corresponding transverse planes. More importantly, the maximum transverse brightness for an

electron bunch with repetition frequency 𝑓 , can be described as

𝐵𝑛
𝑓

=

𝑚𝑐2𝜖 𝐸𝑐𝑎𝑡ℎ𝑜𝑑𝑒
2𝜋𝑘 𝐵𝑇

(3.3)

where 𝐸𝑐𝑎𝑡ℎ𝑜𝑑𝑒 is the electric field at the cathode surface. Here, it is evident that finding

materials with low emitting transverse thermal energy 𝑘 𝐵𝑇, and increasing the electric field at the

cathode’s surface, is essential for maximizing the obtainable beam brightness [19].

It is worth noting that the emittance of electron beams have a minimum theoretical value

determined by the starting values of the beam’s dispersion and the optical functions of the magnets

in the structure, as described by [24], [25]. This value must be individually calculated for every

system [21].

Luminosity, in the context of accelerator science, is a term that combines the characteristics of

the incident beam and the target to determine the interaction rate per unit cross-section, and has

units of cm−2s−1. This is,

𝐿 = 𝑛1𝑛2𝑙

24

(3.4)

where 𝑛1 is the number of particles per second in the beam, while 𝑛2 is the particle number

density of the target and 𝑙 is the length of the target. Using this, we can define the rate of interactions

per second as

𝑑𝑅
𝑑𝑡

= 𝜎𝐿

(3.5)

where 𝜎 is the cross section for the type of interaction concerned, with units cm2 [26]. We

have here assumed that the target has homogeneous particle density 𝑛2 and its cross-sectional area

is larger than the incoming beam’s.

3.1.2 Photoinjectors

For the application at hand, it is said that a source is DC if the duration is longer than 1 𝜇s.

This must not be confused with Continuous Wave (CW), which denotes that for each period of

an RF cycle a bunch of electrons is present [19]. It is well known that RF accelerating structures

can achieve higher acceleration gradients than a DC acceleration structure due to the lower voltage

breakdown threshold of the later, which allows RF accelerating systems to be more compact and

energy efficient. Yet, at the initial stages of beam extraction, DC acceleration offers a simple and

stable configuration that can yield lower emittance [27]. Additionally, only DC guns can be used

for extraction of spin polarized electron beams, a possible upgrade for the system. For this reason,

here I will discuss a hybrid system that uses a high voltage Direct Current to extract the electrons

from the cathode, which are then accelerated by one or more RF acceleration cavity.

3.1.2.1 Advanced DC guns

In this section we present two advanced photocathode electron guns. The first gun, an enhanced

version of a Jefferson Lab’s earlier model, incorporates several upgrades to optimize its performance

and reliability. These improvements include a more effective vacuum system, refined motion

mechanisms for the photocathode, and a novel approach to cesium deposition for high voltage

operations. Additionally, a unique shield door mechanism is introduced to protect the cathode from

damage during the high voltage conditioning. The second gun, developed by Cornell, features a

cryogenically cooled design with a Cs3Sb photocathode. It was constructed primarily from stainless

steel and it showcases a unique electrode configuration and a specialized insulator design. The

25

gun’s baking and cleaning procedures, both needed to achieve an optimal vacuum environment, are

described in details, and the gun’s operational characteristics at various temperatures and voltages

are also mentioned. Both guns could potentially be used in electron elastic scattering experiments.

Jefferson Lab’s high average current gun

Figure 3.1 depict a newer model of a Cs:GaAs photocathode gun that was made based on the

Thomas Jefferson National Laboratory (JLab) 1 kW Demo IR FEL gun [28]. Note that the optical

set up is external, as per usual for photocathode guns, and the laser is directed onto the cathode

through one of the viewing ports of the cathode chamber.

In the newer model, they decreased

the vacuum level, added motion mechanisms for better photocathode performance and developed

a new method for depositing Cs to allowed a more reliable high voltage operation. One of the

new features for photocathode handling included a swing shield door to protect the cathode from

back-ion bombardment damage during high voltage conditioning. A detailed schematic inside the

ball cathode, that shows the shielding, is presented in Figure 3.2. This gun demonstrated up to

9.1 mA CW beam for a 122 pC bunch with 75 MHz repetition rate [29].

Figure 3.1 DC photocathode gun schematic. (a) vacuum cham-
ber, (b) instrumentation, (c) anode, (d) photocathode, (e) ball
cathode, (f) NEG pumps, (g) ball support tube, (h) ceramics, (i)
shield door actuator mechanism, and (j) stalk retracting mecha-
nism [29].

The exposed area of the photocathode has a diameter of 2.54 cm, much greater than the laser

spot of 0.8 cm diameter, and it is illuminated off center to reduce QE degradation through back-ion

26

Figure 3.2 Detail of the inside of the ball cathode with the stalk
retracted. The shield door is closed only during high voltage
conditioning. (a) Stalk retracted, (b) roller guide unit for stalk
end - sapphire rollers, (c) charge collector, (d) cesium channels,
(e) cathode shield closed, and (f) cathode shield retracted [29].

bombardment. The cathode is a single crystal GaAs Zn-doped wafer, mounted on a molybdenum

disk brazed to one end of the stalk. Before use, the cathode goes through a hydrogen and heat

cleaning cycle, followed by baking of the full system, a high voltage conditioning, and finally

the photocathode activation to increase QE. The equipment was baked at 500 C for 45 minutes,

followed by a longer baking at 250 C until there was no significant vacuum improvement. The

electrodes were conditioned to 420 kV, with the stalk retracted and the shield door closed, for

about 80 hours to reliably operate at 350 kV. They followed a standard cesium-oxygen activation

procedure, as described in [30]. They tested two exact wafers, from the first one a total of 1300

Coulombs were extracted, 100 Coulombs per re-cesiation and 5% QE. With improvements in the

vacuum condition and reduction in the beam halo (they installed more pumps and anodized the

outermost part of the photocathode), they increased the extracted charge from the second wafer to

500 C per re-cesiation and 7% QE [29].

Cornell’s cryo-cooled gun

Another model for high average current electron gun was recently developed by Cornell. It is

a high voltage cryogenically cooled photoemission gun with Cs3Sb photocathode. The gun walls

were made out of stainless steel, and pure helium gas was used during high voltage conditioning.

27

The system allows for back-illumination of the photocathode, although only standard forward

reflecting illumination has been tested. The gun uses two main electrodes, a screening electrode

and a spherical shell one. The first one consist of a thin tube that shields against high field emissions

at the triple junction of the insulator, High Voltage (HV) stalk and the vacuum, and prevents thermal

conduction from the insulator to the cryogenic electrode. The later spherical electrode is made of

two joined hemispheres of 316SST vacuum remelt with an internal copper electrode structure, and

is where the photocathode is located. The system also has a grounded flat mesh anode to create

a uniform electric field in the gap between the cathode and the anode, it is a 0.01 cm wire with

22 lines per cm, and it is located 2 cm away from the cathode and parallel to the flat surface of

the spherical electrode. This grid has two holes, one allows the laser beam to reach the cathode,

and the other allow the extracted electrons to fly out. They used an inverted insulator made of

AL2O3 ceramic with a measured resistance of 16 GW [31]. Diagrams and pictures of the system

are displayed in figures 3.3 and 3.4.

Figure 3.3 Left: A 3D model of the gun. Right: The in-
ternal structure of the gun. The two different types of sub-
strate holders (H) are shown. The standard puck (top), is the
INFN/DESY/LBNL type used in the FLASHX-FEL gun and on
the APEX photoinjector. The modified transmission puck (bot-
tom), was designed and realized to allow the operation of the
photocathode in the transmission mode [31].

28

Figure 3.4 3D model of the beamline of the Cornell Cryogenic DC gun.The electron beam enters
from the left as indicated by the arrow. The inset shows a picture of the pre-assembled beamline
[31].

To achieve a static vacuum pressure of 2 × 10−11 Torr, they baked chambers and fittings

individually in air at 400 °C for 5 days, then thoroughly cleaned the components, and carried

out a final bakeout, after assembly, at 120 °C for three days. The electrodes were polished with

different materials, the finest being diamond suspension of 0.25 𝜇m for the electrode. The HV

conditioning was carried out both at room temperature and after the gun was cryogenically cooled

down. During room temperature conditioning, highly pure He was introduce into the gun vessel

until vacuum was raised above 10−5 Torr with ion pumps off, while the voltage of the gun was

increased up to ∼ 270 kV. The He gas was introduced periodically. Because the introduction of

He gas would significantly increase the temperature at the gun, no gas was introduced during the

cryogenic conditioning where ∼ 300 kV was reached. They set up automatic limits to shut off

the HV power supply in order to avoid major voltage breakdowns. The triggers included: four

radiation monitors with threshold at 2 mR/h, vacuum level larger than 10−8 Torr in the gun vessel,

and excess current larger than 300 𝜇A from the supply controller. The temperature for the cathode

29

was 43 ± 1 K in the first sample, and 39 ± 1 K after replacing the photocathode puck with a polished

one. The cooling took 22 to 30 h, with the time required to reach thermal equilibrium steadily

increasing [31].

The cathode is grown and transported under vacuum, avoiding possible contamination from air

exposure. It was shine with 500 nm wavelength laser at 78 MHz repetition rate and 0.5 mW power,

to produce 0.1 pC bunches with 10𝜇 A average current and 5% QE. At room temperature, the gun

was operated at 230 kV, corresponding to 11.5 MV/m extraction electric fields, and had a beam

root mean square (rms) of 𝜎𝑥 = 1.31 ± 0.07 mm and 𝜎𝑦 = 1.03 ± 0.07 mm. At 43 K, the gun was

operated at 190 kV, corresponding to 9.5 MV/m extraction electric fields, and had a beam rms of

𝜎𝑥 = 1.23 ± 0.07 mm, 𝜎𝑦 = 1.10 ± 0.07 mm [31].

3.1.2.2 Considerations and Challenges

As of today, semiconductors such as GaAs (the only known cathode that can generate spin

polarized electrons), Cs2Te, GaN, and K2CsSb, are the most commonly used materials with highest

QE and lowest emittance used for photo-production of electron beams. Particularly, QE of up to

20% have been measured for GaAs and K2CsSb using 520 nm light [19]. High power lasers for

this particular wavelength have been commercially available for a long time, with multiple sources

offering systems at reasonable prices and high performance [32],[33],[34]. The same cannot be said

for Cs2Te and GaN, which require Ultra Violet (UV) light, a spectrum for which no laser system

has yet been able to produce enough average power. It is important to notice that even for these

materials there is a trade off between QE and thermal emittance [19]. GaAs has demonstrated to be

the semiconductor with the lowest thermal emittance [35]. Unfortunately, this minimum happens

near it’s band gap, as shown in Figure 3.5, where the QE also reaches a minimum lower than 1%.

After deciding the material with which the cathode will be made of, one must caution against

the production of "accidental" electrons from scattered light hitting the cathode outside the desired

spot. This has become one of the greatest factors limiting the lifetime of cathodes, as these electrons

scatter around the chamber producing more electrons, light and ultimately degrading the vacuum

conditions. GaAs is one of the most sensible material, requiring vacuum levels under 1×10−9 Pa for

30

Figure 3.5 Transverse thermal energy for the electrons pho-
toemitted from GaAs as a function of wavelength. The red
points were measured using multiple laser spot sizes while the
blue points were measured using a single spot size. The dashed
line shows a fit over the linear region. kT is the electron’s effec-
tive transverse energy normal to the cathode’s surface [19].

successful operations. A solution to this problem is to inactivate the outer area of the cathode. This

has been done by coating the cathode with an oxide layer, or masking the surface while activating

with cesium [36]. Other mechanisms that limit the lifetime of the cathode are chemical poisoning

and ion back-bombardment. These problems can be overcome by the same coating method of the

cathode surface. Although this coating could potentially lower the QE, the use of such coating has

been demonstrated to deliver higher total currents over the lifetime of the cathode [37]. Improving

the vacuum level could significantly extend its lifetime as well.

One of the main sources of vacuum contamination comes directly from the walls of the system,

a process known as outgassing. To reduce the hydrogen outgassing, systems are baked at 150 °C

for at least 24 hours. The pieces are baked at even higher temperatures and for longer periods of

time before assembly. Another way to minimize contamination in the system is to clean each part

with high pressure (5 × 105 to 7 × 106 Pa) ultra pure de-ionized water or gas (nitrogen or dry-air)

and allowing it to dry in a clean room [19]. A normal spectrum for residual gas found inside a

photoinjector system after baking is found in Figure 3.6.

31

Figure 3.6 Residual gas spectrum after bake-out and non-
evaporable getter pump activation, showing hydrogen, methane,
carbon monoxide and carbon dioxide. The total measured pres-
sure was 6 × 10−10 Pa [19].

Given the constant high voltages during the operation of electron guns, voltage breakdowns can

become a significant problem. In the earlier years of photoemission, cesium activation (needed

to increase the quantum efficiency) was performed in the same chamber where the experimental

extraction was carried on. This contaminant made breakdowns more likely, as it lowers the work

function of the electrodes as much as it lowers the cathodes’. Thus, more recent systems include a

dedicated-isolated vacuum cavity for the purpose of activation, and from there the cathode is taken

into the experimental area by means of a load-lock. Additionally, guns are processed to 10-20%

above the operating value, in some cases up to 750 kV maximum voltage, to reduce field emission

and arcing.

Another factor that influences the beam’s emittance and the behavior of trapped contaminants

are stray magnet fields from the system’s walls. To minimize these, stainless steel is preferred for

vacuum chambers and flanges. Other materials may be used as well, such as molybdenum, copper,

or other pure metals with low outgassing rates. For cathodes and anodes, experiments suggest

that the best materials are niobium, stainless steel, or molybdenum for the former, and titanium or

beryllium for the later.

It is worth noting that using turbo pumps or cryopumps is advice against, since the vibration

32

caused by these can cause disturbances in the beam during extraction. Instead, ion pumps together

with getter pump strips are more suitable for photoemission guns. Every component of the system,

buncher, RF cavities, load-lock, or any other, must be maintained under the same vacuum condition

as the gun chamber to avoid contamination [19].

The development and optimization of photocathodes for electron beam production have made

significant strides, particularly with semiconductors like GaAs and alkali antimonides. These

materials, offering high quantum efficiency and low emittance, are central to advancing the field

of electron injection technology. The balance between quantum efficiency and thermal emittance

remains a key area of research, especially in the context of their operational environments and

the challenges posed by factors like vacuum contamination and voltage breakdowns. Innovations

in cathode design and system maintenance, such as specialized coatings and vacuum techniques,

continue to play a critical role in enhancing the performance and longevity of these vital components

in electron guns.

3.1.3 Thermionic gun

Another alternative for the electron injection are thermionic guns. These have a simpler design,

can be bought from commercial sources at moderate prices, and typically have longer lifetimes than

photo-guns. Furthermore, recent advancements in this technology have led to the achievement of

remarkably high electron currents, enhancing their applicability in various experimental settings,

including elastic scattering experiments.

One notable example of a high current thermionic gun is the Low Energy Electrons from

a Thermionic Cathode at High Intensity (LEETCHI) system, designed and tested by European

Council for Nuclear Research (CERN). LEETCHI is a Pierce-type gun that has the electrode of

the cathode holder and the anode electrode at 30°and 45°with respect to the cathode surface. This

arrangement prevents beam divergence within the accelerating gap [38].

It uses a commercial

cathode assembly that includes a 10 mm radius planar thermionic emitter and two grids that serve

to limit the beam current exiting the cathode. All together, it is capable of delivering a current

density of 6 A/cm2 [39]. The gun output current is limited by the cathode temperature and by

33

space-charge effects between the cathode and the first grid. During operation, the extracted beam

is accelerated up to 140 keV using DC voltages, then, it is focused and measured by an Optical

Transition Radiation (OTR) diagnostic.

It is stated that the machine can maintain a constant

emission of 5 A for pulses as long as 150 𝜇s, yet, all the reported optical measurements were

limited to 6 𝜇s pulses to prevent the deterioration of the OTR. When they measured the rms radii

of the beam for electron currents of 0.5 A and 5.5 A, they got minimum values of roughly 2.5 mm

and 4.1 mm respectively. The normalized rms trace-space emittance was estimated using computer

simulations to be between 𝜖𝑟𝑚𝑠 = 35 mm mrad and 𝜖𝑟𝑚𝑠 = 70 mm mrad. All of this was achieved

with a vacuum of only 10−8 mbar across the full system [38]. A diagram of the beam line can be

seen in Figure 3.7, together with measurements and calculations of the rms radii of the collected

beam at the OTR.

[a]

[b]

Figure 3.7 [a] Diagram of the layout for the LEETCHI system. [b] Experimental and simulated
rms radii for 0.5 A (black) and 4.5 A (red) as a function of the coil current 𝐼𝑠, for a gap between
the grid and the anode aperture of 45 mm [38].

Other systems have been made to deliver up to 100 A/cm2 current densities over 10,000 hours of

operational lifetimes, using advanced cathodes and/or high temperatures at vacuum levels between

10−8 to 10−7 Torr [40]. Some studies about this using simulations have indicated that guns equipped

with cathodes made of LaB6 can generate a 10 mA average current electron beam with 100 keV

energy and normalized emittance of 40 mm mrad, at a filament temperature of 1760 K [41].

Thermionic electron guns play a pivotal role in various high-current applications, with their

design and operational efficiency constantly under improvements. Although this work has mainly

34

focused on the use of photoguns, given the luminosity requirements and the prospect of extracting

currents in the order of 1 A using thermionic guns, it is worth taking the time to consider them.

3.2 Acceleration cavities

In this section we will explore electron bunchers, give some general information about RF

acceleration cavities, and provide detailed experimental results obtained from an advanced cold

copper acceleration cavity. Each of these components plays a pivotal role in the intricate dance

of accelerating electrons to near-light speeds. Electron bunchers, serving as the bridge between

initial electron extraction and high-speed acceleration, are instrumental in grouping and preparing

electrons for the RF cavities. Their ability to modulate particle velocities and densities is essential

for reducing energy spread, thereby ensuring that electrons are suitably prepared for acceleration.

This process is critical in maintaining the integrity and efficiency of the entire system. Moving to

classical RF cavities, because of the high acceleration gradients needed for rapid energy gain, these

structures need to deal with the daunting challenge of managing RF breakdowns, a phenomenon that

can cause significant permanent damage. Understanding and mitigating these breakdowns are vital

for the longevity and effectiveness of the system. Finally, the development of the Cryo-Cu-SLAC,

a high gradient X-band cryogenic copper accelerating cavity, represents a remarkable achievement

in this field. Its ability to deliver substantial accelerating gradients while maintaining a manageable

breakdown rate is a testament to the ingenuity and collaborative efforts of researchers. Each

component discussed in this section – from the precise bunching of electrons to the management of

RF breakdowns and the groundbreaking development of the Cryo-Cu-SLAC – represents a critical

piece in the complex puzzle of building an electron elastic scattering systems.

3.2.1 Electron buncher

In a typical set-up, electron accelerators consist of a DC gun injector, where electrons are

extracted from the cathode, followed by a buncher, that groups the particles into longitudinal

bunches and accelerate them into almost the speed of light, and finally, the bunches are accelerated

to the desired energies by an RF cavity. It is not uncommon for the buncher to be coupled with the

RF elements as one single accelerator assembly. Electrons that come from the gun have uniformly

35

distributed phases and the same energy. Unlike DC acceleration, that is not dependent on phase,

once the particles enter the RF cavity, their energy gain is dependent on their phase, as given by

(??). Reducing the energy spread before entering the accelerating 𝛽 = 1 section is the main purpose

of grouping the electrons into small phase spread bunches, separated in space by one wavelength

in the waveguide [42].

The simplest bunching technique is known as "ballistic" or "klystron". It uses a stand-alone

RF cavity that modulates the particles’ velocities, resulting in density modulation after a drift.

The cavity is physically separated from the accelerator by a drift space. Another common type of

buncher use an accelerating waveguide section with constant phase velocity, which bunches the

beam using phase motion. These are known as "tapered bunchers", and are very efficient since they

allow for simultaneously accelerate and bunch. They provide the best capture rate and allow for

specific beam parameters to be achieved, but because of their complexity, it is the most challenging

design [42].

In a system from Los Alamos National Laboratory, as an example, the initial bunch length after

exiting the gun is approximately 30 − 40 ps, then it is compressed to 5 − 10 ps using a normal

conducting RF buncher cavity, and it is further compressed while passing through several super

conducting RF cavities to achieve a bunch length of 1 − 2 ps. The system also includes several

solenoid focusing magnets that control the beam size and emittance [19].

3.2.2 Considerations and challenges of RF cavities

RF breakdowns is one of the major limiting factors for higher accelerating gradients. We call

an RF breakdown when the transmission and reflection power abruptly and significantly changes

while emitting a burst of X-rays and visible light. The specific change of RF power for Standing

Wave (SW) and Traveling Wave (TW) are different, but both cause damage to the accelerating

structure, as well as the RF components and sources. In the case of TW, the transmitted power

drops and up to 80% of the incident power is absorbs. On the other hand, the input RF power

is reflected for SW structures [43]. Typically, RF breakdowns are separated into trigger and

secondary. While the secondary breakdowns appear to be caused by the damage caused by the

36

trigger breakdown, the trigger breakdown is understood to be dependent on material properties and

structure geometry. Some of the things that can affect the rate of breakdowns are pulsed surface

heating, peak electric and magnetic fields, peak Poynting vector, hardness of the cavity material,

among others. Peak RF pulsed heating is more important for SW, while RF electric peak fields

matter most for TW structures [44].

When measuring the RF breakdown rates, the RF power into the structures is slowly increased.

The number of breakdowns is recorded and the official breakdown rate is obtained once the rate

stabilizes and remains constant for hours. Variability in the original state of the metal surface, due

to different manufacturing methods and surface preparations, make the initial state of breakdowns

irreproducible. The breakdown rate nonetheless can be reproduced, and it is dependant on the

structure geometry and material, as well as working conditions [43].

It has been empirically found, with some theoretical models agreeing, that the breakdown rate,

𝐵𝐷 𝑅, depends on the accelerating gradient. 𝐺, as 𝐵𝐷 𝑅 ∝ 𝐺30 for many TW structures [44].

Numerous studies have been carried out to increase the accelerating gradient while decreasing the

RF breakdown rate. One study found that heat-treated structures (those that used brazing or diffusion

bonding for manufacturing) result in significantly higher breakdown rates [45]. Other methods like

wielding and electroforming (deposit copper onto aluminum mandrel in an electro-chemical bath

and subsequently removing the aluminum by etching) have been suggested as alternatives for high

gradient accelerating structures [43].

3.2.3 Cold copper acceleration cavity

SLAC National Accelerator Laboratory, in collaboration with the University of California - Los

Angeles and Cornell University, have developed a high gradient X-band cryogenic copper acceler-

ating cavity, Cryo-Cu-SLAC, that has proven to deliver up to 250 MV/m at 45 K with 108 RF pulses,

the highest achieved accelerating gradient with it’s reported RF breakdown rate. Furthermore, this

accelerating system reached the highest accelerating gradient for X-band RF structures maintaining

the same breakdown rate of other structures with lesser accelerating gradient. It must be noted

however that by running the machine at gradients higher than 150 MV/m considerable structural

37

degradation was observed, and therefore a trade-off must be made to preserve its operational lifetime

[44].

Since it is currently understood that breakdowns are caused by movement of crystal defects,

induced by periodic mechanical and/or thermal stress, it was theorized that cryogenic temperatures

could decrease such mobility and stress, thus decreasing the total RF breakdown rate. This was the

primary motivation of the group to develop the Cryo-Cu-SLAC. Reducing the cavity temperature

to below 77 K decreases the RF surface resistance and coefficient of thermal expansion, and it

increases the yield strength and thermal conductivity. These changes decrease the pulsed surface

heating and mechanical stress experienced by the cavity. The decrease in RF surface resistance has

been extensively studied, and in cryogenic copper the phenomena can be described by the theory of

anomalous skin effect [46]. Unluckily, there are not many studies of copper cavities at temperatures

below 100 K with high input RF power corresponding to fields greater than a few MV/m. More

so, one study showed a decrease of the intrinsic quality factor, 𝑄0, with increasing fields at 3 GHz,

77 K and surface electric fields of up to 300 MV/m. This study did not reported the observed

breakdown rate [47].

The structure developed by SLAC, Cryo-Cu-SLAC, is made of three cells with the highest

field in the middle cell, and it is shaped to mimic the properties of longer periodic structures. It

has a small aperture of 𝑎 = 2.75 mm radius and elliptical shaped irises of 2.0 mm thickness. It

was designed to be critically coupled at 96 K with 𝑄0 = 19, 100, under-coupled at 293 K and

over-coupled at 45 K, using a TM01 mode launcher as the RF power coupling. This cavity didn’t

have field probes, as they distort surface fields and degrade high power performance.

Instead,

they measured the input/forward RF power, reflected RF power, and signals from current monitors

that intercept the field emission currents, to determine the electric field in the cavity. Its resonant

frequency is 11.424 GHz at 150 K, and 11.4294 GHz at 45 K [44]. Other parameters are shown in

Table 3.1.

The cavity was in contact with the head of the pulsed cryocooler Cryomech PT-415, and placed

inside a vacuum cryostat. A heat shield of 0.015 inch thick copper foil was placed in between the

38

Parameter
Q-Value
Shunt impedance [MΩ/m]
Hmax [MA/m]
Emax [MV/m]
Eacc [MV/m]
HmaxZ0=Eacc
Losses in a cell [MW]
Peak pulsed heating (150 ns) [K]
a [mm]
a=𝜆
Iris thickness [mm]
Iris ellipticity

293 K
8,590
102.891
0.736
507.8
250
1.093
7.97
86.9
2.75
0.105
2
1.385

45 K
29,000
347.39
0.736
507.8
250
1.093
2.36
21.9
2.75
0.105
2
1.385

Table 3.1 Parameters of periodic accelerating structure with 𝜋 phase advance per cell and dimensions
of the Cryo-Cu-SLAC middle cell at 45 K and 293 K. Fields are normalized to 𝐸𝑎𝑐𝑐 = 250 MV/m
for 𝑓0 = 11.424 GHz. Peak pulsed heating is calculated for a pulse with 150 ns flat gradient [44].

SW structure and the rest of the cryostat, before the current monitor, to prevent contamination of

the accelerating structure from gases of the vacuum of the cryostat. A diagram of the system can

be seen in Figure 3.8. The source for the RF power was a SLAC 50 MW XL-4 klystron, with

repetition rate of 5, 10, or 30 Hz and pulse length of up to 500 ns [44].

The studies with Cryo-Cu-SLAC found that lowering temperatures allowed to sustain larger RF

surface electric fields with decreased probability for breakdowns. Still, 𝑄0 decreased with increase

in the accelerating gradient due to dark current beam loading. The processing history of the cavity

for high power measurements, recordings of the acceleration gradient and breakdown accumulation

as a function of pulses, is presented in Figure 3.9. For the accelerating gradient after 70 × 106

pulses, they measured an RF trigger breakdown rate of 2 × 10−4 /pulse/m and a total breakdown

rate of 2 × 10−3 /pulse/m for a shaped pulse with 250 MV/m, corresponding to 507 MV/m peak

surface electric field, and 150 ns flat gradient. These rates were measured for periods of 1-3 million

pulses where the gradient and rate of trigger breakdowns were relatively constant [44]. We can see

how the breakdown rates for the Cryo-Cu-SLAC compares with measurements of other equivalent

shape (2.75 mm aperture radius) accelerating structures of different materials at room temperatures

is presented in Figure 3.10.

39

Figure 3.8 Solid model of the cryostat zoomed-in. (1) Cold head of cryocooler; (2) current monitor;
(3) brazed metal foil; (4) Cryo-Cu-SLAC; (5) RF flange; (6) thermal shield; (7) Cu-plated stainless
steel waveguide; (8) RF input [44].

3.3

Ion trap

For the discussion of ion traps, we will focus on the pioneering work conducted at The Institute

of Physical and Chemical Research (RIKEN), Japan. We first set the stage by detailing the compo-

nents of RIKEN’s sophisticated electron-ion elastic scattering system, which includes specialized

equipment such as a race track microtron, an electron storage ring, and a unique electron-beam

driven rare isotope separator, among others. We then dive into a detailed exploration of the SCRIT

device, its trapping efficiency, the overall setup and details of the electron beam performance. The

facility’s capacity to achieve a high luminosity for stable ions was only the beginning of their

trajectory to become the first facility to perform nuclear charge radius measurement on short-lived

40

[a]

[b]

Figure 3.9 The processing history for Cryo-Cu-SLAC. [a] Purple is the number of accumulated
breakdowns. Black and red points are the calculated average gradient. Red indicates the time
period where breakdown rate was measured. [b] Zoom in of the same data for the period where
breakdown rate was measured [44].

Figure 3.10 Total breakdown rate as a function of acceler-
ating gradient for different structures [44].

nuclei; isotopes that have been out of the reach for electron scattering for decades. This section

not only showcases RIKEN’s technological prowess, but also underlines the importance of such

advanced systems in pushing the boundaries of nuclear physics research.

The Self-Confining Radioactive Ion Target (SCRIT) system at RIKEN was designed to perform

precise measurements of the atomic charge radii for short-lived radioactive isotopes, and has already

demonstrated it’s capabilities with the first ever measurement of electron elastic scattering off 137Cs

last year. Measurements showed up to 1027 cm−2s−1 luminosity for stable ions with 250 mA electron

beam current, and average luminosity of 0.9 × 1026 cm−2s−1 for unstable isotopes [48],[49]. The

system design consist of a 150 MeV race track microtron (RTM) for electron beam generation, a

repurposed electron storage ring (SR2) donated from the National Institute of Advanced Industrial

41

Science and Technology for electron charge accumulation, an Isotope Separation On-Line (ISOL)

system, an electron-beam driven rare isotope separator (ERIS), a DC to RF converter (FRAC), a

luminosity monitor (LMon) and a window-frame spectrometer for electron scattering (WiSES).

A schematic of the system is presented in Figure 3.11. The storage ring is a second generation

synchrotron, composed of a four dipoles and two straight 3.5 m sections for a total circumference

of 20 m. These components work in synchrony to generate, accelerate, and accumulate electron

beams, as well as to produce and select rare isotopes for experimentation. The ring takes the

150 MeV injected electron beam and can accelerate it up to 500 MeV [7].

Central to the SCRIT system’s operation is its innovative ion trapping mechanism located inside

the electron ring and illustrated in Figure 3.12. This setup allows the system to hold up to 108 ions

in a 500 mm length [50]. When the ions are dumped into the SCRIT device, at energies of about

6 keV, they essentially form a three-dimensional gas target. The recirculating electrons provide

transverse focusing, while simultaneously serving as probe for elastic scattering, and a mirror

potential, described in more details below, provides longitudinal trapping [7]. For an electron beam

current of 250 mA, used normally during data collection, the transverse potential depth produced is

roughly 50 V [48]. Data is taken for 1 to 2 s, after which the ions are dumped and the accumulation

of electrons starts again.

The SCRIT device has one main electrode structure, with dimensions 99(h) × 115(w) ×

780(d) mm3, and two sub-electrodes. The main composite electrode structure is made of a

combination of two 3 mm thick electrodes at the top and bottom, and thin 0.1 mm thick mesh

electrodes on both side walls. In order to allow the scattered electrons to enter the detectors, no

mesh was placed horizontally over the 35 mm center of the vertical direction. Outside here, the

mesh was extended for 8 mm in the horizontal and 5 mm in the vertical direction respectively. The

sub-electrodes, used to produce the 6 kV barrier potential, are made of a 2 mm diameter wire in

a birdcage-shaped racetrack at both ends of the main electrode [50]. The careful design of these

electrodes, including the use of mesh and wire components, facilitates the trapping of ions while

allowing scattered electrons to reach the detectors.

42

Figure 3.11 Schematic of the SCRIT electron scattering facility. A picture of
the SCRIT device in the vacuum chamber is included in the figure [50].

Figure 3.12 Ion trapping concept of SCRIT [7].

During injection, the electron beam power is around 0.4 W at a 2 Hz repetition rate, the

repetition rate is later increased for the ion beam production to 20 Hz, which increases the power to

20 W. The peak current for the microtron is almost 3 mA, while at the storage ring 300 mA current

is maintain for 1 h lifetime. The cooler buncher system consists of six quadrupole electrodes and

a set of einzel-lens and other electrodes for a total length of 950 mm. This buncher converts the

continuous ion beam into a 500 𝜇s pulsed beam [48]. The measured electron beam size is about

2 mm in the horizontal and 0.4 mm in the vertical direction. Once accumulation is completed, the

electron beam is deflected before entering the ring to irradiate a uranium carbide target to produce

43

rare isotopes through photo-fission reactions. The rare isotope beam are selected based on mass

and accumulated at FRAC and subsequently dumped at SCRIT where they are trapped and ready

for data collection. Investigations of the efficiency of the trapping system have yield a maximum

trapping efficiency, as defined in (3.6), of 𝜖𝑡𝑟𝑎 𝑝 = 42%, and a overlap efficiency, defined in (3.7), of

𝜖𝑜𝑣𝑒𝑟 = 42%, for currents between 200 − 230 mA [50].

𝜖𝑡𝑟𝑎 𝑝 = 𝑁𝑡𝑟𝑎 𝑝/𝑁𝑖𝑛 𝑗

𝜖𝑜𝑣𝑒𝑟 = 𝑁𝑐𝑜𝑙/𝑁𝑡𝑟𝑎 𝑝

(3.6)

(3.7)

In these equations, 𝑁𝑐𝑜𝑙 refers to the number of ions colliding with the electron beam, 𝑁𝑡𝑟𝑎 𝑝

refers to the number of trapped ions, and 𝑁𝑖𝑛 𝑗 is the number of injected ions [50].

3.4 Safety considerations

3.4.1 Radiation shielding

When highly energetic particles are bent, accelerated or stopped, they emit different forms

of radiation. Of particular interest to us are electrons with energies above a few tens of MeV.

Almost every component of an electron acceleration system is capable of producing radiation

either by design or due to beam losses. When radiation fields are generated by beam losses

they’re called prompt radiation. When the electrons hit a target material, they will generate an

electromagnetic cascade and those secondary particles will result in prompt radiation. At the

energies here considered, the radiation field mainly consist of photons and neutrons.

For a material made of an element with atomic number 𝑍, the critical energy, that is the

energy at which the electron collision losses equal the radiation losses, can be approximated as

𝐸𝑐 [MeV] = 800/(𝑍 + 1.2). With most accelerators structures made of metals such as copper and

stainless steel, with 6 ≤ 𝑍 ≤ 42, our electrons with energies above 150 MeV will be well beyond this

limit. This implies that the electrons colliding with the material will loose energy mostly through

bremsstrahlung photon generation. These photons are generated after an average distance 𝜒0

44

[g/cm2] (radiation length) and will take about 63% of the initial electron’s energy. After a distance

of about 9/7𝜒0, the high energy photon will produce an electron-positron pair and both of these will

generate bremsstrahlung photons again after another 𝜒0 distance, thus the cascade effect. This will

continue until the energy of the produced electrons fall bellow 𝐸𝑐 and the dominating energy loss

process becomes Compton scattering. Photons from bremsstrahlung radiation is the most important

hazard at the first line shielding and its emission is forward peaked. The approximate angle at which

the intensity drops to half of that at 0°is approximately 𝜃1/2 = 100/𝐸 for an electron energy 𝐸 in

MeV. Neutron radiation, on the other hand, does not usually have a predominant angle of radiation

emission. When the nucleus absorbs a photon and subsequently emits a neutron, a process called

giant resonance production, all the angular information is lost, leading to nearly isotropic radiation

fields. Beyond 140 MeV, production of pions and other particles becomes energetically possible

and must be considered as well [51].

There are numerous codes, both open source as well as available for purchase, that can help with

the design of appropriate shielding. Because of the random nature of scattering events involved

in the generation of photons inside the material, the Monte-Carlo technique is often employed to

simulate the radiation production. Some of the available codes that employ three -dimensional

geometries are FLUKA[52], EGS[53], MCNPX[54] and MARS[55]. Other simpler codes that

make use of semi-empirical models also include SHIELD11[56] and SKYSHINE[57].

3.4.2 High voltage insulation

In order to maintain a vacuum of < 1 × 10−9 Pa and bear the high temperatures involved in

the baking process, Alumina alloys, with concentrations ranging from 92% up to 99.5%, is the

most commonly used choice for insulation. We present the properties of different concentrations

in Table 3.2. The insulator must be brazed to a metal ring and then either brazed or welded to a

vacuum vessel or flange for vacuum preservation. Much consideration must be made to the brazing

geometry and material used for joint.

One can do a rough estimate of the needed insulation diameter by using the formula for a coaxial

cylinder, (3.8), although simulation is needed for the final design.

45

Property
Compressive Strength [MPa]
Tensile Strength [MPa]
Young’s Modulus
Thermal Expansion [40-800 °C, ×10−6 °C−1]
Thermal Conductivity @ 20 °C [W (m K)−1]
Dielectric Constant
RF Loss Factor (×10−4) @ 1 MHz
Volume Resistance @ 20 °C [Ωcm]
Volume Resistance @ 300 °C [Ωcm]

92% 96% 99% 99.5%
2300
2350
180
280
7.8
18
9.0
54

2160
241
360
8.0
29
9.9
20

193
320
7.9
24
9.4
38

370
8.0
32
9.9
10

> 1014 > 1014 > 1014 > 1014
1013
1012

1010

1010

Table 3.2 Properties of alumina with varying alumina content [19].

𝐸 =

𝑉/𝑅𝑖𝑛𝑛𝑒𝑟
𝑙𝑛(𝑅𝑜𝑢𝑡𝑒𝑟/𝑅𝑖𝑛𝑛𝑒𝑟)

(3.8)

where 𝑉 is the voltage on the central conductor, 𝑅𝑜𝑢𝑡𝑒𝑟 and 𝑅𝑖𝑛𝑛𝑒𝑟 are the outer and inner radii for

the conductor respectively, and 𝐸 is the electric field on the inner conductor. The outer conductor

is part insulator and part conductor [19].

46

CHAPTER 4

PRELIMINARY STUDIES

4.1 Electron accelerator measurements and modeling

In order to better understand electron accelerator systems, we have studied the Accelerator

Test Facility (ATF) at Brookhaven National Laboratory (BNL). We obtained the blueprint of the

original design as well as experimental images of the electron beam’s transverse profile at different

locations. With the first information, we constructed an online version of the system using the

open software for particle accelerator simulations OPAL. We then compared the results from this

simulation to the experimental images to verify the validity of the simulation. Although some

numbers seemed to be close to the actual data, further tuning of the simulation needs to take place

before predictions from the model are reliably accurate at all instances. From the images, not only

the radius and shape of the beam could be seen, but also the emittance was measured using four

images from subsequent beam monitors. Some samples were chosen for these sections to showcase

general trends. The reader is referred to Appendix A if interested in observing more data.

4.1.1 Capabilities and specifications of ATF

The Accelerator Test Facility (ATF) electron linac at Brookhaven National Laboratory (BNL)

is capable of delivering an electron beam of energies up to 65 MeV, in 0.1 − 2.0 nC bunches

with repetition rate of 1.5 Hz. This beam’s normalized emittance has been measured to 1 µm

at 0.3 nC. As can be seen in Figure 4.1 from right to left, the system has a photoinjector gun,

followed by two S-band acceleration cavities, a beam diagnostics and focusing section, and at last

the experimental hall with different beam-lines.1 The photonic gun matches the linac section, that

is, the copper-elliptical 1.6 cell is operated in the S-band region (2.586 GHz), at a base pressures

of 1 × 10−10 Torr [58].

In Figure 4.2, we can see more detailed pictures of the different sections of this system. At

the top, Figure 4.2 [a], we have the specific section that we are simulating for, and where all the

1This image is from a previous model that hasn’t been updated with the latest modifications to the experimental
hall. Currently, the beam-line at the bottom has been removed, and the middle beam-line bending dipole has been
redirected towards the lower left corner.

47

experimental data was taken. The acceleration sections consist of two TW linear acceleration

structures from Stanford Linear Accelerator Center (SLAC). The drive power for both cavities

comes from a single XK5 klystron tube, also from SLAC. This klystron provides macro-pulses

of 3 𝜇s with up to 25 MW, 1 to 6 times per second, and those macro-pulses accelerate electron

micro-pulses with repetition rates of up to 81 MHz. The other klystron present in the drawings

provides the driving power for the SW photoinjector electron gun and it is from Thales. The gun

area is magnified in Figure 4.2 [b]. Electrons are extracted from a copper cathode, located in

the center of the removable back flange, using a 266 nm laser at around 30 𝜇J. The QE for this

arrangement has been measured to be around 0.01 %. The Thales klystron is connected to the

1.6 elliptical copper cell gun cavity as seen in the image. It can deliver up to 10 MW, although

it is typically run at 2.5 MW in 2.5 𝜇s repetition rate, and has a designed accelerating field of

100 MV/m. Following the gun cavity is a 6 coil pack solenoid that compensate for initial thermal

emittance while keeping zero longitudinal magnetic field at the cathode. Through this solenoid runs

a 100 A current cooled by water. After the solenoid there is one electron beam monitor right before

the accelerator cavities. This monitor is called "LPOP", where L stand for low energy and the POP

designation was given in the software used to manipulate these cameras. Lastly, Figure 4.2 [c] at

the bottom shows the area right after the beam is accelerated. This section is used for emittance

optimization and beam manipulation before it is sent to the experimental area where the users add

their probes for experiments. Depicted in this image are three "focusing triplets", Q1, Q2 and Q3.

Each of these triplets consist of two 10cm long quadruple magnets, with one 20cm long quadrupole

in between them. While the data was taken for the analysis presented here, Q1 and Q2 were off,

thus creating a long drift space in which emittance could be calculated. In this picture, there are six

electron Beam Position Monitors (eBPM) named "HPOP’s" where the "H" stands for high energy.

The bunch compressor shown in the image shortens the beam longitudinal spread to around 100 fs

with up to 1 kA peak current.

48

Figure 4.1 Schematic layout of the ATF full system. From right to left, the blue box encapsulates the
electron gun, where the electrons are emitted from cathode and focused once, after which follows
the acceleration cavities in the green rectangle. These, and the driving klystrons, are all part of the
bigger purple rectangle of all the components included in the OPAL simulation. At the lower left
corner, the experimental hall is highlighted in a yellow rectangle.

4.1.2 Experimental measurements

Using the Charge-Coupled Device (CCD) cameras with phosphor screens at different locations

of the beamline, observed in Figure 4.2, we obtained current intensity images for the transverse

profile of the electron beam. These images were then cut to the approximate radius of the beam

using an automated python script (presented in Appendix B), in some cases ignoring halo effects

and the tail. We determined the edges of the beam by summing over all the rows and columns

of each image, thus projecting the beam in the 𝑦 and 𝑥 axis respectively, and cutting wherever

the summed intensity was lower than 65% of the average highest intensity. This method at this

intensity empirically proved to be very successful at determining the radius for the images after

acceleration used for the emittance calculation, but failed to provide an accurate representation of

the beam edges for some images taken at LPOP, before acceleration, when the shape of the beam

was particularly dim. For that reason, we modified the code to specifically treat images at LPOP1

with a reduced threshold of 35%. Some images were chosen to represent the common trends and

are presented in Figure 4.3 with their respective trimmed version. The lines in the images were

drew into the CCD cameras by the operators to help them during alignment procedures. The

estimated radius obtained with the code for the beam before acceleration ranged from 0.52 mm to

49

[a]

[b]

[c]

Figure 4.2 Zoomed-in areas of the ATF system. [a] Area simulated using the OPAL soft-
ware and the locations at which experimental data was collected using eBPM (HPOP’s).
[b] Components of the electron gun, from extraction up to before acceleration. [c] Sys-
tems for beam manipulation and observation after the RF acceleration of the beam.

50

up to 10 mm, while the range after acceleration was from 0.13 mm (only in the vertical axis while

being more prolonged in the horizontal) and up to 1.4 mm.

We observed that to focus the beam at location LPOP, the current solenoid needed to be

increased to 120 A or 125 A, the specific one dependant upon the phase of the RF gun. For

this location, we analyzed images with RF phases ranging from -19° up to 23°, crossing through

0°. The requirement of either 120 A or 125 A for precise focus did not depended linearly to the

phase, but instead oscillated between these two numbers as the phase was varied. As the solenoid

current deviates from this number, the beam radius increases drastically in size, with decreasing

changes maintaining a more homogeneous shape while increasing changes made the beam look

more defused. This is because as the current of the solenoid is increased, the location at which

the beam is focus gets closer and after which it defocus again. For the beam to be focused in the

vicinity of locations HPOP1 through HPOP5, the solenoid current needed to be reduced to around

100 A. In these images post-acceleration, we can see the projection of the "tail" (examples shown in

Figures 4.3 [c] and [d]). Such tail is caused by wakefield effects in the acceleration sections. This

effect is mainly created at the irises of the linac section, which are connecting around 100 cells of

10 cm in diameter through holes of about 10 mm diameter. As the beam enters the linac, it posses a

longitudinal extension of a few hundred femto-seconds. This not only causes different particles to

perceive different RF fields, but as the beam crosses each iris, the field from the front-most part of

the beam is reflected in the front wall of the cell and pushes the back-end particles further away. If

we could look at the beam from a longitudinal cross-section, one would observe a "fish"-like shape.

The upmost front part of the beam would be focus into a Gaussian-like distribution, followed by

a thin connection and finally a dispersed extended tail of particles that are being lost due to this

wakefield effect. More images showcasing all of the observed attributes are presented in Appendix

A.

Another interesting set of data obtained from ATF were alignment images. During normal

operation, a roughly homogeneous circular laser profile, depicted in Figure 4.4 [a], is used to create a

correspondingly homogeneous circular laser beam. If the laser is not directed exactly perpendicular

51

[a]

[b]

[c]

[d]

Figure 4.3 Transverse beam profiles of the electron system at ATF
unprocessed (left) and after process for beam radius trimming (right),
with an RF voltage of 120 V in the gun. [a] and [b] were both taken
from the first camera, LPOP1. The former was under an RF phase of
-9.0°, solenoid current of 105 A and laser power of 30.44 𝜇J; while
the later was under the conditions of -19°RF phase, solenoid current of
120 A and laser power of 31.95 𝜇J. [c] and [d] were taken at locations
HPOP2 and HPOP4 respectively, both with RF phase of 31°, solenoid
current of 100 A and laser power of 30.72 𝜇J. Intensity scale is for
beam current and is given in arbitrary units relative to self.

52

to the cathode surface, the image will be distorted and will cause unwanted deformations in the

beam shape. In order to verify the alignment of the laser, a mask is placed over the laser [b]) and

beam is checked for deviations. As can be seen in Figure 4.4 [b], this mask is made of various

points that make a circle with line toward the center. If the laser is coming at the cathode with an

angle, some points of the circle would appear elongated while the other side would look shortened.

In Figure 4.5 we can see a reasonable symmetric beam that was focused onto monitor LPOP by

varying the solenoid current.

Figure 4.4 Images of the laser illuminating the cathode at ATF. [a] Is the normal operation
distribution, while [b] is the mask used during the alignment process.

Figure 4.5 Alignment process of ATF. Images were taken with laser power around 1.5 𝜇J, RF
phase of 23° at 120 V in the gun, and solenoid currents of [a] 135 A, [b] 130 A and [c] 125 A.
All images come from location LPOP.

4.1.3 Emittance calculation and comparison

We saw before that the beam emittance is by definition the area enclosed by the beam in the

phase space of space-angle, mathematically expressed as in (4.1). Here 𝛾, 𝛼 and 𝛽 are constants

53

known as the twiss parameters.

𝜖𝑥 = 𝛾𝑥2 + 2𝛼𝑥𝑥′ + 𝛽𝑥′2

(4.1)

According to [59], one can measure the beam emittance using images from the transverse

profile of the beam at three different locations where the beam wasn’t accelerated or magnetically

manipulated, since any electric or magnetic manipulation can and will change the beam’s emittance

and the equations herein used to describe the beam’s emittance will not apply. Equation (4.2) gives

the beam’s width based on the measured emittance 𝜖, and beta function of the linac 𝛽(𝑧).

𝜎(𝑧) = √︁𝛽(𝑧)𝜖

(4.2)

Suppose that we have a electron acceleration system for which we would like to determine the

beam emittance that follows the diagram in Figure 4.6. To do the calculation at the location of the

dark blue vertical line, we would need three subsequent beam profile monitors at a known distance

𝐿𝑎, 𝐿𝑏 and 𝐿𝑐 from the reference point,with measured beam width 𝜎𝑎, 𝜎𝑏 and 𝜎𝑐 respectively.

These measurements are taken in drift space, where the beam is not affected by any external electric

nor magnetic field. We can use to our advantage the relation (4.3), and solve for two of the twiss

parameters and the emittance in the linear equations (4.4).

𝛾 =

1 + 𝛼2
𝛽

𝑎 = 𝛽𝜖 − 2𝐿𝑎𝛼𝜖 + 𝐿2
𝜎2

𝑎𝛾𝜖,

𝜎2
𝑏 = 𝛽𝜖 − 2𝐿𝑏𝛼𝜖 + 𝐿2

𝑏𝛾𝜖,

𝑐 = 𝛽𝜖 − 2𝐿𝑐𝛼𝜖 + 𝐿2
𝜎2

𝑐𝛾𝜖

(4.3)

(4.4)

Using this approach, we were able to calculate the emittance of the beam at ATF. To find the

radius of the beam, we trimmed the images using the sum of the intensities in each column and row

for the y-axis and x-axis projections respectively as described earlier. We evaluated the emittance

in both axes, 𝜖𝑥 and 𝜖𝑦, as well as for the diagonal, 𝜖𝑚𝑠, where the radii for 𝜖𝑚𝑠 is the diagonal

54

Figure 4.6 Diagram of measurement scheme for emittance in an electron linac. The light blue
shade represents the beam’s profile, the dark blue vertical line is the reference point where the
emittance will be calculated, and the red vertical lines are three positions with beam profile
monitors from which we will acquire the data.

formed between the x and y axes, as described in (4.5). We took four consecutive beam profile

images from HPOP2, HPOP3, HPOP4 and HPOP5, and made parabolic fittings for (4.6) to obtain

the real or geometric emittance. Here We know the actual values of 𝜎𝑖 at the locations 𝑧𝑖 and we

are using the fitting to find the predicted emittance 𝜖, and the value and location of the minimum of

the beta function, 𝛽0 and 𝑧0 respectively. Some results of these fittings are presented in Figure 4.7.

We obtained normalized emittance ranging from 0.767 mm-mrad to 5.63 mm-mrad, in agreement

with previous measurements [60].

𝑟𝑚𝑠 =

√︃

𝑥 + 𝑟 2
𝑟 2

𝑦/2

(cid:118)(cid:117)(cid:116)

(cid:32)

𝜖 𝛽0

1 +

𝜎𝑖 (𝑧𝑖) =

(cid:33)

𝑧𝑖 − 𝑧0
𝛽2
0

(4.5)

(4.6)

4.1.4 Simulation using OPAL framework

The OPAL-t open source tool was used to simulate charged electron particles from the ATF

linac. The software records the bunch spatial distribution, the energy spread, the emittance, among

other parameters [61]. In Figure 4.8, we show the model used to simulate ATF electron beamline.

It included the gun, the 6 coil solenoid, the two acceleration cavities, the first focusing triplet and

the CCD cameras from LPOP up to HPOP2. For the emission distribution, we used a Gaussian

flattop of 2.8 mm radius. The initial phase space distribution in the 𝑥-𝑝𝑥 plane is portrayed in

55

[a]

[b]

[c]

Figure 4.7 Parabolic fittings for measurements of the radii in [a] the x-axis, [b] the y-axis, and
[c] the diagonal radius calculated from (4.5), for the operational conditions of RF phase of 31°,
RF voltage of 120 V at the gun and a solenoid current of 100 A for space charge correction.
Notice that the x-axis in this graphs, ’s’, is the position in the beamline in meters, while the
y-axis ’sigma’ refers to the electron beam radius also in meters.

Figure 4.9 [a]. The simulation accounts for the space charge effect created by the electron beam

bunch using self-consistent electrostatic approximation, and determines the equation of motion

from the relativistic Lorentz equation of these particles as they travel through the electromagnetic

fields created in the acceleration system. OPAL has tools to account for wakefield effects, but we

did not incorporated them into our simulation because of the intense computing power required for

the calculation, which would have extended by a factor of 15 the time for each simulation.

We used 3 different field maps for this simulation; two "AstraDynamic" for the electron gun and

the acceleration sections, and one "AstraMagnetoStatic" for the solenoid in the gun. These field

maps describe the magnetic or electric profile of the element along the beam axis, and are used by

OPAL to calculate a Fourier series from which values off-axis can be determined. For all the fields

in this simulation, only the first 𝑁 = 40 Fourier terms were used. The field maps are generic ones

taken from online examples. The parameters shown in Table 4.1 were used for the simulation from

56

Figure 4.8 Visual representation of the beamline created using OPAL. From left to right: the
first black rectangle is the cathode, the yellow circle is the gun acceleration, the blue box is the
solenoid, the flags are CCD cameras, the two joint thin black rectangles are the acceleration
sections (compacted) and the later three red rectangles are more quadrupole magnets.

which all the images in this section were taken.

Parameter
RF gun phase
RF gun voltage
Linac phase
Linac power
Beam current
Solenoid current
Quadrupole current

Value
3.5
100
19
10
2.46×10−8
96
7

Units
degrees
volts
degrees
volts
amperes
amperes
amperes

Table 4.1 Variable parameters used for the OPAL simulation.

In Figure 4.9 [b] we can see the energy gain of the particles along the beamline with the end

of the linac section marked with a dash line. This graph was the first sign that the field maps used

for the simulation may not be a realistic representation of the system. We can see that the beam

gains energy after the linac sections, particularly, the quadrupoles seem to be providing acceleration

to the particles. This was proven by changing the current; with increasing current we observed

increasing energy. On the other hand, the energy gained by the particles inside the acceleration

cavities accurately represents the energy of the beam measured at the facility.

57

[a]

[b]

Figure 4.9 Simulation using OPAL codes for [a] initial phase space and [b] the energy gain of
the electron beam. In [b] the vertical blue dash line marks the end of the acceleration sections.

We can observe the transverse beam profile images of the simulated electron beam at different

locations in Figure 4.10. We first notice that the image after acceleration doesn’t posses any tail or

halo’ it is perfectly symmetric. This is because space charge effects only cause the beam to spread,

and all deformations from a circle come from the wakefield effect ignored in this simulation. The

second thing to notice is that the beam profile after exiting the solenoid (this corresponds to the

location of LPOP) has immediately been compressed. In real life, we know that with a current

of 96 A the beam should be focusing at a minimum radial shape around HPOP4, much further

down the beamline. This is the second clue that suggested errors within the magnetic field maps.

Furthermore, notice that right after exiting the quadrupoles the beam profile has increased almost

four times its size. This is the contrary effect of what is expected. Quadrupole fields are supposed to

focus the beam in one direction while defocusing in the perpendicular axis. It is well known, though,

that the focusing effect is much stronger than the defocusing, such that after various successive

quadrupoles positioned in perpendicular to each other, the net result should be a compressed

beam. The emittance of the beam was estimated to be around 51 mm mrad before acceleration

and 14 mm rad after acceleration. This discrepancy with experimental results is believed to be

correlated to the magnetic fields as well since we know that solenoids and quadrupoles are used for

emittance compensation.

Upon concluding this investigation we have decided that refinement of the magnetic field

maps for the elements in this simulation needs to take place in order for the simulation to match

58

[a]

[c]

[b]

[d]

Figure 4.10 Simulation using OPAL codes for the beam profile images [a] before
entering the solenoid, [b] right after exiting the solenoid, [c] after exiting the linac
sections and [d] right after exiting the first focusing quadrupole triplet (Q1).

experimental results. Other simulation software will be used to generate them.

4.2

Ion trap and design strategy

4.2.1 Capabilities and specifications of D-line

FRIB is capable of producing ion beams from an assortment of elements, ranging from H

to 238U, with energies of over 200 MeV in current beams as high as 3.7 p𝜇A2 of approximately

0.5 mm radius. This beam is produced from the collision of a primary-very-fast stable-isotope beam

(generated using electron-cyclotron resonance sources) with a selected (also stable and usually light,

such as beryllium or carbon) target which causes projectile fragmentation, and the resulting unstable

2"The electrical current in nanoamperes (10−9 A) that would be measured if all beam ions were singly charged"

[62].

59

isotopes follow particle and energy selection processes [63]. After selection, the rare isotopes can

be stopped in helium gas-cells for low energy experiments. For our interest, we take a closer look at

a later portion of the beamline named D-line, where the ion beam’s energy is significantly reduced

to about 30 keV and can be cooled to a momentum spread as small as Δ𝑝

𝑝 ≈ 10−5.

4.2.2 Calculations for rare isotope and electron interactions

The number of ions per unit of time, 𝑁𝜏, that can be confined inside the trap can be estimated

from the known parameters of the electron beam. If we assume that the electron beam is fully

neutralized with the ions in the target, such that the total charge for each is the same, we can

calculate 𝑁𝜏 as in (4.7). In this equation, 𝐼𝑒 is the electron current, 𝑣 is the velocity of the beam, 𝐿

is the length of the trap , and 𝑒 is the unit charge of an electron 𝑒 = 1.602 × 10−19 C.

𝑁𝜏 =

𝐼𝑒 𝐿
𝑒𝑣

(4.7)

If our electrons have energies of about 150 MeV, they are relativistic and we can calculate their

velocity from (2.36) giving approximately 𝑣 𝑐, and from this one can estimate the density capacity

of the trap. We present such estimates in Table 4.2 for an electron current ranging from 1 mA to

1 A, assuming 100% efficiency for a trap length of 0.5 m (the length of SCRIT) and assuming that

all of our ions are confined in a 2.5 mm radius from the center of the electron beam.

Ion density in 1/cm2 Luminosity in 1/s/cm2

Current in A
0.001
0.01
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

# of ions
1.04 × 1007
1.04 × 1008
1.04 × 1009
2.08 × 1009
3.12 × 1009
4.16 × 1009
5.20 × 1009
6.24 × 1009
7.28 × 1009
8.32 × 1009
9.37 × 1009
1.04 × 1010

# of electrons
6.24 × 1015
6.24 × 1016
6.24 × 1017
1.25 × 1018
1.87 × 1018
2.50 × 1018
3.12 × 1018
3.74 × 1018
4.37 × 1018
4.99 × 1018
5.62 × 1018
6.24 × 1018

5.30 × 1007
5.30 × 1008
5.30 × 1009
1.06 × 1010
1.59 × 1010
2.12 × 1010
2.65 × 1010
3.18 × 1010
3.71 × 1010
4.24 × 1010
4.77 × 1010
5.30 × 1010

3.31 × 1023
3.31 × 1025
3.31 × 1027
1.32 × 1028
2.98 × 1028
5.29 × 1028
8.27 × 1028
1.19 × 1029
1.62 × 1029
2.12 × 1029
2.68 × 1029
3.31 × 1029

Table 4.2 Predicted luminosity and number of ions for a given electron beam current.

60

According to [64], the electric space charge potential produced by an electron beam can be

calculated by (4.8), with the linear charge density as in (4.9). A description of the variables used

in these equations, with the assumed values, can be found in Table 4.3, and the resulting potential

is presented in Figure 4.11.

𝜙 =

𝑄𝑒
2𝜋𝜖0





(cid:104) 1
2

(cid:16)1 − 𝑟 2

𝑏2

(cid:17)

+ ln (𝑎/𝑏)

(cid:105)

,

0 ≤ 𝑟 ≤ 𝑏,

ln (𝑎/𝑟) ,

𝑏 < 𝑟 ≤ 𝑎,

𝑄𝑒 =

−𝐼𝑒
𝛽𝑐

≤ 0

(4.8)

(4.9)

Parameter
Electron beam radius 𝑏
Beam tube radius 𝑎
Permittivity of free space 𝜖0
Electron current 𝐼𝑒
Relativistic electron velocity 𝛽
Speed of light 𝑐

Unit
m
m

Value
0.005
0.25
8.85 × 10−12 F/m
0.5
1.00
3.00 × 108

A
unit-less
m/s

Table 4.3 Variables used for the calculation of the electric potential produced by an electron beam
of 150 MeV.

Figure 4.11 Predicted electric potential inside the ion trap.

61

CHAPTER 5

SUMMARY AND OUTLOOK

The Facility for Rare Isotope Beams (FRIB) stands at the forefront in the quest to deepen our under-

standing of the nuclear structure, by providing beams of isotopes with sub-milliseconds lifetimes

in the greatest densities ever seen. This thesis aimed to investigate the feasibility of integrating an

electron beam with FRIB’s capabilities to perform electron elastic scattering experiments, thereby

contributing to the precise measurement of nuclear charge radii and the charge density distribution

of rare isotopes, complementing the existing reach of its BECOLA facility. A comprehensive re-

view of the technology necessary to perform electron elastic scattering off online ions was presented

in pair with the theory of the experiment. We discussed the details from the cathode, explored the

option of using either photoemission or thermionic guns, presented the latest advanced compact RF

acceleration cavity, explained the technology implemented at RIKEN for the simultaneous trapping

and scattering of electrons off ions, and presented information of the science that allows precise

measurement of the angular and energy distributions. High voltage insulation and radiation safety

aspects were also discussed since ensuring the safety of the system and the personnel is always a

priority.

The study of the BNL electron linac test facility helped us understand the structural and

operational requirements of an electron beam facility. With the measurements of the transverse

beam profile at multiple locations, we were able to extract the geometrical and normalized emittance.

The measurement of the emittance, as well as the reconstructed beam radius, were in agreement

with previous observations. The effect of the RF phase on the beam was evaluated from the profile

images. Our computational model needs further improvements, one of which being a revision of

the field maps. Additional upgrades will include wakefield effects and RF cavities with higher

accelerating gradient such as the cold copper cavity developed by SLAC.

We also analyzed the electron-ion trapping potential and made an estimate for the achievable

luminosity of the system. The ability to trap ions effectively and achieve high luminosity is vital

for conducting successful scattering experiments. Our findings indicated that with an electron gun

62

that can provide 100 mA, it is possible to attain a luminosity of up to 3 × 1027 cm−2s−1, sufficient

for high-precision measurements of absolute nuclear charge radius. In order to achieve the required

luminosities, thermionic guns offer the best beam properties (high average currents with small

radius) for a one-pass system. If a photocathode gun is desired, for the potential to be upgraded

to allow a spin polarized electron beam, a re-circulation scheme is required in order to obtain the

necessary average current. The cold copper structure for particle acceleration developed by SLAC

would be the best fit for a compact system that could achieve the desired energies since it has

demonstrated the highest accelerating gradients to date.

It is possible to use other accelerating

structures, but these would lead to a higher space requirement since their accelerating gradients

are significantly smaller. As for the interaction region, a modified version of the SCRIT concept

developed by RIKEN to trap the ions while simultaneously carrying out both scattering experiments

and laser spectroscopy is a good starting point.

Figure 5.1 Possible schematic concepts for the electron acceleration systems that would allow
charge radius and charge density distribution experiments with radioactive isotopes at FRIB. At
the top is the idea of a single-pass system and below the recirculating electron beam for current
accumulation. Courtesy of Paul Gueye.

Figure 5.1 depicts two schematics of the concept developed in this thesis, showing the essential

components of the system while detectors, cooling systems, power supplies and other supporting

equipment are omitted. The dual use of laser spectroscopy and electron scattering will provide

63

cross-reference measurements. The diagram at the top shows a single pass before the electrons are

dumped. This design can only be achieved using thermionic guns because of the high average current

required to observe beyond the first minimum of the form factor. If a photoemission gun is chosen,

one would need to accumulate enough electrons before performing the experiments in order to have a

sufficiently high luminosity. This concept of re-circulating electrons is implemented at RIKEN and

also with slightly different implementation at the Continuous Electron Beam Accelerator Facility

(CEBAF) at Jefferson National Laboratory. Using energy recovery linacs would be ideal for either

single-pass or re-circulating systems in order to reduce cost.

The precise measurement of nuclear charge radii for radioactive isotopes will fill a critical gap

in our current knowledge of the nuclear matter, providing essential data for theoretical models.

These models are foundational for understanding various nuclear phenomena and for applications

in nuclear medicine, astrophysics, and energy research. While this thesis lays the groundwork

for integrating an electron acceleration system with FRIB, several avenues for future research

and development remain.

In the future, the computational model must be expanded to achieve

an electron beam with energies above 150 MeV, and potentially the design of the gun should be

modified to accommodate for higher currents and subsequently higher brightness. After the electron

beam is modelled with the desired parameters, the next step would be to build a small prototype of

the ion trap, and potentially the electron gun with minimum energy. This will involve collaborative

efforts with national labs and universities, and multi-disciplinary teams to overcoming complex

technical hurdles.

64

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70

APPENDIX A: BEAM PROFILE IMAGES

Figure A1 Images for the beam profile at location LPOP with 120 V gun voltage and
10° phase, an average laser power of ≈30 𝜇J and varying solenoid current as indicated
in the title of each image.

71

Figure A2 Images for the beam profile at location LPOP with 120 V gun voltage and
-19° phase, an average laser power of ≈30 𝜇J and varying solenoid current as indicated
in the title of each image.

72

Figure A3 Images for the beam profile at location LPOP with 120 V gun voltage and
0° phase, an average laser power of ≈30 𝜇J and varying solenoid current as indicated in
the title of each image.

73

Figure A4 Images for the beam profile at location HPOP1 with 120 V gun voltage and
25° phase, an average laser power of ≈28 𝜇J and varying solenoid current as indicated
in the title of each image.

74

Figure A5 Images for the beam profile at location HPOP1 with 120 V gun voltage and
35° phase, an average laser power of ≈28 𝜇J and varying solenoid current as indicated
in the title of each image.

75

Figure A6 Images for the beam profile at location HPOP2 with 120 V gun voltage, an
average laser power of ≈28 𝜇J, and varying solenoid current and RF phase as indicated
in the title of each image.

76

Figure A7 Images for the beam profile at different locations indicated in the title of each
image with 120 V gun voltage and 31° phase, an average laser power of ≈30 𝜇J and 96 A
solenoid current.

77

Figure A8 Images for the beam profile at different locations indicated in the title of each
image with 120 V gun voltage and 31° phase, an average laser power of ≈30 𝜇J and
100 A solenoid current.

78

Figure A9 Images for the beam profile with a masked laser (part of the alignment process)
at location LPOP with 120 V gun voltage, an average laser power of ≈1.8 𝜇J, and varying
solenoid current and RF phase as indicated in the title of each image.

79

APPENDIX B: CODES

OPAL template file for simulation

Title, string="ATF Gun and Linac";

REAL bf = 2856.0;
REAL IHQ1 = 7;
REAL IHQ2 = -6;
REAL IHQ3 = 6;

REAL gunphase = 0.05236;
REAL accelphase = 25.0;
REAL accelvolt = 120.0;
REAL beamcurrent = 2.459345657376521e-08;
REAL solcurrent = 96.0;

"OPT1": option,autophase=4.0,psdumpfreq=50.0,statdumpfreq=1.0,version ⌋
↩→

=10900.0;

"D1": DRIFT,l=0.18;
"D2": DRIFT,l=0.042;
"D3": DRIFT,l=0.05;
"D4": DRIFT,l=0.20;
"D5": DRIFT,l=0.17;
"D6": DRIFT,l=0.20;
"D7": DRIFT,l=0.20;
"D8": DRIFT,l=0.17;

"LPOP1": MONITOR, OUTFN="LPOP1f";
"HPOP1": MONITOR, OUTFN="HPOP1f";
"HPOP2": MONITOR, OUTFN="HPOP2f";

"ATF_RFGUN": RFCAVITY,fmapfn="DriveGun.T7",freq=BF,l=0.15,lag=gunphas ⌋
↩→

e,type=STANDING,volt=100.0;

"ATF_SOLENOID":
↩→

SOLENOID,fmapfn="ATFsolenoidFmap.T7",ks=solcurrent,l=0.42756;

"GUNSOURCE": SOURCE;

80

"LINACRF_T1": TRAVELINGWAVE,apveto=true,designenergy=25.0,fmapfn="tws ⌋
↩→

_slac_s.T7",freq=2856.0,l=3.0,
lag=accelphase,mode=0.6666666666,numcells=84.0,volt=accelvolt;

↩→

"LINACRF_T2": TRAVELINGWAVE,apveto=true,designenergy=55.0,fmapfn="tws ⌋
↩→

_slac_s.T7",freq=2856.0,l=3.0,
lag=accelphase,mode=0.66666666,numcells=84.0,volt=accelvolt;

↩→

"HQ1": QUADRUPOLE, L=0.1, K1=IHQ1*1.2264;
"HQ2": QUADRUPOLE, L=0.2, K1=IHQ2*1.2264;
"HQ3": QUADRUPOLE, L=0.1, K1=IHQ3*1.2264;

"GUNSOURCE#0": "GUNSOURCE",elemedge=0;
"ATF_RFGUN#0": "ATF_RFGUN",elemedge=0;
"ATF_SOLENOID#0": "ATF_SOLENOID",elemedge=0.15;
"D1#0": "D1",elemedge=0.57756;
"LPOP1#0": "LPOP1",elemedge=0.75756;
"D2#0": "D2",elemedge=0.75756;
"LINACRF_T1#0": "LINACRF_T1",elemedge=0.79956;
"D3#0": "D3",elemedge=3.79956;
"LINACRF_T2#0": "LINACRF_T2",elemedge=3.84956;
"D4#0": "D4",elemedge=6.84956;
"HPOP1#0": "HPOP1",elemedge=7.04956;
"D5#0": "D5",elemedge=7.04956;
"HQ1#0": "HQ1",elemedge=7.21956;
"D6#0": "D6",elemedge=7.31956;
"HQ2#0": "HQ2",elemedge=7.51956;
"D7#0": "D7",elemedge=7.71956;
"HQ3#0": "HQ3",elemedge=7.91956;
"D8#0": "D8",elemedge=8.01956;
"HPOP2#0": "HPOP2",elemedge=8.18956;

Injector: LINE=("GUNSOURCE#0","ATF_RFGUN#0","ATF_SOLENOID#0","D1#0"," ⌋
↩→

LPOP1#0","D2#0",
"LINACRF_T1#0","D3#0","LINACRF_T2#0","D4#0","HPOP1#0","D5#0","HQ1 ⌋
#0","D6#0","HQ2#0","D7#0","HQ3#0","D8#0","HPOP2#0");

↩→

↩→

//"FILTR1": FILTER, TYPE="RELATIVEFFTLOWPAS", THRESHOLD=0.01;
//"WAKEFIELD": WAKE, TYPE="1D-CSR-IGF", NBIN=20, FILTERS="FILTR1";

"FS1": fieldsolver,bboxincr=1.0,fstype="FFT",mt=32.0,mx=16.0,my=16.0, ⌋
↩→

parfftx=true,parffty=true;

81

"DIST1": distribution,ekin=0.4,emissionmodel="ASTRA",emissionsteps=20 ⌋
↩→

.0,emitted=true,
maxstepssi=500.0,nbin=5.0,sigmat=0.0006,sigmax=0.0028,sigmay=0.00 ⌋
28,tfall=2.45e-12,
tpulsefwhm=5.04e-12,trise=2.45e-12,type="GUNGAUSSFLATTOPTH";

↩→

↩→

↩→

"BEAM1": beam,bcurrent=beamcurrent,bfreq=BF,npart=45000.0,particle="E ⌋
↩→

LECTRON",pc=P0;

"T1": track,beam=beam1,dt=1e-11,line=Injector,maxsteps=30000.0,zstop= ⌋
↩→

10.0;

run,
↩→

beam=beam1,distribution=Dist1,fieldsolver=Fs1,method="PARALLEL-T";

endtrack;

Python code to generate simulated data from experimental parameters

"dir_expData",

simulations for unique

#Reading the parameters for the simulation from files saved in the folder
↩→
# under the names of "folder_name_experimentalData", and executing
↩→
# sequences of the variables ['ID_run', 'rfphase', 'rfpower', 'sol', 'laser'].
#The file "nameofSimulationFile", that must be saved in the same
↩→
# will be taken as the template for the OPAL simulation ".in" file.

directory as this running file,

#IMPORTS
import pandas as pd
import os
import re
import numpy as np
from math import pi, floor, log10
import shutil
from datetime import datetime
from pytz import timezone

#GLOBAL VARIABLES ------------------------------------------------------- ⌋
↩→

-----------------

82

#example for experimental files: 20240321104718_hpop2_rfphase_31.0deg_rfp ⌋
↩→

ower_0.00dB_sol_96.0A_laser_30.98uJ_LT1HX_0.350A_0

#Constants related to OPAL files
searchword1 = "accelphase = 25.0"
searchword2 = "accelvolt = 120.0"
searchword3 = "solcurrent = 96.0"
searchword4 = "beamcurrent = 2.459345657376521e-08"
searchword5 = "LPOP1f"
searchword6 = "HPOP1f"
searchword7 = "HPOP2f"

samplefilename = "/home/sirepo_user/OPAL/Ambar/fromExperim/Simulation_202 ⌋
↩→

#****************PATH FOR REFERENCE OF ATF

40523/ATFlinac.in"
OPAL FILE********************

↩→

#Constants related to experimental file manipulation
x_variables = [] #we will store the numbers found in the files here,
x_variables = [['ID_run', 'rfphase', 'rfpower', 'sol', 'laser', #'POP',
↩→
'POP'#]]

↩→
folder_name_experimentalData = ['20240301',
↩→

'20240304','20240305','20240321'] #The first two, 2024030 and
20240304, contain only LPOP, thus will be
# used for average beam profile and radii comparison. The third has
# before and after acceleration images, and will be used for
↩→
# and beam profile, in both locations. Finally, 20240321, has only
# measurements after acceleration that will be used for emittance
↩→

comparison of radii

calculation.

↩→

c = 299792458 #m/s
mkg = 9.1093837e-31 #kg
h = 6.62607015e-34 #J/Hz
q_e = 1.602176634e-19 #C

#
#Start loop to read different groups of folder. See more information
↩→
for l in range(len(folder_name_experimentalData)):

further up.

*****

83

dir_expData = '/home/sirepo_user/MeasurementsATF/' +
↩→

str(folder_name_experimentalData[l])
REFERENCE********************

#****************PATH

'/home/sirepo_user/OPAL/Ambar/fromExperim/Simulation_20240523/'

↩→
outfiledir =
↩→

#FUNCTIONS ------------------------------------------------------------ ⌋
↩→

-------------------

def Get_experimentalNumbers(file_name, ID):

keywords

([\d.]+)"

# Regular expression pattern to extract numeric values after specific
↩→
pattern1 = r"rfphase_([\d.]+).*rfpower_([\d.]+).*sol_([\d.]+).*laser_ ⌋
↩→
pattern2 = r"rfphase_-([\d.]+).*rfpower_([\d.]+).*sol_([\d.]+).*laser ⌋
↩→
pattern3 = r"([hl])pop(\d+)"
pattern4 = r"([HL])POP(\d+)"
values1 = re.search(pattern1, file_name)
if not values1:

_([\d.]+)"

values1 = re.search(pattern2, file_name)

if values1:

values2 = re.search(pattern3, file_name)
if not values2:

values2 = re.search(pattern4, file_name)

if values2:

numeric_values = ([ID] + [values1.group(j) for j in range(1,
↩→

5)] + [values2.group(1), values2.group(2)])

else:

numeric_values = 0

else:

numeric_values = 0
return numeric_values

def SearchforANDchageWord(filename, searchword, newword):

with open(filename, 'r') as file:

data = file.read()
data = data.replace(searchword, newword)

with open(filename, 'w') as file:

file.write(data)

84

return

def GenerateNewFile(samplefile, newname):

src = samplefile
dest = newname
doCopy = shutil.copyfile(src,dest)
return

def Average_oneVarinList(variable_list, n):

new_list = variable_list.copy()
# Extract the number from each sublist
numbers = [sublist[n] for sublist in variable_list]
# Calculate the average
if len(numbers) > 0:

average = sum(numbers) / len(numbers)

else:

average = None

for i in range(len(new_list)):

new_list[i][n] = average

return new_list

------------------

#GETTING VARIABLES ---------------------------------------------------- ⌋
↩→
#The laser variation is used to account for statistical error (we don't
↩→
# of electrons). We will make an average and save that for all sets in
↩→
# unique permutations to simulate.

the list, and trim again the set of

want simulations to vary in number

#Reading files

#****************PATH REFERENCE******************** (missing in
↩→

putty format)
dir_list = os.listdir(dir_expData)
i = 0
for file_name in dir_list:

if file_name.endswith(".asc"):

x_variables_thisFile = Get_experimentalNumbers(file_name, i)
if x_variables_thisFile:

x_variables.append(x_variables_thisFile)
i+=1

85

print("Exracted array of variables:\n", x_variables)

in x_variables

#Code to filter and save only the unique sequences in the lists saved
↩→
x_var_IDonly = []
for i in range(len(x_variables)):

x_var_IDonly.append(x_variables[i][1:5])

unique_sequences = set()
x_unique = []
for seq in x_var_IDonly:

seq_tuple = tuple(seq)
if seq_tuple not in unique_sequences:

unique_sequences.add(seq_tuple)
x_unique.append(seq)

num_files = len(x_unique)

#Converting units from saved in file to used by OPAL
for i in range(num_files):

x_unique[i][1] = 10**(float(x_unique[i][1])/20) * 120
↩→

#conversion from dB to volts dB = 20*log(volts/ref_volts), where
ref_volts = 120 MV

float(x_unique[i][3])*1e-6/E_phot #The energy of laser is

↩→
E_phot = c*h/265e-9
num_phot =
↩→
x_unique[i][3] = (num_phot * 0.0035) * q_e
↩→

provided in microJoules

laser power to current; assuming QE = 0.0035 = #elect/#photons,
AND E_phot=c*h/lambda, lambda = 265nm
C unit charge

AND q_e=1.602176634e-19

↩→

↩→

#conversion from

#The index number for the beam current (in x_unique) to be set

as constant for the simulation of this folder

n = 3
↩→
x_unique = Average_oneVarinList(x_unique, n)
unique_sequences = set()
x_unique_new = []
for seq in x_unique:

seq_tuple = tuple(seq)
if seq_tuple not in unique_sequences:

unique_sequences.add(seq_tuple)
x_unique_new.append(seq)

num_files = len(x_unique_new)
x_unique = x_unique_new
print("Number of unique permutations of variables:", num_files)
print("Unique data sets to be simulated:\n", x_unique)

86

TABLE -------------

#LOOP THAT GENERATES INPUT FILES, RUN OPAL SIMULATION AND SAVE INFO TO
↩→
referencefile = "ReferenceTable.txt"
hs = open(referencefile,"a")
tz = timezone('US/Eastern')
current_time = datetime.now(tz)
line = str(current_time) +'\n'
hs.write(line)
line = "LinacPhase1 \t LinacPhase2 \t GunPhase \t SolenoidCurrent \t
↩→
hs.write(line)
hs.close()

Filename \n"

for i in range(num_files):
#Generating new names
dataID = str(x_unique[i][0])+"_"+str(x_unique[i][1])+"_"+str(x_unique ⌋
↩→

[i][2])+"_"+str(np.format_float_positional(x_unique[i][3],
2,fractional=False))

↩→
dataID = dataID.replace('.', 'dot')
dataID = dataID.replace('-', 'pow')
infilename = "ATFlinac" + dataID + ".in"
newphase = "Accelphase = " + str(x_unique[i][0])
newpower = "Accelvolt = " + str(x_unique[i][1])
newScurrent = "solcurrent = " + str(x_unique[i][2])
newBcurrent = "beamcurrent = " + str(x_unique[i][3])
newmonitor1 = searchword5 + dataID
newmonitor2 = searchword6 + dataID
newmonitor3 = searchword7 + dataID

#Running functions
GenerateNewFile(samplefilename, infilename)
SearchforANDchageWord(infilename, searchword1, newphase)
SearchforANDchageWord(infilename, searchword2, newpower)
SearchforANDchageWord(infilename, searchword3, newScurrent)
SearchforANDchageWord(infilename, searchword5, newmonitor1)
SearchforANDchageWord(infilename, searchword6, newmonitor2)
SearchforANDchageWord(infilename, searchword7, newmonitor3)

#Run opal simulation

87

outfilename = outfiledir + "outfile.out"
command = "opal "+ infilename + ' > ' + outfilename
print("\nOpal simulation started, with command: ", command)
os.system(command)

print("Opal simulation ended, count: ", i+1, ", with values")
print("\t linac phase:", x_unique[i][0], ",\t linac power:",
↩→

x_unique[i][1], ",\t solenoid current:", x_unique[i][2], ",\t
beam current:", x_unique[i][3])

↩→
print("\tThe corresponding unique ID:", dataID)

#Save details to table
hs = open(referencefile,"a")
line = str(x_unique[i][0])+"\t"+str(x_unique[i][1])+"\t"+str(x_unique ⌋
[i][2])+"\t"+str(x_unique[i][3]) + "\t" + newmonitor1 + ".h5
↩→
\n"

↩→
hs.write(line)
hs.close()

Python code to read experimental data and determine the beam’s radii

by rf

#Script to sequentally read a collection of ascii monitor data (filename
# example: #ID_{EXP_CONDTS}_HPOP#ID.amf, where the #ID of the images for
# emittance measuremeants in folder 202403, and {EXP_CONDTS} is replaced
↩→
# phase, rf power, laser power, solenoid current). It then crops to
↩→
# the beam edges, assuming that is wherever the count is less than
# current_tolerance% of maximum average count. Here the count is taken as
# electron current.

images at

#IMPORTS
import numpy as np
import pandas as pd
from math import pi,sin, cos, exp, sqrt, floor
import matplotlib.pyplot as plt
import matplotlib.image as im
import scipy

88

from google.colab import drive
import os
import shutil
import h5py as h5
import collections.abc
import re

---------------------------------------------------------------

#GLOBAL VARIABLES
↩→
#example file: 20240321104718_hpop2_rfphase_31.0deg_rfpower_0.00dB_sol_96 ⌋
↩→

.0A_laser_30.98uJ_LT1HX_0.350A_0

#boolean to decide to plot of not the trimmed radius

plot_bool = 1
current_tolerance = 0.35 #*** We are assuming that the beam ends where
the everage pixel count drops 65% from the average maximum
↩→
x_variables_names = ['ID_run', 'rfphase', 'rfpower', 'sol', 'laser',
↩→

'JPOP', 'POPN']
RF phase and power for the acceleration cavities, the current at the
focusing solenoids, and the laser power in the photocathode
↩→
y_variables_names = ['r_x', 'r_y', 'r_ms', 'center_x', 'center_y']
x_variables = []
y_variables = []
folder_names = ['20240301']

#the unique ID by which each run is saved as, the

↩→

c = 299792458 #m/s
mkg = 9.1093837e-31 #kg
meV = 0.5109989461e6 #eV

#FUNCTIONS
↩→

----------------------------------------------------------------------

def get_avergMax(n, data):

#assuming data is pandas dataframe
values_series = data.stack()
highest_values = values_series.nlargest(n).tolist()
average =

np.mean(highest_values)

89

return average

current in the file

def find_edges(image, percent):
len_y, len_x = image.shape
steps_y, steps_x = floor(len_y/3), floor(len_x/3)
currMax = get_avergMax(100, image) #Saving the average high beam
↩→
tolerance = currMax * percent #We are assuming that the whatever signal
↩→
beam_x = [] #Here we'll save the indeces for which the current in that
↩→
beam_y = [] #Here we'll save the indeces for which the current in that
↩→
for i in range(steps_x):

is below percent (70%) of the maximum is background

column (vertical) was above tolerance

row (horizontal) was above tolerance

n_largest_values_i = image.iloc[:, i*3].nlargest(10)
max_value_column = np.mean(n_largest_values_i.values)
if max_value_column >= tolerance:

beam_x.append(i*3)

for j in range(steps_y):

n_largest_values_j = image.iloc[j*3, :].nlargest(10)
max_value_row = np.mean(n_largest_values_j.values)
if max_value_row >= tolerance:

beam_y.append(j*3)

return beam_x, beam_y

def edges_are_different(image, tolerance):

boolean = 0
lenx, leny = image.shape
avrg_ax1_0 = np.mean(image.iloc[:, 5])
avrg_ax1_n = np.mean(image.iloc[:, leny-5])
avrg_ax2_0 = np.mean(image.iloc[5, :])
avrg_ax2_n = np.mean(image.iloc[lenx-5, :])
ratio_ax1 = avrg_ax1_0/avrg_ax1_n
if ratio_ax1 > 1:

ratio_ax1 = avrg_ax1_n/avrg_ax1_0

ratio_ax2 = avrg_ax2_0/avrg_ax2_n
if ratio_ax2 > 1:

ratio_ax2 = avrg_ax2_n/avrg_ax2_0

if ratio_ax1 <= tolerance or ratio_ax2 <= tolerance:

boolean = 1
return boolean

def cut_unmatched_ax(image, tolerance, n):

#finding average values of current at each end of each axis
lenx, leny = image.shape

90

avrg_ax1_0 = np.mean(image.iloc[:, 5])
avrg_ax1_n = np.mean(image.iloc[:, leny-5])
avrg_ax2_0 = np.mean(image.iloc[5, :])
avrg_ax2_n = np.mean(image.iloc[lenx-5, :])
ratio_ax1 = avrg_ax1_0/avrg_ax1_n
if ratio_ax1 > 1:

ratio_ax1 = avrg_ax1_n/avrg_ax1_0

ratio_ax2 = avrg_ax2_0/avrg_ax2_n
if ratio_ax2 > 1:

ratio_ax2 = avrg_ax2_n/avrg_ax2_0

#cutting axis below tolerance at the edge with lowest average current
↩→
if ratio_ax1 <= tolerance:

value

if avrg_ax1_0 < avrg_ax1_n:

beam_image_cut = image.iloc[:, n:]

else:

beam_image_cut = image.iloc[:, :leny-n]

if ratio_ax2 <= tolerance:

if avrg_ax2_0 < avrg_ax2_n:

beam_image_cut = image.iloc[n:, :]

else:

beam_image_cut = image.iloc[:lenx-n, :]

return beam_image_cut

def get_y_variables(beam_image, ax1, ax2, Cam_resol):

int((beam_x[0]+beam_x[-1])/2)* Cam_resol

#Getting radius, center_x, center_y
center_x, center_y = int((beam_y[0]+beam_y[-1])/2)* Cam_resol,
↩→
len_x, len_y = beam_image.shape
r_x, r_y = (len_x/2*Cam_resol), (len_y/2*Cam_resol)
radius = sqrt((r_x**2 + r_y**2)/2)
y_var = [r_x, r_y, radius, center_x, center_y]
return y_var

#DATA HANDLING
↩→

------------------------------------------------------------------

drive.mount('/content/drive')

for this_folder in folder_names:

dir = '/content/drive/My Drive/Codigos/ExperData/' + this_folder

91

dir_list = os.listdir(dir)

i = 1
for file_name in dir_list:

specific keywords

if file_name.endswith(".asc"):
#Load individual image data
full_path = dir + '/' + file_name
beam_image = pd.read_csv(full_path)
#Regular expression pattern to extract numeric values after
↩→
pattern1 = r"rfphase_([\d.]+).*rfpower_([\d.]+).*sol_([\d.]+).*lase ⌋
↩→
pattern2 = r"([hl])pop(\d+)"
pattern3 = r"([HL])POP(\d+)"
pattern4 = r"rfphase_-([\d.]+).*rfpower_([\d.]+).*sol_([\d.]+).*las ⌋
↩→
values1 = re.search(pattern1, file_name)
if not values1:

er_([\d.]+)"

r_([\d.]+)"

values1 = re.search(pattern4, file_name)

if values1:

#Looking for beam edges and cutting
beam_x, beam_y = find_edges(beam_image, current_tolerance)
cols, rows = len(beam_x), len(beam_y)
beam_image_cut = beam_image.iloc[beam_y[0]:beam_y[-1],
↩→
#Verifying for outlier
cut_lenx, cut_leny = beam_image_cut.shape
#abs(beam_x[0]-beam_x[-1]), abs(beam_y[0]-beam_y[-1])
if cut_lenx > 250 or cut_leny > 65:

beam_x[0]:beam_x[-1]]

edges_ratio_tolerance = 0.4
if edges_are_different(beam_image_cut, edges_ratio_tolerance):

cut_unmatched_ax(beam_image_cut,edges_ratio_tolerance, 25)

print("This image was trimmed twice")
beam_image_cut =
↩→
beam_x, beam_y = find_edges(beam_image_cut, .50)
beam_image_cut = beam_image_cut.iloc[beam_y[0]:beam_y[-1],
↩→
cut_lenx, cut_leny = abs(beam_x[0]-beam_x[-1]),
↩→

abs(beam_y[0]-beam_y[-1])

beam_x[0]:beam_x[-1]]

#Evaluate and save values for x_variables and y_variables
values2 = re.search(pattern2, file_name)
if not values2:

values2 = re.search(pattern3, file_name)

92

↩→

x_variables_thisFile = ([i] + [float(values1.group(j)) for j in
↩→

range(1, 5)] + [values2.group(1), values2.group(2)])
#x_variables = ['ID_run', 'rfphase', 'rfpower', 'sol', 'laser',
'JPOP', 'POPN']

↩→
if x_variables_thisFile[5] == 'h' or
↩→

x_variables_thisFile[5] ==
#every pixel has a resolition of approximately 15e-6

'H':
[m] for HPOPs and...

↩→

y_variables_thisFile = get_y_variables(beam_image_cut, beam_x,
↩→

#y_variables = ['r_x', 'r_y', 'r_ms',

beam_y, 15e-6)
'center_x', 'center_y']

↩→

elif x_variables_thisFile[5] == 'l' or
↩→

x_variables_thisFile[5]
#...approximately 50e-6 [m] pixel resolution for

== 'L':
LPOPs

↩→

y_variables_thisFile = get_y_variables(beam_image_cut, beam_x,
↩→

beam_y, 50e-6)

x_variables.append(x_variables_thisFile)
y_variables.append(y_variables_thisFile)
i+=1

#Print the information and plot the image
print('\n')
table = 'Values for' + file_name + ':\n\t'
for var in range(len(x_variables_names)):

table = f'{table}{x_variables_names[var]} \t
↩→

{x_variables_thisFile[var]}\n\t'

for var in range(len(y_variables_names)):

table = f'{table}{y_variables_names[var]} \t
↩→

{y_variables_thisFile[var]}\n\t'

print(table)

if plot_bool:

imshow = plt.imshow(beam_image_cut)
plt.colorbar(imshow)
title = "Automated crop around beam edges [x:" + str(cut_lenx)
↩→
plt.title(title)
plt.show()

+ ', y:' + str(cut_leny) +']'

Python code to determine emittance from experimental data

#This script assumes that one of the codes in the previous section has
↩→
#It takes:

been run.

93

# x_variables = ['ID_run', 'rfphase', 'rfpower', 'sol', 'laser', 'JPOP', 'POPN']
# y_variables = ['r_x', 'r_y', 'r_ms', 'center_x', 'center_y']
# and calculates the emittance.

#IMPORTS
import numpy as np
import matplotlib.pyplot as plt
from math import pi,sin, cos, exp, sqrt, floor
from scipy.optimize import minimize
import re

---------------------------------------------------------------

#GLOBAL VARIABLES
↩→
#We assume that the minimum of beta is located somewhere between the
↩→

HPOP3 and HPOP4

#boolean to decide to plot of not the trimmed radius

#location of the

HPOP[2,3,4,5] in the beamline in meters

plot_bool = 1
si_HPOPs = {2:8.545, 3:12.052, 4:13.176, 5:14.653}
↩→
J = 5
r = [2, 3, 4]
↩→
r_measured = ['x', 'y', 'rms']

dictionaries -> key:[JPOP,POPN,{y_variables}]

#location in the x_variable for JPOP

#locations for the r measurement in the value arrays of

all_sigma_s = {}
↩→

#all the 4-tuple combinations of the beam radii and

↩→

position data { ID:[[{rx_data}], [{ry_data}], [{rms_data}]], ID:[[]],
...}; each ri_data has lx4 number of [sigma,s] pairs, where l
correspond to the number of minimum data arrays in the NPOP dictionary

↩→
all_emmitance_eq = {}
↩→

for all the 4-tuple pictures of the beam.

#the results of optimized [emittance, beta0, s0]

c = 299792458 #m/s
mkg = 9.1093837e-31 #kg
meV = 0.5109989461e6 #eV

94

#FUNCTIONS
↩→
def emittance_functn(params, sigma, s):

----------------------------------------------------------------------

emittance, beta0, s0 = params
predicted_sigma = np.sqrt(abs(emittance * (beta0 + (s - s0)**2 /
↩→
return np.sum((sigma - predicted_sigma)**2)

beta0)))

def find_min_number(lst):

if all(x == lst[0] for x in lst):

return lst[0]

else:

return min(lst)

they are reading

----------------------------------------------------------------------

#CALCULATING
↩→
#Saving the results of radii (sigma) and location (s), in groups of 4, as
↩→
# measured experimentally.
grouped_x_y = []
↩→
dict_x_y = {}
↩→
n0, n1, n2, n3 = 1, J-1, J, (J + len(y_variables_names))
↩→
# ID = x[n0:n1],

'rfpower', 'sol', 'laser', 'JPOP', 'POPN','radius', 'center_x', 'center_y']]

ID of run and the new y_variables which include JPOP and NPOP

'laser']:[['JPOP', 'POPN','radius', 'center_x', 'center_y']]}

#{[ID]:[values],...} = {['rfphase', 'rfpower', 'sol',

#x_variables + y_variables = [['ID_run', 'rfphase',

values = x[n2:n3]

#values for

AND

#Create grouped array
for l in range(len(x_variables)):

grouped_x_y.append(x_variables[l] + y_variables[l])

#Create dictionary
for x in grouped_x_y:

ID = tuple(x[n0:n1])
value = x[n2:n3]
if ID in dict_x_y:

dict_x_y[ID].append(value)

else:

dict_x_y[ID] = [value]

95

print("Unique IDs in the data:")
print(dict_x_y.keys())

#Sorting information into groups of 4 consecutive measurements
for key in dict_x_y.keys():
y_arrays = dict_x_y[key]
↩→
NPOPs_dict = {}
for sublist in y_arrays:

0.0008100000000000001, 0.0008611946644052087], [], ... [] ]

#y_arrays = [ ['h', '2', 0.0002925,

N = int(sublist[1])
if N in NPOPs_dict:

NPOPs_dict[N].append(sublist)

else:

NPOPs_dict[N] = [sublist]

len_Ns = []
for N in NPOPs_dict.keys():

len_Ns.append(len(NPOPs_dict[N]))

if len(len_Ns) == 4:

l = find_min_number(len_Ns)
sigmas_s = {ri:[] for ri in r_measured}
for i in range(l):

for N in NPOPs_dict.keys():

Ni_array = NPOPs_dict[N]
Ni_array = Ni_array[i]
s = si_HPOPs.get(N)
sigmas_s[r_measured[0]].append([Ni_array[r[0]], s])
sigmas_s[r_measured[1]].append([Ni_array[r[1]], s])
sigmas_s[r_measured[2]].append([Ni_array[r[2]], s])

all_sigma_s[key] = [sigmas_s[r_measured[0]], sigmas_s[r_measured[1]],
↩→

sigmas_s[r_measured[2]]]
#Print result of grouping of information
print("\nResult for all_sigma_s")
for key in all_sigma_s.keys():

KEY:', key)

print('
print(' values:', all_sigma_s[key])
value = all_sigma_s[key]
print(' rx: values[0][0]:', value[0][0])
print(' ry: values[1][0]:', value[1][0])
print(' rms: values[2][0]:', value[2][0])
sigmas =
sigmas = [item[0] for item in sigmas]
print("

First 4-tuple of sigmas:",sigmas)

value[0][0:4]

#Perform optimization and save results of fitted parabola
num_of_samples = int(len(value[0])/4)

96

for key in all_sigma_s.keys():
values = all_sigma_s[key]
for i in range(len(r)):

for j in range(num_of_samples):

sigma = values[i][j:j+4]
sigma = np.array([item[0] for item in sigma])
s = values[i][j:j+4]
s = np.array([item[1] for item in s])
emittance_guess = np.mean(sigma**2)
beta0_guess = np.min(s)
s0_guess = np.mean(s)
initial_guess = [emittance_guess, beta0_guess, s0_guess]
print("\nInitial guess:", initial_guess)

method='Nelder-Mead')

result = minimize(emittance_functn, initial_guess, args=(sigma,s),
↩→
emittance_opt, beta0_opt, s0_opt = result.x
if key in all_emmitance_eq:

all_emmitance_eq[key].append(result.x)

else:

all_emmitance_eq[key] = [result.x]

print("Result of fitting (emittance_opt, beta0_opt, s0_opt):",
↩→

result.x)

#Optional plotting
if plot_bool:

s_fit = np.linspace(min(s), max(s), 1000)
sigma_fit = np.sqrt(emittance_opt*(beta0_opt + (s_fit -
↩→

s0_opt)**2/beta0_opt))

plt.figure(figsize=(5, 3))
plt.scatter(s, sigma, label='Measured Points')
plt.plot(s_fit, sigma_fit, label='Fitted Function', color='red')
plt.xlabel('s')
plt.ylabel('sigma')
#plt.title('Measured Points and Fitted Function')
plt.legend()
plt.grid(True)
if i == 0:

print('Case for r_x')

elif i == 1:

print('Case for r_y')

elif i == 2:

print('Case for r_ms')

plt.show()

97

print("\n\n\nRESULTS FOR EMITTANCE OPTIMIZATION:")
for ID in all_emmitance_eq.keys():

print("
ID:", ID)
sets = all_emmitance_eq[ID]
for i in range(len(sets)):
print('\t', sets[i][0])

98