of the transfer line of the ESSnuSB project

TVE-F 19020

Examensarbete 15 hp Juni 2019

Numerical methods for design

of the transfer line of the ESSnuSB project

Independent Project in Engineering Physics

Karl-Fredrik Kylesten

Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Numerical methods for design of the transfer line of the ESSnuSB project

Karl-Fredrik Boholm Kylesten

ESS neutrino Super Beam (ESSnuSB) is a project that aim to create a high energy beam of neutrinos and anti-neutrinos to study the phenomenon neutrino oscillation and learn more about symmetry violations in quantum mechanics. To create the neutrino beam, negative Hydrogen ions must be transported from the ESS linear accelerator at 2.5 GeV, to a proton accumulation ring. This is done through a transfer line, that shall direct the ion beam while preserve the beam as much as possible. In this

project, there was an attempt at finding a design for this transfer line. Preferably, the line consists of a long main line of FODO cells and two matching sections at each end. A simulation of the beam was done that gives the progression beta and dispersion functions, statistical measurements of the particle distribution, through a part of the transfer line. A design for the main line was found. For tuning the quadrupole magnets, an iterative method using the system's response matrix was used. However, it could not match more than four parameters at the time, while six was required for complete matching. Because of this, it is not able to match the

dispersion.

Tryckt av: Uppsala

ISSN: 1401-5757, UPTEC F19020 Examinator: Martin Sjödin

Ämnesgranskare: Rikard Emanuelsson Handledare: Maja Olvegård

Contents

1 Introduction 4

1.1 Conservation and symmetry . . . 4

1.2 ESS neutrino Super Beam . . . 5

1.3 Objective of this project . . . 5

2 Linear beam optics 6 2.1 The 6D Phase space . . . 6

2.2 Transfer Matrices . . . 8

2.3 Description of particle distribution . . . 11

2.4 Dispersion . . . 12

2.5 FODO . . . 13

2.6 Lorenz Stripping . . . 14

2.7 Parameter optimization. . . 15

3 Method 16 3.1 General strategy . . . 16

3.2 Simulation 1 using random particles . . . 17

3.3 Simulation 2 using the covariance matrix . . . 17

3.4 Simulation 3 using twiss vector . . . 17

3.5 Simulation of dispersion . . . 18

3.6 Finding symmetric Twiss functions for transfer line . . . 18

3.7 Tuning start section . . . 19

3.8 Geometric design . . . 20

3.9 Testing the design of transfer line . . . 22

4 Result 23 4.1 Simulation methods . . . 23

4.2 Symmetric solution of middle cells . . . 24

4.3 Geometric design . . . 24

4.4 Matching parameters in the main line . . . 26

5 Discussion 28 5.1 Simulation and initial values for symmetric solutions . . . 28

5.2 Matching parameters . . . 29

5.3 Conclusion and final design . . . 29

1 Introduction

1.1 Conservation and symmetry

Symmetry and conservation laws are fundamental parts within all fields of physics.

Momentum, energy and charge are examples of quantities that always are conserved in any event. It is also postulated that the total charge of the universe is zero, which means there must be a fundamental symmetry between positive and negative charges.

Yet, symmetry and conservation is not always found when expected.

Matter and anti matter are in theory very similar, as every fundamental particle has an anti particle with the same mass but opposite quantum numbers, such as charge and spin. That means when a particle and its anti particle annihilates into energy, these quantum numbers are still conserved. Likewise, a photon with sufficient energy can create a particle anti particle pair. However, we know that there exists an unknown asymmetry between matter and anti matter. A simple proof for this is that we live in a world where atoms and molecules are made out of matter, while anti matter is rare. This is an example of a CP-violation, where a symmetry with respect to particle conjugation and space is violated.

Another example of violation of conservation laws and symmetry is neutrino osscila- tions. Neutrinos are light, electric neutral leptons. The leptons can be divided into generations and each generation consist of one charged lepton and one neutrino with the same flavour quantum numbers, also called lepton numbers. For example, we have generation 1, which consists of the electron and the electron neutrino, that both have the electron number 1, while the other lepton numbers are 0. All leptons and their lepton numbers are listen in table 1. Neutrino oscillations is when neutrinos

Generation 1 2 3

charged e⁻ µ⁻ τ⁻

(l_e, l_µ, l_τ) (1, 0, 0) (0, 1, 0) (0, 0, 1)

Neutrino ν_e ν_µ ν_τ

(l_e, l_µ, l_τ) (1, 0, 0) (0, 1, 0) (0, 0, 1)

Table 1: The three generation of leptons and their flavour quantum numbers.

travels long distances and some of them change lepton flavour. For example if a beam of neutrinos consists of both myon-neutrinos and electron neutrinos with the mixed state

|ψi = a|νµi + b|νei, (1)

the ratio a/b will change as the beam travels a long distance. This violates the law of lepton number conservation, which in the standard model applies in all interactions.[1]

Linear Accelerator Target Building

Accumulation Ring

Target Detector

Garpenberg L = 540 km

Figure 1: Sketch over the planned ESSnuSB experiment.

1.2 ESS neutrino Super Beam

The project ESS nuetrino Super Beam (ESSnuSB) intends to create two strong nuetrino beams, one consisting of muon neutrinos while the other of anti-muon neu- trinos. To achieve this, proton pulses from the European Spallation Source in Lund (ESS) shall be extracted from the linear accelerator at the energy 2.5 GeV and then transported through a transfer line into an accumulation ring. The accumulation ring will merge several pulses into one with more total energy an then let it hit a target.

At the target, the protons will produce pions π^±. Pions are electrically charged and can be maneuvered using electromagnets until they decay. The decay products are among other particles muon neutrinos. The neutrino beams then get to travel a long distance, 540 km untill they reach a detector in Garpenberg. The purpose of ESS- nuSB is to compare neutrino osculations between a particle and anti particle pair and thus examine CP-violations. A sketch over the setup is shown in figure 1.[3]

1.3 Objective of this project

The objective of this project was initially to design the transfer line between the ESS linear accelerator and the accumulation ring. The transfer line should consist of a start section, a main line and an end section. The main line should consist of many identical so called FODO cells and transfer the beam the majority of the way. By using many identical parts for the line, it they become easier and cheaper to manufacture. The start section should take the beam extracted from the accelerator and inject it into the main line and the end section should transfer the beam from the main line into the accumulation ring. Some other requirements for the transfer line are that:

1. The beam size σx must be no larger than 3 cm.

2. The energy of the ingoing particles from the accelerator shall be 2.5 GeV .

3. The rate of particle loss in the tunnel must be no more than 1 W/m.

The design should include geometry of the line as well as properties for the electro- magnets used. To test and optimize the transfer line, simulation software are needed.

Despite the fact that such software already exists, new programs was written using Matlab in order to learn more about the physics and numerical methods behind the simulations.

However, these goals had to be changed because of various reasons. ESSnuSB is still in its planning state. That is why many of requirements for the transfer line is still uncertain. Furthermore, as stated in Results, tuning quadrupole magnets was more difficult than planned. Thus, the design of the start and end section could not be completed. Instead, the focus of this rapport became to write a simulation program for the transfer line, design the main line and evaluate a numerical method for tuning quadrupoles.

2 Linear beam optics

2.1 The 6D Phase space

Consider a particle traveling along a given path, see figure 2. An ideal particle path can be practiced over time by ~r₀(t) = (x₀(t), y₀(t), z₀(t))^T. The ideal particle also has a momentum, ~p0(t), that always align with the given path. At any moment it is possible to define the state of other particles with six coordinates,

X = (x, x⁰, y, y⁰, δ, z)^T (2) in a coordinate system with the ideal particle in the origin. In figure 3 a particle is shown at time t₀ in the x-z plane. Our first coordinate x, is simply defined as the horizontal offset of the particle. The particle has also a momentum, ~p = (p_x, p_y, p_z), that can be projected onto the plane. Then, the second coordinate x⁰ is defined as the angle between the momentum and the ideal path, which is

x⁰ = tanp_x pz

(3) Likewise, the third and the forth coordinate is the vertical offset y and

y⁰ = tanp_y

p_z (4)

respectively. The fifth coordinate, δ describes the particles momentum and thus its energy. It is defined as the relative momentum of set,

δ = ∆p

p₀ = p − p₀

p₀ (5)

The sixth coordinate correlates to the the offset of a particle along the line. In some cases it will represent time or just a phase angle, but in this project, z is a distance.

[6]

p

z x y

(x

,y

,z

)

t=t

Figure 2: A particle traveling along the ideal path

x'

p

p

p x

z

Figure 3: A representation of a particles coordinates in the xz-plane.

2.2 Transfer Matrices

The phase space representation is very useful in order for follow a particle that has a little offset from the ideal path. In one case a particle may have initial coordinates at X₁ = (x₁, x⁰₁, y₁, y⁰₁, δ₁, z₁)^T. If the horizontal movements is of interest the vector may be shortened to (x1, x⁰₁, δ₁, z₁)^T. After that the particle have traveled distance s along the path in a drift space, a distance in vacuum without external forces, the particle have have the new coordinates X₂ = (x₂, x⁰₂, δ₂, z₂)^T. The new coordinates can be written as















x2 = x1+ s tan(x⁰₁) ≈ x1+ sx⁰₁ x⁰₂ = x⁰₁

δ2 = δ1+_γ^s2 R

z₂ = z₁

(6)

[6] The variable γ_Rtakes relativistic affects into account.[4] To simplify and get a linear system, γ_R² can be set to the numerical value 3.664 · 10² in this project. Equation (6) can be rewritten as a matrix equation

X₂ =







 x₂ x⁰₂ δ₂ z₂









=









1 s 0 0 0 1 0 0 0 0 1 _γ^s2 R

0 0 0 1















 x₁ x⁰₁ δ₁ z₁









= M_DriftX₁ (7)

where M_{Drif t} is the transfer matrix of the drift space. The full 6 × 6 drift space transfer matrix is written as

M_Drift =

















1 s 0 0 0 0 0 1 0 0 0 0 0 0 1 s 0 0 0 0 0 1 0 0 0 0 0 0 1 _γ^s2 R

0 0 0 0 0 1

















(8)

[5] Another kind of magnet in the transfer line is the dipole magnet. The dipole magnet produces a homogenize magnetic field ~B. Because of the Lorenz force the particle with the charge e and the velocity c will move in a circular curve whith the curvature radius ρ,

1 ρ = eB

pc (9)

. In the transfer tunnel, we can describe a dipole magnet solely with ρ and the distance s the particle travels in the field. A short dipole can be described as a small change in the direction of the particles momentum. In (9) its implied that this change depends on the size of the momentum. Then, we got an effect called Dispersion D, where the particles in the beam are split up based on their momentum (See figure 4).[4] Dispersion will be further discussed later on. The transfer matrix M_Dipole for

Figure 4: Three particles traveling in a dipole field. One of them has the ideal momentum p₀ and is following a path. The other particles, despite having the same initial position and traveling approximately the same distance s_Dipole, will differ at the end of tunnel in terms of x and x⁰ coordinates.

the dipole magnet is obtained by adding new elements to M_{Drif t}. It is assumed that the field is in the y-direction and thus only affect the movement in the x-plane for the particles. Furthermore, it is assumed that movements in the x and y plane are completely independent. This approximation is typical within linear optics and is motivated by the fact that both x and y are small numbers. By neglecting cross terms, a linear model is obtained that is easier to work with and the transfer matrix M_Dipole can be written as:

χ = s ρ

MDipole=



















cos χ ρ sin χ 0 0 0 ρ(1 − cos χ)

−^{sin χ}_ρ cos χ 0 0 0 sin χ

0 0 1 0 0 0

0 0 −^{tan χ/2}_2ρ 1 _γ^s2

R

0

−^{sin χ}_ρ −ρ(1 − cos χ) 0 0 1 0

0 0 0 0 0 1



















(10)

[4,6] The last kind of magnet is the quadrupole. As the name suggests, a quadrupole has four poles. Thus, it has a magnetic field that is 0 in the middle and stronger at the sides, see figure 5. The quadrupole works as a focusing lens in one plane, but as a de-focusing lens in another. A quadrupole is defined by its length s and the field gradient,

k = ∂B_y

∂x = −∂B_x

∂y (11)

that the magnet is designed to keep constant.

N

N S

S

+ +

+ x

y F

F

F k

Figure 5: A slice of a focusing quadrupole magnet with particles "×" traveling in the direction into the paper. The magnetic field is zero in the middle and stronger further out.

Therefore, particles experience stronger Lorenz force further away from the middle. The x-offset of particles result in a force into the middle, proportional to the offset, while an offset in y-direction gives a force of same magnitude but opposite direction.

As the particles are moving in the z-direction, the Lorenz force on them are propor- tional to the distance from the center. The x- component in this can be written as F_x = kx. This expression is similar to Hook’s law for springs. Therefore it is possible to write a differential equation

∂²x

∂t² = kx (12)

which is solved by

x(t) =

(A cos t√

k + B sin t√

k, k ≤ 0 C cosh t√

k + D sinh t√

k, k > 0 . (13)

Because the time a particle is in the magnet is related to its length, t can be replaces by s in (13). In this project a convention is used that a quadrupole is called focusing or QF and has k < 0 if its focusing in the x-direction. Thus, the transfer matrix can be written as

Ω = sp|k|,

M_QD =



















cosh Ω ^{sinh Ω√}

k 0 0 0 0

√

k sinh Ω cosh Ω 0 0 0 0

0 0 cos Ω ^{sin Ω√}

k 0 0

0 0 −√

k sin Ω cosh Ω 0 0

0 0 0 0 1 _γ^s2

R

0 0 0 0 0 1



















(14)

for a de-focusing quadrupole with k > 0 and

M_QF =



















cos Ω ^{sin Ω√}

k 0 0 0 0

−√

k sin Ω cos Ω 0 0 0 0

0 0 cosh Ω ^{sinh Ω√}

k 0 0

0 0 √

k sin Ω cos Ω 0 0

0 0 0 0 1 _γ^s2

R

0 0 0 0 0 1



















(15)

For a focusing quadruple with k < 0. [6, 4]

2.3 Description of particle distribution

When analyzing the transfer tunnel, it is desired to know how a group of particles behaves, not just the properties of one single particle. That is why it is necessary to analyze the particles as a distribution.

All particles can be completely described by the 6-dimensional vector X. It is rea- sonable to assume all properties of a particle to be normally distributed. Therefore, X is a multi variable normal distribution and can be described with its co variance matrix Σ, where σ_i,j = V ar[X_i, X_j]. Because of how the phase space is defined, the expected value for all X_i is 0 and can therefore be neglected in calculations. Thus, the elements in Σ is

σ_ij = hX_i· X_ji. (16)

[7] Like in (7), it is possible to calculate Σ for the next step in the tunnel. If Σ₀ is the co-variance matrix before a step described by M , the next co-variance matrix Σ₁ is calculated as

Σ₁ = MΣ₀M^T. (17)

To analyze Σ, it is common practice in accelerator physics to divide it into three 2 × 2 matrices, Σ_x, Σ_y and Σ_δz. It is already stated that co-variance between x and y are neglected. Therefore, all cross terms σi,j, |i − j| > 1 can be neglected and assumed to be 0. The matrix Σ_x describes the correlation of x and x⁰, that is, the movement in x direction. In accelerator physics, it is common to divide Σ_x into so twiss parameters that gives useful information abut the properties of the beam.

Σx=  β_x −αx

−α_x γ_x



(18) where the emmitance  is defined as

²_x = det(Σ_x) (19)

And describes the transverse area of the beam. The beta function βx(z) is a normal- ized measurement for the focus of the beam and α, gives the correlation between x and x⁰, which determines how the beam will propagate. It can be shown that

α_x(z) = d

dzβ_x(z) (20)

and that

γ_x = 1 − α²_x βx

. (21)

Because of (17), we know that β₁, α₁ and γ₁ after a step with transfer matrix M can be written as a linear combination of β0, α0 and γ0 before the step. Then, it is possible to express it as a matrix equation,

B₁ = M^∗_xB₀, (22)

where

B₀ = (β₀, α₀, γ₀)^T. (23) in this report is called T wisss The matrices M^∗_x and M^∗_y have the dimensions 3 × 3 and can be computed from a transfer matrix M as and

























 M_x =







m²_1,1 −2m1,1m1,2 m²_1,2

−m1,1m2,1 m1,1m2,2+ m1,2m2,1 −2m2,2m1,2

m²_2,1 −2m2,2m2,1 m²_2,2







M_y =







m²_3,3 −2m3,3m3,4 m²_3,4

−m3,3m4,3 m3,3m4,4+ m3,4m4,3 −2m4,4m3,4

m²_4,3 −2m4,4m4,3 m²_4,4







. (24)

[6]

2.4 Dispersion

As previously mentioned, a particles movement in a dipole magnet depends not only by its position and speed, but also its momentum. The dispersion is a property defined by the tunnel, not the particle itself, and is a way to describe how much the offset x of the particle depends of its momentum offset. In certain places in a transfer line or an accelerator we want the dispersion to be as low as possible, ideally zero. One example is in detectors, where it’s undesirable that ⁰ affects the results. Therefore, it is important to know the dispersion function Dx(z) of the tunnel. That way, it is possible to have moments that requires precision, like measurement, extraction or injection, where the dispersion is low.[6]

The dispersion function and its derivative in the beginning of a dipole are called D0

and D₀⁰ respectively. All relevant elements in the transfer matrix of the dipole M can be compressed into a 3 × 3 matrix S.

S =





M_1,1 M_1,2 M_1,6 M2,1 M2,2 M2,6

0 0 1



 (25)

QF/2 D QD D QF/2

Figure 6: A FODO-cell consists of two half length focusing quadrupoles QF/2, two dipoles D and one defocusing quadrupole QD. The distance between the magnets counts as drift space. If multiple FODO-cells is combined, they make up a line switching between focusing and de-focusing quadrupoles. The shape of the magnets in this picture is merely to show the similarity between the magnets and optic lenses.

By constructing a vector D_vec = (D, D⁰, 1) it is possible to find the values of D and D⁰ after the matrix by

D_vec,1 = SD_vec,0 (26)

[4] We have previously showed that is possible to compute the beta function from a Sigma matrix. If both the beta function and the dispersion are known it is possible to go the other way around and compute the beam size σ_x as

σ_x =p

β_x+ δ²D_x² (27)

The beam size is a measurement of the actual radius of the particle beam. It is important to keep this function limited as a particle size larger than the tunnel radius will cause a large particle loss. [6]

2.5 FODO

The difficulties with building a transfer tunnel is bending the particle beam without losing too much focus and energy. The most common solution is building repetitive blocks where the bending dipole magnets are mixed with quadrupoles that acts as thin lenses. One solution is a FODO cell. A FODO cell starts and ends with a half focusing quadrupole and has a de-focusing quadrupole in the middle, see figure 6.

The benefit of using FODO-cells is that all the drift spaces between the magnets are relatively small. Thus, the beam size is easy to control. [8]

As stated in the introduction, a transfer line consists of a start section, a main line and an end section. FODO cells are particular useful in the main line, as it’s possible to obtain functions for the twiss parameters that are periodic on each cell if the right initial twiss parameters are chosen. That is to say, if M^∗_FODO and S_FODO are matrices that describes the total changes is the cell, there are solutions D_vec and B that solves

(B = M^∗_FODOB

D_vec= S_FODOD_vec (28)

One such solution is the symmetric, where the derivatives of all functions must be zero in the beginning and the end of each FODO-cell. That implies that





 α₀ = 0 γ₀ = 1/β₀ D₀⁰ = 0

(29)

Which specifies the equation 27 further into































 β₀

0 1/β₀





= M^∗_FODO





 β₀

0 1/β₀











 D₀

0 1





= SFODO





 D₀

0 1







(30)

Thus, we are able to solve for the values at the beginning and the end of each FODO- cell, which is











β_x,0 =q

−^m_m^1,2m2,2

2,1m1,1

β_y,0=q

−^m_m^3,4m4,4

4,3m3,3

D0 = −^m_m^2,6

2,1

(31)

[6]

2.6 Lorenz Stripping

In the ESSnuSB project, its is not actually protons, but H⁻-ions, one proton with two electrons, that will be generated by the Linear accelerator and transported through the transfer tunnel.The reason for this is that the storage ring is full of protons.

Therefore, if a protons would arrive to the ring from the tunnel, they would simply be repelled by the protons already inside before they are captured by the storage ring’s magnetic field. Negatively charged H⁻-ions will instead be attracted by the other protons until the electrons are stripped away and the new proton is inside the ring. The second electron is very loosely bounded to the proton. Therefore, the magnetic fields within the tunnel will strip away the electrons before the ion has reached the ring. Figure 6 shows the relationship between the strength of the dipole magnet and the rate of Lorenz stripping. The loss rate of the tunnel cannot be more than 1 W/m. Thus, the strength of the B-field cannot be stronger than 0.15 T . As the dipole strength is related to its bending radius ρ it is possible to set a minimum bending radiusρ to

ρmin = 73.45 [m] (32)

(All this information is obtained from the ESSnuSB group at Uppsala University)

0.12 0.13 0.14 0.15 0.16 0.17 0.18 magnetic field B (T)

10^-10 10^-8 10^-6

loss rate per meter

Figure 7: The graph shows the interval the magnets in the transfer tunnel can have that has an acceptable rate of Lorenz stripping. The Graph is made by Maja Olvegård

2.7 Parameter optimization

An important step of designing a transfer tunnel is choosing right properties of the FODO cell to get desired twiss parameters of outgoing beam. Let the vector function F contain all these twiss parameters~

F = (f~ ₁, f₂, . . . , f_n)^T (33) where fi for any i could be any twiss parameter, including βx, αx and Dx at the end of the section. All these values can be written as functions

f_i = f_i(x₁, x₂, . . . , x_m) (34) where xj is any parameter of the FODO cell. Most commonly, the gradient of the quadrupoles are adjusted. Of course, f_i must not be analytically functions, bet can be results of simulations. Here a vector containing all parameters is called

X = (x₁, x₂, . . . , x_m)^T. (35) In order to optimize these parameters, there is a iterative method that takes one vector X_k and finds a better set of parameters X_k+1. The step involves finding the Jacobian matrix of ~F :

J_k=











∂f1

∂x1

∂f1

∂x2 . . . _∂x^∂f1

∂f2 m

∂x1

∂f2

∂x2 . . . _∂x^∂f2 .. m

. ... ...

∂fn

∂x1

∂fn

∂x2 . . . _∂x^∂fn

m











X=Xk

(36)

All partial derivatives can be calculated as

∂f_i xj

= f_i(x₁, . . . , x_j+ ∆x), . . . , x_m) + f_i(x₁, . . . , x_j − ∆x), . . . , x_m)

2∆x (37)

If the twiss parameters desired att the end of the section are stored in a vector called F~_Ideal, the next set of parameters solve the equation

J_kX_k+1 = ~F_ideal− ~F (X_k). (38)

In the beginning an initially guess has to be made for X. Ideally, this methods can be repeated until we converge to best possible outcome. If the solution doesn’t converge or gives nonphysical values, the initial guess can be adjusted. [6]

If there is a cell with n tunable parameters and m twiss parameters that we want to match, the equation system is over determined. It is still possible to solve equation (38) by Gausian elimination in matlab’s equation solving operator if n = m. On the other hand, if m > n and the system is over determined, but there is still a possibility to find a solution by using the matrix J^−1∗, the pseudo inverse matrix of J

J^−1∗= (J^TJ )⁻¹J^T (39)

Using The pseudo inverse matrix, the iteration becomes

X_k+1 = X_k+ J^−1∗(F_Ideal− F(X_k)) (40)

[9]

3 Method

3.1 General strategy

The first thing that was done in the project was to create a program for simulating a beams progression trough a FODO cell. For this, an example cell designed by Volker Zieman was used to test the programs. The properties of that FODO cell is found in table 2 and initial values of the twiss parameters are found in table 3.[8] Three different simulation methods was tested.

In order to simulate the cell, all matrices was written in mat-lab as functions of their respective parameters.











Mf q = Mquad(kf, sQ, γ²) Mdq = Mquad(kd, sQ, γ²) MDip = MDipole(r, sDipole, γ²) MDrif t = MDrif t(sDrif t, γ²)

(41)

However, both the dipoles and the quadrupoles were divided in two separate magnets of half length so that M = M₁· M₂. This is so we can evaluate the parameters at more point and see what is happening halfway through the magnet.

When a working simulation method was obtained, it could be used to write a program for tuning the quadrupoles in the start and end sections. As previously stated, this step was not successful. Subsequently, the method was only tested on the start section.

Furthermore, the geometry of the ESS facility was taken into account when designing FODO cells.

3.2 Simulation 1 using random particles

The code begins by generating 5000 random, normal distributed particles with Mat- lab’s randn() function given the initial values in table 2. Then, all of them was mul- tiplied with the transfer matrices of the FODO-cell. After each step in this presses, Σ was calculated by assuming the expected value of all phase vectors was approximately equal the mean value. Then,

σ_i,j = mean(X_i· X_j) (42)

by using the Matlab function mean. Then, using equation (19), we can calculate the emittance by the determinants

_x =    

σ_1,1 σ_1,2 σ_2,1 σ_2,2    

, _y =    

σ_3,3 σ_3,4 σ_4,3 σ_4,4    

(43) Then, the twiss function values could be calculated as

(β_x = ^σ_^1,1

x

β_y = ^σ_^3,3

y

(44)

3.3 Simulation 2 using the covariance matrix

In the second simulation no particle vectors were generated. Instead, the Σ matrix was computed from the initial values and and is then propagated through the tunnel as described in equation (17). Again, the twiss function values are calculated as in equation (42) and (43).

3.4 Simulation 3 using twiss vector

In the third method, we simulate the FODO cell by propagating the vector B like in equation (21). This method is simple to use as it gives the relevant values directly.

The fodo cell was designed to give symmetric propagation of the twiss parameters.[5]

Before the FODO-cell, the twiss parameters had the values given in table 2. [5]

Table 2: This table show the properties of an FODO cell that was examined Design parameter value

k_QF -0.2 [T /m]

k_QD 0.2 [T /m]

ρ 3.879[m]

s_QF 0.5 [m]

s_QD 1.0 [m]

sDipole 1.0 [m]

s_{drif t} 2.0 [m]

Table 3: Twiss parameters of the incoming beam in the designed FODO-cell Twiss parameter value

β_x 14.36 [m]

β_y 5.57 [m]

α_x 0

α_y 0

 0

3.5 Simulation of dispersion

In order to find the dispersion function, The vector D_vec= (D₀, 0, 1) was constructed and then multiplied with the S matrix from equation (26) for every step in the FODO cell. The results from all simulation of this standard FODO cell are presented in the first section of results.

3.6 Finding symmetric Twiss functions for transfer line

As previously stated, We wanted to use identical FODO-cells in a row to transport the beam and what makes this possible is to have symmetric and periodic functions for the Twiss parameters. To find the initial values of those functions, Equation (31) was used. First, all transfer matrices in the FODO cell was multiplied to create one new transfer matrix R for the wole cell.

R = M_FQLM_DriftLM_DipLM_DriftLM_DQLM_DriftLM_DipLM_DriftLM_DQL (45) Then the initial values for symmetry is, as previously stated,











β_x,L =q

−^R_R^1,2R2,2

2,1R1,1

β_y,L =q

−^R_R^3,4R4,4

4,3R3,3

D_L = −^R_{R 2,1}^2,6

(46)

Thereafter, the FODO cell was simulated using the found values and the third simu- lation method.

Table 4: Design parameters set in Global_Parameters.m Main line design

sDipL Length of dipoles

s_QL Length of all quadrupoles s_{Drif tL} Length of driftspace

k_{F Q} Gradient in focusing quadrupole k_DQ Gradient in defocusing quadrupole r_Dip Bending radius of dipole.

Matching FODO design s_DipL sQL

s_{Drif tL} r_Dip Parameters of incomming beam

β_xLinac Beta parameter out of Linear accelerator βyLinac Beta parameter out of Linear accelerator α_xLinac Alpha parameter out of Linear accelerator α_yLinac Alpha parameter out of Linear accelerator DxLinac Dispersion out of Linear accelerator

D⁰_xLinac Derivative of dispersion out of Linear accelerator Table 5: Properties of incoming beam

Twiss parameter value

β_x 28.8788 [mm · π · rad]

αx 0.5193

β_y 51.9032 [mm · π · rad]

α_y -1.9732

ε 0.1281077381

3.7 Tuning start section

And for the beam going into the accumulationion ring the desired properdies are given in table ZZZ. The start section had the purpose of taking in a beam from the accelerator and match it to the matched beam parameters calculated for a symmet- ric solution in the FODO-cells of the main line. This was done as another matlab script that has access to both the parameters from the accelerator and the matched parameters from our FODO-cell. In the end of the function there is a function F(X) that gives the twiss parameters of the beam after the matching cell given the initial parameters and the the vector ~X, that contains gradients for all quadrupoles in the cell. The calculations in F(X) are made using the Third simulation method. We want to tune the quadroploes using the iterative method described in equation (38).

Thus, the response matrix J is required. Using equation (36) and (37), J is calculated in Matlab as

Table 6: Desired of outgoing beam beam Twiss parameter value

β_x 28.8788 [mm · π · rad]

α_x 0.5193

βy 51.9032 [mm · π · rad]

α_y -1.9732

ε 0.1281077381

1 F_x1=(F([X(1)+dx;X(2);X(3);X(4)])− ...

F([X(1)−dx;X(2);X(3);X(4)]))./(2*dx);

2 F_x2=(F([X(1);X(2)+dx;X(3);X(4)])− ...

F([X(1);X(2)−dx;X(3);X(4)]))./(2*dx);

3 F_x3=(F([X(1);X(2);X(3)+dx;X(4)])− ...

F([X(1);X(2);X(3)−dx;X(4)]))./(2*dx);

4 F_x4=(F([X(1);X(2);X(3);X(4)+dx])− ...

F([X(1);X(2);X(3);X(4)−dx]))./(2*dx);

5 J=[F_x1,F_x2,F_x3,F_x4];

The iteration shall stop when X converges. To measure convergense, we defined µ as

µ_k=p

(X_k− X_k+1) · (X_k− X_k+1)^T (47) And run the simulation until µ is less than a certain value, in our case 10⁻⁴. One difficulty with this method is that it requires a start guess and that it not always con- verges. If not, new, improved start values for X₀ can be chosen, but the entire Cell might have to be redesigned since the possibilities exist that no solution exist, given the current design. One must also be critical to the values given if a solution is ob- tained, since the quadruple gradients may converge into nonphysical or unrealistically large values.

3.8 Geometric design

Last, the geometric properties of the transfer tunnel must be considered. Figure 8 show a map over a part of the ESS site with the transfer line and accumulation ring drawn in approximately where they are meant to be. A dipole magnets bends the beam away from the accelerator at the point where its energy is 2.5 GeV . In fact, a higher energy would be preferred, but then the transfer line would interfere with the accelerator building. In this section, we will primarily discuss the main line when doing measurements. The plan for the site is still undergoing changes and it is not possible to com up with the final design for all FODO cells. However, it is prectical to get realistic proportions for the FODO cells. When adding the matching cells, some of the DODO cells in the main line can simply be replaced or drift space be added to the matching cells. The distance d between the beginning and the end of the transfer line is known to be 185 m. If the transfer line as a whole has the bending radius R

R b

s

Figure 8: Mat over the ESS site with the transfer line and accumulation ring

and bends the beam an angle θ, a trigonometric relationship can be written as R cosθ

2 = d

2 (48)

The length of the circle section s_cthat follows the transfer line can be written as

sc= Rθ (49)

However, our transfer line is not a perfect circle section. It consists of many FODO cells and its total length will therefore differ from the length of the circle section.

Figure 8 shows a sketch of a half FODO cell that bends the ray the angle ϕ. In the sketch, the length a is the length of one drift space s_{Drif t} and and one half quadrupole sQ/2. With trigonometry it can be shown that

H = a + h + a cos ϕ (50)

which can be rewritten as

R sin ϕ = ρ sin ϕ + a(1 + cos ϕ) (51) We call the quota of the tunnel that are dipole magnets p_Dip, then

pDip = ρϕ

2a + ρϕ = 1

1 + _ρϕ^2a (52)

The expression a/ρϕ can be rewritten with help of equation (44) to get 2a

ρϕ = sin ϕ(R − ρ)

(1 + cos φ)ρφ (53)

Since each ϕ is small the approximations sin ϕ ≈ ϕ and cos ϕ ≈ 1 can be used. Thus, the expression simplifies to

2a

ρϕ = 2ϕ(R − ρ)

2ρϕ = R

ρ − 1. (54)

ρ H h

a

a

φ φ

φ

Figure 9: A circle section with radius R and angle ϕ is replaced by a circle section with smaller radius but the same angle and two lines of length a. The height of the big section is H and of the small h.

Consequently,

p_Dip = 1

1 + (^R_ρ − 1) = 1

R ρ

= ρ

R (55)

and depends only on ρ and R for small ϕ. It is suggested by the ESSnuSB team the about 2/3 = q of the whole transfer line should consist of dipole magnets. Too much dipoles and the beam will expand, but too little and the construction will become very big before turning the beam. Thus, (55) gives that R ≈ 110 m. Then (48) can be rewritten as

θ = 2 arccos d

2R ≈ 114^◦. (56)

Using a Matlab script, different number of FODO cells were tested. If we use 33 cells, We get that each dipole becomes 2.2 m, the total cell about 6 m and a 0.5 m. Moreover, it must be taken into account that it should be practically possible to build all parts. Usually, the length of a quadrupole is about ∼ 0.5 m. Furthermore, it should be possible to fit measurement equipment into the drift spaces.[8] A few designs were tested and the resulting main line was drawn using a Matlab script.

3.9 Testing the design of transfer line

When testing a design, the developed methods was used in systematically. First, the design parameters of both the FODO cells of the main line and the start section was loaded using the script GobalVariables.m. Secondly, the symmetric solutions for the Twiss parameters of the main line was solved for using the script FindSym- metry.m. Last, the quadrupoles of the start section was tuned using the script StratSectionTuning.m. All code is to be found in the Appendix.

Simulation 1

0 2 4 6 8 10 12

Z [m]

1.6 1.8 2 2.2 2.4 2.6 2.8 3

Dispersion [m]

Dx Simulation 2

Simulation 3

0 2 4 6 8 10 12

Z [m]

1.6 1.8 2 2.2 2.4 2.6 2.8 3

Dispersion [m]

Dx Simulation Dispersion

Figure 10: Different simulations of a FODO cell described in table 3 with an incoming beam with parameters listed in table 4. Simulation 1 was made using simulation of particles, simulation 2 by propagating the covariance matrix Σ and simulation 3 by propagating the B vector trough M^∗ matrices. The fourth simulation shows the dispersion throughout the cell.

4 Result

4.1 Simulation methods

In figure 9 are all graphs from the evaluation of different simulation methods. Simula- tion 1 was made by simulating 5000 individual particles, simulation 2 by propagating the co variance matrix Σ and simulation 3 by multiplying the β, α, γ vector. As we see, they all giver the same result. That is a good thing since it means all the method works for simulating a particle beam. Since they give equivalent results, we can choose the most efficient simulation, which is the third. The first method takes 5000 as long as any of the other method, since we used 5000 particles, and then calculate expected values of them. Out of the methods that do not use particles, the second method is the slowest, since at each step, three 6 × 6 matrices need to be calculated. The third method gives the next values only by multiplying a 3 × 3matrix with a vector. One

-10 -5 0 5 10 x

-4 -2 0 2 4

x'

Phase space at: (5)

-15 -10 -5 0 5 10 15

y -2

0 2 4

y'

Figure 11: Plot of 5000 particles in the xx⁰ and the yy⁰ plane right after the first dipole.

The beam is focusing in x and defocusing in y.

benefit width the first simulation however, is that we are able to display the particles at a given point along the line. We can here clearly see that the particles as expected are normally distributed. In figure 10 the particles are displayed directly after the first dipole magnet. In the figure, the phase elipse is drawn in as well. The phase ellipse marks 1 standard deviation away from the center. The ellips has a negative tilt in the xx⁰- plane and a positive tilt in the yy⁰-plane, witch means that the beam is focusing in x and defocusing in y. This cohere to the fact that βx is in decreasing and β_y is increasing after the first defocusing quadrupole, as seen in figure 9.

4.2 Symmetric solution of middle cells

The script FindSymmetry.m was used on the design of Ziemann’s FODO cell. The resulting symmetric solution is displayed in figure .

4.3 Geometric design

After that a few possible designs were tested, the following design was decided. This

0 2 4 6 Z [m]

7 8 9 10 11 12 13 14

[m]

x y

0 2 4 6

Z [m]

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

Dispersion [m]

Dx

Figure 12: Result of fitting of symmetric twiss parameter functions in a FODO cell. Left:

β_x and β_y, Right: Dispersion D_x.

Table 7: Geometric parameters for the main line.

parameter [m]

s_DipL 2.0 s_QL 0.4 s_{Drif tL} 0.3

design gives the total length LL = 6.0 m for each FODO cell, and each cell is bending the beam ∆θ = 3.4603^◦. All geometrical parameters are listed in table 7. We can choose a design where the main line consists of 35 such FODO cells. As the start and the end sections, we can use a modified FODO cell with 4 qudropoles.

(DQ/2 − −F Q/2 − −Dip − −DQ − −Dip − −F Q/2, Matching in

F Q/2 − −Dip − −DQ − −Dip − −F Q/2 − −DQ/2, Matching out (57) Thus, we have four quadropoles that we can tune. In these modified FODO cells, we can use the lengths

Table 8: Final design parameter [m]

s_DipM 1.7 s_QM 0.4 s_{Drif tM} 0.3

-150 -100 -50 0 50 x [m]

-160 -140 -120 -100 -80 -60 -40 -20 0

y [m]

Figure 13: Map over the transfer line

4.4 Matching parameters in the main line

If choosing kF = −0.5 T /m and kD = 0.5 in the main line cells, the beta and dispersion functions becomes symmetric for the initial values given in table . The result from its simulation is shown in figure 13. The four quadropole in the start section was tuned and converged after 10 iterations into the values given in table 9.

Results from a simulation of the matching cell is presented in figure 14. There, we see that the cell is matching the beta function to the initial values for the main line.

The matching cell also had to make βx and betay switch places. That is the reason why a defocusing quadrupole was put in front of a fodo cell.

However, if the program was changed to also match the Dispersion function, it did not converge and was unable to find a solution

Table 9: Gradient values for matching cell between accelerator and tunnel.

parameter value kd1 2.6589 k_{f 1} -5.2019 k_d2 2.0713 kf 2 -2.5732

0 2 4 6 Z [m]

7 8 9 10 11 12 13 14

[m]

x y

0 2 4 6

Z [m]

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

Dispersion [m]

D_x

Figure 14: The symmetric β and Dispersion functions of the designed FODO-cell

0 1 2 3 4 5 6

Z [m]

0 10 20 30 40 50 60 70 80

[m]

x y

0 1 2 3 4 5 6

Z [m]

0 0.05 0.1 0.15 0.2

Dispersion [m]

D_x

Figure 15: Graphs of the Beta and Dispersion parameters through the first matching cell.

Table 10: Design of transfer line z [m] First matching cell

0.0 - 0.4 Defocusing quadropole k = 2.6589 T /m 0.7 - 0.9 Half focusing quadropole k = −5.2019 T /m

1.2 - 2.2 Dipole r = 73.45 m

2.5 - 2.9 Defocusing quadropole k = 2.0713 T /m

3.2 - 4.9 Dipole r = 73.45 m

5.2 - 5.4 Half focusing quadropole k = −2.5732

z [m] Main line FODO ×35

0.0 - 0.2 Half focusing quadropole k = −0.5 T /m

0.5 - 2.5 Dipole r = 73.45 m

2.8 - 3.2 Defocusing quadropole k = 0.5 T /m

3.5 - 5.5 Dipole r = 73.45 m

5.8 - 6.0 Half focusing quadropole k = −0.5 T /m

5 Discussion

5.1 Simulation and initial values for symmetric solutions

By using three different methods for simulating the progression of a beam through a transfer line it is possible to compare them. In method number one, the beam was simulated by calculating the path of a large number of particles and then evaluating statistical measurements, while in method two and three, the statistical measurements was used directly to calculate the next state of the beam. When used to simulate the same scenario, method two and three gave the same result as method one, which is an indication that they working, despite not simulating real particles.

The benefits of method two and three are that many times faster than the first.

Primarily, we only have to progress an array object through the line once, but we also don’t need to use time computing mean values of all the phase space vectors. Saving computer power is essential in the simulations. Not only can transfer lines consist of a huge number of transfer matrices, but in algorithms lite quadruple tuning, the line must be simulated many times.

What distinguish method two from method three is that is uses the co-variance matrix directly and propagate it through the transfer matrices, while in method three, the transfer matrix is rewritten into two 3x3 matrices, which are used two multiply with a vector containing the alpha, beta and gamma function. Since the co variance is a familiar concept it can be easier to understand this method, however, multiplying 3 6 × 6 matrices takes more computing power than multiplying a 3 × 3 matrix with a vector. Moreover, the structure of the program becomes a lot simpler when the state of the beam is represented at a given time by a vector and not a matrix. Thus, the extra step of rewriting the transfer matrices are worth it.

The method for computing initial values of the beta and dispersion functions that gives symmetric solutions in a cell was successful. This is essential for the design of the transfer line as a symmetric and periodic solutions enables to use identical FODO-cells and building a long line, which simplifies the work. However, we could see int the result that the dispersion was never equal to zero within the line. This is unfortunate as measurements are best done when the dispersion is zero, since variations in momentum, should not influence the measurements.

An improvement of the design could therefore be to add sections consisting of two symmetric matching cells that takes the dispersion to zero in the middle and then back up again. This however requires that we have are able to tune quadropoles in matching cells.

5.2 Matching parameters

The program for tuning quadropoles managed to successfully match the beta func- tions. Unfortunately, the method was unstable and seldom converged. When the method converged, it was only for certain length proportions of the cell, carefully cho- sen initial guesses and never when matching more than four quadrupoles. A matching cell must be able to match six parameters if we want both the beta functions and the dispersion in one direction, since both the functions and their derivatives must be matched.

Therefore, more work has to be put into designing the matching cells. Then, other numerical methods should be implemented in order to make the tuning more sta- ble.

5.3 Conclusion and final design

Using linear, numerical methods in Matlab, a transfer tunnel could be simulated and tested. The suggested design of the transfer line is presented i table 9. The Main line cells can transport a beam with symmetric twiss parameter function and a beam width that is less than 3 cm. Further more wont the magnets cause too much Lorentz stripping.

However, it shall be noted that this design needs improvement. The numerical method for could note match more than 4 twiss parameters at the time, while 6 is the number required. Also, sections with zero dispersion needs to be added. In order to fix this, another numerical tuning method could be used. An interesting project would be to develop a more advanced tuning program and study the circumstances required for convergence of the iteration. For ESSnuSB however, a program like MadX can be used for designing the matching sections.

References

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[2] Wildner E, Martini M, Schonauer H, Burgman A, Cederkall J, Christiansen P, Ekelof T, Olvegard M. The opportunity offered by the ESSnuSB project to exploit the larger leptonic CP violation signal at the second oscillation maximum and the requirements of this project on the ESS accelerator complex. 2015. Cornell University, arXiv:1510.00493v3 [2019-03-25]

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[6] Wille K, The Physics of ParticleAccelerators. 2nd edition. McFall, translator. Ox- ford: Oxford University Press; 2000. chap. 3, Linear Beam Optics;

[7] Rade L, Westergren B, Mathematics Handbook for Science and Engineering. 5th edition. Lund: Studentliteratur AB; 2004. chap. 17 Probability Theory; p. 430 [8] Ziemann V, Using FODO Channels in ESS Target Beam Lines. Uppsala: Uppsala

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