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SALİM OĞUR VELİ YİLDİZ

Bogazici University, Department of Physics

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Outline

1. Emittance Why we need to measure the emittance at the exit of

the ion source ?

2. Quadrupole Variation Method Our calculations and results

3. Forward Method Forward Method using Quadrupole Forward Method using Solenoid

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1. Emittance

A particle’s properties can be figured out via Lagrange Mechanics, however a system of particles had better be defined in Hamiltonian Mechanics which requires Phase Space.

Phase Space includes position and (combination of) derivative of position (velocity, momentum).

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1. Emittance

In Accelerator Physics, we define 3 phase spaces for 3 dimensions (x,y,z). Again, phase space includes position(i.e. x) and its gradient (x’).

where α,β,γ are twiss

parameters which enables

me to define the area uniquely.

A= . π ϵ

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1. Emittance

Emittance with determined , , values α β γgives information about the beam as a whole, and this information is used through the beam propagation simulations. This is why we had to figure out the emittance and twiss parameters.

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2- Quadrupole Variation Method

This figure was drawn by a multiparticle simulation program PATH, and considered as the beam exiting the ion source. Due to radial symmetry at the exit of the ion source both xx’ and yy’ have the same emittance and twiss parameters.

ϵ_rms = 1.0000 π.mm.mrad β=0.2000 mm/π.mradα=-2.0001

P.S: Beam pipe is taken with infinite radius such that all created particles can travel and are not annihilated. Also, no effect of space charge !

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2- Quadrupole Variation Method

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2- Quadrupole Variation Method

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Beam Dynamics

Transfer Matrices

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2- Quadrupole Variation Method

We have 10 cm of quadrupole with effective length and 20 cm of drift space. Therefore our transfer matrix R(k) can be calculated, and notice that although focusing quadrupole stands before the drift , in transfer matrix they are in backwards sequence.

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2- Quadrupole Variation Method

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2. Quadrupole Variation Method

So far we have seen that the emittance and twiss parameters are very close those ones simulated, but this is the case for 0 mA (No space charge effect). Normalized solutions are given for trials with equations more than the unknowns.

Denemeler emittance [pi*mm*mrad] Beta [pi/mm*mrad] alpha Error in em[%] Error in b[%] Error in a[%]

Gerçek Değerler 153,15 0,2000 -2,0001 - - -

1.deneme (1,7,11) 155,95 0,1943 -1,9988 1,8252 2,83 0,0650

2. Deneme (3, 7, 9) 156,30 0,1950 -2,0055 2,0537 2,49 0,2700

3. Deneme (1, 3, 7, 9, 11) 157,30 0,1924 -1,9775 2,7066 3,78 1,1299

4. Deneme (1, 3, 7) 155,98 0,1926 -1,9907 1,8448 3,69 0,4700

5. Deneme (1, 2, 3, 5, 7) 155,84 0,1928 -1,9927 1,7533 3,6125 0,3700

6. Deneme (bütün ölçümler) 157,16 0,1927 -1,9807 2,6152 3,64 0,9700

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2. Quadrupole Variation Method

In 2 mA space charge effect does not play such an important rule but in 10 mA, the beam emittance measurement blows up dramatically. What about beam pipe with physical length ?

Trials emittance [pi*mm*mrad] Beta [pi/mm*mrad] alpha Error in em[%] Error in b[%] Error in a[%]

Real Values 153,15 0,2000 -2,0001 - - -

0 mA (ölçüm 3,7,9) 156,30 0,1950 -2,0055 2,05370 2,49 0,2699865

2 mA (ölçüm 3,7,9) 156,3 0,19502 -2,0052 2,05370 2,49 0,2549873

10 mA (ölçüm 3,7,9) 170,56 0,17296 -1,9758 11,36455 13,52 1,2149393

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2. Quadrupole Variation Method

Table is meant to give a taste about what is happening in the case of a 3 cm radius of beam pipe. Notice for small gradient x_rms values are close.

Gradient (T/m) x_RMS [m] (no constraint) x RMS [m] (constraint) Transmission (%)

2,50 4,82E-03 4,84E-03 42

3,45 6,37E-03 6,37E-03 34

2,53 4,75E-03 4,78E-03 42

2,96 4,67E-03 4,71E-03 38

2,77 4,47E-03 4,51E-03 40

1,89 7,78E-03 7,66E-03 47

1,29 1,21E-02 1,10E-02 51

1,04 1,42E-02 1,19E-02 52

What is the effect of the beam pipe?

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3. Forward Method

In QVM, we calculate emittance via analytical way such that from x_rms and gradient values, we go backward by transfer matrices and find out the emittance and twiss parameters.

In FM, we have an inception beam, and we feed this emittance into simulations to get close to the x_rms and y_rms values using TRAVEL.

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3. Forward Method (Quadrupole)

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3. Forward Method (Quadrupole) Now we insert our parameters to create a beam after analyses We draw the beam with PLOTWIN

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3. Forward Method (Quadrupole)

Beam Parameters Real Beam Inception Beam Forward Method

alpha_x -2 -2,1767 -2,0401

beta_X 0,2 0,1948 0,203

emittance_X pi.mm.mrad 1 1,0851 1,007

alpha_y -2 -2,1213 -2,0201

beta_y 0,2 0,2024 0,202

emittance_y 1 1,0319 0,9748

error_alpha_x 8,835 2,005

error_beta_X 2,6 1,5

error_emittance_X pi.mm.mrad 8,51 0,7

error_alpha_y 6,065 1,005

error_beta_y 1,2 1

error_emittance_y 3,19 2,52

We used inception beam in simulations and the beam from FM is at good agreement with real beam. Notice that inception does not fit well with real beam.

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3. Forward Method (Solenoid)

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3. Forward Method (Solenoid)

We have good agreement with real beam parameters by using solenoid, as well.

Beam Parameters Real Beam Inception Beam Forward Method

alpha_x -2 -2,1767 -2,0381

beta_X 0,2 0,1948 0,2026

emittance_X pi.mm.mrad 1 1,0851 1,0045

alpha_y -2 -2,1213 -2,0183

beta_y 0,2 0,2024 0,2017

emittance_y 1 1,0319 0,9752

error_alpha_x 8,835 1,905

error_beta_X 2,6 1,3

error_emittance_X pi.mm.mrad 8,51 0,45

error_alpha_y 6,065 0,915

error_beta_y 1,2 0,85

error_emittance_y 3,19 2,48

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Conclusion

Quadrupole Variation Method is useful and gives accurate results at 20 keV and 2 mA with shortcomings in beam transmission rate for 3 cm of radius beam pipe.

Forward Method are both applicable with Quadrupole and Solenoid magnets. Moreover, transmission rate does not follow since the particles do not hit solenoid surface.

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Thanks for your kind attention …