Testing the Angular Acceptance and Ion Optics
Calculations for the St. George Recoil Mass Separator
Michael Rix
A dissertation submitted to the Department of Physics at the University of Surrey in partial fulfilment
of the degree of Master in Physics
January 2013
Abstract
Recoil mass separators have been employed in experimental nuclear physics to study radiative capture
reactions of astrophysical importance through the technique of inverse kinematics[1]. The St. George
recoil mass separator at the University of Notre Dame has been designed to study such low energy
(α,γ) reactions[2] induced by stable beams, it was tested using a sealed Americium-241 source. The
St. George consists of a helium gas jet target, 11 quadrupole focussing magnets, 6 dipole bending
magnets and a Wien Filter. The alpha particle source was used to test the energy angular acceptance
and to validate the ion optics calculations for the St. George. Those calculations rely on the COSY
Infinity Code[3].
The ion optics calculations were validated for the first two quadrupole magnets. The count-rate,
energy distribution and position of the alpha particles all closely matched the predicted results. The
measurements of the position and energy distribution after the first dipole magnet were less
compatible with the simulation. Multiple combinations of magnetic fields in the dipoles provided
measurements which seemed to be compatible with the calculations. While it is possible that a fault in
the calculations was discovered it is far more likely that this discrepancy was due to the technical
limitations of the experiment and that a measurement using an accelerated beam will be required to
reach a final conclusion.
i
Acknowledgements
I would like to thank my supervisors Dr. M. Couder, Prof. M. Wiescher and Dr. P. H. Regan I would
also like to thank Prof. E. Stech, Prof. J. Hinnefeld, Dr. D. Robertson, B. Mulder, Prof. W. Tang, J.
Lingle, M. Sanford, J. Holdemann and S. Lyons for their invaluable assistance and advice throughout
the project
ii
Contents
Abstract .................................................................................................................................................... i
Acknowledgements................................................................................................................................. ii
Introduction ............................................................................................................................................ 1
Chapter 1................................................................................................................................................. 2
Background Theory ............................................................................................................................. 2
1.1
Stellar Helium Burning ........................................................................................................ 2
1.1.1 Triple-Alpha Process .......................................................................................................... 2
1.1.2 The Alpha-Process .............................................................................................................. 3
1.2
Inverse Kinematics .............................................................................................................. 5
1.3
The St. George Recoil Mass Separator ................................................................................ 7
1.3.1
Gas Target ................................................................................................................... 9
1.3.2
The Wien Filter .......................................................................................................... 10
1.3.3
Ion Optics Calculations .............................................................................................. 12
Chapter 2............................................................................................................................................... 14
Simulation of the St. George ............................................................................................................. 14
2.1 The COSY Infinity Code............................................................................................................ 14
2.2 Using COSY Infinity to Simulate the St. George ...................................................................... 15
Chapter 3............................................................................................................................................... 20
Experimental Setup ........................................................................................................................... 20
3.1 Apparatus and Equipment ...................................................................................................... 20
3.2 Magnetic Components ............................................................................................................ 23
3.2.1 Quadrupole Magnets ....................................................................................................... 23
3.2.2 Dipole Magnets ................................................................................................................ 27
3.3 Silicon Diode Detector ............................................................................................................ 32
Chapter 4............................................................................................................................................... 39
Experimental Measurements and Analysis ....................................................................................... 39
4.1 Position 1 – Q2B1 .................................................................................................................... 40
4.2 Position 2 – B1B2 .................................................................................................................... 47
4.3 Position 3 – B2Q3 .................................................................................................................... 50
References ............................................................................................................................................ 56
Appendix A ............................................................................................................................................ 58
Clean_alpha.fox ................................................................................................................................ 58
Introduction
The radiative capture of alpha particles is an important reaction mechanism for stellar helium
burning[4]. The experimental study of low energy (α,γ) reactions is normally conducted using intense,
low energy alpha beams directed at a heavier target[5]. The reaction cross-sections can then be
measured by detecting the gamma rays emitted from the reaction products. At stellar energies the
reaction cross-sections are extremely low; in addition to this the gamma rays from the reaction
products are easily overwhelmed by background radiation from cosmic rays, natural radiation from
the environment and also beam induced radiation caused when the beam hits the target.
An alternative method is to use inverse kinematics, where an intense heavy ion beam is directed at a
helium gas target. This has several advantages firstly it makes it possible to measure the reaction
induced gamma radiation and when the reaction products and beam are directed into a recoil mass
separator the reaction products can be separated and measured as well. To conduct these experiments
the St. Ana 5MV vertical particle accelerator and the St. George Recoil Mass Separator have been
built at the University of Notre Dame. Due to delays with the construction of the St. George it has yet
to be used, and while extensive ion optics calculations have been made it has so far been impossible to
experimentally validate them and commission the separator. In order to do so we have used an
Americium-241 source with a diameter of 2mm to replace a particle beam entering the separator. By
positioning two 1-dimension position sensitive silicon detectors in the gas target at the entrance of the
separator it was possible to make measurements of the position in the horizontal plane and energy of
the alpha particles which were then compared to the ion optics calculations using an adapted COSY
simulation of St. George. The primary goal was to validate the ion optics calculations and identify any
errors in the simulation; a further objective was to demonstrate that it was actually possible to test the
recoil mass separator using a radioactive source in place of a particle beam.
1
Chapter 1
Background Theory
1.1 Stellar Helium Burning
When a star has burned all of its hydrogen via the proton-proton chain and/or the CNO cycle the
decrease in energy production causes the core to contract as the radiation pressure from the fusion
reactions is overcome by the gravitational pressure brought about by the star’s mass. As the core
contracts the temperature increases, at this point if the star has a mass greater than approximately 0.4
solar masses it will begin fusing helium[6]. Smaller stars (less than 8-11 solar masses) fuse helium via
the triple-alpha process, resulting in a carbon core. Larger stars can go on to burn carbon by
repeatedly fusing heavier elements with helium nuclei; this is known as the alpha-process or alphaladder.
1.1.1 Triple-Alpha Process
When the core of a star contracts after hydrogen burning its temperature rises, if the mass of the star is
greater than approximately 0.4 solar masses the temperature can increase to around 10 8K at which
point the Coulomb barrier for 4He-4He fusion can be overcome[4].
8
Be is extremely unstable and decays back into two 4He atoms in approximately 10-16 s. However at
around 108K helium nuclei will be fusing often enough to result in an equilibrium concentration of
8
Be, this allows further helium fusion to take place resulting in stable 12C.
This process would have an extremely low probability of occurring, however as predicted by Fred
Hoyle 12C has a 0+ state which induces a resonance at 7.65MeV for the reaction between 8Be and 4He,
2
both of which have a spin-parity of 0+. This state of carbon is known as the Hoyle Resonance and it
allows the rate of production of 12C to be much greater than would otherwise be possible. As a result
the production of 12C via helium fusion is one of the most important and interesting nuclear reactions
in stellar nucleosynthesis.
At temperatures which allow the triple-alpha process to occur, helium can also be involved in other
nuclear reactions to produce heavier elements as follows.
These reactions are the start of the alpha-ladder, a process which creates a large number of elements
with atomic mass numbers that are multiples of 4.
1.1.2 The Alpha-Process
For stars with a mass greater than approximately 8-11 solar masses carbon burning is possible. Once
the carbon has been consumed the core contracts and the temperature increases further, allowing the
fusion of heavier and heavier elements. At these temperatures alpha particles, liberated by the
photodisintergration of other atoms can be involved in further nuclear reactions, producing a range of
heavier nuclei. The so-called alpha ladder consists of the following reactions.
3
The final reaction,
56
Ni(α,γ)60Zn consumes energy, as a result the alpha process ends here. These
reactions have very low rates due to their increasing Coulomb barriers and do not produce a
significant amount of energy within the star. However, they do have a significant impact on the
isotopic abundance of the elements in the universe.
Alpha capture is not limited to only producing elements on the alpha-ladder, the CNO cycle produces
isotopes of carbon, nitrogen and oxygen. 13C and 14N in particular can fuse with 4He, these reactions
both result in the emission of neutrons.
14
N(α,γ)18F results in a reaction with two branches once the 18F decays t 18O
4
These reactions greatly impact the elemental composition of the universe, however at low energies
they have very small cross-sections and it is extremely difficult to experimentally measure their rates
of reaction.
1.2 Inverse Kinematics
The conventional experimental method of direct kinematics would be to accelerate helium nuclei
towards a solid target of the chosen isotope[5]; however at low energies background radiation often
makes the measurement of reactions with low cross-sections very difficult. One solution to this is to
carry out the experiment deep underground; another option is to use the alternative method of inverse
kinematics. The principle of inverse kinematics is to swap the roles of target and projectile in
experimental nuclear reactions[7]. A heavy ion beam is accelerated towards a simple target, typically
hydrogen, helium or deuterium. For hydrogen and deuterium the target can be composed of chemical
compounds generally with the target nuclei bonded to carbon, alternatively they can be used in a gas
target. For helium either a gas or liquid target must be used. The majority of the beam passes straight
through the target but a small proportion of the beam collides with the light nuclei and the reaction
products are emitted in a narrow cone with very forward angles in the laboratory frame of reference.
There are a number of advantages to this method. Firstly, quite often the heavy nuclei will be
inherently unstable, with lifetimes so short that preparing a target is practically impossible. Another
significant advantage is that it reduces beam-induced radiation. Low Z impurities in a target will emit
unwanted background radiation upon collision with a particle beam, a gas target of hydrogen or
helium has no such impurities.
Most importantly, inverse kinematics makes it possible to measure both the induced gamma radiation
as well as the reaction products of the collision as opposed to only measuring the induced gamma
radiation, which could easily be handicapped by background radiation. This is especially true at low
energies when the reaction cross-sections are so low that the events of interest will occur very rarely.
5
The momentum of the original beam and resulting reaction products are almost identical, the variation
in momentum of the reaction products can be approximated as
(1.1)
Where Pcms is the momentum associated with the centre of mass and Pr is the recoil momentum. The
angular acceptance of the kinematic cone of recoil particles is given by
(1.2)
Figure 1.1 Angular opening of reaction products in inverse kinematics[8]
The ability to measure the recoil products in coincidence with the gamma rays allows for more precise
measurements than would be possible by only detecting the gamma rays, however to detect the
comparatively small number of recoil particles necessitates the use of a recoil mass separator to reject
the far more intense original particle beam.
6
1.3 The St. George Recoil Mass Separator
The fundamental purpose of a recoil mass separator is to separate the recoil particles from the far
more intense particle beam that passed through the target with no reaction. In addition to this the
recoil mass separator can remove any contaminant particles that may be present in the kinematic cone
of recoil particles. These contaminants can have many sources, whether they originate from fusioninduced fission or the particle accelerator that produced the beam. The separation of the recoil
particles is typically achieved by directing the beam and reaction products through electric and
magnetic fields in order to separate the ions by their charge state, in addition to this all recoil mass
separators dedicated to studying radiative capture reactions use either one or multiple Wien filters,
these have perpendicular electric and magnetic fields, and are able to deflect charged particles based
on their velocity.
The St. George (Strong Gradient Electro-magnetic Online Recoil separator for capture Gamma Ray
Experiments) is a recoil mass separator designed and built by the University of Notre Dame’s Nuclear
Science Laboratory for the purpose of studying low energy (α,γ) reactions for stable beam masses up
to approximately A=40[2]. It has been built to work with the laboratory’s new St. Ana 5MV vertical
particle accelerator. Whereas previous recoil mass separators such as DRAGON[14] at TRIUMF have
been designed to study a more limited set of nuclear reactions, the St. George has been specifically
designed for a wider range of reactions.
Because the St. George has been designed to study a large number of different reactions there are a
number of challenges that must be overcome. To cope with the range of reactions of interest the
required angular acceptance of the St. George has been calculated to be ±40 mrad with an energy
acceptance of ±7.5%[2].
7
Figure 1.2 The St. George recoil mass separator[2]
The St. George consists of eleven quadrupole magnets (Q1-11), six dipole magnets (B1-6) and one
Wien Filter, in addition to the vacuum chambers, pumps and cooling systems. The system is separated
into three sections. The gas target is positioned at the start of the separator and the detector chamber is
at the end. In total it is 20.29 metres long[2].
The first section, extending from the target to just before Q3, selects the desired charge state by
magnetic analysis. The intense beam and recoil particles with the selected charge state then pass
forward into the rest of the separator while all particles with other charge states are collected in
faraday cups positioned near B2.
The second section, extending from Q3 to just after the Wien filter, separates the intense beam from
the recoil particles based on the difference in their mass. The recoil particles will be heavier than the
original beam and therefore slower since the momentum of each is almost identical. This is the most
important part of the St. George.
The third section serves two purposes, firstly it matches the phase space of the recoil particles to what
is required by the detector further downstream, and secondly it bends the remaining particles through
8
another 52o in an attempt to further reduce background noise. Beyond Q11 the detectors are houses in
the final vacuum chamber.
There is an abundance of access ports throughout the vacuum chambers so that beam stops, slits or
other equipment can be inserted into the beam line. In addition to this all of the dipoles have access
ports for slits at the entrances and exits as well as zero-degree ports for alignment purposes. The beam
line is separated by individual valves which allow different vacuum chambers to be pumped down
while others are open, to reduce vibrations magnetically levitating turbo-pumps are used to maintain a
vacuum of approximately 10-7 Torr.
1.3.1 Gas Target
The windowless, supersonic gas target system HIPPO[9] (HIgh Pressure POint like target) has been
designed specifically for the St. George. It is able to produce a jet with a full width at half maximum
of 2.1mm and a maximum thickness of (2.67±0.16)x1017 atoms/cm2[9]. The gas target has an
extremely compact design with the intention of maximising the detection of gamma rays emitted
when the beam collides with the gas jet. The small width of the jet makes it possible to measure the
angular distribution of gamma rays by positioning detectors around the target.
Figure 1.3 Top view of gas target chamber[9]
9
One slight problem with the compact design is that with an improperly tuned beam it is possible for
the projectile particles to come into contact with the chamber, producing a large quantity of gamma
rays, however this is easily compensated for by correctly tuning the beam and being able to detect
induced gamma rays at the target is extremely advantageous.
Figure 1.4 Cross-section of main chambers of gas target as seen from the side[9]
A remarkable feature of the gas target is that while the jet is under high pressure, the pressure outside
the pumping region remains low, as a result a vacuum is maintained around the jet and no window is
needed. In addition to this previous experiments have demonstrated that a supersonic jet is not
adversely affected by intense beam currents, which will be required once the St. George is
operational.
1.3.2 The Wien Filter
The Wien filter used in the St. George plays the role of a mass filter. After the particle beam collides
with the gas target the resulting recoil particles have a momentum equal to the beam particles, but
with a mass increase of four amu since they have fused with a helium nucleus. As a result they are
travelling slightly slower, this difference in velocity can be exploited.
The principle of a Wien filter is that when a magnetic field and an electric field exist perpendicular to
one another, when a charged particle moves through the two fields they will each exert a force in
opposite directions.
10
(1.3)
As a result, any charged particle with velocity v=E/B will experience no force when travelling
through the Wien filter. It is therefore possible to calculate the velocity of the recoil particles of
interest and then alter the magnetic and electric fields of the Wien filter so that only particles with that
velocity can pass through without being deflected.
Figure 1.5 Electric and magnetic fields in a Wien Filter
A potential problem with this Wien filter is that due to the physical dimensions of the device as well
as the properties of magnetic dipoles, the magnetic fringe field extends further than the electric fringe
field. This can prevent the Wien filter from functioning as a velocity filter. In order to compensate for
this adjustable field clamps were added for the purpose of reducing the length of the magnetic field
along the beam axis. The electrodes have also been shaped in order to extend the electric fringe field
to closely match the magnetic fringe field[2].
11
Figure 1.6 Top view of the Wien Filter. Electrostatic dipole is mounted inside the magnet[2]
1.3.3 Ion Optics Calculations
Extensive ion optics calculations have been conducted to ensure that the St. George could meet the
requirements that it was designed for. In addition to being able to accommodate the multiple reactions
of interest it was also required to achieve a beam suppression of order greater than 10 15[2]. The design
was also constrained by the building space available, as a result the St. George needed to be as
compact as possible, the resulting eleven quadrupoles was the absolute minimum number possible
while still achieving the desired results. While the first and last two dipoles perform specific
functions, B3 and B4 exist to prepare the beam for the Wien filter and also because a wall was in the
way.
The calculations were conducted using the COSY Infinity code[3]; aberrations in the magnetic fields
were calculated up to the fourth order and were reduced by altering the boundaries of the individual
dipole magnets. The results[2] of the final ion optics calculations are shown below, the beam mass is
A=36 and the recoil particles have mass A=40, both have a charge of Q=11
12
Figure 1.7 Ion optics for the St George in the horizontal plane[2]
Rays 1,2,3,9 and 11 demonstrate the most extreme rays the St. George is designed to accept in both
angular acceptance and energy acceptance. Rays 5 and 10 take into account any extensions in the size
of the target; ray 5 enters with the minimum possible angle while ray 10 enters at the maximum angle
of acceptance. Rays 4 and 6 are both emitted at the minimum possible angle but with extreme
energies, they deviate shortly after B1 but still remain within the acceptance of the St. George. Rays 7
and 8 are like ray 6 but they are emitted with wider angles to further test the acceptance of the
separator.
Figure 1.8 Starting values for the rays shown in figure 1.7[2]
13
Chapter 2
Simulation of the St. George
2.1 The COSY Infinity Code
Computer modelling and numerical simulations are routinely employed in the design and analysis of
optical systems including particle accelerators, spectrometers and recoil mass separators[16]. Multiple
codes have been developed for the purpose of modelling particle beam optics, broadly speaking they
are separated into two categories[17].
The codes in the first category use the ray tracing method in which numerical integration is used to
calculate the equations of motion for every individual particle being modelled. It is then possible to
determine the trajectories of these particles through the magnetic fields of the optical system in
question[17]. Because this method is performed for every single ray the code can be very slow when
dealing with large numbers of particles, in addition to this extensive knowledge of the magnetic fields
involved is required. This method is very suitable when the optical system includes specialised optical
elements or if events such as scattering and energy loss need to be taken into account[18]. Codes in
the second category describe the optical elements of a system in the form of transfer-matrices, or
transfer maps, and are known as map codes. Taylor expansions are then computed to describe the
action of the optical system in phase space[17]. This method is generally faster than the ray-tracing
method and is more suitable for systems involving standard optical elements such as dipole and
quadrupole magnets. While it is possible to gain more insight into the optical system with this method
its accuracy can be inadequate since mapping codes are typically written for very simplified magnetic
fields and are only able to approximate more complex systems such as fringe fields and higher order
optical aberrations[17].
Differential algebraic techniques[16] provide an alternative which incorporates the advantages of the
previous two categories of code by computing the Taylor expansion of the final position of a
14
simulated particle with respect to the original position and conditions of the particle such as mass and
charge.
(2.1)
The map M relates the final coordinate zf with the initial coordinate zi; the coordinates contain the
position and momenta of the particle while the vector δ contains the other parameters which affect the
particle such as energy and mass[20]. The vector can also include parameters for the optical system
such as magnetic field strengths. Computing the Taylor expansions of map M produces a set of
differential equations of increasing complexity, while conventional map codes can only solve the
equations to third order[16] differential algebraic techniques are able to solve them to fifth order.
Using this method it is possible to produce transfer maps for complicated optical elements to high
order; it is then possible to calculate the trajectories of rays passing through the system using
numerical integration in a similar method to the ray tracing codes. While this would presumably result
in the same limitation in speed that affects the ray tracing method, differential algebra can generate
high order numerical integrators while only requiring the computational power needed for low order
integrators[17].
The COSY Infinity code is an object oriented code based on differential algebraic techniques for the
purpose of the design and study of particle beam optics. COSY Infinity provides a versatile set of
tools[19] to compute and manipulate transfer maps for optical systems and is able to interface with
both C++ and Fortran 90[17]. Fundamentally, COSY Infinity interprets a file written in
COSYScript[17] describing a set of optical elements, it then calculates the transfer map that describes
the transformation of every particle passing through the optical elements.
2.2 Using COSY Infinity to Simulate the St. George
For the purposes of this experiment it was possible to treat COSY as a “black box”. Simulated
particles were generated with individual sets of starting parameters using Monte Carlo techniques in
C++ and the optical system parameters were written in COSYScript[17] these were then fed into the
15
COSY “black box”, the output of which was the ray trajectories and final parameters of the particles
which were stored in ROOT[21] files for analysis.
The critical files involved in the simulation were generateprofile_alpha.cpp, profile.fox,
saveprofile.cpp and clean_alpha.fox. See appendix A for the clean_alpha.fox file as an example of
COSYScript used in this project, for the sake of brevity the other files will not be included.
Generateprofile_alpha.cpp, written in C++ source code, was responsible for generating the initial
parameters for an arbitrary number of simulated particles produced in an inverse kinematic reaction at
the gas target. These parameters included the energy distribution, angular distribution, charge, mass,
momentum and starting coordinates of the particles. For general experiments several run-time files
containing tables of isotopes and data for nuclear reactions are called upon to provide the necessary
information to calculate the particle parameters. In this experiment since no reaction was taking place
the mass, charge and energy distribution of the alpha particles were coded directly into the file. In
order to model the radioactive source it was assumed that the 2mm diameter source emitted particles
randomly in all directions. A random number generator was used to pick a random value for the
azimuthal angle to choose the starting coordinates on the surface of the source. Having done this the
polar angle was randomly selected in the same way to determine the initial trajectory of the particle.
The energy of the particles was modelled as a Gaussian distribution with a mean of 4.66MeV.
Initially the modifications made to the file resulted in a number of bugs, specifically the angular
distribution of the simulated particles was uneven. A simple Fortran 95 program was written to model
the trajectories of the emitted particles and attempt to reproduce the results of the compiled
generateprofile_alpha.cpp, as a result it was possible to identify the errors in the code.
Profile.fox written in COSYScript contained the parameters of the optical systems in the St. George.
These included the magnetic rigidities of the optical elements, the drifts lengths between the elements,
the bending radii of the dipoles as well as the magnetic field strengths and field lengths of the
quadrupoles.
16
Figure 2.1: ASCII dump of the variables contained in the ROOT file
17
Figure 2.2: Plot of coordinates of simulated particles between dipoles B1 and
B2, all distances in metres. The red rectangle shows the area represented in
figure 2.4.
Figure 2.3: Plot of energy against position of the simulated particles in the
horizontal plane between dipoles B1 and B2, distances in metres and energy in
MeV. The red rectangle shows the area represented in figure 2.4.
18
Figure 2.4: Energy(MeV) histogram for the region marked in figures 2.2 and
2.3.
The COSYScript file clean_alpha.fox contained the optical system parameters of the St. George as
well as 27 predetermined rays designed to represent the entire range of possible parameter values.
Rather than providing the range of information stored in the ROOT files this gave a quick
demonstration of the ion optics.
Figure 2.5: COSY output with clean_alpha.fox
19
Chapter 3
Experimental Setup
In order to validate the ion optics calculations for the St. George we needed a source of charged
particles, and the ability to detect them as they travelled through the recoil mass separator. Ideally we
needed a high count rate as well as very good energy resolution; however without a particle
accelerator it is very difficult to achieve both. A sealed 1μCi Americium-241 source was selected to
be used in this experiment; it consisted of a disk of Americium-241 sealed behind a thin mylar
window. It is worth mentioning that the source in question was not designed to be used in laboratory
experiments, as such the quality of the source was questionable. The primary reason for choosing this
source was its activity; while the laboratory possessed several radioactive sources designed for use in
experiments and therefore had far better energy resolutions, their activity was far lower and it was
decided that having a higher count rate would be more advantageous. For the detection of the alpha
particles two four-channel P-N junction silicon detectors were loaned to us by Professor Wanpeng
Tang. By positioning the source in the gas target at the position of the jet behind a collimator we
could limit the opening cone of the alpha particles to the maximum angular acceptance of the St.
George.
3.1 Apparatus and Equipment
An extremely important condition for the experiment to be successful was that the source needed to
be placed perfectly in line with the centre of the entrance into the St. George. Misalignment would
cause the alpha particles to be deflected incorrectly by the magnetic fields. In order to ensure that the
alpha source could be positioned correctly a source holder was designed which was able to be
precisely manipulated before being sealed in the gas target.
20
Figure 3.1 Design of source holder
Originally the holder was designed to have the collimator directly attached to it as shown above,
however we found that it was far more effective to attach the collimator directly to the exit of the gas
target in order to ensure that it was optimally positioned. The final arrangement of the source and
collimator is shown below.
Figure 3.2 Diagram of source suspended within the gas target behind the double collimator
It was decided that a double collimator should be used in order to reduce Rutherford scattering caused
by the alpha particles striking the first collimator. While some of the alpha particles would strike the
wall inside the gas target, with a diameter of 6mm the hole was large enough that it would have no
discernible effect on the particle beam leaving the final collimator.
21
Due to the size and shape of the detector there were a limited number of positions where it would be
possible to insert the detector into the beam line.
Figure 3.3 Possible positions for the detector within the St. George
An ISO-63 flange or larger port on the side of the beam line was required in order to ensure that the
detector could be centred inside the vacuum chamber. The only exception to this was position 1 where
a valve was removed so that the detector could be placed in the entrance of the first dipole magnet,
B1. The significant distance between positions 3 and 4 was problematic, especially because it made it
impossible to measure the effects of the central dipole magnets B3 and B4 separately.
Each detector was connected via a preamp to an individual power supply so that it was possible to
provide the required positive voltage of 30V in order to reverse bias the P-N junction. The detector
channels themselves were connected via an 8-channel feed-through to a biased preamp which in turn
was connected to a 12V power supply. The biased preamp was then connected via another 8-channel
feed-through to an amplifier at which point the signal was split, part of it going to the ADC while the
fast signal went to the CFD, logic fan and gate generator.
22
Figure 3.4 Schematic and image equipment
3.2 Magnetic Components
The electromagnets used in the St. George are supplied with highly stable direct currents[2], in order
to minimise the saturation of the magnets the iron components are all composed of solid, soft iron.
The wire coils are hollow copper conductors, water cooling is used to keep the coil temperatures
beneath 55ºC.
3.2.1 Quadrupole Magnets
The purpose of the quadrupole magnets is to focus the particle beam within the recoil mass separator,
it is vital that they do not contribute in any way to steering the charged particles. Due to the desire to
reduce the number of quadrupoles required[2] each focussing magnet has been build to unique
parameters determined by the ion optics calculations, they can be seen below.
Figure 3.5 Quadrupole design parameters[2]
23
Most of the quadrupole magnets require a wider horizontal good-field region than their vertical goodfield region, the physical dimensions of the vacuum chambers within the magnets have been designed
with this in mind.
Figure 3.6 Cross-section of Q10 including vacuum chamber, all lengths in mm[2]
The property of a quadrupole magnet which enables it to be in used in particle beam focussing is that
the magnitude of the magnetic field produces increases with the radial distance from the longitudinal
axis of the magnet. As a result any charged particles passing through the quadrupole will be deflected
towards the centre of the magnet, either vertically or horizontally. Of the two quadrupole magnets
studied during this experiment, Q1 focuses vertically and Q2 horizontally. The relationship between
magnetic field strength and radial distance from the centre of the magnets are shown below.
24
0.4
0.3
0.2
50A
0.1
42.5A
0
-0.08
-0.06
-0.04
-0.02
0
-0.1
0.02
0.04
0.06
0.08
35A
15A
-0.2
-0.3
-0.4
Figure 3.7 Field strength in tesla versus radial distance from magnet centre in metres for Q1
0.4
0.3
0.2
0.1
90A
0
-0.15
-0.1
-0.05
-0.1
76.5A
0
0.05
0.1
0.15
63A
27A
-0.2
-0.3
-0.4
-0.5
Figure 3.8 Field strength in tesla versus radial distance from magnet centre in metres for Q2
The hysteresis effect in the quadrupole magnets is very small[2], as such it is possible to accurately
set the magnetic field by changing the current supplied to each of the magnets. To facilitate this the
relationship between the current and magnetic field was found for the quadrupole magnets used in the
experiment.
25
0.25
y = 0.0045x + 0.0066
R² = 0.9998
0.2
0.15
0.1
0.05
0
0
10
20
30
40
50
60
Figure 3.9 Field strength in tesla versus current in amps for Q1
0.35
y = 0.003x + 0.0114
R² = 0.9982
0.3
0.25
0.2
0.15
0.1
0.05
0
0
20
40
60
80
Figure 3.10 Field strength in tesla versus current in amps for Q2
Where I is the current supplied to the magnet and the field strength is in Tesla.
26
100
3.2.2 Dipole Magnets
The dipole magnets are responsible for bending the particles around the recoil mass separator and
rejecting particles with unwanted charge states. The dipoles used in the St. George are H-type
magnets, while each is built to different specifications they all have a bending radius of 750mm and
each magnet bends the particles through an angle of 26º[2].
Figure 3.11 Field vector presentation in C (left) and H (right) dipole magnets[10]
H-type magnets restrict access to the vacuum chamber however they have a smaller fringe field and
are far more rigid than C-type dipole magnets[10].
The dipoles have been designed so that the pole faces serve as the top and bottom of the vacuum
chamber, in order to measure the magnetic field produced central port extending through the inner
yoke is able to hold an NMR or Hall probe without requiring the chamber to be vented. In addition to
this ports are located at the entrance and exit of each dipole which can hold adjustable slits or other
diagnostic equipment in order to stop or measure the beam as well as measuring any background
radiation. For alignment purposes and any other miscellaneous access requirements 0º ports are in
place in both directions[2].
27
Figure 3.12 Top view of dipole B1 and vacuum chamber[2]
The dipole magnets can produce a maximum magnetic field of 0.6T in order to bend particles with
magnetic rigidities of up to 0.45Tm. The vertical gaps of all the dipoles except for B5 are 70mm, B5
has a gap of 80mm. The horizontal good-field regions of magnets B1 to B4 are 200mm while those
for B5 and B6 are 140mm. Additional specifications for the magnets are shown below[2].
Figure 3.13 Design parameters for dipole magnets [2]
In order to improve the reliability of the magnetic field Rose shims have been placed on the sides of
the magnet poles. These are iron strips machined to the desired thickness and size which reduce
28
variations in the magnetic field; they ensure that the field is constant to within dB/B < 2x10 -4 in the
good-field region, the inner and outer return yokes shield against the fringe field and contribute to the
field produced by the coils of the dipole. In order to ion optic aberrations the physical shape of the
magnetic poles was carefully designed in accordance with the results of the ion optics calculations,
this can be seen in the diagram above where the asymmetrical shape of the poles at the entrance and
exit of the magnet is clearly visible.
The central port for NMR and Hall probes makes it possible to measure magnetic field without
blocking the path of the particle beam. As a result it is impossible to position the probe directly in the
centre of the dipole but it is possible to measure the field while the recoil mass separator is in
operation, this is particularly important because while the effect of hysteresis on the quadrupoles was
small enough to disregard, the effect on the dipoles was more significant. Fortunately the probe port is
still deep enough to reach the good-field region within the dipoles.
B1 Magnetic Field
7000
6000
Magnetic Field (Gauss)
5000
120 Amps
4000
96 Amps
3000
72 Amps
48 Amps
2000
24 Amps
1000
0
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
Position from Centre (in)
Figure 3.14 Field strength in gauss versus distance from magnet centre in inches. The distance
the Hall probe reached is marked by the arrow.
29
The three Hall probes available for use during this experiment were able to reach a position 6.548
inches from the centre of the magnet, comfortably within the good-field region of the dipole even at
higher currents. Unfortunately it was discovered that the Hall probes themselves were not entirely
consistent with each other.
15
Dif f erence in measure f ield strength (Gs)
10
5
0
Probe 1-Probe 2
0
20
40
60
Current (A) 80
100
120
140
Probe 1-Probe 3
-5
Probe 2-Probe 3
-10
-15
-20
-25
Figure 3.15 Difference in measured field strength (in gauss) between probes from 0A to 120A
While the three Hall probes measured approximately the same magnetic field strength there were
clear differences. While some deviation was inevitable since a “perfect” Hall probe could not
technically exist it was expected that any difference in measurements would be uniform, for example
a reasonably constant difference between any two probes. As can be seen above, our second Hall
probe would read a field 20 gauss lower or 15 gauss higher than the other probes, and the difference
changed as the current applied to the magnet changed. It was decided that this was most likely due to
a mechanical fault since the probe itself appeared to be slightly crooked. While the variation between
probes one and three was regrettable, it was far more consistent than with the second probe.
While changing the current supplied to the dipoles and measuring their magnetic fields, it was
important to take into consideration the effect of hysteresis. In a ferromagnetic material unaffected by
any magnetising force the magnetic dipole moments are disordered, as such the material does not
contain an electromagnetic field. When a magnetising force is applied, in this case by current flowing
30
through the coils of wire, the magnetic dipole moments align until the magnetisation of the material
reaches its maximum point, known as the material’s saturation magnetisation[11]. When the
magnetising force is reduced to zero the material retains a considerable degree of magnetisation
depending on the retentivity of the material. With soft ferromagnetic materials such as those used in
the St. George magnets the magnetisation will eventually drop to zero, hard ferromagnetic materials
like hard steel will retain their magnetisation indefinitely unless it is removed by an external
demagnetising field or through heat treatment.
Figure 3.16 Hysterisis loop
By reversing the magnetising force the magnetic field will continue to decrease towards zero, the
force required to bring the magnetic field back to zero depends on the coercivity of the material. Once
the coercive force has been reached the ferromagnet will eventually reach its saturation magnetisation
in the opposite direction, at which point if the original force is applied once again the process will
reverse.
In this experiment we did not reverse the magnetising force as that would involve reversing the
current applied to the dipoles; as such we were only concerned with the upper right quadrant of the
diagram shown above. In order to reproduce the same magnetic fields we simply increased the current
supplied to its maximum of 120A and then slowly reduced the current until the desired magnetic field
was reached. After finishing a measurement the current was dropped to zero and the magnets were
then given time to lose their magnetization before increasing the current again.
31
3.3 Silicon Diode Detector
The principle of a semiconductor radiation detector is that when ionising radiation passes through the
detector it deposits charge, electrons from the valence band of the detector move up to occupy the
conduction band leaving holes in the valence band. If the semiconductor is placed between two
electrodes the electric field causes the electrons and holes to travel to opposite electrodes, this
produces a pulse of current that can be measured in a circuit. The energy needed to produce an
electron-hole pair should already be known and is not dependant on the energy of the radiation that
produced it, as a result it is possible to measure the intensity of the incoming radiation.
An extremely important property of a detector is its signal to noise ratio[12]. Maximising the signal
requires a small band gap so that the ionisation energy is low and electron-hole pairs can be formed
easily. However with a small band gap electrons from the valence band of the detector material can
occupy the conduction band at room temperature producing electron-hole pairs that exist for a limited
time before recombining. A thermal equilibrium is reached between the production and recombination
of the electron-hole pairs, this is known as the intrinsic carrier concentration[13].
i
e
h
g
Where ni is the intrinsic carrier concentration, me and mh are the effective masses of an electron and
hole respectively and Eg is the energy gap between the conduction band and the valence band[13].
The consequence of the intrinsic charge carriers is of course noise. The electron-hole pairs produced
by ionising radiation are indistinguishable from those that are produced without any external input. To
function as a detector it is necessary to remove the intrinsic charge carriers, this is accomplished
through a reverse biased P-N junction[12].
A P-N junction is formed by the interface of a P-doped semiconductor and an N-doped
semiconductor. Doping involves the replacement of a small number of atoms within the atomic lattice
32
(3.1)
of a semiconductor with either group 3 or group 5 atoms, in other words atoms with either 3 or 5
valence electrons. This results in increased conductivity in the doped semiconductor since it
introduces more charge carriers into the material. Whereas the pure semiconductor had an equal
number of holes and electrons due to the intrinsic charge carriers, doped semiconductors have either
more electrons or more holes depending on whether it is P-doped or N-doped.
Replacing a silicon atom with a group 5 donor atom such as phosphorous results in an additional free
valence electron in the atomic lattice. The energy level of the donor is only slightly lower than that of
the conduction band, as a result at room temperature most of the electrons are transferred to the
conduction band, leaving the phosphorous atoms as positively charged ions and raising the Fermi
level of the semiconductor.
Weakly bound
electron
Figure 3.17 The energy of the donor atom is very close to that of the conduction band, as a result the
additional electron is very weakly bound. Figure from [13]
33
Figure 3.18 At room temperature the donor atoms are easily ionised and the weakly
bound electrons occupy the conduction band, this raises the Fermi level. [13]
On the other hand replacing a silicon atom with a group 3 acceptor atom such as boron results in an
extra hole in the atomic lattice. The energy level of the acceptor atom is only slightly higher than that
of the valence band, as a result at room temperature most of the empty energy levels in the acceptor
atom are filled with electrons leaving holes in the valence band and giving the boron atom a negative
charge. This results in the Fermi level of the semiconductor dropping.
Figure 3.18 The energy of the acceptor atom is very close to that of the valence band, the
electrons require a very small amount of energy to fill the open bond. [13]
34
Figure 3.19 At room temperature most of the acceptor levels are occupied by electrons
from the valence band leaving holes behind, this lowers the Fermi level. [13]
A P-N junction is made from a single semiconductor crystal P-doped and N-doped in separate
regions[12]. In the N region the majority of the charge carriers are electrons with an equal
concentration of positively charged donor atoms whereas in the P region the majority of charge
carriers are holes with an equal concentration of negatively charged acceptor atoms. At the interface
between the two regions the difference in Fermi levels causes the charge carriers in each side to
experience a diffusion force and they move to the opposite region.
However as the charge carriers are exchanged the positively charged donors and negatively charged
acceptors remain in place. This causes the region around the interface to become charged, negatively
on the P-side and positively on the N-side, producing a space charge and resulting in a potential
difference between the two regions. The electric field produced by the potential difference exerts an
opposing force on the charge carriers preventing them from diffusing further and resulting in
equilibrium between the two sides. The space charge region around the interface is known as the
depletion zone.
35
Figure 3.20 At the interface the difference in Fermi levels results in diffusion of the charge carriers
across to the opposite side until an equilibrium is reached and the Fermi level is equal. The ions around
the boundary produce a space charge and the resulting electric field prevents further diffusion. [13]
To reverse bias the P-N junction an external voltage must be applied with the anode attached to the N
region and the cathode to the P region. This results in the charge carriers being pulled away from the
depletion zone towards their respective electrodes, widening the depletion zone. This increases the
potential difference between the two regions further suppressing any diffusion across the interface.
With sufficient reverse biasing there will be very little passage of current across the junction (known
as leakage current) and therefore very little noise due to the intrinsic charge carriers in the detector.
Figure 3.21 Reverse biasing the junction widens the depletion zone, further reducing the flow
of current across the junction.[13]
With the intrinsic charge carriers effectively removed the detector can function correctly, when
ionising radiation passes through the detector the deposited charge carriers will be drawn to their
36
respective electrodes and the resulting brief pulse of current can be used to determine the intensity of
the incident radiation.
P+ type
Charge carriers
-
N type
N+ type
Incident radiation
Figure 3.22 Incident radiation deposits charge within the detector, the charge flows to the
corresponding electrode and produces a brief pulse of current which can be measured. Note that
the + and – stands for more or less intensely doped, the surfaces of the detector require the most
donor or acceptor atoms.
The detectors used in our experiment consisted of a segmented P-type silicon crystal on the front
implanted on a solid N-type crystal, each detector had four 4cm long, 1cm wide channels and could be
arranged in either a 4x2 grid pattern or in a 1x8 line. While the 4x2 grid would certainly have given us
the ability to detect the alpha particles in two dimensions it would have given us far less precision
horizontally, instead we chose to use the 1x8 line, sacrificing the ability to measure in two dimensions
for greater precision in the horizontal dimension.
Figure 3.23 Possible arrangements for the detector, each strip is 4cm long and 1 cm wide.
37
Unfortunately since the detectors were originally designed to be arranged in the 4x2 grid, our decision
meant that there was a slight gap between the central channels as can be seen below. In addition to
this the detector had been used in multiple experiments beforehand and as a result had suffered some
superficial damage.
Figure 3.24 The two detectors used in the experiment. The gap between the central channels is 4mm
wide.
The detectors were designed to be operated with a reverse bias of 30V, however over the course of the
experiment we found that one of the detectors could not hold the full voltage without the leakage
current rising sharply and uncontrollably, the most likely cause of this was a short-circuit somewhere
in the detector. Lowering the biasing voltage improved the leakage current and while the energy
resolution of the affected detector was severely degraded the signal strength was still good enough
that there was no apparent loss in signal. Other inconveniences with the detector included the leakage
current occasionally rising uncontrollably for brief periods of time seemingly at random and electrical
noise from the building interfering with the signal, we also found that the outer channels picked up
less signals than the central ones, however the signal loss was consistent and could be compensated
for. These problems were generally manageable through noise reduction techniques and did not
severely affect the measurements taken.
Note that within the separator detector channel 1 is furthest beam-right, and channel 8 is furthest
beam-left.
38
Chapter 4
Experimental Measurements and Analysis
The measurement goals of the experiment required us to be able to determine the energy of the
charged particles incident upon the detector. Silicon detectors provide electrical signals in response to
incident charged particles, it was therefore necessary to calibrate the detectors by determining the
relationship between the electrical signals and the particle energy. To calibrate the detector it was
placed in a vacuum chamber facing an unsealed calibration source containing a 10nCi Gadolinium148 source and a 10nCi Americium-241 source positioned approximately 17cm from the detector.
Ganadium-148 emits alpha particles of energy 3182.69keV while Americium-241 emits alpha
particles of energy 5485.56keV (84%) and 5442.8keV (13.1%). The energy resolution of our detector
was not good enough to distinguish between the two Americium peaks; we also discovered that
detector strip 1 was not functional.
Figure 4.1: Calibration spectra for the detector. Channels 2 to 8 see the 5485.56keV Am241 peak and the 3182.69keV Gd-148 peak on either side of the experimental source
Am-241peak. Note the much larger energy resolution of the experimental source
compared to the calibration sources.
39
Having successfully calibrated the detector with the unsealed calibration source we replaced it with
the 1μCi Americium-241 source. The mean energy of the alpha particles emitted by the source was
measured to be 4.66MeV with a resolution of 0.6MeV FWHM upon reaching the detector. Due to the
thickness of the source it was expected that energy would be lost due to scattering within the source
itself, while this regrettable it was an unavoidable consequence of the high activity of the source.
4.1 Position 1 – Q2B1
With the detector calibrated it was moved to position 1 (see fig. 3.3) at the exit of quadrupole Q2. The
source was placed in the gas target with the collimator attached directly to the source holder; the gas
target had been aligned with the St. George prior to this.
In order to study the solid angle of the detector and to verify that we understood the role of the
collimator initial measurements were taken without a magnetic field in the quadrupoles, the entrance
to B1 was the only position where it would be possible to detect the alpha particles without using the
magnetic fields and therefore with as few variables affecting the measurements as possible. With an
angular distribution of ±40mrad the unfocussed alpha particles reached a width of approximately
16cm, as a result they came into contact with the walls of the vacuum chamber. We found that the
alpha particles scattering off the walls had a significant impact on our measurements by increasing the
number of counts recorded, particles which would otherwise have missed the detector were deflected
towards the strips after scattering from the vacuum chamber walls.
40
Figure 4.2: Cross section of unfocussed alpha particles with angular distribution of
±40mrad at the exit of Q2, the vacuum chamber walls are marked in red. Distances in
metres.
With the collimator attached directly to the source holder we were not confident that the alpha particle
source was aligned with the St. George. In an attempt to discover whether or not the alpha particles
were entering the separator correctly the quadrupoles were set to produce magnetic fields close to
those determined by the ion optics calculations. In order to compare results from different runs the
counts per detector strip were normalised to the counts on strip 3. The results were not easily
distinguishable from those without magnetic fields. With the exception of detector channel 8 the
normalised counts per channel for the focussed run were within the error bars of the normalised
counts per channel of the unfocussed run. By comparison the focussed and unfocussed runs had far
less agreement with their corresponding simulations. The spread of the alpha particles relative to the
size of the detector meant that it was impossible to determine whether or not the alpha particles in the
focussed run were entering the magnetic fields incorrectly, which would have resulted in the alpha
particles being deflected, or steered, away from the centre of the beam line by the quadrupoles. It was
41
expected that steering would be visible on the detector, however the two runs were effectively
identical.
2600
2500
Normalised counts
2400
2300
Magnetic field in Q1 Q2
2200
No Field
2100
2000
1900
0
1
2
3
4
5
6
7
8
9
Detector channel
Figure 4.3: Normalised counts per channel for position 1 with no magnetic field in the quadrupoles
and with magnetic field supplied to the quadrupoles (32.418A in Q1 producing a field of 0.152
Tesla, 54.794A in Q2 producing a field of 0.164 Tesla). Note that channel 1 is not shown because
it was not functional.
2600
2500
Normalised counts
2400
2300
No Field
2200
Simulation no field
2100
2000
1900
0
1
2
3
4
5
6
7
8
9
Detector channel
Figure 4.4: Normalised counts per channel for position 1 with no magnetic field in the quadrupoles
and the corresponding simulation
42
2600
2500
Normalised counts
2400
2300
Magnetic field in Q1 Q2
2200
Simulation with field
2100
2000
1900
0
1
2
3
4
5
6
7
8
9
Detector channel
Figure 4.5: Normalised counts per channel for position 1 with magnetic field in the quadrupoles
and the corresponding simulation. The increased counts in the central detector channels were most
likely due to scattering within the vacuum chamber. This should not have occurred on the focussed
run since the alpha particles should have been within the vacuum chamber. The clear difference
between the simulation and results suggested that the alpha particles were not being focussed
correctly, the asymmetry of the counts per detector channel was also concerning, however it
appeared to be matched by the simulation.
In an attempt to determine whether or not the alpha particles were centred on the detector it was
decided to increase the current in Q2 in order to focus them horizontally as much as possible and
increase the counts in the central detector channels.
43
2900
2800
2700
2600
2500
2400
Max field in Q2
2300
Simulation
2200
2100
2000
1900
0
2
4
6
8
10
Figure 4.6: Normalised count per channel for position 1. With the maximum safe field in Q2 (95A
giving a field strength of 0.2964 Tesla) and the corresponding simulation. The clear asymmetry of
the counts per detector channel in comparison to the previous runs demonstrates that the alpha
particles were being steered by the quadrupoles.
The uniformity of the simulations demonstrated that with an angular acceptance of ±40mrad we were
unable to obtain useful data with our detector at this location, the large angular opening in
combination with the large energy spread of the alpha particles meant that it was only possible to
obtain an achromatic focus (where the size is independent of the particle energy) at a few positions in
the separator. In the St. George the point of achromatic foxus was after the Wien Filter[2]. Decreasing
the angular acceptance of the collimator to ±20mrad would produce a narrower spread of alpha
particles which would be possible to focus at multiple points throughout the separator.
Having demonstrated that the alpha particle source was not aligned with the St. George it was decided
that attaching the collimator directly to the source holder was a mistake; instead a new double
collimator was attached directly to the gas target.
Initial measurements with the new collimators were encouraging, a short run with no magnetic fields
demonstrated that the alpha particles were clearly hitting the middle of the detector. Further
measurements with varying magnetic fields in the quadrupoles demonstrated the reliability of the
44
simulation. Having yet to pass through the dipole magnets the energy distribution of the alpha
particles had not changed enough to provide useful information.
The narrower angular acceptance prevented scattering from within the vacuum chamber, additionally
there was a clear difference in the counts picked up by each detector channel which made it relatively
simple to determine that the alpha particles were not being visibly steered by the quadrupoles.
1800
1600
Normalised counts
1400
1200
1000
800
3050
600
Simulation
400
200
0
0
2
4
6
8
10
Detector channel
Figure 4.7: Normalised counts per channel for position 1 with 30A (0.1416 Tesla) in Q1 and 50A
(0.1614 Tesla) in Q2.
45
3500
3000
Normalised counts
2500
2000
4080
1500
Simulation
1000
500
0
0
2
4
6
8
10
Detector channel
Figure 4.8: Normalised counts per channel for position 1 with 40A (0.1866 Tesla) in
Q1 and 80A (0.2514 Tesla) in Q2.
200
180
160
Normalised counts
140
120
100
2070
80
Simulation
60
40
20
0
0
2
4
6
8
10
Detector channel
Figure 4.9: Normalised counts per channel for position 1 with 20A (0.0966 Tesla) in Q1 and 70A
(0.2214 Tesla) in Q2. This run was very short, as a result the statistical error in the results is far
greater than in the previous runs.
46
4.2 Position 2 – B1B2
Measurements taken from position 2 onwards were initially impaired by the gas target becoming
misaligned. At some point during the experiment the gas target shifted by approximately 1mm, as a
result the alpha particles leaving the collimator did not enter the magnetic fields of the quadrupoles
correctly. The result was the alpha particles being steered by the focussing magnets and then entering
the dipoles incorrectly. If the misalignment occurred while the detector was at position 1 the steering
was not noticeable, however after the dipoles it became very problematic. The steering was not
initially discovered because at the detector it appeared that the alpha particles were correctly centred,
rather than finding the correct field in the dipoles we had been steering the alpha particles back to the
centre of the beam line. Eventually the path the alpha particles were taking caused them to come into
contact with the vacuum chamber walls, secondary peaks appeared in the spectra which could not
have been the secondary Americium-241 peak since the energy resolution of the source was so large
that it was impossible to discriminate between the two peaks, instead the secondary peaks were
caused by the alpha particles scattering on the vacuum chamber walls and reaching the detector with
lower energy.
Figure 4.10: The secondary peak caused by scattering beside the primary peak.
47
Having determined that the alpha particles source and collimator were correctly aligned and that the
simulation could reliably predict the angular distribution of the alpha particles with a range of
magnetic fields supplied to the quadrupoles the detector was moved to position 2, between dipoles B1
and B2. The magnetic field required in the dipole magnets is determined by the momentum, and
therefore the energy as well as the charge of the particle of interest, it is given by the following
equation,
(4.1)
Where ρ is the bending radius of the dipole magnet. For the 4.66MeV alpha particles in this
experiment the required field was calculated to be 0.4154 Tesla.
For the initial measurements the magnetic field in dipole magnet B1 was set to 0.4154 Tesla as
measured by the Hall probe while Q1 was set to 33.065A (0.1554 Tesla) and Q2 was set to 51.224A
(0.1651 Tesla).
18000
16000
Normalised counts
14000
12000
10000
Simulation
8000
0.4154 T
6000
4000
2000
0
0
1
2
3
4
5
6
7
8
Detector channel
Figure 4.11: Normalised counts per channel for position 2 with 33.065A (0.1554 Tesla) in Q1,
51.224A (0.1651 Tesla) in Q2 and 0.4154 Tesla in B1 as measured by the Hall probe and the
corresponding
simulation.
The
measurements
clearly
do not
the simulation.
point
This was not an
unexpected
result;
while the Hall
probe
had match
been zeroed
correctly At
wethis
knew
thatinittime
had
problems with the leakage current on both detectors meant that it was not possible to accurately
not been compared to an absolute
known
so of
it was
almostparticles.
certainly not reading the true field. As
measure
thefield,
energy
the alpha
48
such it was decided to gradually increase the field in B1 until the results matched the simulation. At
length it was found that at 0.4211Tesla as measured by the Hall probe the results seemed to most
closely match the simulation.
12000
10000
Energy (keV)
8000
6000
0.4211 T
Simulation
4000
2000
0
0
1
2
3
4
5
6
7
8
Detector channel
Figure 4.12: Normalised counts per channel for position 2 with 33.065A (0.1554 Tesla) in Q1,
51.224A (0.1651 Tesla) in Q2 and 0.4211 Tesla in B1 as measured by the Hall probe and the
corresponding simulation.
4840
4820
Energy (keV)
4800
4780
4760
4740
0.4211 T
4720
Simulation
4700
4680
4660
0
2
4
6
8
Detector channel
Figure 4.13: Measured energy in keV per detector channel and the corresponding simulation. Note that
due to leakage current problems channels 1 to 4 could not be used, also channel 8 had become
4.3 unreliable
Position
3 –point.
B2Q3
at this
The result being only three of the eight detector strips could be used to
accurately measure the energy of the alpha particles.
49
4.3 Position 3 – B2Q3
From these results it appeared that 0.4211 Tesla as measured by the Hall probe was the correct
magnetic field for the quadrupoles, this was not an unreasonable value at 1.4% from the original
calculated field strength. To test this the detector was moved to position 3 between dipole B2 and
quadrupole Q3. Initial measurements with 0.4211 Tesla in each dipole did not produce the expected
results.
6000
Normalised counts
5000
4000
3000
0.4211_0.4211 T
2000
Simulation
1000
0
0
2
4
6
8
Detector channel
Figure 4.14: Normalised counts per channel for position 3 with 33.065A (0.1554 Tesla) in Q1,
51.224A (0.1651 Tesla) in Q2 and 0.4211 Tesla in both dipoles as measured by the Hall probe and
the corresponding simulation. The beam is clearly bent too far right.
50
4950
Energy (keV)
4900
4850
4800
0.4211_0.4211 T
4750
Simulation
4700
4650
0
2
4
6
8
Detector channel
Figure 4.15: Measured energy in keV per detector channel and the corresponding simulation.
Further experimentation to find combinations of magnetic fields that centred the beam on the detector
produced two notable results. Initially the goal was to keep the fields in each dipole identical, then it
was decided to keep the field in B1 at 0.4211 Tesla and find the corresponding field in B2 that would
centre the beam. It was of course very possible that we would simply be steering the beam towards the
target.
4000
Normalised counts
3500
3000
2500
2000
0.4172_0.4172 T
1500
Simulation
1000
500
0
0
2
4
6
8
Detector channel
Figure 4.16: Normalised counts per channel for position 3 with 33.065A (0.1554 Tesla) in Q1,
51.224A (0.1651 Tesla) in Q2 and 0.4172 Tesla in both dipoles as measured by the Hall probe, and
the corresponding simulation.
51
4900
Energy (keV)
4850
4800
0.4172_0.4172 T
4750
Simulation
4700
4650
0
2
4
6
8
Detector channel
Figure 4.17: Measured energy in keV per detector channel and the corresponding simulation.
These results were very promising, 0.4172 Tesla was extremely close to the calculated field strength
of 0.4154 Tesla, a difference that small could be explained by the energy distribution of the source
being uneven, which was likely to begin with. However 0.4172 Tesla was definitely not enough in
position 2 to centre the beam on the detector.
1800
1600
Normalised counts
1400
1200
1000
800
0.4211_0.4071 T
600
Simulation
400
200
0
0
2
4
6
8
Detector channel
Figure 4.18: Normalised counts per channel for position 3 with 33.065A (0.1554 Tesla) in Q1,
51.224A (0.1651 Tesla) in Q2, 0.4211 Tesla in B1 and 0.4071 Tesla in B2 as measured by the Hall
probe, and the corresponding simulation.
52
4880
4860
4840
Energy (keV)
4820
4800
4780
0.4211_0.4071 T
4760
Simulation
4740
4720
4700
4680
0
2
4
6
8
Detector channel
Figure 4.19: Measured energy in keV per detector channel and the corresponding simulation.
These two combinations of magnetic fields produced results which closely matched the
simulations; in addition to this the fields used were extremely close to the calculated value,
the greatest difference was 2%, this could be explained by the fact that the Hall probe used
had not been calibrated with an absolute known field, but the fact that at position 3 more than
one combination of magnetic fields was found to match the simulation is alarming.
For position 2, between B1 and B2 there should only be one possible field in B1 that could
bend the particles correctly, it was measured to be 0.4211 Tesla. While ideally the dipoles
should all require exactly the same magnetic field to bend the alpha particles a slight
difference was expected. From our results it seems likely that 0.4211 Tesla in B1 and 0.4071
Tesla in B2 was the correct solution, or at least very near to it. Unfortunately the poor energy
resolution of the radioactive source used in this experiment meant that it is not possible to be
certain, experience gained when the gas target was not aligned with the St. George
demonstrated how a very slight misalignment of the charged particles entering the magnetic
fields of the optical elements can significantly impair any measurements taken, especially
when it is only possible to detect the charged particles at one point in the separator.
53
Unfortunately it was not possible to progress beyond B2. Between positions 3 and 4 were five
quadrupoles, two dipoles and the Wien Filter, with the time remaining it was not considered
worthwhile to attempt to detect the alpha particles at the next location. In particular this was
because it was impossible to position the detector between dipoles B3 and B4, we had already
found that the dipoles required slightly different magnetic fields in order to centre the alpha
particles. Without being able to test their effects on the ion optics individually we would have
had no choice but to make an educated guess as to the combination of the magnetic fields.
Since the alpha particles would need to pass through two quadrupoles and the Wien Filter
before reaching the detector any steering caused by an incorrect field in one or both of the
dipoles would almost certainly have prevented the alpha particles from reaching the detector.
With the equipment used in this project it is most likely not possible to progress any further.
In order to accurately determine the fields required in the dipoles it is necessary to be able to
measure the position of the charged particles at both the entrance and exit of the dipoles. This
can be accomplished by inserting adjustable slits into the entrance and exit ports of the
dipoles, an intense focussed particle beam could then be sent through the separator and the
magnetic fields adjusted until the beam passed through both entrance and exit slits without
being detected.
While this could theoretically be accomplished with a radioactive source the comparatively
low activity is a severe handicap in terms of the time required to make any useful
measurements. This could be compensated for by choosing a more active source, however
that would result in a worse energy resolution which would make it much harder to focus the
charged particles.
In order to fully validate the ion optics of the St. George it will be necessary to use the St.
Ana particle accelerator. While it is technically possible to use a radioactive source and a
54
silicon detector the practical limitations of this method would make it very difficult to
progress further.
55
References
[1].
C. Neish. Novae in a test-tube. TRIUMF, 2003
[2].
M. Couder, G.P.A Berg, J. Gorres, P.J. LeBlanc, L.O. Lamm, E. Stech, M. Wiescher, J.
Hinnefeld. Design of the recoil mass separator St. George. University of Notre Dame,
Indiana University South Bend, 2007
[3].
M. Berz. COSY Infinity. http://www.bt.pa.msu.edu/index_files/cosy.htm
[4].
K.S. Krane. Introductory Nuclear Physics. John Wiley & Sons, 1987
[5].
C. Iliadis, A.E. Champagne. Nuclear Astrophysics: Direct Measurements with Stable Beams.
Nucl. Phys. A758, 73, 2005
[6].
D. Clayton. Principles of Stellar Evolution and Nucleosynthesis. University of Chicago Press,
1984
[7].
W. Mittig, P. Roussel-Chomaz. Results and techniques of measurements with inverse
kinematics. GANIL, 2000
[8].
E. Traykov, et al. Production of radioactive nuclides in inverse kinematics. Kernfysisch
Versneller Instituut, 2007
[9].
A. Kontos, et al. HIPPO: A Supersonic Helium Jet Gas Target for Nuclear Astrophysics.
University of Notre Dame, INFN Sezione di Napoli, University of Surrey
[10].
S. Russenchuck. Design of Accelerator Magnets. CERN
[11].
I.S. Grant, W.R. Philips. Electromagnetism. John Wiley & Sons, 1990
[12].
M. Krammer. Silicon Detectors. Institute of High Energy Physics, Vienna
[13].
C. Kittel. Introduction to Solid State Physics, Eighth Edition. John Wiley & Sons, 2005
[14].
D.A Hutcheon, et al. The DRAGON facility for nuclear astrophysics at TRIUMF-ISAC:
design, construction and operation. TRIUMF, 2002
[15].
G. F. Knoll. Radiation Detection and Measurement, Second Edition. University of
Michigan, Ann Arbor, Michigan (1989)
56
[16].
B. Erdelyi. Applications of Differential Algebraic Methods in Beam Physics. NIU,
ANL
[17].
M. Berz, K. Makino. COSY INFINITY 9.1 Beam Physics Manual. MSU Report
MSUHEP 060804, Michigan State University, June 2011
[18].
J. Saren. The ion-optical design of the MARA recoil separator and absolute
transmission measurements of the RITU gas-filled recoil separator. University of
Jyvaskyla, 2011
[19].
M. Berz. The code COSY INFINITY. NSCL, Michigan State University
[20].
M. Berz. Differential Algebraic Description of Beam Dynamics to Very High Orders.
Lawrence Berkley Laboratory, University of California, 1988
[21].
G. Christian. MoNA ROOT Guide. NSCL, 2010
57
Appendix A
Clean_alpha.fox
Clean_alpha.fox is the COSYScript file for modelling the ion optics of the St. George with 27
predetermined rays.
INCLUDE
'/afs/crc.nd.edu/user/n/nsl/nuclear/x86_64_linux/install/COSY9.1/COSY';
PROCEDURE RUN ;
VARIABLE ENERGY 1;
{Recoil mean energy}
VARIABLE D_ENERGY 1;
{Recoil Delta energy}
VARIABLE MASS
1;
{Recoil mass}
VARIABLE CHARGE 1;
{Recoil charge state}
VARIABLE MOMENTUM 1;
{Recoil momentum will be calculated from
ENERGY MASS}
VARIABLE ST_BRHO 1;
VARIABLE
VARIABLE
VARIABLE
VARIABLE
{Standard Brho from initial calculation}
B_ENERGY 1;
{Beam mean energy}
B_SIG_ENERGY 1;{Beam sigma energy}
B_MASS
1;
{Beam mass}
B_CHARGE 1;
{Beam charge state}
{Quad magnetic field}
VARIABLE Q1 1;
VARIABLE Q2 1;
VARIABLE Q3 1;
VARIABLE Q4 1;
VARIABLE Q5 1;
VARIABLE Q6 1;
VARIABLE Q7 1;
VARIABLE Q8 1;
VARIABLE Q9 1;
VARIABLE Q1B 1;
VARIABLE Q2B 1;
{Quad magnetic field correction factor}
VARIABLE QF 1;
{Quad length}
VARIABLE RQ1 1;
VARIABLE RQ2 1;
VARIABLE RQ3 1;
VARIABLE RQ4 1;
VARIABLE RQ5 1;
VARIABLE RQ6 1;
VARIABLE RQ7 1;
VARIABLE RQ8 1;
VARIABLE RQ9 1;
VARIABLE RQ1B 1;
VARIABLE RQ2B 1;
VARIABLE LQ1 1;
58
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
LQ2 1;
LQ3 1;
LQ4 1;
LQ5 1;
LQ6 1;
LQ7 1;
LQ8 1;
LQ9 1;
LQ1B 1;
LQ2B 1;
{Dipole curvature radius}
VARIABLE RADIUS 1;
{Dipole pole face curvature}
VARIABLE B1N 1 5;
VARIABLE B1S1 1 4; VARIABLE
VARIABLE B2S1 1 4; VARIABLE
VARIABLE B3S1 1 4; VARIABLE
VARIABLE B4S1 1 4; VARIABLE
VARIABLE B5S1 1 4; VARIABLE
VARIABLE B6S1 1 4; VARIABLE
{Drift length}
VARIABLE DL1 1;
VARIABLE DL2 1;
VARIABLE DL3 1;
VARIABLE DL4 1;
VARIABLE DL5 1;
VARIABLE DL6 1;
VARIABLE DL7 1;
VARIABLE DL8 1;
VARIABLE DL9 1;
VARIABLE DL10 1;
VARIABLE DL11 1;
VARIABLE DL12 1;
VARIABLE DL13 1;
VARIABLE DL14 1;
VARIABLE DL15 1;
VARIABLE DL16 1;
VARIABLE DL17 1;
VARIABLE DL18 1;
VARIABLE DL19 1;
VARIABLE DL20 1;
VARIABLE DL21 1;
VARIABLE DL22 1;
{Wien filter parameter}
VARIABLE WFRADIUSA 1;
VARIABLE WFALENGTH 1;
VARIABLE VELOCITY 1;
VARIABLE WFMAGNETA 1;
VARIABLE WFELECTRA 1;
B1S2
B2S2
B3S2
B4S2
B5S2
B6S2
1
1
1
1
1
1
4;
4;
4;
4;
4;
4;
{Variable related to drawing and to calculation of particular
trajectories}
{Recoil}
VARIABLE X 1;
VARIABLE A 1;
VARIABLE Y 1;
VARIABLE B 1;
59
VARIABLE
VARIABLE
{Beam}
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
{FLAGS}
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
K 1;
Q 1;
B_A 1;
B_B 1;
B_K_MEAN 1;
B_K_SIG 1;
B_K 1;
B_KS 1;
B_G 1;
B_Q 1;
X_FLAG 1; {Recoil X Drawing flag}
Y_FLAG 1; {Recoil Y Drawing flag}
XQ_FLAG 1; {Recoil X-Q Drawing flag}
YQ_FLAG 1; {Recoil Y-Q Drawing flag}
B_X_FLAG 1; {Beam X Drawing flag}
B_Y_FLAG 1; {Beam Y Drawing flag}
B_XPLUSQ_FLAG 1; {Beam X+Q Drawing flag}
B_YPLUSQ_FLAG 1; {Beam Y+Q Drawing flag}
B_XMINUSQ_FLAG 1; {Beam X-Q Drawing flag}
B_YMINUSQ_FLAG 1; {Beam Y-Q Drawing flag}
{Order and Fringe field}
VARIABLE FF 1;
VARIABLE ORDER 1;
{Dummy variable}
VARIABLE PLOT 1;
VARIABLE DUMFIT 6;
VARIABLE TT1 6;
VARIABLE TT2 6;
{Mass separation}
VARIABLE MRES 1;
{######################################################################
######}
{######################## END OF VARIABLE DEFINITION
########################}
{######################################################################
######}
PROCEDURE XSIZEmin;
VARIABLE SRAY 1;
VARIABLE XMAX 1;
VARIABLE XMIN 1;
VARIABLE I 1;
XMAX:=-10000.0; XMIN:=10000; DUMFIT:=0;
LOOP I 2 NRAY;
VELGET RAY(1) I SRAY;
IF (((SRAY)<XMIN)); XMIN:=(SRAY); ENDIF;
ENDLOOP ;
DUMFIT:=XMIN;
ENDPROCEDURE;
PROCEDURE XSIZEmax;
VARIABLE SRAY 1;
VARIABLE XMAX 1;
VARIABLE XMIN 1;
VARIABLE I 1;
60
XMAX:=-10000.0; XMIN:=10000; DUMFIT:=0;
LOOP I 2 NRAY;
VELGET RAY(1) I SRAY;
IF ((SRAY)>XMAX); XMAX:=(SRAY); ENDIF;
ENDLOOP ;
DUMFIT:=XMAX;
ENDPROCEDURE;
PROCEDURE XSIZERmax;
VARIABLE SRAY 1;
VARIABLE XMAX 1;
VARIABLE XMIN 1;
VARIABLE I 1;
XMAX:=-10000.0; XMIN:=10000; DUMFIT:=0;
LOOP I 2 28;
VELGET RAY(1) I SRAY;
IF ((SRAY)>XMAX); XMAX:=(SRAY); ENDIF;
ENDLOOP ;
DUMFIT:=XMAX;
ENDPROCEDURE;
{######################################################################
######}
{######################### MASS SEPARATION PROCEDURE
########################}
{######################################################################
######}
PROCEDURE MASS_SEPARATION;
VARIABLE B_MAX 1;
VARIABLE B_MIN 1;
VARIABLE RECOIL_SIZE 1;
VARIABLE A1 1;
VARIABLE A2 1;
VARIABLE A3 1;
VARIABLE DUM 1;
B_MAX:=0; B_MIN:=1E20;
{RECOIL_SIZE := X*ABS(ME(1,1))+A*ABS(ME(1,2))+K*ABS(ME(1,6));}
XSIZERmax;
RECOIL_SIZE := DUMFIT;
LOOP A1 -1 1;
LOOP A2 -1 1;
LOOP A3 -1 1;
DUM:=A1*X*ME(1,1)+A2*B_A*ME(1,2)+(B_K+A3*B_KS)*ME(1,6);
IF (DUM>B_MAX); B_MAX:=DUM; ENDIF;
IF (DUM<B_MIN); B_MIN:=DUM; ENDIF;
ENDLOOP;
ENDLOOP;
ENDLOOP;
MRES:=ABS(ME(1,7))/(RECOIL_SIZE+(B_MAX-B_MIN)/2);
ENDPROCEDURE;
{######################################################################
######}
{############################# DRAWRAY PROCEDURE
############################}
61
{######################################################################
######}
PROCEDURE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
VARIABLE
DRAWRAY;
I1 1; VARIABLE I2 1; VARIABLE I3 1;
I4 1; VARIABLE I5 1; VARIABLE I6 1;
I7 1;
J11 1; VARIABLE J21 1; VARIABLE J31
J41 1; VARIABLE J51 1; VARIABLE J61
J71 1;
J12 1; VARIABLE J22 1; VARIABLE J32
J42 1; VARIABLE J52 1; VARIABLE J62
J72 1;
1;
1;
1;
1;
VARIABLE V1 1; VARIABLE V2 1; VARIABLE V3 1;
VARIABLE V4 1; VARIABLE V5 1; VARIABLE V6 1;
VARIABLE V7 1;
VARIABLE COLOR 1;
VARIABLE DUM 1;
VARIABLE COUNT 1;
COUNT := 0;
I1:=0;I2:=0;I3:=0;I4:=0;I5:=0;I6:=0;I7:=0;
J11:=0;J21:=0;J31:=0;J41:=0;J51:=0;J61:=0;J71:=0;
J12:=0;J22:=0;J32:=0;J42:=0;J52:=0;J62:=0;J72:=0;
V1:=0.;V2:=0.;V3:=0.;V4:=0.;V5:=0.;V6:=0.;
IF X_FLAG=1;
V1 := X;
V2 := A;
V3 := 0;
V4 := 0;
V5 := K;
V6 := 0;
V7 := 0;
COLOR :=1;
ELSEIF Y_FLAG=1;
V1 := 0;
V2 := 0;
V3 := Y;
V4 := B;
V5 := K;
V6 := 0;
V7 := 0;
COLOR :=1;
ELSEIF XQ_FLAG=1;
V1 := X;
V2 := A;
V3 := 0;
V4 := 0;
V5 := K;
V6 := 0;
V7 := Q;
COLOR :=2;
ELSEIF YQ_FLAG=1;
V1 := 0;
V2 := 0;
62
V3 :=
V4 :=
V5 :=
V6 :=
V7 :=
COLOR
ELSEIF
V1 :=
V2 :=
V3 :=
V4 :=
V5 :=
V6 :=
V7 :=
COLOR
ELSEIF
V1 :=
V2 :=
V3 :=
V4 :=
V5 :=
V6 :=
V7 :=
COLOR
ELSEIF
V1 :=
V2 :=
V3 :=
V4 :=
V5 :=
V6 :=
V7 :=
COLOR
ELSEIF
V1 :=
V2 :=
V3 :=
V4 :=
V5 :=
V6 :=
V7 :=
COLOR
ELSEIF
V1 :=
V2 :=
V3 :=
V4 :=
V5 :=
V6 :=
V7 :=
COLOR
ELSEIF
V1 :=
V2 :=
V3 :=
V4 :=
V5 :=
V6 :=
Y;
B;
K;
0;
Q;
:=2;
B_X_FLAG=1;
X;
B_A;
0;
0;
B_K;
B_G;
0;
:=3;
B_Y_FLAG=1;
0;
0;
Y;
B_B;
B_K;
B_G;
0;
:=3;
B_XPLUSQ_FLAG=1;
X;
B_A;
0;
0;
B_K;
B_G;
B_Q;
:=3;
B_XMINUSQ_FLAG=1;
X;
B_A;
0;
0;
B_K;
B_G;
B_Q;
:=3;
B_YPLUSQ_FLAG=1;
0;
0;
Y;
B_B;
B_K;
B_G;
B_Q;
:=3;
B_YMINUSQ_FLAG=1;
0;
0;
Y;
B_B;
B_K;
B_G;
63
V7 := B_Q;
COLOR :=3;
ENDIF;
IF V1#0 ; J11:=1; J12:=1; ENDIF ;
IF V2#0 ; J21:=1; J22:=1; ENDIF ;
IF V3#0 ; J31:=1; J32:=1; ENDIF ;
IF V4#0 ; J41:=1; J42:=1; ENDIF ;
IF V5#0 ; J51:=1; J52:=1; ENDIF ;
IF V6#0 ; J61:=-1; J62:=1; ENDIF ;
IF V7#0 ; J71:=1; J72:=1; ENDIF ;
IF B_XPLUSQ_FLAG=1; J71:= -1; ENDIF;
IF B_XMINUSQ_FLAG=1; J72:= -1; ENDIF;
IF B_YPLUSQ_FLAG=1; J71:= -1; ENDIF;
IF B_YMINUSQ_FLAG=1; J72:= -1; ENDIF;
LOOP I1 -1.*J11 1.*J12;
LOOP I2 -1.*J21 1.*J22;
LOOP I3 -1.*J31 1.*J32;
LOOP I4 -1.*J41 1.*J42;
LOOP I5 -1 1.*J52;
LOOP I6 -1*J61 1.*J62;
LOOP I7 -1.*J71 1.*J72 2;
DUM := COLOR ;
IF (COLOR=2)*(I7=0); COLOR:=1 ; ENDIF;
IF (COLOR=2)*(I7=1); COLOR:=4 ; ENDIF;
IF (COLOR=3)*(I6=0); COLOR:=1 ; ENDIF;
IF (V5#0)*(V6#0);
SR I1*V1 I2*V2 I3*V3 I4*V4 0. V5+I5*B_KS I6*V6 I7*V7 COLOR;
ELSEIF V6=0;
SR I1*V1 I2*V2 I3*V3 I4*V4 0. I5*V5 I6*V6 I7*V7 COLOR;
ENDIF;
COLOR := DUM ;
COUNT := COUNT +1;
ENDLOOP;
ENDLOOP;
ENDLOOP;
ENDLOOP;
ENDLOOP;
ENDLOOP;
ENDLOOP;
ENDPROCEDURE;
{######################################################################
######}
{########################### RECOIL_LINE PROCEDURE
##########################}
{######################################################################
######}
PROCEDURE RECOIL_LINE;
FR FF; {Fringe field flag}
DL DL1;
{ SA -0.01 0.;}
MQ LQ1 Q1 RQ1 ;
DL DL2 ;
MQ LQ2 Q2 RQ2 ;
DL DL3 ;
WRITE 6 Q1;
WRITE 6 Q2;
WRITE 6 QF;
64
PS 0.1;
MC RADIUS 26. 0.035 B1N B1S1 B1S2 4;
PS 0.1;
{ DL 0.51;}
DL DL4 ;
PS 0.1;
MC RADIUS 26. 0.035 B1N B2S1 B2S2 4;
PS 0.1;
DL DL5;
PS 0.024;
DL DL6 ;
MQ LQ3 Q3*QF RQ3;
DL DL7;
{ MQ LQ4 Q4*QF RQ4;
DL DL8;
MQ LQ5 Q5*QF RQ5;
DL DL9;
MC RADIUS 26. 0.035 B1N B3S1 B3S2 4;
DL DL10;
MC RADIUS 26. 0.035 B1N B4S1 B4S2 4;
DL DL11;
MQ LQ6 Q6*QF RQ6;
DL DL12 ;
MQ LQ7 Q7*QF RQ7;
DL DL13 ;
IF (FF=2);FR 3; ENDIF; CB;
WF WFRADIUSA WFRADIUSA WFALENGTH 0.061 ; CB;
IF (FF=2);FR FF; ENDIF;
DL DL14 ;
PS 0.06;
MASS_SEPARATION;
WRITE 6 MRES;
XSizemin; TT1:=dumfit;
XSizemax; TT2:=dumfit;
dumfit:=TT2-TT1;
WRITE 6 dumfit;
DL DL15;
MQ LQ8 Q8 RQ8;
DL DL16;
MQ LQ9 Q9 RQ9;
DL DL17;
MC RADIUS 26. 0.04 B1N B5S1 B5S2 4;
DL DL18;
MC RADIUS 26. 0.035 B1N B6S1 B6S2 4;
DL DL19;
MQ LQ1B Q1B RQ1B;
DL DL20;
MQ LQ2B Q2B RQ2B;
DL DL21; PS 0.02;
DL DL22; PS 0.025;}
ENDPROCEDURE;
{######################################################################
######}
{########################### DEFINITION AND COMMAND
#########################}
65
{######################################################################
######}
ORDER := 5;
FF:=2;
OV ORDER 3 3 ;
{For higher order go to linux store the map and apply the in the next
step}
WSET 1.5E-3;
ENERGY := 4.67 ;
{Recoil mean energy}
D_ENERGY := 0.2 ; {half of FWHM(400kev)} {Recoil Delta energy}
MASS
:= 4.;
{Recoil mass}
CHARGE := 2. ;
{Recoil charge state}
A := 0.020;
{Maximum horizontal divergence}
B := 0.020;
{Maximum vertical divergence}
B_ENERGY := 5.5;
{Beam mean energy}
B_SIG_ENERGY:=0.003;
{Beam sigma energy}
B_MASS:=4;
{Beam mass}
B_CHARGE:=2;
{Beam charge state}
X := 0.001 ;
{Maximum horizontal extension}
Y := 0.001 ;
{Maximum vertical extension}
K := D_ENERGY/ENERGY ;
{Maximum Energy of recoil distribution}
Q := 1/CHARGE ;
{Recoil charge state +/-1}
B_A := 0.002 ;
{Maximum horizontal divergence}
B_B := 0.002;
{Maximum vertical divergence}
B_K := (B_ENERGY-ENERGY)/ENERGY ;
B_KS:= 3*B_SIG_ENERGY/ENERGY;
B_G := (B_MASS-MASS)/MASS; {Mass difference between beam and recoil}
B_Q := 1/CHARGE ;
{Beam charge state +/-1}
{STANDARD MAGNETIC RIGIDITY - CONSTANT !!!!}
ST_BRHO := 99.23570738315271;
{<--- NOT CHANGE THIS!!!}
RQ1:=0.05; RQ2:=0.085;
RQ3:=0.10; RQ4:=0.10; RQ5:=0.10;
RQ6:=0.085; RQ7:=0.085;
RQ8:=0.07; RQ9:=0.09;
RQ1B:=0.09; RQ2B:=0.07;
LQ1 :=0.25; LQ2 :=0.25;
LQ3 :=0.35; LQ4 :=0.35; LQ5 :=0.35;
LQ6 :=0.25; LQ7 :=0.25;
LQ8 :=0.25; LQ9 :=0.25;
LQ1B:=0.25; LQ2B:=0.25;
{below are field at pole tip before scaling factor applied}
Q1:=-(33-1.4473)/206.44166;
Q2:=(51+3.603)/307.36682;
{Q1:=-(30-1.4473)/206.44166;
Q2:=(50+3.603)/307.36682;}
Q3:=0.9411411244187678E-001; Q4:=-.1288532063545663;
Q5:=0.1063765496207145;
Q6:=0.4693739906022038E-001; Q7:= 0.000000000000000;
Q8:=-.9779179171475419E-001; Q9:=0.1743962745915836;
Q1B :=0.2109222841425732;
Q2B:=-.1396235508515609;
WFELECTRA:=1050000.000000000;
B1S1(1):=0.23; B1S1(2):=0.05; B1S1(3):= 2.86; B1S1(4):= 0.0;
66
B1S2(1):=0.23; B1S2(2):=0.25; B1S2(3):= 0.0 ; B1S2(4):= 0.0;
B2S1(1):=0.13; B2S1(2):=-.74; B2S1(3):= 0.0 ; B2S1(4):= 0.0;
B2S2(1):=0.13; B2S2(2):=-.59; B2S2(3):= 0.0 ; B2S2(4):= 0.0;
B3S1(1):=0.23;
B3S2(1):=0.23;
B4S1(1):=0.23;
B4S2(1):=0.23;
B5S1(1):=0.23;
B5S2(1):=0.23;
B6S1(1):=0.10;
B6S2(1):=0.10;
B3S1(2):=0.09;
B3S2(2):=-.61;
B4S1(2):=-.67;
B4S2(2):= 0.0;
B5S1(2):= 0.0;
B5S2(2):= 0.0;
B6S1(2):= 0.0;
B6S2(2):= 0.0;
DL1 :=0.600; DL2 :=0.200;
DL5 := 1.20; DL6 :=0.500;
DL9 :=0.500; DL10:=0.510;
DL13:=0.500; DL14:= 1.59;
DL17:=0.860; DL18:= 1.28;
DL21:=.400; DL22:=0.500;
B3S1(3):= 0.0;
B3S2(3):= 0.0;
B4S1(3):= 0.0;
B4S2(3):=-.32;
B5S1(3):= 0.0;
B5S2(3):= 0.0;
B6S1(3):= 0.0;
B6S2(3):= 0.0;
DL3 :=0.580;
DL7 :=0.350;
DL11:=0.650;
DL15:= 1.24;
DL19:=0.560;
B3S1(4):=
B3S2(4):=
B4S1(4):=
B4S2(4):=
B5S1(4):=
B5S2(4):=
B6S1(4):=
B6S2(4):=
0.0;
0.0;
0.0;
0.0;
0.0;
0.0;
0.0;
0.0;
DL4 := 1.00;
DL8 :=0.210;
DL12:=0.200;
DL16:=0.330;
DL20:=0.300;
RADIUS := 0.75;
MOMENTUM := SQRT(2*931.5*MASS*ENERGY);
VELOCITY := MOMENTUM/(931.5*MASS);
VELOCITY := VELOCITY*299792458;
QF:=(MOMENTUM/CHARGE)*1/ST_BRHO;
WFALENGTH:=1.10 ;
WFRADIUSA:=4.348457677339926;
WFMAGNETA:=3.356E-3*(MOMENTUM/(CHARGE*WFRADIUSA));
{(ENERGY*1E6/(CHARGE*WFRADIUSA))*0.122) Voltage on each WF plate in
volt}
{(VELOCITY:=MOMENTUM*1E6/(299792458*CHARGE*RADIUS)) Field in dipole T}
{***************************************************}
{Initializing flags}
X_FLAG:=0;
{Recoil X Drawing flag}
Y_FLAG:=0;
{Recoil Y Drawing flag}
XQ_FLAG:=0;
{Recoil X-Q Drawing flag}
YQ_FLAG:=0;
{Recoil Y-Q Drawing flag}
B_X_FLAG:=0; {Beam X Drawing flag}
B_Y_FLAG:=0; {Beam Y Drawing flag}
B_XPLUSQ_FLAG:=0; {Beam X+Q Drawing flag}
B_YPLUSQ_FLAG:=0; {Beam Y+Q Drawing flag}
B_XMINUSQ_FLAG:=0; {Beam X-Q Drawing flag}
B_YMINUSQ_FLAG:=0; {Beam Y-Q Drawing flag}
{***************************************************}
RP ENERGY MASS*PARA(1) CHARGE*PARA(2) ;
PTY 0;
UM; CR; NRAY:=0;
X_FLAG:=1;
DRAWRAY; X_FLAG:=0;
{ B_X_FLAG:=1; DRAWRAY; B_X_FLAG:=0;}
{ B_XPLUSQ_FLAG:=1; DRAWRAY; B_XPLUSQ_FLAG:=0;}
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{ XQ_FLAG:=1;
BP;
RECOIL_LINE;
EP;
PP -10 0. 0.;
DRAWRAY; XQ_FLAG:=0;}
PTY 0;
UM; CR; NRAY:=0;
B_Y_FLAG:=1; DRAWRAY; B_Y_FLAG:=0;
Y_FLAG:=1; DRAWRAY; Y_FLAG:=0;
BP;
RECOIL_LINE;
EP;
PP -10 0. 90.;
ENDPROCEDURE ;
RUN ;
END ;
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