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Ion optics
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Ion Optics .
M. Valentina Ricciardi
********** Elements in ion optics ***********
There are mainly 4 ion-optical elements, which compose ion-optical devices.
1) Drift section
The drift section is an empty space (vacuum) where no forces are acting. The particle just goes through in a
straight line. It is important because it can change the phase space of the particle (it changes the position r
but not the momentum p).
2) Magnets
The equations that rule the motion of a particle (with charge q and mass m) are given combining the 2nd law
of dynamics and the Lorentz force:

d
m  v   m0  d   v   q  v  B
dt
dt
Note that m    m0 with  
1   
2 1
and   v c is velocity dependent.
Since the force in a magnetic field is perpendicular to the velocity, then the net effect is a change in the
particle direction, but the particle is not accelerated or decelerated.


If B and v are uniform and perpendicular to each other than it comes:
B 
m0    v p

q
q
Circular trajectory
where  is the bending radius of the trajectory. The ratio p/q ( = B) is called magnetic rigidity and it is a
characteristic of a particle with a certain mass, charge and velocity.
2.1.) Dipoles
Here we consider two different types of dipoles, represented by two examples: ALADIN and a dipole
magnet of the FRS.
ALADIN= A Large Acceptance DIpole magNet
ALADIN is a big rectangular magnet with a large opening with respect to the length. The magnet can accept
at once many particles with a large range of magnetic rigidities and different incoming angles; they will
follow different paths and exit on different positions. So, no "standard" or "optimum" bending angle is
defined. By detecting at least three points (two before and one after or vice-versa) of their trajectory, their
magnetic rigidity can be deduced.
Ion optics
ALADIN
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FRS
FRS
In the FRS, a dipole is a curved long tunnel with an opening much smaller than its length. Therefore, only
particles with a limited range of bending radii, centered around 0, can pass. The acceptance in angle is
rather limited, too. The binding radius 0 is defined by the geometry of the magnet.
(picture)
Use of the dipoles:
a) to identify a charged particle
Once B is know by measuring the trajectory of the ion, the identification of the A/Z of a charged particle is
easily done if an independent measurement of the velocity can be done. We exploit this equation:
A
B

Z c   
with cβ known from another source
b) to evaluate the velocity of a fixed charged particle (selection of momentum)
Once B is know by the measurement of the trajectory of the ion, the evaluation of the velocity can be done
if the A/Z of the charged particle is already known. We exploit the usual equation:
c    
B
AZ
How the magnetic rigidity is connected to the measurement of the position?  Dispersion
The dispersion is a quantity that connects the variation of the position of an ion, bent by a magnet, with the
variation of its magnetic rigidity. So two identical ions (i.e. same Z and A) that enter the magnet with
different momenta (velocities) will follow different paths and land in different positions of a given plane.
The dispersion can be defined as the transversal distance between a reference trajectory and the trajectory of
a particle with (B) = (B)/(B)= 1 %. It is expressed in cm/%.
Actually in the most general case (real magnet, not focussing, with border effects, etc. etc.) the position
where an ion lands depends on many factors (e.g. the angle, the path through the B). For this reason it is
more correct to define the dispersion as the partial derivative with respect to the variation of the momentum
of the ion:
D,,B,... 
x
p p
Ion optics
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higher p
A, Z
fixed
p
x
lower p
NB: in "real life" we can say that:
- the dispersion does not depend on the value of B
- the dispersion depends on the plane where it is measured
- the dispersion depends on the entrance angle but generally this dependence is not so strong
The dependence on the angle is not so important also because every dipole has got in reality a focussing1
power. At the ends of the dipole the magnetic field does not end sharply, so at the edges B is curved and by
this one gets also higher order effects, which act like a lens. So, any dipole has got a focussing power and
one can work on this by shaping the edges of the magnet.
2.2.) Quadrupoles
In an experiment it is very much convenient to know which were the particles (which had passed through a
dipole by entering it with different angles) which had the same magnetic rigidity. In a magnet like ALADIN
this work is very tedious because every trajectory has to be reconstructed and from that the motion of the ion
(and then its p/q) deduced. So, it is very practical to have a device in which the particles with the same
magnetic rigidity but different incoming angle land in the same position (so that a measure of the position is
directly a measure of the magnetic rigidity).
To focus means to collect particles that are equal in rigidity but not in angle.
This is done with the help of a "lens" (or, vice-versa, we can call "lens" a magnetic device that focuses ions).
How can be a lens done?
A focalizing lens is realized with two quadrupoles, rotated of 90 and separated by some distance (drift
section).
A quadrupole is a magnetic element with 4 poles, shaped in such a way that the intensity of the magnetic
field increases linearly from center to the pole.
So: |Bx|  |x| and |By|  |y|; the field is zero in the center. The ion propagates along the z-axis.
In the 1st quadrupole, the net result is that, when the ion crosses a point of the x-axis, it will be focussed
towards the center and the more far the ion is from the center the stronger is the force acting on it. When it
crosses a point of the y-axis it will be defocused. When it cross any point of the xy-plane it will be focussed
in the x direction and defocussed in the y direction. In the second quadrupole the opposite thing will happen:
The y axis is focussing the x axis is defocussing.
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We will see later what "to focus" means.
Ion optics
v
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F
B
v
F
B
If we set the two (rotated) quadrupoles one after the other in a very short distance, the net result is null. The
trick is then to insert a drift section between the two quadrupoles in such a way that the total focalization is
stronger than the defocalization. So, thanks to the drift section and to the fact that the intensity of the force
increases linearly with the distance from the center, the combined effect is focalization. The focussing power
is proportional to the length of the drift section and to the strength of the magnetic fields.
Note: these ion-optic properties of the two quadrupole magnets are completely symmetric, so the focalization
acts in both direction (i.e. both from left to right than from right to left).
3) Electric fields
Electric fields are used both to deflect and to accelerate or decelerate a particle. A particle is deflected when
the direction of the electric field is perpendicular to the velocity of the charged particle. This application is
realized in SHIP. In this case, the trajectory is a parabola. In an accelerator, the field is parallel to the
velocity.
4) Degrader (interaction with matter)
An energy degrader, realized as a thick layer of matter, is not a "classical" ion-optic element. In classical ion
optics the interaction with matter was always considered as a disturbing unavoidable effect to be minimized.
In some optical devices, as in the FRS, the interaction with matter is exploited and pieces of matter act really
as ion-optical elements. The effect of a degrader is to slow down the particle. The shape of the degrader can
also be adjusted in a convenient way, e.g. as a trapeze, to have larger energy loss (slowing down) in one
place than in another. In principle this ion-optic element should affect only the velocity of the particle (not its
charge, mass or trajectory). In practice unfortunately the interaction with matter brings also unavoidable
undesired effects, like angular straggling, energy straggling or nuclear reactions. So the thickness and the
material of the degrader is chosen as a compromise between desired and undesired effects.
Ion optics
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************ Optical systems ************
How can we describe a particle in an ion-optical device?
The particle is characterized by 6 variables, which give its position and its momentum in the 3-D space:
P = (x, px, y, py, z, pz)
so, its features are given at any time by a point in the phase space; this point defines all the properties of the
particle for the future. So, in principle, if we know the forces acting inside an ion-optical device, we can
exactly reconstruct the classical2 equation of motion of the particle through the elements of the optical
system. However, "to follow" at any time the life of a particle through its whole path across the optical
device is not so practical. What is actually of interest, is to know its characteristic at the exit.
In this sense, an optical system can be regarded as a box in which the relation between the properties at the
entrance and the properties at the exit is defined. The system transforms the input values of the particle into
some output values and vice-versa. The function which defines the transformation is biunique. So, all what is
happening in between the entrance and the exit can be put in a system of equations. The solution of this
system of equations gives a function f:
 x exit 
 x entrance 
 exit 
 entrance 
 px 
 px

 exit 
 entrance 
 y   f  y

entrance
 p exit



p
y
y
 exit 
 entrance 
z 
z

 p exit 
 p entrance 
 z 
 z

The solution of the system of equations can be very complicated. One can develop it in series of powers and
decide to which order of precision to extend the calculation. If one stops at the first order, one does not have
mixed terms, that means the function f can be expressed as a 66 matrix of coefficients aij:
 a11

 a21
a
f   31
 a41
a
 51
a
 61
a12
a13
a14
a15
a22
a23 a24
a25
a32
a33
a34
a35
a42
a43 a44
a45
a52
a53
a54
a55
a62
a63
a64
a65
a16 

a26 
a36 

a46 
a56 
a66 
that gives, for example, :
xexit  a11  xentrance  a12  pxentrance  a13  yentrance  a14  pentrance
 a15  z entrance  a16  pzentrance
y
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The nucleus wavelength is much smaller than the dimensions of our detectors. (The position is measured with an
uncertainty of 1-2 mm, which is 13 order of magnitude bigger than the quantum-mechanical uncertainty of the nucleus).
So in our experiments the kinematics is that one of a classical, relativistic motion.
Ion optics
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In the first-order calculation, aij are just coefficients; in higher-order calculations aij are functions of (x, px, y,
py, z, pz) 3.
According to Liouville's theorem, if the forces acting inside the ion-optical system are conservative, then the
volume in phase space occupied by the particles is conserved. This volume is called emittance. The
magnets, electric fields, the drift sections are conservative, but the degraders are not. The analysis of the
system can be anyhow divided in steps: in those sections where all the elements are conservative the
emittance does not change, in those pieces where there are non-conservative forces one has to calculate how
the emittance is changed. This new emittance goes as an input for the next section.
In certain circumstances some sub-volume in phase space (for example, the volume of (x, px)) can stay
constant. In that case it is possible to define the sub-emittance (or, for example, "emittance in x").
A real ion-optical apparatus in general does not transmit all (for example, it cuts in angle or in momentum)
and the volume in phase space which can pass4 is called acceptance. If the emittance is smaller than the
acceptance then the transmission is 100%.
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4
The ion-optical calculations for the FRS are made by MOCADI or GICO. GICO goes up at least to the 3 rd order.
This is independent of the particle; it is a property of the apparatus.