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This work is about ion traps. An ion trap is an important tool used in performing experiments throughout the natural sciences. An important use of ion traps is mass spectrometry, with which the ratio between charge and mass of particles can be

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Structural Defects with Trapped Ions in Multipole Traps
V. Isarov,
S. Y. Agnon Highschool, Spinoza st., Netnaya, Israel
1. Introduction
This work is about ion traps [1]. An ion trap is an important tool used in performing experiments throughout the natural sciences. An important use of ion traps is mass spectrometry, with which the ratio between charge and mass of particles can be measured. Using such measurements it is possible to analyze samples of different materials and identify their constituents. In physics, ion traps are used for explorations of quantum-mechanical phenomena [2], for very precise measurements, and for studying fundamental physical laws. Using advanced techniques of laser cooling, ions can be cooled to very low temperatures such that their motion is very limited, and they crystallize near the minimum configuration of the potential energy. Ion traps can be loaded with single ions or clouds of tens of thousands of ions. In this work I examine multipole traps [3], in which the effective trapping potential is of the form
2 2
k
V r r
, where
2 2
r x y
is the distance to the srcin of the
xy
plane, and
2
k
is the power of the multipole. The total trapping potential energy is given by
(,) =
(
−
/2) +
. In addition, this research deals with a phenomenon which occurs for the minimum configuration of trapped ions, the appearance of “defects” [5] when one ion or more move from their location in the global minimum of the potential and sit at a point which corresponds to a local minimum of the potential. The interest in these configurations stems mainly from the properties of the normal modes of oscillation of the ions about the minimum configuration locations, in the approximation of small harmonic motion. The focus in this work is on characterizing the ions configuration in a multipole trap for different parameters, finding the normal mode frequencies and the
components of the ions oscillations, and searching for defects in the minimum configuration. Matlab is used for numerical simulations. I try to characterize the trapping configuration and the normal mode properties, in a manner most useful for an experimental investigation of these configurations.
2. The Potential and Normal Modes in a Multipole Trap
The total effective potential energy for ions in a multipole trap is [4]
2 2 22 2 2 2 202 202 1/22 2 2
1 / 22 162
k z i i ik ii j i j i ji j
k e V V m z r r m r e x x y y z z
where
2 2
r x y
is the distance from the srcin in the
xy
plane,
e
the electron charge,
m
the ion mass,
z
the trapping frequency resulting from DC electrodes along the
z
axis,
0
V
the potential between the multipole electrodes,
the rf frequency for the multipole electrode’s driving,
0
r
the internal radius and
k
is the multipole power. We define natural units of length by
=
and change to nondimensional coordinates given by
.
/
i i
x x d
d y y
ii
/ ~
d z z
ii
/ ~
After dividing the potential energy by
, we get
V = V
= [12(
−
/2 ) +
!
1"#
$
%
&
'+ 12[*
− *
+ -*
− -*
+
−
'
!.0
Defining the strength of the radial trapping by
=
!
1"#
$
%
&
we get the nondimensional potential energy of the ions in the form
V = [12(
−
/2 ) +
'+ 12[*
− *
+ -*
− -*
+
−
0
'.
For the rest of this work I will denote the coordinates simply by
, ,
i i i
x y z
, and focus on octupole traps, i.e.
4
k
. When the ions are cooled in the trap, they crystallize in a certain configuration which is close to a minimum of the trapping potential. Expanding the motion of the ions about the minimum configuration locations, in a Taylor series in all the ion coordinates, the first (linear) term of the series is equal to zero (by definition of a minimum), and the second-order (harmonic) terms can be written in the form
ji j jii
qK qV
,,
21
where the coordinates of the ions are written in vector form as
111
nnn
x y zq x y z
and components of the matrix
K
are given by the sum of the second derivatives of the trapping part and the Coulomb part, with the following expressions
1,5 35 3
3 ,3 ,
Coulomb i jm n j im nn m n m n m n mn j n j n j n j j n
d K R RdR dR R R R R R R R R m n R R R R R R R R m n
! "
and for the trapping part (suppressing the ion index here)
10)()1(22 / 1)()2)(1(4
)()2)(1(4
)()1(22 / 1)()2)(1(4
2223222
3222223222
#####
####
#####
dzdzdV dzdydV dydzdV dzdxdV dxdzdV y xk y xk k y
dydydV y xk k y x
dydxdV dxdydV y xk y xk k x
dxdxdV
k k k k k
Where
i
R
denotes the
component of
i
R
,
2 2 2
,
ii i i j i j i j i ji
x R y R R x x y y z z z
and
10
ij
i ji j
!
. A Matlab routine is used to diagonalize the matrix, and thus find the frequencies and vector components of the normal modes of oscillation of the ions about each minimum configuration.
3. The Minimum Configuration
Running simulations to find the minimum configuration of the ions in the trap, at low radial trapping strength (low
) the distances along the
z
axis are very small, of the order of
1
, and Fig. 1 shows a top view of the trapping configuration.
The distances in the figure and for the rest of the work are in nondimensional natural units.
Figure 1 : Top view on the
xy
plane for the minimum energy configuration with 32 ion, at
= 34 1
and
5 = 6
. The distances in this figure and all other figures are in natural units.
In Fig. 1 it is easy to see that the configuration is a planar ring in the
xy
plane. Increasing the radial trapping strength, at some point the different configuration of Fig. 2 appears. In Fig. 2 one can see that the planar ring has split into two rings at different heights above the
z
axis [4], with the ion locations forming a “zigzag” between the two chains (each ion is located exactly between the two adjacent ions in the plane above or below it). For stronger radial trapping, the two rings are found to split into three rings or more, and there are also more complex configurations, not consisting of planar rings only.

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