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The Effect of an Optical Lattice on Ions
Jonathan E. Daron, Tuan Le, and Eric M. Need
Department of Physics, Wabash College, Crawfordsville, IN 47933
(Dated: March 17, 2015)
We trapped ceramic microparticle ions and tracked their motion in a Linear Electrode Ion Trap and compared this motion to that of the ion with an optical lattice
in place. We held this ion in an optical lattice created by the interference of two
laser beams with a peak to peak width ∆x = (6.69 ± 0.25)µm(95%CI, gaussian pdf)
as predicted by our model. We observed that the optical lattice constrained the
motion of the ion by comparing the variance (σ 2 ) of the transverse motion of the
ion with and without the optical lattice in place; with values of σ 2 = 0.92 µm2 and
σ 2 = 37.48 µm2 , respectively.
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There are many purposes for the confinement or trapping of microscopic particles. Many
of these include the modeling of colloids, modeling quantum computing, modeling of many
atom systems, measuring Brownian motion, and constraining an ion for any other measurements. Recently, there has been a lot of research done on the subject of Brownian motion.
Brownian motion is the random motion of a particle that results from the interaction between the ion and particles of the fluid surrounding it[1][2][9]. There has also been research
investigating how a particles motion can be affected by lasers. Lasers have been used to
create optical forces[3] which are then used to trap microscopic particles suspended in fluid.
Research has also been done on a particle in an optical lattice[7]. This optical lattice is
a standing wave created from the superposition of two interfering laser beams[7]. The ion
will not experience a hard constraint on its position if the optical lattice is not present or
does not influence the ion due to its properties. If so the ion is still able to move about
randomly due to Brownian motion. The optical trap works due to the momentum transfer
from the laser beam to the particle[5][3]. In order for this to work the particles that we
are trapping, ceramic microparticles, must allow the laser beam to pass through them (This
can be seen in Fig. 3). Volpe published a paper about the Brownian motion of a particle
in an optical trap[6]. Research has also shown another way to trap a small particle. A
planar linear electrode ion trap can be used to trap microscopic ions[4][8]. In general, the
electrode ion trap uses alternating potentials to hold onto the ions. In the last two years,
Skowronski investigated the Brownian motion of an ion in a Paul trap[10], which is a form
of an alternating potential ion trap. In this paper we will discuss how we combined a linear
electrode ion trap and an optical lattice to examine the motion of an ion. For our set up
the ions are interacting with the air molecules[2]. This motion will cause the ion to move
from one site in the constraining optical lattice to another neighboring site where it will
remain trapped for a bit before again moving[7]. This movement is only accomplished if the
amplitude of the optical lattice is small enough that the Brownian motion excites the ion
enough to allow it to “hop” to the next site.
To set up the optical lattice we first came up with a model to produce the desired optical
lattice for the particles we want to trap. To do so we took into consideration that we wanted
to make our optical trap from two interfering laser beams and so our model was then based
off a similar model as the classic double slit interference. The optical lattice we aimed to
create is the product of two Helium Neon laser beams, λ = 632.8nm, interfering and creating
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a standing wave. The first step to developing our model was to consider the intensity of the
standing wave created from the interference of the two identical HeNe laser beams. This
intensity is proportional to the amplitude of the two beams electric fields. For simplicity
purposes we considered the case where the intensity is at its minimum such that
I ∝ |A1 − A2 |2 ,
(1)
where A1 and A2 are the amplitudes of the electric fields of the first and second incoming
beam respectively. So, now rewriting this and going through the algebra we have that
~
~
I = I0 |ei(k1 ·r~1 ) − ei(k2 ·r~2 ) |,
which by rewritng as the complex conjugate is
∗ ~
~
~
~
I = I0 ei(k1 ·r~1 ) − ei(k2 ·r~2 )
ei(k1 ·r~1 ) − ei(k2 ·r~2 ) ,
and then taking the complex conjugate we have
~
~
~
~
I = I0 2 − ei(k2 ·r~2 −k1 ·r~1 ) − e−i(k2 ·r~2 −k1 ·r~1 ) ,
where I0 is some constant of proportionality, k~1 =
2π ˆ
k,
λ 1
k~2 =
2π ˆ
k,
λ 2
and r~1 and r~2 are
the directional vectors of the HeNe laser beams (See Fig. 1). Simplifying this further and
rewriting this without the complex notation and factoring out the constant terms into the
I0 we have
I = I0 1 − cos [k~2 · r~2 − k~1 · r~1 ] .
(2)
Upon arriving at this solution for the intensity we now need to consider our set up again.
For our set up r1x = −r2x . Using this we now have that
k~2 · r~2 = −k sin (θ)x + k cos (θ)y
(3)
k~1 · r~1 = k sin (θ)x + k cos (θ)y,
(4)
and
with θ being the angle between the perpendicular of the initial beams and the outgoing
beams from the beam splitter and the mirror (See Fig. 1). Then using equations 2, 3, and
4 we have that
I = I0 1 − cos [2k sin (θ)x] .
(5)
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y
Δx
x
r1
Optical Lattice
Plaser =75 mW
λ =632.8 nm
2θ
r2
θ
θ
flense =50 cm
D
flense =50 cm
d
Figure 1. The parameters of the model we will use to predict the “size” of our optical trap and
the layout requirements are shown here. The beam splitter and mirror are seperated by a distance
d = (5.30 ± 0.20)cm(95%CI, gaussian pdf). The angle of the HeNe laser beams relative to the
linear ion trap’s perpendicular is about θ = 2.71 degrees. The spacing between our “tweezing”
apparatus and the linear ion trap is D = (56.00 ± 0.20)cm(95%CI, gaussian pdf). The peak to peak
separation of the optical lattice created from the HeNe beams interference model is ∆x = (6.69367±
0.00055)µm(95%CI, gaussian pdf). The focal length of the two lenses used is flense = 50cm. Lastly,
the parameters of the HeNe laser used in the setup are the wavelength , λ = 632.8nm, and the
max power output, Plaser = 75mW, of the laser. The relationships and methods used to derive the
equations relating the parameters are the same as for the classic double slit experiment.
Now since we wanted to solve for the required parameters of our set up we found the
conditions needed to do so. To do this we needed to consider the minimum value of the
intensity since this is what we began with in equation 1. According to equation 5 this
means that cos [2k sin (θ)x] = 1, so that I = 0, its minimum value. To meet this condition
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2k sin (θ)x = 2π, where x is the distance between the nth order destructive interference fringe
and the the n ± 1 order destructive interference fringe. This distance is the same distance
between two consecutive constructive interference fringes or two consecutive destructive
interference fringes. Similar to the double slit experiment. So for our set up x is ∆x, the
peak to peak optical lattice separation (See Fig. 1). Thus we have
2k sin (θ)x =
4π
sin (θ)∆x = 2π
λ
(6)
which must be satisfied. From this condition we can find multiple relationships between the
parameters of our set up. The first of these is the relationship between ∆x, λ, and θ
h λ i
θ = arcsin
,
2∆x
(7)
where λ = 632.8nm is a fixed parameter for our set up. Using this and an arbitrary value
of ∆x = 10µm we find that θmin = 0.9◦ = 0.2 radians, which allowed us to know that we
could build the required set up and it would not be too difficult. The next relationship we
needed to find was a relationship involving the limiting spacial parameters we had to set
up the optical tweezers. These were d, the spacing between the 50:50 beam splitter and the
mirror, and D, the distance between the point of transmission and interference of the two
laser beams. To find this relationship we considered the geometry of the set up. This yeilds
that
tan θ =
d
.
2D
(8)
Which by combining with equation 7 gives a relationship between the spacial parameters
and the optical lattice spacing,
∆x =
h
d i−1
λ
sin arctan
.
2
2D
(9)
Using this relationship and the limitations of our set up now we can find the optical lattice
spacing we expect. For the spacial limitations due to the optical table we are using we have
D = (56.00 ± 0.20)cm(95%CI, gaussian pdf) and d = (5.30 ± 0.20)cm(95%CI, gaussian pdf).
This, along with λ = 632.8nm (with a negligible uncertainty), leads to an optical lattice
spacing of ∆x = (6.69367 ± 0.00055)µm(95%CI, gaussian pdf) and angle of approximately
θ = 2.71◦ = 0.047 radians.
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After arriving at our model we next began constructing the appropriate set up. First
we needed to set up the linear electrode ion trap. The electrode ion trap was used to hold
the ion for a long period of time so that we could test our optical tweezers (See Fig.‘2). A
diagram of the linear electrode ion trap’s parameters can be seen in Fig. 2. To get an ion
into the linear electrode ion trap, we brushed a thin nonconducting rod with ions on it over
the electrodes of the trap. If multiple ions were in the trap, we would adjust the potentials
of the electrodes to try to move all but one ion out of the trap. Once this was accomplished
we then set up the microscope and a camera so that we could track and see any ions that
we had trapped. After all this was accomplished we began our data collection and analysis
of a single ion in the electrode trap.
Figure 2. Above is a diagram of the cross section of the Linear Electrode Ion Trap that we used
to trap the ions in order to test our optical tweezers. The different lines lying flat in the xy-plane
are the electrodes. Note, the force vectors acting on the ion above the depicted trap oscillate over
time at a set frequency, ftrap = 150.0Hz. These force vectors are created from the electric field of
the Linear Ion Trap’s electrodes, which also oscillates at frequency ftrap . The voltages used on our
trap were Vtrap = 2.73V and Vground = Vside = 0V.
Next, we began building our optical tweezers to put the optical lattice in place that our
model predicted. To build our optical tweezers we used a Helium Neon (HeNe) laser, a 50:50
beam splitter, multiple mirrors, and two focusing lenses with focal lengths of flense = 50cm.
After we finished building the optical tweezers we then tested that when we had the beams
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aligned correctly we got interference patterns on the wall. We then individually aligned the
focus of the two beams from the HeNe laser on the ion in the linear electrode trap. To do
this, we blocked the beam from the mirror and adjusted the angle of the beam splitter. We
then blocked the beam from the beam splitter and adjusted the mirror. When a beam was
on the ion, the image of the ion in the microscope became very bright. Once the beams had
been aligned, we placed a filter in front of the microscope that filtered out the red light from
the HeNe laser, reducing the glare, and allowed the green light through. The microscope was
used to track the motion of the ion. In order to track the ion’s motion we illuminated the
ion through the use of a green laser shown down the x-axis of the electrode trap just above
its surface. We would alternate taking data with the optical trap on and off, adjusting the
alignment of each HeNe laser beam to ensure the ion was in the optical lattice each time.
Each video lasted for two minutes. When we took data of the ion’s motion we made sure
to have the lights out in the room, be as quiet as possible, and not move at all because the
ion’s motion was very sensitive to the slightest bit of wind, even though it was isolated in a
clear and enclosed box.
The ions that we are trapping, ceramic microparticles, allow the laser beam to pass
through them (This can be seen in Fig. 3). The beam changes direction because the particles
have a different index of refraction, nion , from air, nair . This change in direction leads to a
change in momentum, moving the particle due to the forces created. By these mechanisms
the restoring forces are created and the ion can be constrained to a position with good
probability of being found at its central location[5]. The ions position, however, does not
follow a hard constraint as a baseball in a bucket would, but rather its constraint exhibits
a more Gaussian like distribution of the ions position (See Fig. 6).
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Figure 3. The optical force model that allows the optical tweezers to work uses conservation of
momentum through the principles of light diffraction and interference. The index of refraction
of air is nair = 1.0003 and the index of refraction of the ceramic microparticle, nion , which is
unknown.[5] The particle that is being trapped must be clear and spherically symmetric for this
model to work well. However, in our set up we are trapping ionic ceramic mircoparticles of which
we do not know the clarity, shape, or nion and so our model will most likely not work well.The goal
of the optical tweezers is to create a transfer in momenta such that we can create a restoring force
proportional to the intensity of the incoming laser beams that will always point toward a central
location.
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Figure 4. Depicted here is the raw position data from the tracking of the ceramic microparticle
ion. The two different colors here are for two separate cases. The orange colored graphs are for
the situation where the optical tweezers were off. Likewise, the blue graphs are for the situation
in which the optical tweezers were on. a.) Here we see that without the optical tweezers the
ion does not favor any specif location along the x-axis of the linear electrode trap. The variance
in the x direction for this data is σ 2 = 37.48 µm2 . With the optical tweezers on we can clearly
see two separate favored positions for the ion. The first and second favored position both have
a range of about 5µm, which is very close to the ∆x = (6.69 ± 0.25)µm(95%CI, gaussian pdf)
value that we expect for our optical lattice due to the model. It is also important to note that
the “stair stepping” pattern here tells us the ion moved from one site to a neighboring site in the
optical lattice. The parameter that tells us the ion was indeed constrained is the variance of the
motion. For this data the variance is σ 2 = 0.92 µm2 for the peak at 0 µm and σ 2 = 0.55 µm2
for the peak at about 6 µm. b.) Using a histogram of the ion’s position data we see that there
was no preferred location without the optical tweezers in place. The histogram of the position
data with the optical tweezers in place shows us that the ion has two preferred locations. It also
shows us the Gaussian distribution of the ion about these locations which we would expect from
our model. Most importantly the histogram shows us a peak to peak separation of these two sites
is about ∆x = 6.3µm. Overall, the data presented in this figure shows that the ion is indeed being
constrained by our optical tweezers and that those constraints closely follow the model we set forth
with our optical lattice spacing of ∆x = (6.69 ± 0.25)µm(95%CI, gaussian pdf).
Upon building the optical tweezers we then began to test our model. To do so we took
videos using a camera connected to the microscope in our set up. After taking the video of
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the ions position in the trap we then imported it into an analysis program that we wrote to
track the center of intensity of each frame giving us a data set for the ions position in the
cameras frame over time. we then take this data and check to see that the ion is not moving
in the y-direction as it should not since the electrode ion trap is constraining in this direction
(See Fig. 2). After making sure that the ion was indeed constrained in the y-direction we
analyze the x-position over time of the ion. Graphs of some examples of these data sets can
be seen in Fig. 4 and Fig. 5. We took data with the optical lattice both on and off. The
data with the lattice off serves as a measure of our “background”; it gives us a reference of
what the ion does in the trap on its own. We can then compare this data to the data taken
with the optical lattice in place.
From our model’s predictions if the ion was trapped in the optical lattice we would expect
to see distinct x-positions occupied by the ion, these positions we call optical lattice sites.
To confirm the trapping of an ion we took record of the variance, σ 2 , in the ions transverse
position. The variance is simply the square of the standard deviation of the ion’s position.
The variance of the ions position should be much smaller than the variance of the position
with the optical lattice off if the ion is trapped. When an ion is trapped another indication
of this is the “stair stepping” pattern we observe when the optical lattice was on for the data
shown in Fig. 4 a.) and Fig. 5 b.). Furthermore, we can analyze these sites and find the
spacing between each site, ∆x. We found these values by fitting the histogramed data with
a Gaussian distribution and measuring the peak to peak separation of the fits (See Fig. 6).
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Figure 5. Here we have depicted an example of the raw data from a trial where we reduced the
power of the incoming HeNe laser beam to 4.22 mW. We can see that by doing this we have allowed
the ion to more easily move from one lattice site to another. From the data we see there are four
lattice sites that the ion moved in and out of. a.) The x-position vs time graph tells us that the
ion was first in the site at about -25µm then moved to -15µm and then to -7µm then finally to
0µm. After the ion had arrived at the 0µm site we see that it jumped over the site at -7µm to
the -15µm site. These jumps from one site into another are good evidence of two things. First,
this shows that our ion is indeed trapped by the optical tweezers. Secondly, this shows that the
particles motion with the ion trap on is primarily due to Brownian motion from the surrounding
fluid, air. b.) Again here we see the Gaussian like distribution in the histogram of the particles
position, as we should. We also see from this a equal spacing between the four separate sites.
Doing so for one of our trials, with the power of the incoming laser beam turned down to
Plaser = 4.22mW, we found the peak to peak separation between the first and second lattice
sites to be ∆x = (7.653 ± 0.065)µm (95%CI, gaussian pdf). Similarly we found a peak to
peak spacing of ∆x = (8.74 ± 0.39)µm (95%CI, gaussian pdf) between peaks 4 and 3 and
∆x = (7.48 ± 0.15)µm (95%CI, gaussian pdf) between peaks 3 and 2. These ∆x values
should all agree with one another according to our model. The last two values agree with
each other while the first one does not agree with either. This could be due to the very
small amount of data for this lattice site; which is clearly shown by the x-position vs time
graph in Fig. 6. All three ∆x values for this trial, however, do not agree with our expected
value of∆x = (6.69 ± 0.25)µm(95%CI, gaussian pdf).
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Figure 6. Here we have depicted the Gaussian fits of the histogram data for the multiple lattice
sites in Fig. 5. The Gaussian fits are used to find the peak to peak separation distances between
two adjacent peaks, ∆x. This data shows a peak to peak spacing of ∆x = (8.74 ± 0.39)µm (95%CI,
gaussian pdf) between peaks 4 and 3, ∆x = (7.48 ± 0.15)µm (95%CI, gaussian pdf) between peaks
3 and 2, and ∆x = (7.653 ± 0.065)µm (95%CI, gaussian pdf) between peaks 2 and 1. These ∆x
values should all agree according to our model. The last two values agree with each other while
the first one does not agree with either. This could be due to the very small amount of data for
this lattice site; which is clearly shown by the x-position vs time graph on the left. All three ∆x
values for this trial, however, do not agree with our expected value of∆x = (6.69±0.25)µm(95%CI,
gaussian pdf).
After careful inspection of all the data we gathered from our trials we arrived at the
conclusion that our ∆x peak to peak separation values we measured do not agree with the
predictions put forth by our model. Our model gives a predicted value of ∆x = 6.694µm.
Some of our measured values are ∆x = (8.74 ± 0.39)µm (95%CI, gaussian pdf) between
peaks 4 and 3, ∆x = (7.48 ± 0.15)µm (95%CI, gaussian pdf) between peaks 3 and 2, and
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∆x = (7.653±0.065)µm (95%CI, gaussian pdf) between peaks 2 and 1 for the data set shown
in Fig. 5 and Fig. 6. Our hypothesized reason for our measured values not agreeing with the
model is that the ceramic microparticle ion is not a perfectly clear nor perfectly spherical
particle. If this was the case then the ion would not react as strongly to the optical lattice
due to the differences in the way the transfer of momentum occurred. This would explain
why our ∆x values are not only in disagreeance but also larger. More importantly we have
demonstrated that an ion can be constrained using an optical lattice. This conclusion is
brought about by the comparison of the variances of the transverse motion of the ion. For
the data set shown in Fig. 4 the variance of the ceramic microparticle with the optical lattice
in place was σ 2 = 0.92 µm2 and without the lattice σ 2 = 37.48 µm2 for the first peak at
position 0 µm. Future work involving the optical lattice would be to test the model in the
exact same fashion for ionized amino polystyrene spheres. This would allow the potential
for any discrepancies between the model and data to be explained based on the particles
properties. Other work will be to find the variances as a function of the optical lattice
size/strength.
[1] Daniel T. Gillespie, Fluctuation and dissipation in Brownian motion, American Journal
of Physics, 61, 12, 1993.
This paper delves into a mathematical representation of Brownian motion described by the
Langevin Equation. It goes through two analyses. The first being the use of continuous Markov
processes and Newton’s second Law. The latter analysis taking the approach of using probabilities on the molecular level, jump Markov processes, and an analysis of the velocities behavior
to arrive at the same conclusions. Ultimately stating that the effect of a surrounding fluid on
a particle that is causing Brownian motion can be described by a simple superposition of a
steady dissipative drag force and a zero-mean temporally uncorrelated fluctuating force.
[2] Dongdong Jia, The time, size, viscosity, and temperature dependence of the Brownian motion of polystyrene microspheres, American Journal of Physics, 75, 2, 2007.
There are many demonstrations using a suspended Brownian particle in a fluid to analyze
the motion in undergraduate labs. In this paper, the author refers to an apparatus in which
automated computer based video microscopy is used to track a polystyrene sphere, providing
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a huge amount of data. This method allows analyzing Brownian motion as a function of
time, micro sphere size, fluid viscosity, and the temperature. Sample suspensions are made by
suspending the micro spheres into the liquid at a concentration such that only one sphere is
observed in the optical field. Then they used the computer to track the path of the sphere,
which is Brownian motion. The model of the average time can be obtained using Langevin
formula and Stoke flow for a sphere. The data from the computer and the predicted model
are then shown in a same graph, which appears to match well with the predicted model. This
is one of the apparatuses being used in graduate labs to analyze Brownian motion.
[3] D.N. Moothoo, Beth’s experiment using optical tweezers, American Journal of Physics,
69, 3, 2001.
Optical tweezers are used here in the presentation of how circularly polarized light can be
used to transfer angular momentum to trapped micro-sized particles and how this in turn
causes rotations replicating Beth’s experiment. A full breakdown of what the optical tweezers
used consisted of and how they were constructed, including many diagrams/figures on this
subject, are present in his discussion. Furthermore, important facts about how the optical
tweezers work are brought forth in his discussion. Overall, the goal of this paper is to provide
an undergraduate student with the knowledge and capability of constructing optical tweezers
for use in the analysis of the transfer of angular momentum from circularly polarized light to
a small particle.
[4] C. E. Pearson, Experimental investigation of planar ion traps, The American Physical
Society, 1-12, 2006.
The stable ion-trap plays an important role in quantum computing and other fields. The iontrap design we are using is proposed by Chiaverini using a planar ion trap. The electrodes all
lie in a plane and ions are trapped above the plane of electrodes. There are five electrodes in the
ion-trap. The center and outermost electrodes are held at rf ground while the remaining two
electrodes are biased with a rf potential for radial confinement. Either the center electrode or
the outermost two electrodes can be segmented and dc biased for axial confinement to provide
axial confinement and to shuttle the ions along the trap axis.
[5] Stephen P. Smith, Inexpensive optical tweezers for undergraduate laboratories,
American Journal of Physics, 67, 1, 1999.
Optical Tweezers work because there is a momentum transfer from the light to the particle.
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The forces on the sphere can be calculated by finding the momentum change of incoming rays
of light. The strength of an optical trap increases with an increase in power. The strength
will also increase from a decrease in the focal point size. Helium-Neon lasers are commonly
used for optical tweezers. It is possible to create optical tweezers using a single laser, mirrors,
lenses, and and a beam splitter. When setting up the mirrors it is best to center the beam
on the mirror so there is more room for adjustment. Once the mirrors are in place, the lenses
should be positioned, again centering the beam.
[6] Giorgio Volpe and Giovanni Volpe, Simulation of a Brownian particle in an optical
trap,American Journal of Physics, 81, 224, 2013.
Brownian motion can be easily shown using a microscopic particle in a fluid. The random
motion arises from the particle colliding with the molecules of the surrounding fluid. Mathematically, Brownian motion uses the differential equation for an over-damped harmonic oscillator with an added white noise term. The white noise is from the fluctuating forces caused by
the molecules surrounding the particle. Optical traps can be used to hold Brownian particles
because there is a momentum transfer from the laser light to the particle. Scattering of the
light results in the particle being trapped in an elliptically shaped area.
[7] Martin Siler and Pavel Zemanek, Particle jumps between optical traps in onedimensional optical lattice, NEW Journal of Physics, 6, 12, 2010.
[8] H.G. Dehmelt, Radiofrequency spectroscopy of stored ions I: Storage, Adv. Atom.
Mol. Opt. Phys. 3, 53–73 (1967).
[9] Albert Einstein, Le Mouvement Brownien et la Réalité Moleculaire,Ann. Chimi. Phys.
18, 5-114, (1909)
[10] M.J.Madsen and A. D. Skowronski, Brownian Motion of a trapped microsphere ion,
American Journal of Physics, 82, 10, 2014.