Simulation of Uniform Heating
of Wires Attached to Reduced
Mass Targets
A THESIS
Presented in Partial Fulfillment of the Requirements for the Degree Master of
Science in the Graduate School of The Ohio State University
By
Danielle K. Kelly, B.S.
Graduate Program in Physics
The Ohio State University
2014
Master’s Examination Committee:
R. Freeman, Advisor
C. D. Andereck
K. Akli
c Copyright by
⃝
Danielle K. Kelly
2014
Abstract
A simulation of the temperature changes of wires connected to reduced mass targets is
presented. This simulation looks at thin copper wires and determines the temperature
changes due to thermal diffusion, electrical heating, and losses from radiation using material
properties from SESAME equation of state and Lee-More-Desjarlais models. The effects of
changing the current, and therefore the electrical heating, were simulated. This simulation
finds that for a gaussian current with an exponential fall-off tail the copper wire is heated
to a uniform temperature in 0.5 ps and stays at that temperature for several picoseconds
before thermal diffusion and radiation effects add a noticable temperature gradient to the
wire.
ii
To my mom and dad. You gave my life, supported me, and loved me. Thank you for
molding me into the woman that I’ve become.
iii
Acknowledgments
First I would like to acknowledge Dr. John Morrison for his assistant, time, and patience
with this project. I will also acknowledge Dr. Chris Orban, Matt McMahon and Mark
Schillali for getting me the data tables that were crucial to this work and the program they
wrote which provided the data for the work presented here. Stephen Hageman and Ginny
Cochran for taking the time to edit my paper and supporting me emotionally. Finally I
would like to thank Prof. Richard Freeman for supporting my work and believing that I
can understand physics.
iv
Vita
Ocotber 20, 1989 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Born—Cincinnati, OH
June 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Park City High School
June 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. in Engineering Physics with Honors
Research Distinction, The Ohio State University, Columbus, Ohio
August 2012 to June 2014 . . . . . . . . . . . . . . . . . . . . . .Graduate Teaching Associate, Department of Physics, The Ohio State University
August 2012 to present . . . . . . . . . . . . . . . . . . . . . . . . Graduate Research Associate, Department of Physics, The Ohio State University
Publications
“Proton Cancer Therapy: Using Laser Accelerated Protons for Radiation Therapy” Danielle
K. Kelly, The Ohio State University, June, 2012
Presentations
“Selective Target Normal Sheath Acceleration Deuteron Beam Production for Applications”
D. K. Kelly, J.T. Morrison, , F. Aymond, M. Storm, E. Chowdhury, K.U. Akli, S. Feldman,
C. Willis, R.L. Dadalova, P. Belancourt, M. Engle, E. McCary, P. Schiebel, T. Ditmire,
L. Van Woerkom, and R.R. Freeman, International Conference on High Energy Density
Sciences, Yokohama, Japan, (2013).
v
Fields of Study
Major Field: Physics
Studies in:
High Energy Density Physics
Prof. R.R. Freeman
vi
Table of Contents
Abstract . . . . . .
Dedication . . . .
Acknowledgments
Vita . . . . . . . .
List of Figures .
List of Tables . .
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Page
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. viii
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ix
Chapters
1 Introduction
2 Model
2.1 Thermal Heating . . .
2.2 Electrical Heating . .
2.2.1 Case A . . . .
2.2.2 Case B . . . . .
2.2.3 Case C . . . .
2.3 Radiative Energy Loss
2.4 Initial Conditions . . .
1
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3 Results
3.1 Effects of Thermal Diffusion . . . . .
3.2 Effects of currents in cases A, B, and
3.2.1 Case A . . . . . . . . . . . .
3.2.2 Case B . . . . . . . . . . . . .
3.2.3 Case C . . . . . . . . . . . .
3.3 Effects of Emissivity . . . . . . . . .
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4
9
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C
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16
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19
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4 Conclusion
21
Bibliography
25
vii
List of Figures
Figure
Page
1.1
Aluminum phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Cartoon drawing of target . . . . . . . . . . . . . . . . .
Plots of electrical and thermal conductivities for copper
Plot of specific heat for copper . . . . . . . . . . . . . .
Steady state thermal dispersion . . . . . . . . . . . . . .
Trajectory of electrons from laser plasma interaction . .
Initial conditions of simlation . . . . . . . . . . . . . . .
XUV image from 5 J laser pulse . . . . . . . . . . . . . .
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5
7
8
10
12
14
15
3.1
3.2
3.3
Temperature along wire for current in case A . . . . . . . . . . . . . . . . .
Temperature along wire for current in case B . . . . . . . . . . . . . . . . .
Temperature along wire for current in case C . . . . . . . . . . . . . . . . .
17
18
19
4.1
4.2
XUV image of target hit with a 5 J laser . . . . . . . . . . . . . . . . . . . .
Aluminum reflectivity from Milchberg et al. . . . . . . . . . . . . . . . . . .
22
23
viii
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3
List of Tables
Table
2.1
Page
List of variables used in simulation . . . . . . . . . . . . . . . . . . . . . . .
ix
6
Chapter 1
Introduction
Since the advent of lasers in the mid 1960’s substantial research endeavors have focused on
improving the energy, intensity, time scale, and other properties of laser pulses. Research
is also focused on examining and understanding the complex interactions that occur when
laser light interacts with solid metals and plastics, and the plasmas that form from said
interaction. The laser plasma interaction of high-intensity short pulse lasers has been used
to study ion acceleration [2,16,29], x-ray generation [19,30], and laboratory astrophysics [21].
The vast majority of high intensity laser experiments are focused on studying these laser
plasma interactions, however in the following simulation and the focus of this paper, will
be on the heating of metal adjacent to the laser plasma interaction.
Recent experiments at the Scarlet laser facility have focused on creating a uniform
temperature and density plasma by shooting a high intensity laser at reduced mass targets.
The goal of the experiment is to determine how to tune laser and target parameters in order
to achieve the desired uniform temperature and density of the target. The minimization
of density and temperature gradients in the plasma would allow measurements of opacity,
thermal conductivity, and other parameters and lead to more accurate models of plasmas at
the temperatures and densities produced. The reduced mass targets in this experiment are
100 µm diameter discs held in place by three grounding wires connected to an outer loop
to make mounting the target in place easier. The ability to reach uniform temperature and
densities is an essential step to measuring any material properties. One region of particular
interest is called the warm dense matter (WDM) regime.
The WDM regime consists of matter near solid densities, approximately 0.1 to 10 g/cm3
and temperatures ranging up to 10 eV. This regime is crucial for the understanding of what
occurs in many focus areas from astrophysics, specifically the inner cores of gas giants such
as Jupiter and Saturn [3, 8, 27], to inertial confinement fusion [9, 13, 24].
When matter is excited to temperatures in the WDM regime the electron temperatures
become significantly higher than the ion temperatures due to their much smaller mass.
The large temperature discrepancy between the ion and electrons has led to questions of
1
if local thermodynamic equilibrium can be applied in this regime. The characterization of
WDM properties is essential for understanding how materials transition from condensed
matter to plasmas but measurements of this regime are particularly difficult to make. The
measurements are challenging because the material properties must be measured under
uniform and local thermodynamic equilibrium conditions in order to be compared to theory.
If they are not measured under these conditions then comparison can only be made through
simulations that try to account for the effects of gradient and non equilibrium conditions [7].
A second reason why the measurement is difficult is because during the transition from
condensed matter to plasma the material becomes partially ionized; this requires that the
ions, free electrons, and molecules be treated quantum mechanically [10].
The many models that give equation of states for materials are based in regions outside
of the WDM regime. The models most commonly used in the WDM regime are Lee-More
and Spitzer [36], however these models are not always accurate [9]. A plot of the modeling
capabilities for different temperature and density phase space of aluminum is shown in fig.
1.1. The shaded region in the middle corresponds to the WDM regime and has no equation
of state model. In order to approximate a model the SESAME equation of state tables
interpolate across different models. The ACTEX or activity expansion equation of state is
used for weakly coupled plasmas at very high temperatures [22, 23]. This model becomes
less accurate as the condensed matter regime is approached. The Saha model [25, 26] is the
equation of state representing a mixture of interacting electrons, ion, atoms, and molecules of
a plasma and is accurate over moderate temperatures. The soft sphere model is for expanded
liquid and vapor and the model is based on Monte Carlo computer calculations of fluids.
The Gray equation of state uses a combination of theoretical models, experimental data,
and analytic fits to the experimental data to calculate the equation of state for monatomic
materials in the solid, liquid and melt regions [17]. The Thomas-Fermi-Kirzhints theory adds
nuclear corrections for the condensed matter region not covered by the Gray model [14].
The APW or augmented plane wave region uses electron band theory for solid aluminum
at zero temperature at highly compressed densities [15].
Although the purpose of the performed experiment is to reach uniform temperatures
and densities of the target, the purpose of the following simulation is to determine if the
grounding wires attached to the target are able to reach uniform temperatures in the WDM
regime. Reaching the WDM regime in the grounding would ease calculations because of the
ability to ignore the laser plasma interaction while focusing on the grounding wire.
In section II of this thesis the physics taken into consideration and the simulation setup
is discussed. Section III will discuss the results of the simulation and section IV the conclusions.
2
Figure 1.1: Theoretical codes used in different temperature density phase space regions of
aluminum to construct the equations of state in SESAME [17].
3
Chapter 2
Model
We wish to model the time-dependant temperature rise in reduced mass targets. The
reduced mass targets used in the Scarlet laser experiment are shown in fig. 2.1. The discs
are 100 µm in diameter and have three grounding wires connected to an outer loop used
to mount the target. The targets in the experiment consisted of 1, 3, or 6 µm copper
sandwiched between 1 µm of aluminum on front and back. The grounding wires are 16
µm wide and approximately 100 µm long. The simulation written to examine the effect of
different currents moving down a grounding wire was written in C++. The wires have a
simulated dimension of 16 µm by 100 µm and are assumed to be fully ionized at the start of
the simulation. The wire’s area is broken into a mesh of 0.1 x 0.1 µm2 for the computation
with a cell depth of 1 µm. Each mesh point holds information about the temperature
and energy fluctuations for each time step. The effects of thermal diffusion and electrical
heating, as well as radiative energy loss are simulated.
The properties of copper needed for the calculation are the density, emissivity, thermal
conductivity, electrical conductivity, and specific heat. For the symbols, units and ranges
of these values see Table 2.1. The values for thermal conductivity, electrical conductivity,
and specific heat change with temperatures. The density, ρ, is that of solid density copper,
8.96 g/cm3 , this is used in order to simplify the simulation (for changing density see [28]).
The emissivity of an ionized plasma is a very complex problem [6]. In order to simulate
the exact emissivity all process that can cause radiation to emit must be considered including
emissions from collisional excitation and bremsstrahlung emission by high energy particles.
For this experiment the emissivity was set to a constant in order to simplify the computation.
The electrical conductivity, σ, and thermal conductivity, κ, is found using the Lee-MoreDesjarlais (LMD) model for dense plasmas [5,36]. The LMD model for electrical and thermal
conductivity was used because the model more accurately predicts the conductivities near
solid densities. An alternative popular model for determining the conductivities of materials
is the Spitzer formula [31, 32], however Spitzer is known to give results that error on the
order of a factor of 10 or more near solid densities. The LMD model solves the Boltzmann
4
Figure 2.1: The reduced mass target used in the Scarlet laser experiments. The inner disc
has a diameter of 100 µm. The width of each wire is 16 µm along the circumference of
the disc. The outer circle is for stability of the target. The targets consisted of 1, 3, or 6
µm of copper sandwiched between 1 µm of aluminum on the front and back of the target.
(Drawing not to scale)
5
Name
Electrical Conductivity
Thermal Conductivity
Specific Heat
Stefan-Boltzmann Constant
Current
Emissivity
Symbol
σ
κ
csh
σSB
I
ϵ
Unit
Ω−1 cm−1
W cm−1 K −1
J g −1 K −1
W µm−2 eVT−4
C f s−1
Range
5.95 · 105 − 2.45 · 106
2.92 - 404
0.475 − 541
0.01176
Dependent on specific case
0.05 - 0.7
Table 2.1: List of variables used their symbols, units, and range when applicable. The
subscript on eV for the Stefan-Boltzmann constant denotes a temperature and not energy.
equation,
e ⃗ ∂f
∂f
f = fo − τ v •
− E•
∂⃗r
m
∂⃗v
(2.1)
to find the conductivities, where f is the electron distribution function, τ is the electron
relaxation time, ⃗r and ⃗v are the position and velocity vectors and fo is the local electron
distribution function. The LMD model requires a calculation of the momentum transport
cross sections by using the Coulomb cross sections. The electrical and thermal conductivites
are found using the electrical current and energy fluxes from the Boltzmann theory, solved
using Fermi-Dirac integrals with appropriate upper and lower cut-off parameters on the
integration.
The upper cut-off limit, or maximum impact parameter, is determined by the DebyeHückel theory [4]. The Debye-Hückel theory is valid for high density ionized plasmas, with
solid density (8.96 g/cm3 ) considered a sufficiently high density. The theory considers that
on average the plasma is charge neutral, this leads to a characteristic distance where the
ions are shielded from the Coulomb potential of ions outside the characteristic length. For
the calculation the electrons in the plasma are considered to have a Boltzmann thermal
distribution. For time averaged systems with smooth potentials near the ions the Poisson
equation can be solved to find the charge distribution near an ion. This charge distribution
is an exponential screening with a characteristic scale λD shown in equation 2.2.
s
λD =
ϵ0 kB
≊
P
−1
2
2 n T −1
e ne Te + e2 m Zm
m m
r
ϵ0 kB Te
e 2 ne
(2.2)
The variables in the Debye shielding characteristic length; ϵ0 is the permittivity of free
space, kB Boltzmann’s constant (not to be confused with the Stefan-Botzmann constant),
e the fundamental charge, ne and nm the density of electrons and ions respectively, and Te
and Tm the temperature of the electrons and ions respectively. This Debye length is used
for the maximum impact parameter.
The lower cut-off, or minimum impact parameter can be determined either classically
6
Figure 2.2: The left figure (a) shows how the electrical conductivity changes with temperature. The electric conductivity decreases until a minimum is reached and then increases
linearly once the material is fully ionized. The right figure (b) shows how the thermal conductivity changes with temperature. The thermal conductivity increases as the temperature
increases.
or quantum mechanically. For the classical determination the distance of closest approach
method is used bmin = Z ∗ e2 /mv 2 , where Z ∗ is the ionization state, e the fundamental
charge, and v is the velocity. This classical determination is essentially how close the
charges can come together before they are repulsed by the Coulomb potential. The quantum
mechanical method is determined by the uncertainty principle bmin = λ/2 = h/2mv, where
λ is the de-Broglie wavelength and h is Planck’s constant.
The electrical and thermal conductivity as a function of temperature is found using a
MATLAB script written by Chris Orban and Matthew McMahon which implements the
LMD model. The values from the program are input into the C++ program and for each
calculation the electrical conductivity is found for the current temperature of the specific
cell mesh being analyzed.
The electrical and thermal conductivities are measure of the material properties. The
electrical conductivity measures the ability of a material to conduct electric current. A plot
of the electrical conductivity is shown in fig. 2.2a. The electrical conductivity decreases
until reaching a minimum. This minimum occurs because the minimum scattering length
of free electrons does not depend on the specific processes in the material [18]. Once the
material is fully ionized the electrical conductivity increases. Thermal conductivity is a
measure of how well a material is able to connect heat. As the copper heats up the ability
of it to conduct heat increases, see fig. 2.2b. This means as the wire in the simulation heats
up the ability to move heat through thermal diffusion increases.
The specific heat, csh , is given by the SESAME equation of state tables distributed by
Los Alamos National Lab. Equation of state tables are a list of values relating properties
7
Figure 2.3: Specific heat versus temperature for solid density copper. Once the copper is
ionized the specific heat increases linearly with temperature.
such as volume, internal energy, and temperature of different materials based on a specified density. The tables give the values for the ion and electron internal energies versus
temperature for different densities. The specific heat is found by taking a derivative of
the internal energy with respect to the temperature [12], (or csh/ele =
dUele
dT )
This calcula-
tion was preformed by Mark Schillaci. The total heat capacity is found by the equation
csh = cion + Zbar cele , where Z bar is the average ionization state of copper, 4.688.
Specific heat is a measure of how much heat is added to a material for a given temperature change per unit mass. The specific heat increases with temperature linearly because
the material is ionized (fig. 2.3). Once a material is ionized there are no more degrees of
freedom to be excited so all heat put into the material will go directly into heating the
material.
8
2.1 Thermal Heating
The standard equation to find the diffusion of heat through a two dimensional rectangle is
given by
∂T
∂t
2
2
= k( ∂∂xT2 + ∂∂yT2 ) [1], where k = κ/(ρcsh ) in µm2 /s (κ is thermal conductivity, ρ
is density, csh is specific heat). In order to solve this equation computationally an explicit
calculation is performed for each cell mesh. This leads to equation 2.3, where the subscripts
represent the matrix element of the cell mesh, ∆h is the mesh size, and ∆t is the time step
between computations.
n+1
Ti,j
=
△t ⋆ k n
n
n
n
n
n
(Ti+1,j + Ti−1,j
+ Ti,j+1
+ Ti,j−1
− 4Ti,j
) + Ti,j
△h2
(2.3)
In order to verify that the calculation for heat diffusion performed as expected, the final
answer for a simulation with only heat diffusion was compared to the well known steady
state diffusion problem for a two dimensional plate. For this simulation a grid of 1 x 3 µm
of copper was simulated with a mesh size of 0.01 µm. The thermal conductivity and specific
heat are held constant at 400 W/mK and 0.386 J/gK respectively. For this verification the
temperature of the hot end is set to 300 K while the temperature of the other three sides
is set to 270 K and the wire is also initialized to 270 K.The steady state answer is shown
in equation 2.4 [1],
∞
X (−1)n+1 + 1
2
nπy
nπx
nπW
T = T1 + (T2 − T1 )
sin(
)sinh(
)/sinh(
)
π
n
L
L
L
(2.4)
n=1
where W is the width and L the length of the plate and T1 and T2 are the set to 270 K
and 300 K respectively. The results of the simulation are shown in fig. ??.
The figure shows the plate slightly warming on the left side, where the temperature
is set to 300 K. The hottest temperature is centered up and down with the temperature
decreasing to 270 in all directions because the top, bottom and left are set to 270 K. The
solution of the simulation after sufficient time, when the temperature is no longer changing
with time, agrees with the steady state equation within one one-thousandth of a percent.
2.2 Electrical Heating
The electrical heating is caused by the increase of energy from the well known equation
dE
dt
= I 2 R [11], where I is the current, R is the resistance and E is the energy. This equation
is written
∆E = I 2 R ∆t
The resistance is found using R =
l
σ SA ,
(2.5)
where σ is the electrical conductivity found
using the LMD model, l is the length the current travels and SA is the cross sectional
9
Figure 2.4: The resultant thermal dispersion of a copper plate. The temperature on the
top, bottom, and right of the figure are held constant at 270 K and the temperature on the
left is held at 300 K. The simulated region is 1 µm tall and 3 µm long, the temperature
gradient for these temperature is approximately 1.5 µm long with the majority of the plate
staying at the inital 270 K.
10
surface area. The surface area term depends on the depth of the cell, 1 µm.
There are three currents used to find the heating of the wire. All of the currents assume
the same amount of initial electrons in the disk at the same temperature. For these results to
be compared to experiment a 5 J laser with intensity 5x1020 W/cm2 , and 0.8 µm wavelength
are used in the calculations. To calculate the average electron
qtemperature Wilks scaling
law
is used [35]. The equation for Wilks scaling law is,ϵp =
1 + Iλ2µ /1.37x1018 − 1 m0 c2 ,
where ϵp is the electron temperature in eV, I is the intensity of the laser in W/cm2 , λµ is the
wavelength in µm, m0 is the mass of the electron, and c is the speed of light. For a laser with
intensity of 5x1020 W/cm2 and wavelength 0.8 µm, the average electron temperature is 7.3
MeV. Using the conversion efficiency of laser energy to heated electrons ( [35]) there is an
assumed 1.28x1012 free electrons in the disc from the laser plasma interaction. Both currents
assume that all electrons have a temperature of 7.3 MeV and that there are 1.28x1012
electrons trapped in the central disc. Electrons of 7.3 MeV have a gamma factor of 15 so
can be approximated as moving the speed of light.
Once the current is calculated, details for the specific currents given in sections 2.2.1,
2.2.2, and 2.2.3, resistive stopping is used to maintain energy conservation. As the current
travels down the grounding wire the current will dampen due to the voltage build up along
the wire. When this voltage buildup is greater than the energy of an electron the electron
will stop traveling down the wire and the current will decrease. In order to find how much
the current is reduced the resistive stopping is calculated to find the voltage buildup for a
specific section of the current and using a Maxwell-Boltzmann distribution the percentage
of current that survives is calculated.
The resistive stopping is calculated by V = IR, where the voltage is equal to the
current, in amps, multiplied by the resistance, in ohms. When finding the resistive stopping
the current and resistance from the previous time steps is entered into a matrix. This matrix
travels with the current to keep track of the voltage buildup for individual sections of the
current. The Maxwell-Boltzmann distribution of electron is equation with kT found from
Wilks equation,
r
−E
E
e kT
(2.6)
3
kT π
This distribution is the energy probability for one electron with the average energy < E >=
dN (E)/dE = 2
3
2
kT . The percentage of initial current that makes it down the wire is found is
Z∞
%I = 100 ·
β(E − V )
dN (E)
dE
dE
(2.7)
0
where β is the relativistic velocity found from E = (γ − 1)mc2 , substituting in γ =
p
√
1/ 1 − β 2 and solving gives β(E) = E 2 + 2Emc2 /(E + mc2 ). Substituting these val11
Figure 2.5: The laser hits the target from left to right at the center of the disc. This causes
electrons to heat from the laser energy and the electrons are sprayed in a 90◦ cone. The
electrons sprayed directly forward down the wire are contributing to the current in case A.
ues into equation 2.7 gives the final current percentage,
r
Z∞ p
−E
(E − V )2 + 2(E − V )mc2
E
%I = 100 ·
·2
e kT dE
2
3
(E − V ) + mc
kT π
(2.8)
0
2.2.1 Case A
The current for the first case is found by modeling the laser pulse hitting the target and
spraying electrons towards the wire as seen in fig. 2.5. If the electrons are considered to
be sprayed in a cone of 90◦ [20] then fifteen percent will be trapped in the copper wire, or
1.875x1011 electrons. Particle-In-Cell (PIC) simulations in large scale plasma (LSP) simulations have shown that the current through a plane behind the laser pulse is approximately
a gaussian with a slightly longer full width half max (FWHM) than the laser pulse [34]. The
laser pulse in the experiment is approximately 40 fs FWHM because the pulse will slow in
the material a gaussian of 50 fs FWHM is used for the simulated electron current [34]. This
2 /900
leads to a current equation of 3.53x109 e−t
units
2
5.66x10−10 e−t /900
electrons per femtosecond, or in traditional
coulombs per femtosecond.
12
2.2.2 Case B
The current for the second case is modeled by assuming the high energy electrons are
trapped in the disc and will diffuse down the wire if their path happens along an opening
for a wire. In order to calculate the time scale that the electrons diffuse from the disc
one needs to know the average distance traveled, the velocity of the electrons, and the
probability of encountering a wire and not the target edge. The average distance from one
point on the edge of the disc to any other point on the disc is 4 r/π, the radius of the target
is 50 µm giving an average distance of 63.7 µm traveled per electron. The probability of
encountering a wire is the width of the wires divided by the circumference of the disc. This
gives the ratio of 3 · width/(2πr), for a single wire width of 16 µm and a circumference
of approximately 314.16 µm the probability of encountering a wire is 0.15 or 15%. From
this information it is found that on average each electron will travel across the disc 6.67
times. Then the average distance an electron will travel is 424.9 µm, and for electrons
moving approximately the speed of light it will take 1.42 ps to exit the disc. This 1.42 ps
is the characteristic time for the electrons to diffuse into the wire. These numbers give an
equation for the population decrease in the center disc of 1.28x1012 e−t/1.42 electrons per
picosecond. Taking a derivative of the population gives the rate of change, or the current
exiting the disc. To get the current traversing one grounding wire the sign of the current
leaving the disc is inverted and then the equation is divided by three. This gives a final
current of 4.8x10−11 e−t/1420 coulombs per femtosecond.
2.2.3 Case C
The current in the third case is a combination of cases A and B. This current assumes that
initially the current has the same gaussian pulse as A but the exponential fall off of B for
the falling edge of the pulse. This current is more characteristic of what actual currents
look like from particle in cell simulations [33].
2.3 Radiative Energy Loss
The outer surfaces of the two dimensional mesh will lose a small amount of energy from radiative losses. The blackbody radiation equation was used in the calculation. This equation
is computationally written as equation
∆E = A σSB ϵ T 4 ∆t
(2.9)
The area, A, used in the equation is the surface area which is the two times the mesh size
squared or 2∆h2 for the bulk and 2∆h2 + d∆h for the exterior with d being the depth of the
material, 1 µm. The constant σSB is the Stefan-Boltzmann constant, 5.67x10−8 W/m2 K 4 .
13
Figure 2.6: The copper wire 100 µm long and 16 µm wide is initial set to 0.04 eVT . The
outside vacuum and the ground on the right is also set to 0.04 eVT while the central disc of
the target, on the left, is set to 45 eVT .
In the experiment the emissivity was set at 0.05 and then to 0.7, with all other factors
remaining the same, in order to establish how much of an effect it has on the overall
cooling of the wire. The percent difference of any two values varied by no more than 0.08%
leading to the conclusion that the complex emissivity profiles of ionized plasmas has a barely
noticeable effect on this problem.
2.4 Initial Conditions
The set up of the simulation is that the entire wire begins at 0.04 eVT . The sides connected
to air and to ground are also set at that same temperature while the side connected to the
central disc is set to 45 eVT , see fig. 2.6. The temperature of the central disc is set based
on the results from the experiment performed on the Scarlet laser. The central disc of the
target is heated up, almost uniformly, to 45 eVT as shown in fig. 2.7.
14
Figure 2.7: XUV image of a 100 µm diameter target. The target is 3 µm thick with 1 µm
of copper sandwiched between 1 µm of aluminum on top and bottom. The three wires are
highlighted by dashed black lines. The wires show a temperature gradient ranging from 30
eV directly attached to the target to 10 eV 100 µm down the wire. This image was taken
with a wavelength of 68 eV on Scarlet laser at OSU.
15
Chapter 3
Results
For all current cases the wire heated to a uniform temperature in femtoseconds and then
after picoseconds the thermal diffusion effects started increasing the temperature at the
target end of the wire. The effects of thermal diffusion, electric heating and radiative
cooling are all discussed in the following section.
3.1 Effects of Thermal Diffusion
The effect that thermal diffusion has on the simulation is minimal compared to the effect
of electrical heating. The total time needed for the current to travel down the 100 µm long
wire ranged from 440 fs to 1 ps, depending on the current profile. The effect of thermal
diffusion heating up the target end of the wire did not make a noticeable difference until 10
picoseconds after the start of the simulation. This means that there is a few picoseconds after
the wire has been uniformly heated before the target end heats significantly from the thermal
diffusion, more then enough time to take measurements of opacities and conductivites for
a specific experiment.
3.2 Effects of currents in cases A, B, and C
3.2.1 Case A
The current in case A is a gaussian pulse with a FWHM of 50 fs. The pulse is from a laser
shot directly down the wire. The maximum temperature reached by the gaussian pulse is
approximately 10.05 eV. This temperature is uniform across the whole wire as can be seen in
fig. 3.1 . The gaussian pulse has a steep gradient at the leading edge of the pulse compared
to what is seen in the exponentially decaying current (fig. 3.2). The temperature remains
relatively constant after the maximum heating temperature and falls approximately 0.001
eV after 200 fs from the radiative energy loss.
16
Figure 3.1: Plots of the temperature in eV for the gaussian current of case A. The plots are
taken at 10, 200, 300 and 400 fs. The steep rise of the gaussian edge can be seen heating the
wire. The uniform temperature is reached remain 440 fs after the start of the simulation.
17
Figure 3.2: The exponentially decaying current of case B is shown for 10, 300, 600 and 900
fs. The temperature increase is much slower then the gaussian current with a much longer
temperature gradient as the current travels down the wire. The maximum temperature
reached in case B is 3.5 eV.
3.2.2 Case B
The current in case B assumed that hot electrons heated by a laser pulse remained trapped
in the disc and diffused down the wire. The maximum temperature reached by this exponentially decaying current is 3.5 eV. The temperature is uniform across the wire after 2
picoseconds. The temperature increase along the wire can be seen in fig. 3.2. This current
is characterized by a large temperature gradient.
3.2.3 Case C
The current in case C was a hybrid of case A and B of a gaussian pulse with an exponential
tail. The maximum temperature reached by this current is 10.13 eV. The heating occurs
with the same gradient seen in the case A gaussian current as shown in fig. 3.3. The main
difference between case A and case C, other then the slight increase in final temperature,
is that the temperature in the wire stays warmer on a longer timescale then that of case A.
18
Figure 3.3: The current in case C of a gaussian with an exponential tail has the same
heating gradient as the gaussian current in case A. The wire heats up to a slightly higher
temperature of 10.13 eV. This current takes 20 fs longer then the current in case A to heat
the wire to its maximum temperature for a total heating time of 460 fs.
This is due to the higher gaussian tail of case C creating enough heating to counteract the
effect of the radiative cooling.
3.3 Effects of Emissivity
The effect the emissivity has on the cooling of the wire is tested on the gaussian current,
case A, for two different emissivities. The emissivities chosen are 0.05, and 0.7. The low
emissivity of 0.05 is chosen because this is the emissivity of polished copper, the lowest
emissivity copper can achieve. The higher emissivity of 0.7 is chosen because this is the
highest emissivity that solid copper can obtain, this is the emissivity of oxidized copper.
Except for the simulation run where the emissivity was set to 0.7 all other runs have an
emissivity of 0.05.
The results show that the emissivity has a negligible effect on the cooling of copper.
The simulation shows that the wire heats to the same temperature, 10.05 eV, in the same
amount of time, 440 fs. The simulation ran further and was compared at 1 picosecond. The
highest percent temperature difference found for any section of mesh at 1 picosecond is less
then 5 thousandth of a percent.
19
The higher emissivity does have a more noticeable effect on the temperature after the
pulse has traveled down the wire. The emissivity of 0.05 causes the temperature to decrease
approximately 0.001 eV after 200 fs; the temperature decreases by 0.01% in the first 200 fs
after the wire was heated. This rate is higher for the emissivity of 0.7 which decreases the
temperature by approximately 0.01 eV after the same 200 fs; the temperature decreased by
0.1% in the first 200 fs after the target was heated.
20
Chapter 4
Conclusion
The results from this simulation show that for a gaussian current (FWHM of 50 fs), or a
gaussian current with an exponential tail, created by a 5 J laser pulse with an intensity of
5x1020 W/cm2 and FWHM of 40 fs, will heat the grounding wires to temperatures of 10 to
10.15 eV. This heating occurs over a time scale of 0.5 ps after the laser pulse hits the target.
The effect of thermal diffusion from the target did not noticeably heat the connecting wire
until approximately 9.5 ps after the current had finished heating the wire. The decreases in
temperature from radiative decay is also a slow process. For the highest emissivity value of
0.7 the target will lose 0.1% of its temperature in 200 fs to radiative decay. The radiative
energy loss scales as the area exposed to background. This implies that, discounting the
edges of the wire, there will continue to be no temperature gradient along the wire but that
the wire will decreases in temperature at the same rate.
These results match the experimental XUV data in fig. 2.7, shown also in 4.1 without
the wires highlighted to ease the reading of the colors. The XUV shot in this picture is
time integrated and shows the hottest temperature reached. The simulation results show
that the wires are heated uniformly to approximately 10 eV before the thermal diffusion of
temperature from the hot center target diffuses down the wires. The XUV image shows,
especially for the top right and bottom right wires, that the wires are heated to approximately 10-15 eV extending 150 µm from the center target. The first 50-75 µm extending
away from the target have a strong temperature gradient but this can be attributed to the
thermal diffusion from the heated target.
This simulation has shown that there are approximately 9.5 ps after current has heated
the grounding wires when a measurement can be made on a uniformly heated plasma. An
experiment similar to one performed by Milchberg et al. in 1988 [18] can be designed
to measure the electrical and thermal conductivities of the heated wire. The experiment
Milchberg performed heated an aluminum target with a 0-5 mJ energy laser pulse and
measurements of the time integrated reflectivity and frequency shift of the reflected pulse
were taken. The data from the frequency shift was used to correlate the electron temperature
21
Figure 4.1: XUV image of a 100 µm diameter target. The target is 3 µm thick with 1 µm
of copper sandwiched between 1 µm of aluminum on top and bottom. The three wires can
be seen with one pointing to the top right corner of the graph, another the left edge and
the third to the bottom right corner. The wires show a temperature gradient ranging from
30 eV directly attached to the target to 10 eV 100 µm down the wire.
22
Figure 4.2: The results of the experiment performed by Milchberg et al. of the resistivity
of aluminum versus the electron temperature.
in the target with the intensity of the laser. The Drude model was used to relate the collision
frequency to the complex conductivity which is related to the index of refraction through
n2 = 1 + i4πσ/ω, where n is the index of refraction, σ is the conductivity and ω is the laser
frequency via the Fresnel equations. The complex conductivity is Re(σ) = (v/4π)ξ, and
the imaginary Im(σ) = (ω/4π)ξ, where v is the collision frequency ω is the laser frequency
and ξ = ωp2 /(ω 2 + v 2 ) is a dimensionless parameter where ωp is the plasma frequency. The
Fresnel equations are only valid when vacuum material interface is less then a skin depth. If
this is not true then the Helmoltz wave equation is solved numerically to obtain a relation
between the index of refraction, n, and the reflectivity which requires knowledge of the
electron density and an assumption of local thermodynamic equilibrium. The results of
this experiment are shown in fig. 4.2 of the resistivity versus temperature of the aluminum;
resistivity is the inverse of the conductivity, ρ = 1/σ.
The Milchberg experiment can be refined to find more accurate measurements of the
material properties. One experiment that could succeed is to shoot a reduced mass target
with wire extending from the central target. A second laser pulse, or probe beam, can be
23
reflected of the wire at a delayed time to measure reflectivity in the wire. This measurement
will be time resolved and the probe beam can vary in wavelength adding constraints to the
data. The potential for varying the temperature of the wire by changing the laser intensity
would allow for more accure measurements to be taken to better define the material property
of electrical conductivity in the WDM regime.
24
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