CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
OPTICAL PROBING OF THE ELECTRON TEMPERATURE
GRADIENT
T. Ao, L Vollrath, A. Ng
Department of Physics <fe Astronomy, University of British Columbia,
Vancouver, British Columbia .Canada V6T121
Abstract. Previous pyrometric measurements on shock waves in silicon have revealed the existence of
different electron and ion temperature gradients in the shock front. This was attributed to the relatively
low energy equilibration rate between electrons and ions in the strongly coupled plasma that exists in a
shock wave. In this paper, we will describe a new approach to assess this effect based on the
reflectivity and change in phase of a P- and S-polarized probe reflected from a shock front in-flight in
silicon. Predictions from numerical simulations will be presented and discussed.
INTRODUCTION
using a phenomenological electron-ion coupling
coefficient g.
Since no simple analytical
expression for g is available in the regime of
interest, it is left as a free parameter. The energy
equations for the electrons (1) and ions (2) are
given by the following equations,
Ideally, a shock wave is treated as a propagating
discontinuity in all of the thermodynamic variables.
However, finite gradients can exist due to
relaxation processes occurring in the shock
compressed material; namely the energy exchange
or equilibration between electrons and ions. In
this paper, the effect of the equilibration rate
between electrons and ions on the electron thermal
gradient at the shock front is discussed.
In
addition, a new approach using the reflectivity and
reflected phase of P- and S-polarized light to
measure the strength of electron-ion coupling is
described.
-Z-(pE.) = --Z-\ou\E.
- K
dx
(D
P o
P +
— pE,
^ '
a
- — pu
^-+H
2
P
BACKGROUND
Initially, shock compression of a material leads to
heating of the ions. The electrons are subsequently
heated via electron-ion Coulomb collisions.
Thermal equilibrium between electrons and ions
would only be established somewhere behind the
shock front.
However, at the shock front the
temperature of the electrons may deviate from that
of the ions depending on the rate of energy
exchange between the electrons and ions. In this
paper, the process of equilibration is described
(2)
The difference between the electron and ion
temperatures in a shock wave due to a finite
electron-ion energy exchange rate is shown in
Fig. 1. It is evident that at the shock front the
electron and ion temperatures differ substantially.
In addition, there is a noticeable difference in the
temperature gradients between the electrons and
ions. At sufficiently high electron-ion coupling
1243
coefficient values, these temperature profiles will
collapse together and the shocked regime may be
treated as a single-temperature plasma.
g(W/m3-K)
2xl01JS
6xl017
5xlQ16
2xl017
2 IO4
1.5 IO4
*
Method
Spitzer5
Fermi Golden Rule5
Coupled Mode
Modified Spitzer6
TABLE 1. Theoretical calculations of the energy relaxation
rate of 3 eV electrons in Al at solid density with the ions kept at
the melting point,
1 IO4 r
NEW APPROACH
3
5 IO
The following is a description of an alternative
approach for studying the electron-ion equilibration
rate within a shock wave. First of all, at the shock
front a gradient in electron temperature leads to a
gradient in the electrical conductivity, 0^, which in
turn would lead to a gradient in the dielectric
function, E^. Now, consider an electromagnetic
wave reflected from a shock wave in-flight in
silicon. The amplitude and change in phase of the
reflected electromagnetic wave would depend upon
not only the dielectric value, but also the gradient
of EO) at the shock front. In addition, the motion of
the shock front would affect the amplitude and
phase of the reflected electromagnetic wave.
-8.4-8.3-8.2-8,1 -8 -7.9-7.8
x (jLim)
FIGURE 1. Electron temperature (solid line) and ion
temperature (dashed line) profiles of a 10 km/s shock in Si for
g=10l7W/m3-K.
So far, observations of the electron-ion
equilibration rate within a shock wave have relied
upon emittance (spectral radiance) and absorbance
(emissitivity) measurements. The first emittance
measurement was obtained by Celliers et al, in
1992 on laser-driven shock waves in-flight in
silicon.1 For shock speeds of 15 to 20 km/s, the
observed emittance were found to be a factor of 20
to 50 lower than that predicted for an equilibrium
shock state. These results gave the first assessment
of the electron-ion coupling coefficient in a shock
wave in silicon to be about IO16 W/m3-K.
An electromagnetic wave incident upon a surface
may be in either P- or S-polarization. It is known
that P- and S-polarized light have different
sensitivities to the gradient in E^ because of
resonance absorption that occurs in P-polarized
light.
Thus, examinations of the reflected
amplitude and phase of both P- and S- polarized
light may provide insight into the electron-ion
equilibration rate in a shock wave. A plausible
experimental scheme using this alternative
approach is shown in Fig. 2.
More recently, Lower et a/.2'3 were able to
measure simultaneously the emittance and
absorbance of x-ray driven shock waves in silicon
and aluminum. Their results were also consistent
with that of low g values of IO16 and IO17 W/m3-K
in silicon and aluminum, respectively.
Meanwhile, theoretical predictions of electron-ion
equilibration rates in strongly coupled plasmas
show large variations. The results based on the
traditional simple Spitzer model,4 and the Fermi
Golden Rule and coupled mode calculations of
Dharma-wardana and Ferret,5 are presented in
Table 1, Also included is the value given by
More6 who extended the Spitzer formulation by
using a maximum scattering cross-section which is
consistent with the minimum electron mean-free
path.
^/^
800
Probe Laser
P- & S~po!
Shock
Drive r
FIGURE 2, Experimental schematic of new approach.
Silicon is chosen to be the material of study since
unperturbed silicon is nearly transparent at nearinfrared wavelengths. The target would consist of
a sample of silicon sandwiched between an
1244
the Al/Si interface. However, the reflectivity ratio
exhibits an interesting dependence on the electronion coupling coefficient, as shown in Fig. 5.
aluminum pusher layer and anti-reflective layer. A
steady shock wave is launched into the sample
layer, while a probe laser is incident obliquely on
the AR-coated free surface of the sample layer.
O.9....,,,...,ii ..,.,,.,.,.
To illustrate this approach, a 1-D hydrodynamic
code is used to simulate a desired shock wave. In
the 1-D hydrodynamic code, the shock is treated as
a two-temperature electron-ion fluid. The equation
of state used was based on the QEOS model of
More et aL1
The electrical and thermal
conductivities used were based on the dense
plasma conductivity model of Lee and More.8 As
stated earlier, a phenomenological coupling
coefficient g was used to describe the electron-ion
equilibration rate.
0.85
0.8
0,75
0.7
-100
0
100
200
300
400
Time (ps)
FIGURE 4. Ratio of reflectivity of P- and S-polarized 800 nm
probes @ 45° off a 10 km/s shock in-flight in Si as a function of
time, (g=1016 W/ra3-K)
The interaction of an electromagnetic wave with
the shock wave is treated by solving the Helmholtz
equations for P- and S-polarizations.9
The
dielectric function of the shock material is obtained
using the Drude model, where the collision
frequency, vei, is related to the DC conductivity, 0Q.
0.9
0.80.7-
Shown in Fig. 3 are the reflectivities as a function
of time for the P- and S-polarized probes. Due to
the propagation of the shock wave towards the free
surface, the reflectivities steadily increase as less
and less unperturbed silicon remains ahead of the
shock front.
1
0,60.5
10
101
FIGURE 5. Ratio of reflectivity of P- and S-polarized 800 nm
probes @ 45° off a 10 km/s shock in-flight in Si at various
electron-ion coupling,
Fig. 6 shows the reflected phases for the P- and S~
polarized probes. No phase change occurs until the
shock reaches the Al/Si interface, but after that the
motion of the shock dominates the change in the
both of the reflected phases.
0
0.8 1
-50
1OO 200 3OO 4OO
Time (ps)
FIGURE 3. Reflectivity of P- (dashed line) and S-polarized
-1OO
-100
~
(solid line) 800 nrn probes @ 45° off a 10 km/s shock in-flight
in Si. Time zero refers to the point when the shock reaches the
Al/Si interface, (g=1016 W/m3-K)
-15O
-200
-100
The effect of shock propagation is suppressed by
comparing the ratio of the reflectivities, (Rj^Rs),
which remains nearly constant after an initial
transient behavior, as shown in Fig 4, This initial
jump in (Rf/Rs) is due to thermal conduction across
0
100
200
300
400
Time (ps)
FIGURE 6. Reflected phase of P- (dashed line) and S-polarized
(solid line) 800 nm probes @ 45° off a 10 km/s shock in-flight
in Si. Time zero refers to the point when the shock reaches the
AySi interface. (g=1016 W/m3-K)
1245
However, Fig. 7 shows that the difference
between the reflected phases, (6P - 8S), after the
initial transient effect, remains constant during the
shock propagation,
speed, thus giving a unique assessment of the
shock state.
Measurements of the ratio of reflectivities of Pto S-polarized probes to examine the strength of
electron-ion coupling in strongly coupled plasmas
should be viable using impact generated shock
waves. Meanwhile, a high sensitivity diagnostic
such as frequency domain interferometry may be
used to assess the value of g using measurements
of reflected phases of P- and S-polarized probes.
The latter method would be more suitable in laser
generated shock wave experiments.
0,6,, , , i i , i . . . . . . . , . . . . . . . .
0.55
O.5
O.45
0,4
-100
In conclusion, a new approach for an independent
assessment of the thermal equilibration rate in a
shock wave has been presented. This approach is
applicable to other samples, provided they have the
following properties. First, the unperturbed sample
must be characterized by low photo-absorption to
allow the probe light to reach the shock front with
little attenuation. Second, a high reflectance in the
shocked state is required for high signal levels for
reflected probe measurements. Accordingly, an
interesting class of samples to study would be ionic
crystals,
O
1OO 20O 3OO 40O
Time (ps)
FIGURE 7. Differential phase change of P- and S-polarized
800 nra probes @ 45° off a 10 km/s shock in-flight in Si as a
function of time. (g=1016 W/m3-K)
A comparison of this differential phase change
between P- and S-polarizations for various g values
is shown in Fig. 8.
0.7,
. ,......,
, ........
. .......
. , ..
0.65
0,6
REFERENCES
0.55
0,5
1.
0.45
2.
0.4
10 1
10 1
10 1
10 1
10 1
g
FIGURE 8. Differential phase change of P- and S-polarized
800 nm probes @ 45° off a 10 km/s shock in-flight in Si at
various electron-ion coupling.
4.
DISCUSSION AND CONCLUSIONS
5.
It should be noted that temporal variations in the
reflectivities of the P- and S-polarized probes are
governed by the motion of the shock wave due to
the change in the amount of unperturbed material
ahead of the shock front attenuating the probe
signals. On the other hand, temporal variations in
the reflected phases of the P- and S-polarized
probes are dominated by the Doppler motion of the
shock front. Both of the observations can be used
to provide a direct measurement of the shock
6.
7.
3.
8.
9.
1246
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