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
SHOCK WAVES AND PLASMA PHYSICS
A.Ng
Department of Physics & Astronomy, University of British Columbia,
Vancouver, British Columbia, Canada V6T1Z1
Abstract. The impact of shock wave on advances in plasma science is illustrated using studies of
plasma conductivity, electron-ion equilibration rate and pressure ionization. Conversely, these studies
have led to the recognition of the non-equilibrium thermal structure of a shock wave as well as the role
plasma physics plays in the high-pressure regime of shock physics. The convergence of the two fields
also leads to the development of Warm Dense Matter science that incorporates the physics of highdensity plasmas, high-temperature condensed matter and shock processes.
in the strong coupling regime and has led to many
significant advances in that field.
INTRODUCTION
In this paper, I will highlight the coupling
between shock wave and plasma physics. A
plasma is known as the 4th (ionized) state of matter.
Most studies of plasma science has been focussed
on ideal plasmas with relatively low density or
high temperature, where the potential energy of
Coulomb interaction is much less than the thermal
energy. In these plasmas, the electrons and ions
are treated as systems of ideal gases. Charge
screening effects are described by the Debye
length. On the other hand, there is a more complex
regime identified as strongly coupled plasmas with
relatively high density or low temperature, where
the ratio (1^) of the Coulomb potential to the
thermal energy exceeds unity. In such plasmas,
ions are strongly correlated and the electrons are
partially degenerate. They can no longer be treated
satisfactorily by Debye screening or perturbative
approaches. The coupling between shock wave
and plasma physics occurs in this regime.
in 4 _________________
0.1
FIGURE 1. Shock Hugoniot and the regime of strongly
coupled plasma of aluminum described by QEOS.1
IMPACT OF SHOCK WAVE PHYSICS ON
PLASMA SCIENCE
Understanding plasma conductivity
It is well known that in condensed matter
electrical conductivity varies inversely as
temperature as a result of the thermal excitation of
phonons, whereas in a plasma it increases as
temperature to the power 3/2 (Spitzer
conductivity2) due to the decrease in the electronion collision cross-section with electron thermal
speed.
Accordingly, a region of minimum
As illustrated in Fig. 1 the shock Hugoniot is the
loci of well-defined strongly coupled plasma states.
This renders shock waves one of the most powerful
techniques in laboratory studies of plasma physics
53
improved spatial uniformity of the shock front,
reflectivity measurements were made with a
resolution of 7 ps. As shown in Fig. 4,7 the results
showed a similar disagreement between experiment
and theory with calculation based on Sesame
equation of state and Rinker's conductivity.
Significant improvement in agreement is found for
the analysis based on Lee and More's conductivity
and QEOS1 assuming equal electron and ion
temperatures. However, the best agreement is
obtained for a two-temperature plasma and an
electron-ion coupling constant of 1017 W/m3.K.8
conductivity must exist as illustrated in Fig. 2.
This is the region of strongly coupled plasmas.
101
101
101
10 1
103
104
10 5
T (K)
106
107
FIGURE 2. Electrical conductivity of aluminum at normal
solid density from Lee & More's model.3
0.8
The first experimental assessment of plasma
conductivity models was obtained using a shock
wave emerging from the free surface of an opaque
sample.4 Strongly coupled plasmas were produced
in the shock and release waves. Time-resolved
reflectivity measurements then provided a test of
conductivity models for the compressed and
expanded states. The results for aluminum are
presented in Fig. 3 for a temporal resolution of 30
ps. The hydrodynamics of the shock and release
waves were calculated using the Sesame equation
of state for a single temperature plasma.5 The data
clearly showed significant disagreement with
analysis based on Rinker's conductivity.6
However, they provided the first benchmark of Lee
and More's conductivity model3 which is now
widely used in simulation codes.
0.6
0.4
0.2
0
-40
0.7
0.8
TIME
0.9
1jO
0
20
Time (ps)
40
60
FIGURE 4. Comparison of reflectivity data (circles) wtith
calculations using Sesame with Rinker's conductivity (dot-dotdot-dashed line), single-temperature QEOS with Lee & More's
conductivity (dotted line), two-temperature QEOS with Lee &
More's conductivity (solid line)
Equilibration between electrons and ions
The above discussion leads to the important basic
question about the process of thermal equilibration
in a two-component plasma. In the ideal plasma
regime, this is treated in the Spitzer-Brysk model9
which takes into account electron degeneracy but
assumes uncorrelated ions. Energy exchange is
also considered in a single particle scattering
picture. Lacking more sophisticated treatments, we
have used a simple approach for strongly coupled
plasmas based on a phenomenological electron-ion
coupling constant10 defined by:
H 0.1;
0,6
-20
1.1
(ns)
(1)
dt
FIGURE 3. Measured reflectivity of the rear surface of a 50(im aluminum target for a shock speed of 2.2x106 cm/s (solid
lines) compared with results of calculations using Rinker's
conductivity (dashed line), Lee & More's model (dot-dashed
line), and Lee and More's model with Amfp(min)=2r0 (dot-dotdashed line).
- g
•V*?"
Corroborative results were obtained recently in a
x-ray generated shock experiment.7 In addition to
54
(2
>
Theoretical support for the above findings is
provided by the calculation of energy relaxation of
electrons in a background of ions that are kept at
the melting point of aluminum at solid density.12
For electrons with an energy of 3 eV, the Fermigolden-rule approach for energy exchange between
independent electron and ion subsystems yielded a
coupling constant of 6.7xl017 W/m3.K. On the
other hand, when the coupling between electron
and ion density fluctuations was included, the
coupled-mode approach yielded a corresponding
value of 5.7xl016W/m3.K.
where p, u, E, P, T, and K are respectively mass
density, fluid velocity, energy density, pressure,
temperature and thermal conductivity.
Shock wave offers an excellent example of a twotemperature, strongly coupled plasma. Shock
compression gives rise to ion heating. Electrons
then gain energy via electron-ion collisions. At the
shock front, the electron temperature is governed
by the electron-ion energy exchange. Fig. 5 shows
the thermal structure of a shock wave in silicon.
Such non-equilibrium can be detected when the
diagnostic scale length becomes less than the scale
length of the thermal gradient at the shock front.
The electron temperature is determined from the
brightness temperature derived from Kirchhoff s
law of emittance and absorbance.
(b)
~K
14
IS
16
17
18
19
20
21
shock $p««d (km/t)
(a)
MO 7
position cierV
FIGURE 5. Spatial profiles of electron and ion temperatures
in a 6 Mbar shock wave in silicon for g=1016 W/m3.K.
14
The first observation of a two-temperature shock
front was obtained using a laser-driven shock wave
in-flight in silicon10. As shown in Fig. 6, the
observed emittance was a factor of 20-50 lower
than that predicted for an equilibrium shock state.
The results provided the first assessment of the
electron-ion coupling constant of about 10 16
W/m3.K for a strongly coupled plasma of silicon.
15
16
17
IS
19
20
21
shock sp«*d (km/s)
FIGURE 6. Peak intensity of shock emission at (a) 430 nm
and (b) 570 nm. Data (squares), calculations using Spitzer
conductivity (dashed line), Lee and More's conductivity (dotdashed line), non-equilibrium model with g=1017 W/m3.K
(dotted line), non-equilibrium model with g=1016 W/m3.K (dotdot-dot-dashed line)
Pressure ionization in strongly coupled plasmas
The most basic process that determines the
properties of high-density strongly coupled
plasmas is pressure ionization. The average
ionization, <Z>, is required to evaluate electron
density and transport coefficients such as
conductivities.
In addition, details of ion
abundance, energy and population of atomic levels
are needed for the calculation of equation of state
and radiative opacities. Due to the fundamental
significance of ionization physics, theoretical
models based on various approaches can be found
in the literature. Unfortunately, there are
substantial divergences between results derived
This is corroborated recently by an independent
experiment using x-ray driven shock waves11 and
simultaneous measurements of the emittance and
absorbance of the shock front in-flight. For a 6.6
Mbar shock in silicon, the observed brightness
temperature was 1.4 eV versus the predicted
Hugoniot temperature of 4.3 eV. The results were
also consistent with an electron-ion coupling
constant of 1016 W/m3.K. Further corroboration
was found in a similar experiment on aluminum.7
As discussed by Ao et al, the results for aluminum
were consistent with a coupling constant of about
1017W/m3.K.8
55
uniform plasma whose state can be defined
uniquely by the shock speed and whose thickness
increases linearly with time. Fig. 9 shows the
schematic arrangement of a laser-driven shock
experiment.
from different models as illustrated in Fig. 7 for
aluminum at 12.5 eV. At three-fold compression,
the value of <Z> can vary from 2.6 according to
the widely used Sesame equation of state to 3.2
for the neutral-pseudo-atom density-functionaltheory model13 or the detailed configuration
model,14 to 4.0 given by the also widely used
QEOS.1 Compounding the problem is the lack of
experimental data in the high density regime for
the validation of ionization theory.
Be (lOOA)
Pump Laser
illlllllllllllillllllllllllilllllllllii
Shock
breakout
diagnostics
7
——HH
FIGURE 9. Laser-driven shock for opacity studies.
A
H
S3
V
12
8
4
0
0.01
0.1
1
1?.
10
8
P/P0
4
FIGURE 7. <Z> obtained from QEOS (solid line), Sesame
(dot-dot-dot-dash line), NPA-DFT model (dash line) and
detailed configuration model (circles).
0
-0.5
"'"^""t**"^-
'
50 PS , 10
_Li_
,
i p
S
.... i.....
—• • • [ • • •. • •
J
i
0
15
T
j 10
p
150 ps
... i .....
—i
5
0
1.5
0.5
FIGURE 10.
Snapshots of mass density and electron
temperature profiles of shock wave in silicon.
In general, the relative abundance of ions in a
plasma can be revealed by the photo-absorption
lines associated with the various ions. For Al
plasmas in which the Al"14 ions dominate, its Ka
absorption line becomes a sensitive indicator of
<Z> as shown in Fig.
8.14 If the transition
probability is known, the determination of <Z> is
reduced to the measurement of the opacity of the
photo-absorption line.
1
o.i
50
100
Time (ps)
0.8
FIGURE 11. Calculated normalized transmission at the Al"14
Ka absorption line.
The target is composed of layers of a Be light
shield, a CH ablator, a Si pusher and an Al sample.
Ablation of the CH layer by the laser pulse drives a
shock wave forward. A steady shock is allowed to
form in Si before it is launched into the Al sample.
The shock speed in the sample can be obtained
from shock transit measurements using either
active or passive diagnostics. Fig.
10 shows
snapshots of the mass density and electron
temperature profiles at different times after shock
arrival at the Si-Al interface. The origins of the zaxes are chosen so as to align the Si-Al interface to
x 1<T4 g/cm2
- 3 - 2 - 1 0
1
2
Energy from Line Center (eV)
3
FIGURE 8. Predicted Al*4 Ka absorption line spectral for Al
plasma with an areal mass density of 6.75x10"4 g/cm2 at 11.25
eV and 6.08 g/cm3 with <Z>=3.04 (dotted line), 12.5 eV and
6.75 g/cm3 with <Z>=3.08 (solid line), and 13.75 ev and 7.43
g/cm3 with <Z>=3.15 (dashed line).
It is particularly noteworthy that shock waves
provide a novel and elegant approach for line
opacity studies. This is based on the formation of a
56
4104
facilitate comparison. The results correspond to a
steady 25 Mbar shock in Al. It is evident that an
increasing larger region of uniform Al plasma is
formed apart from minor gradients at the Si-Al
interface and the shock front. The former is due to
the temperature mismatch between Si and Al and
the latter is governed by the electron-ion energy
exchange rate.
3 1C4
1 2104
S>°
1 104
no4
At the frequency v corresponding to the Al*4 Ka
photo-absorption line, the opacity of the Be, CH
and Si layers are relatively small compared with
that of Al. Absorption due to those layers
contributes to a background level only. The
opacity due to the Al layer leads to,
2104
3104
FIGURE 12. Plot of the shock versus particle speed in
aluminum for QEOS (solid line) and Sesame (dotted line).
(3)
where I0 and IT are the source and transmitted
intensities, p the mass density of the shocked state,
(jv the absorption cross-section for shocked
aluminum, d the thickness of the shock compressed
region of aluminum, and Us the shock speed.
Since p(t)d(t)= p0 Us t, Eq. (3) can be written as
Ut}
1 104
2 104
Up (m/8)
0
3104
FIGURE 13. <Z> of shock compressed aluminum as a
function of particle speed for QEOS ((solid line) and Sesame
(dotted line).
(4)
no1
A plot of ln{Ij{f)IIo} versus t is thus a linear graph
whose slope yields a direct measurement of the Ka
photo-absorption cross-section of the Al+4 ions.
This is illustrated in Fig. 11. As expected the
corrections due to the gradients at the Si-Al
interface and the shock front are negligible.
8101
^s 4 101
210 16
IMPACT OF PLASMAS ON SHOCK
PHYSICS
0
From the above discussions, it is clear that shock
waves have and will continue to impact on
advances in the study of high-density plasmas. On
the other hand, it is equally important to recognize
the impact of plasma science on shock physics.
The most direct example is the study of electronion equilibration in a dense, strongly coupled
plasma, which leads to the discovery of nonequilibrium thermal structure in a shock wave.
This is crucial not only for the proper description
of a shock wave but also the interpretation of its
optical properties. In addition, this underscores the
need to understand relaxation processes that may
occur in a shock transition.
1104
2104
,(m/s)
3104
FIGURE 14. Electrical conductivity of shock compressed
aluminum as a function of particle speed for Lee and More
conductivity (solid line) and Rinker conductivity (dotted line).
Equally important is the recognition that the highpressure frontier of shock physics lies in the regime
of strongly coupled plasmas. Here, of fundamental
significance is the understanding of ionization
physics which impacts on every aspect of the
calculation of material properties including
equation of state and transport coefficients as well
as radiative opacities. It is particularly illuminating
to examine results derived from Sesame and
QEOS, two of the most widely used equation of
state models. Fig. 12 shows a plot of shock speed
57
Us versus particle speed UP. It signifies nearly
identical equations of state.
104
103
This would hardly be expected from results of
ionization calculations in each model as shown in
Fig. 13. Substantial divergences are also found in
electrical conductivity based on these ionization
models (Fig. 14) although the relative effects of the
ionization and scattering processes on transport
coefficients are not yet understood.
Similar
divergences would exist in thermal conductivity.
These clearly indicate that understanding shock
waves at very high pressures requires more detailed
knowledge than shock and particle speeds.
102
101
10°
10- *
0.1
1
10
P/P0
FIGURE 15. Warm Dense Matter is the extreme regime of
high pressure science.
CONCLUSIONS
Shock waves have been and will continue to be an
essential means for studying the physics of strongly
coupled plasmas. On the other hand, the physics of
strongly coupled plasmas will be key to the
understanding of shock waves at ultrahigh
pressures. The convergence of these is now
driving advances in the field of Warm Dense
Matter science.
UNIFIED AREA OF WARM DENSE MATTER
The convergence of plasma and shock wave
physics as discussed above has led us to introduce
the notion of Warm Dense Matter as a unified
concept. This was introduced a few years ago to
refer to expanded or compressed states of a solid
with comparable thermal and Fermi energies.
These states are considered to lie in the region 0.1
to 10 times solid density and 1 to 100 eV. The
purpose of introducing this new notion is twofold.
It signifies the convergence of plasma physics and
shock physics (Fig. 1), together with condensed
matter physics and high pressure science (Fig. 15).
Equally important, it points to solid-plasma
transition as a critical missing link in our basic
understanding. This is readily evident by examing
the structure of Sesame, a tour de force in equation
of state. In spite of the use of seven distinct
models, there is a region in the phase diagram that
can only be described by interpolations between
adjacent models. This is the region of Warm
Dense Matter.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
Warm Dense Matter has become an emerging,
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studies of high-density strongly coupled plasmas,
high-temperature condensed matter and high
pressure processes including shocks and chemistry.
It also finds applications in inertial confinement
fusion, planetary physics and astrophysics, as well
as material science at extreme conditions.
9.
10.
11.
12.
13.
14.
58
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