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
THE COUPLING BETWEEN SHOCK WAVES AND CONDENSED MATTER:
CONTINUUM MECHANICS TO QUANTUM MECHANICS
Y.M. Gupta
Department of Physics and Institute for Shock Physics, Washington State University, Pullman, WA. 99164-2814
Abstract. Shock wave compression experiments are well suited to examine condensed matter changes
in real time. A comprehensive understanding of these physical and chemical changes requires timeresolved measurements at a variety of length scales, and theoretical developments to link the
experimental observations in a consistent manner. Two representative examples are presented to
summarize recent progress in coupling continuum and microscopic results.
INTRODUCTION
I am pleased and honored to have been selected
by the shock wave community to receive the 2001
American Physical Society Shock Compression
Science Award. Since the summer of 1970,1 have
had the privilege of working with many
outstanding individuals in this challenging, multidisciplinary field.
My teachers, colleagues,
collaborators, and students have contributed greatly
to my education, enthusiasm, scientific excitement,
and accomplishments.
I owe much to two
institutions, Washington State University and the
Stanford Research Institute, for supporting my
research activities.
For my presentation, I have chosen a theme that
has guided my own research interests and
activities, and that reflects the scientific breadth
inherent in shock wave studies.
Continuum
physics provides the underlying scientific
foundation to analyze and understand large
amplitude or shock wave propagation. Although
the early work (1) focused on gases, many of the
mathematical developments (2,3) are also valid for
large amplitude wave propagation in liquids and
solids. Because of the composition of condensed
matter, complex wave structures are expected in
these materials.
In contrast to wave propagation, fundamental
understanding of condensed matter phenomena
requires scientific developments governed by
quantum physics. Hence, detailed studies of shock
wave induced physical and chemical changes in
condensed matter require that developments in
continuum mechanics and quantum mechanics be
linked in a systematic manner. Scientific issues
related to this linkage are discussed here.
Although historical developments are not the
subject of this paper, two comments are in order.
First, scientific activities related to shock
compression of condensed matter have their origin
in the pioneering developments at Los Alamos in
the U.S., and similar activities in the Soviet Union,
in the years following World War II. (4,5) The
ability to perform well controlled experiments
using high explosives and to obtain high precision
measurements, in these early years, was central to
initiating research into condensed matter response.
Second, the ability to achieve large compression in
shock wave or dynamic high pressure experiments
led to interactions with static high pressure
activities (6) and resulted in a broader interest in
the field.
The purpose of this paper is to point out
scientific issues related to the coupling of shock
waves and condensed matter, and to summarize
representative research activities that highlight this
coupling. None of the material presented in the
specific examples is new and details may be seen
in the references cited in the later sections.
SHOCK WAVE COMPRESSION
Shock waves are supersonic disturbances that
cause near-discontinuous changes in mass velocity,
density and internal energy.
This idealized
definition is valid only if the wave front can
propagate without a change in shape in the material
of interest. Although a very large momentum can
be imparted instantaneously and uniformly to a
large planar surface, the propagation of this jump
will depend on the material response to the initial
loading condition. That is - there is a strong
nonlinear coupling between shock wave
propagation and material response. To discuss
issues related to condensed matter response, it is
useful to consider characteristic features of shock
wave compression:
• pressure range: ~1-104 GPa
• time scales: ~l(T13-10"6s
• directionality of loading
• non-isotropic compression
• large temperature rise (assuming thermal
equilibrium)
Because of these features, shock waves can
produce profound changes in condensed materials
and are well suited to examine matter at extreme
conditions. For condensed matter sciences, two
aspects of shock compression are noteworthy:
macroscopically well defined loading conditions
and the time scales inherent in shock wave studies.
The first aspect is necessary for quantitative
analyses and understanding, and the second is
valuable in probing physical and chemical changes
in real time. As envisioned here, a shock wave is a
mechanical disturbance causing the motion of ions
and nuclei, and the phenomena of interest are
within the framework of the Born-Oppenheimer
approximation.
In discussing condensed materials, issues of
characteristic times associated with a physical or a
chemical change and length scales (or granularity)
inherent to the material need to be considered.
Since length scales were discussed in an earlier
plenary paper, (7) we focus attention on
characteristic time scales.
Shock waves are
characterized by three parameters: amplitude,
duration, and rise time. The latter is determined in
part by the material response. The role of these
parameters in condensed matter changes can be
seen by considering the curve shown in Figure 1.
Shape not important
Stress
or
Pressure
Region of change
th.
Time
Figure 1. Role of stress amplitude and time in time-dependent
condensed matter changes.
All material phenomena, by definition, are timedependent and the curve shown in Figure 1
separates regions of change and no change. Pth and
tth correspond to threshold amplitude and time
values, respectively, for a particular phenomenon
of interest.
In studying the temporal evolution of shock
induced physical and chemical changes, both
amplitude and duration are important parameters.
As shown in Figure 1, different combinations of
amplitude and duration can cause changes in
material response. Furthermore, time-resolution
and duration are important factors in obtaining
credible measurements to examine time-dependent
material phenomena.
As indicated earlier, a comprehensive
understanding of condensed matter phenomena
under shock compression requires a linkage
between continuum mechanics and quantum
mechanics. Typically, this has been achieved by
using ab-initio calculations to generate a complete
equation of state (EOS). The calculated EOS is
combined with the Rankine-Hugoniot jump
conditions to generate the Hugoniot curve which is
then compared to the P-V data generated
experimentally. Thus, starting with a quantum
mechanical description of matter, the continuum
response is obtained.
In our work, we have chosen a somewhat
different approach. Optical spectroscopy was used
to experimentally characterize the shocked state at
the microscopic level, in addition to the continuum
measurements. Quantum mechanical calculations
were carried out to obtain the electronic energies,
relevant to the particular continuum state, in order
to analyze the optical data. This approach is
discussed in the next section
wt. percent or less of chromium) are mechanically
indistinguishable in the elastic range.
Sapphire
Impactor
SELECTED EXAMPLES
As representative examples of the scientific
theme selected for this report, a summary of the
work on R-line shifts of shocked ruby crystals and
the shock wave induced phase change in CdS is
presented here. Each of these studies spanned
several years and involved many talented
colleagues. Although it is not possible to fully
cover these two topics in the space available here, I
hope to convey the essence of the scientific
objectives, issues, and accomplishments. In both
these problems, the combination of continuum and
quantum mechanical developments resulted in a
comprehensive understanding of the experimental
results.
R-lines in Shocked Ruby
This research effort was motivated by two
overall objectives: the development of an optical
stress sensor to permit time-resolved measurements
of large dyanmic stresses; and to use changes in Rline spectra as a microscopic probe of the shocked
crystal lattice. At the start of this work in 1984, it
was not clear that time-resolved measurements of
R-line spectra were even feasible under shock
wave loading. Significant contributions to the
work described here were made by Drs. P. D.
Horn, S. M. Sharma, X. A. Shen, and J. A. Hurt.
The collaboration with Dr. Sharma regarding the
theoretical developments deserves a special
acknowledgment.
Ruby is a sapphire crystal in which some of the
aluminum ions are replaced by chromium ions; a
detailed account of the ruby optical spectra can be
seen in the book by Sugano et al. (8) Since the
pioneering work of Piermarini, (9) the ruby R-lines
are commonly used for pressure calibration in
diamond-anvil-cell (DAC) experiments. To the
best of our knowledge, sapphire and ruby (with 0.1
Figure 2. Experimental configuration to obtain time-resolved
shifts of ruby R-lines under shock loading.
Plate impact experiments were conducted to
subject ruby crystals to uniaxial strain along the
crystal c- and a- axis. (10-12) Figure 2 shows the
experimental configuration used to obtain R-line
spectra in shock wave experiments. 514 nm light
from an Argon ion laser was used to excite the
luminescence R-lines which were dispersed and
recorded using a spectrometer, a streak camera, and
a 2-D detector (OMA/vidicon or a CCD). The final
recorded output consisted of intensity vs. time vs.
wavelength. Spectral and temporal resolutions
were 0.5 A and nanoseconds, respectively. By
varying the flyer and sample thicknesses,
compression or fUll release or tension states could
be achieved in the ruby samples. Representative
data from a tension experiment (12) are shown in
Figure 3. Ambient R-line spectra undergo a red
shift upon the arrival of the compression wave,
followed by a shift back to the ambient value upon
unloading, and the subsequent tensile deformation
causes a blue shift.
6900
6940
Wavelength (A)
Figure 3. Time-resolved shifts of R-lines under compression
and tension.
Before discussing the optical spectra, it is
important to consider the continuum elastic
response of sapphire (or ruby) shocked along c- and
a-axis. Determination of stress - particle velocity
relations (11) for the two orientations, using
second and third order elastic constants of sapphire,
demonstrate that the two orientations are
indistinguishable at the continuum level (the
differences are within 0.5 percent). Shock wave
experiments (13,14) have verified the calculated
elastic response for the two orientations.
Quadratic fit
Hydrostat
a-axis
c-axis
2»
!> o
CD
IS -10
co
5-20
R, Line
-30
-
2
-
1
0
1
2
"tot
Density Compression(%)
•c-axis
R2 Line
3
-
2
-
1
0
1
=
"ambient ~^~"strain
=
-Quadratic fit
-Hydrostat
• a-axis
-
and R2 data show dependence on crystal orientation
and the stress state. An important aspect of the
optical data is the RrR2 separation, discussed later
in this section.
Since the R-line shifts are a consequence of the
changes in the energy levels of the d-electrons in
the chromium ions, (8) the combination of the
continuum and optical results leads to the following
overall conclusion: although the continuum
response of ruby crystals shocked along the c- and
a- axis is nearly identical, the local or microscopic
environment around the chromium ions is very
different. Furthermore, the optical spectra or the
local environment around the chromium ions
cannot be understood in terms of a continuum
description, and quantum mechanical calculations
are required.
The theoretical developments (15) to model the
optical spectra under shock compression consisted
of first writing down the Hamiltonian as:
2
3
Density Compression(%)
Figure 4. RI and RI line shifts versus density compression for
different orientations and loading conditions.
Results from a comprehensive set of R-line shift
measurements are summarized in Figure 4.
Compression and tension data for both RI and R2
lines are shown for crystals shocked along c- and aaxis. Since hydrostatic data are not available in
tension, the compression data were extrapolated
into tension. The optical data in Figure 4 show
many interesting features: (i) In compression, R{
line shifts vary with crystal orientation and depend
on the nature of the stress state, (ii) In contrast to RI
line shifts, the R2 line shifts in compression do not
depend either on the crystal orientation or the
nature of the stress state, (iii) In tension, both R{
"oct ~^~ "trig ~^~ "so ~^~ "strain
The first two terms on the right side refer to the
local site symmetry, the next term refers to spinorbit coupling, and the last term is a consequence of
the imposed deformation. In solving this problem,
we assumed (15) the following: Hso does not
change from the ambient value, only the coupling
of 2E level to the 2T2 level was considered, and
Hstrain has the same symmetry as the external
deformation.
Using these assumptions, we combined the
continuum or stress-strain response of the material,
group theoretical considerations, and quantum
mechanical calculations to obtain the eigenvalues
for H totai. Details may be seen in Ref. 15. In our
approach, only the shock wave compression data
were used to obtain the model parameters. All
subsequent experimental data (uniaxial strain
tension along c- and a-axis, hydrostatic
compression, and uniaxial stress compression along
c- and a-axis) were modeled without any iteration
of the parameters.
Experimental measurements of RrR2 separation
provide a stringent check of the theoretical model.
Figure 5 shows a plot of the RrR2 splitting as a
function of density change for both compression
and tension. Within experimental error, excellent
agreement can be seen between theory and
experiment.
The nonlinear increase in both
compression and tension for uniaxial strain loading
along the a-axis is a consequence of the site
symmetry change due to deformation.
20
U)
10
'5
Density Change (%)
Figure 5. Ri-Rz separation as a function of density change for
crystals shocked along the c- and a-axis.
The success of the ruby work showed the value of
combining continuum and quantum mechanical
concepts. In relating changes in ruby luminescence
spectra to mechanical deformation, several unique
attributes of shock wave loading were particularly
valuable: well defined uniaxial strain along
different crystal axes; non-hydrostatic but uniform
loading of the samples; and the ability to produce
large tensile stresses (-10 GPa) in the sample.
Phase Change in Shocked CdS
Shock wave induced polymorphic phase
transformations in solids represent a topic of long
standing scientific interest. (16,17) Typically, the
measured transformation stresses under shock
loading are comparable to static transformation
pressures. Shock wave studies have the added
benefit that structural rearrangements can be
examined in real time. The challenge is to combine
the time-resolved nature of shock wave
measurements with the ability to probe material
response at different length scales, as is commonly
done in static pressure studies. (18)
The
comprehensive effort on CdS summarized here
demonstrates both the complexity and the
usefulness of shock wave studies in understanding
the dynamics of solid state changes under
compression. Drs. M. D. Knudson, Z. P. Tang, and
S. M. Sharma contributed significantly to the work
described here. The excellent experimental and
theoretical work carried out by Marcus Knudson for
his
Ph.D
thesis
deserves
a
special
acknowledgement. Also, I want to acknowledge
the late Professor A. B. Kunz for his role in the
theoretical effort.
Professor G. E. Duvall
introduced me to the study of shocked CdS.
In 1966, Kennedy and Benedick (19) reported a
two-wave structure in shocked CdS single crystals,
which they attributed to the wurtzite-to-rocksalt
transition (20) observed in static pressure studies.
Because of the expected large HEL of CdS single
crystals, Duvall suggested (21) that the two-wave
structure could not be assigned unambiguously to
the polymorphic phase change.
He further
suggested that, using a method developed by
Dremin, (22) I examine the response of CdS
powders in a PMMA matrix.
These initial
experiments conducted in 1970 were not successful
for a variety of reasons. Many years later, this
approach was revisited using an elastomer matrix
and embedded particle velocity gauges. (23) The
composite results, including in-situ wave profiles,
were modeled well using a time-dependent
continuum description (23) for the phase change in
CdS.
To get detailed insight into the shock induced
phase change in CdS and associated kinetics, a
comprehensive study was undertaken in which the
continuum response of CdS, shocked along c- and
a-axis, was examined in detail. (24,25) Typical
stress-histories at the impact surface and in the
sample interior (24) are shown in Figure 6. Timedependent features and the wave structure
associated with the phase change can be observed.
60
- If '
Shot 8751 4
"^v^g
40 -
20
Impact
Surface
Profile
XUi —
Ua
2
,?p
l\
1
1
1
1
V
>''\
~7 ' -
av ,-''
—•»-—^"*
_4
i Transmitted
j
Profile
0.0
0.2
0.4
0.6
TIME (|is)
0.8
1.0
Figure 6. Stress histories at the impact surface and in the sample
interior in CdS shocked along the c-axis.
Through a judicious combination of shock wave
experiments and analyses for crystals shocked
along the c- and a-axis, (24,25) several important
results were obtained: the onset of transition for
shock compression along the c-axis, unlike
compression along the a-axis, occurred directly
from an elastically compressed state; with
appropriate corrections for strength in the shock
data, good agreement was obtained between the
final states in the shock and static experiments; and
the transition was observed to occur in two steps.
Although suggestions were made about the
atomic mechanisms (25) governing this transition,
continuum results are insufficient to determine
these mechanisms. Static pressure studies, though
providing crystal structures at various P-V states,
cannot provide results regarding transition kinetics
and real time structural rearrangements associated
with the transition.
Ni:Y/G Pimped
Dye Fluorescence
j^p Buffers
L___
j
Projectile
PicoSpectra second MP CCD
meter Steak
Camera
Sample
nU'
Photo
Diode
Figure 7. Experimental configuration to obtain time-resolved
picosecond absorption and transmission measurements.
To gain insight into atomistic mechanisms,
experimental methods were developed (26,27) to
obtain optical absorption data, with -100 ps
resolution, because these data can better address
mechanistic questions (of course, x-ray diffraction
data would be optimal for this objective). Figure 7
shows the overall experimental configuration.
Broad band visible light is transmitted through the
sample as the shock wave propagates into the
sample.
Similar to the ruby measurements
described earlier, the combination of a
spectrometer, a streak camera, and a CCD detector
provides intensity vs. wavelength vs. time results.
A typical record is shown in Figure 8.
Both the wavelength and time-dependent features
associated with the optical absorption data are
observed in the figure.
Figure 8. Absorbence results from CdS shocked to 4.6 GPa
along the a-axis.
The type of data shown in Figure 8 served as the
starting point for a comprehensive analysis
designed to examine and understand optical
absorption measurements in terms of structural
changes. The work by Knudson (28) examined
effects of crystal orientation and peak stress on
time-dependent changes in shocked CdS. Some of
the key experimental results are as follows: at
higher stresses (the particular stress value depends
on the crystal orientation), the absorption
coefficient was wavelength dependent but timeindependent since the transformation reached a
nearly constant state within the -100 ps resolution
time; at lower stresses, the kinetics are quite
different and a steady state is not reached until a
few to tens of ns; the higher stress data show that
the observed absorption changes are caused by a
change from a direct band gap material to an
indirect band gap material with a gap energy of
1.51 ± 0.02 eV in the shocked state.
It is important to emphasize two issues related to
the optical results: analysis of the optical data in
terms of structural changes requires electronic
structure and total energy calculations for various
postulated structures; optical data have to be
analyzed in conjunction with the continuum data to
ensure that both sets of results are interpreted in a
consistent manner. The latter issue is discussed
first.
The indirect band gap material, with a gap
energy of 1.51 eV, is consistent with the expected
rock salt structure (29) for the final shocked state.
However, time-resolved continuum measurements,
(24) despite their lower temporal resolution,
showed a two-step process leading to the final state:
a very rapid first step (faster than the 10 ns
resolution) followed by a slower (100-200 ns)
relaxation to the final rocksalt phase. Thus, the
continuum data would imply that the material state
observed directly behind the shock front in the
optical measurements corresponds to a metastable
structure of CdS during its transformation from the
wurtzite to the rocksalt structure.
This
interpretation would also address the difficulty of
observing a 20 percent volume change associated
with this transition in less than 100 ps.
Ab-initio quantum mechanical calculations can
be used to link the optical or electronic changes to
structural changes in compressed crystals.
Theoretical studies (30) were carried out to
examine the suggestion by Sharma and Gupta (25)
that lattice structures closely related to the wurtzite
structure (through a combination of shifting in the
alternately stacked basal planes and deformation
within the basal plane) be considered as potential
metastable structures. In the ab-initio calculations,
two slightly different structures were considered:
face-centered orthorhombic (fco) and face-centered
tetragonal (fct).
0.5
Wurtzite Phase
Rock Salt Phase
Intermediate Lattice Structure
Between Wurtzite and fct
0.4 -
V
?
0.3 -
i
J
\
Metastable fct Structure
Relaxation of fct Phase
Final Rock Salt Structure
0.2 -
1
CONCLUDING REMARKS
0.1 -
0.0
0.65
energy and electronic structure calculations was
carried out (30) to examine various crystal
structures. It was determined that the most likely
candidate for the metastable structure was the fct
structure and not the fco structure as proposed in
Ref. 25. Figure 9 shows a plot of the total energy
as a function of volume compression for the initial,
metastable and final structures for shocked CdS.
On the basis of time-resolved continuum and
optical measurements in CdS crystals shocked
along c- and a-axis, and ab-initio calculations, a
strong case can be made for the following two-step
atomic mechanism: a very rapid (less than 100 ps)
transformation to a metastable face-centeredtetragonal (fct) structure consisting of an out of
phase shifting of alternately stacked basal planes of
the wurtzite structure, and a two-dimensional
deformation within the basal plane; a slower (tens
of ns) transformation from the metastable fct
structure to the final rocksalt structure, consisting
of a collapse along the crystal c-axis and slight
separation of the cadmium and sulfur sublattices.
The discussion so far has focused on the higher
stress results.
At lower stresses (below
approximately 6.5 GPa and 5 GPa along the a- and
c-axis, respectively), the optical data showed timedependent behavior. Using a nucleation and
growth model, (31) the optical data were simulated
well by considering both absorption and scattering
from the new phase. Details of this work can be
seen in Ref. 32.
A comprehensive understanding of the shock
wave induced phase change in CdS was obtained
by using continuum mechanics and quantum
mechanics concepts to analyze experimental data at
different length scales. This study also pointed out
the importance of obtaining results over a broad
range of time scales since nucleation and growth of
the new phase span a range of time scales.
0.70 0.75 0.80 0.85 0.90
0.95 1.00
V/V0
Figure 9. Calculated total energy as a function of volume
compression for various crystal structures.
A comprehensive and systematic set of total
Shock wave studies are well suited to examine
condensed matter response at extreme conditions
and to probe the underlying physical and chemical
changes in real time. Two representative examples
were summarized to show how experimental
findings at different length scales may be linked to
develop a comprehensive insight into the
underlying processes. The work described here
7.
represents a start and a great deal needs to be done.
An important challenge in the coming years is to
ensure a consistent comparison between
theoretical/numerical calculations and experimental
measurements at comparable length and time
scales. This will require continued advances in
experimental and computational capabilities.
Another important need is to examine and
understand the role of heterogeneities, of varying
sizes, on shock wave induced physical and
chemical changes.
8.
9.
10.
11.
12.
ACKNOWLEDGEMENTS
13.
More than one hundred talented individuals have
contributed immensely to the research activities
that I have been engaged in since 1970. I owe a
special gratitude to my graduate students and
postdocs who have worked hard to combine rigor
and excellence in their studies. In the process, they
have educated me. Support from many federal
agencies (particularly, ONR, DOE, and DNA)
permitted me to pursue the scientific theme
presented here. Finally, I owe a great deal to two
very special individuals. George Duvall introduced
me to this field and was always available to discuss
various issues. The support and encouragement of
my wife, Barbara, has been invaluable to me in my
research work.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
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4.
5.
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