SUPPORTING MATERIAL
I. Determination of reflectivity and temperature
Absolute reflectivity and temperature measurements are usually challenging because they
are based on absolute intensity determinations. In the precompressed targets the throughput from
sample to detector along the optical path is usually poorly known for a variety of reasons. The
most severe problem is that the optical properties of the window (diamond or sapphire anvils) can
be modified during the experiment due to photo-excitation by low-level x-radiation associated
with the laser drive. In the present study, the reflectivity at the shock front and the value of the
thermal emission at 650 nm in shocked He were determined by comparing them against the
reflectivity and emission of the shock front in a quartz reference sample that was placed in situ
with the He sample. For simultaneous temperature and reflectivity measurements we require
both a temperature reference (a surface of known temperature) and a reflectivity reference (a
surface of known reflectivity). This is achieved by using the shock front in quartz for these two
references. The determination of reflectivity and temperature in the sample can then be made
without knowing the throughput of the optical system.
(a) Quartz calibrations
The essential input for doing such a relative determination is a careful calibration of
reflectivity and temperature of shock-compressed quartz along its Hugoniot as a function of
shock velocity.
This has recently been performed[1], yielding fits of the temperature and
reflectivity versus shock velocity for quartz. The reflectivity of quartz at 532 nm was measured
by comparing the intensity of the VISAR signal from the shock front in quartz against the preshock signal from the known aluminum reference. The temperature has also been measured from
the spectral radiance over a 100 nm band centered at 650 nm and in some cases over an
P.M. Celliers et al.
additional band centered at 450 nm. In order to determine the temperature, the throughput of the
optical path together with the sensitivity of the streaked optical detector at both wavelengths were
absolutely calibrated with a NIST-traceable tungsten filament lamp[2]. The quartz calibrations
are plotted as a function of the shock velocities in the two figures below. The fits of these two
data sets, RQ(Us) and TQ(Us) are the input for the present relative reflectivity and temperature
measurements in shocked helium using shocked quartz as a reference.
The width of a strong shock in dense fluids, as here in He, should be of the order of 1-2
nm, and the absorption depth of He under the ionization condition of the present data is estimated
to be greater than 0.1 µm (estimated with the Drude model used below). Hence the shock front
can be considered as a specular reflector amenable to Fresnel analysis, while the temperature
measured is that of shocked He at the thermodynamic conditions just behind the shock front.
Figure 1. Reflectivity and temperature of shock-compressed quartz along the principal Hugoniot
as a function shock velocity. The fit of the data is given in the figure.
(b) Reflectivity determination
The VISAR measures the shock velocity from a fringe pattern shift. It also yields the
reflectivity at the wavelength of the probe laser from the intensity of the fringe pattern. The
reflectivity measurement at the shock front in the sample in contact with quartz can be obtained
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P.M. Celliers et al.
easily from the ratio of intensities of the VISAR fringe pattern in quartz and in the sample. In
both cases these reflectivities were derived from the Doppler-shifted fringe intensities generated
from reflecting shock fronts in these materials, after correcting for ghost reflections (described
below). When the shock front is in the quartz, the measured intensity of the VISAR fringe
pattern IQ is given by:
IQ  IP RQ (Us)(1 RQS ) f T
where IP is the incident probe laser intensity at the shock front, RQ(Us) is the calibrated
 reflectivity of the shock front in quartz, R
QS
is the reflectivity at the quartz/sample interface and fT
is the unknown transfer function of the optical system. When the shock is in the sample, the
intensity of the VISAR fringe pattern is given by the formula:
IS  IP RS fT
where RS is the reflectivity of the sample shock front. Combining these two equations gives

R S  RQ (U S )IS /IQ
The correction of the reflected fraction of the laser at the quartz / sample interface (less than 2%)
 is neglected.
(c) Temperature determination
The streaked optical pyrometer (SOP) images the thermal emission of the propagating
shock front over a small spectral range, first in the quartz and then in the He sample. To
determine the temperature from the measured thermal emission, we assume a grey-body
approximation for the spectral radiance,
I  A( )Exp(hc / kT) 1
1

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P.M. Celliers et al.
where (  ) is the emissivity given by (1-R), R the measured optical reflectivity, A is a system
calibration constant that incorporates the transfer function of the optical system and the response
 of the detector. Inverting this expression to solve for temperature gives,
T  T0 ln( ( )A /I  1) ,
1
where T0 = hc/0kT is related to the wavelength of the passband of the pyrometer (T0 = 1.9 eV at

0 = 650 nm). Since the temperature determination is made relative to the quartz reference, the
temperature in the sample can be determined from the ratio of the signal levels observed in the
quartz and the sample, such that the system calibration constant A drops out of the expression,
TS  T0 ln(exp( T0 /TQ ) 1)(IQ* /IS* )  1
1
with I *  IADU /(1 R) , IADU being the analog-to-digital counts associated with the observed

signal, and R being the reflectivity measured with the VISAR. Knowing the shock velocity in
 quartz, the temperature in quartz, T , is known from the calibrated function T (U ).
Q
s
Q
It should be noted that in pre-compressed targets, the quartz is shocked along a Hugoniot

slightly offset from the principal
Hugoniot. Hence, a correction is added to the TQ and RQ values
associated with the measured Us through the calibrated input, TQ(Us) and RQ(Us).
The
temperature correction has been estimated by using the 7385 SESAME equation of state of SiO2
or by applying the Grüneisen model with a constant value of the Grüneisen parameter, =0.66;
both estimates agree very well. The reflectivity of shocked quartz in the dense fluid state is
primarily temperature dependent, as shown by comparing the reflectivity of quartz and fused
silica. Hence, the evolution of the reflectivity of quartz versus the temperature along the
Hugoniot to a good approximation follows a unique function, RQ(T), independent of the initial
value of the density. Applying the corrected temperature to RQ(T) produces a corrected value for
RQ. In the domain of pre-compression of the present experiment (< 1.25 GPa), these corrections
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P.M. Celliers et al.
amount to adjustments of less than 4% of the temperature values and to less than 15% of the
reflectivity values; both corrections are much smaller than the experimental error bars.
Uncertainties of the fitted reflectivity and temperature are estimated from the confidence
bands of the fits as calculated from the covariance matrix of the fitting parameters at the 1 level.
The fitting uncertainty is propagated in quadrature with the random errors.
(d) Corrections for ghost reflections within the sample cell
The sequence of optical interfaces inside the target tends to generate spurious reflections
from non-moving interfaces, for example from the quartz-He interface prior to shock transit
across that interface, from the He sapphire interface and from the sapphire-vacuum interface. All
of these interfaces are coated with anti-reflection coatings prior to assembly of the samples;
however, such coatings are imperfect and a residual level of ghost reflections is still produced in
the experiments. When these ghost reflections are of amplitude comparable to the Dopplershifted fringes it is possible to incur errors in the velocity and reflectivity determination if a
correction is not applied. We give here the method we have used to correct the data in the
presence of ghost reflections.
We assume that the observed fringe intensity is formed from the sum of a signal, S, and a
ghost, G:
S  Is (1 C cos[kx   s ])
G  Ig (1 C cos[kx   g ]),
where x is the spatial coordinate, k is the fringe frequency, 0 ≤ C ≤ 1 is the intrinsic visibility of
the fringes produced by the interferometer, Is and Ig are the average intensities of the signal and
ghost respectively, and s and g are the phase shifts associated with the signal and ghost
respectively. The observed fringe intensity, O, is the sum of these underlying signals:
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P.M. Celliers et al.
O  S  G  Io (1 Co cos[kx  o ]) .
where Io, Co and o are the observed average intensity, fringe visibility, and phase, respectively.
 The goal of the ghost correction is to extract the underlying values of Is, Ig, and s from the
observed intensity Io = Ig + Is, fringe visibility, Co and fringe phase, o. Through algebraic
manipulation and application of trigonometric identities it is possible to show that Co and o are
related to the underlying parameters through the following expressions:


Co  C
Ig  Is





I
cos[

]

I
cos[

]
s
s

 o  cos1 2 g 2 g

 Ig  Is  2Ig Is cos[ g   s ] 


Ig2  Is2  2Ig Is cos[ g   s ]
From these expressions it is evident that the observed fringe visibility is always less that the
 intrinsic visibility of the VISAR, 0 ≤ Co ≤ C. At this point we make two parameter substitutions,
Co /C  R and Ig /Is  a , where 0 ≤ R ≤ 1 is the ratio of observed fringe contrast to the intrinsic
contrast, and a is the ratio of ghost to signal intensities. In addition we set  g  0 ; this last step


corresponds to the fact that the ghost reflections originate from static (non-moving) interfaces
whose contributions to the data have fixed phase equal to the 
initial reference phase of the
interferogram before the experiment starts. Solving for a and s in terms of the observables R
and o one finds,
a
1 R 2
,
1 R 2  2Rcos[ o ]
1 2Rcos[ 0 ]  R 2 cos[2 0 ]
.
1 R 2  2Rcos[ o ]


 s  cos1
We resolve the sign of s by creating a quadrature construction as follows: define ow  W[o ],

where W is the wrapping function, which wraps the observed (uncorrected) phase into [-, ].

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The quadrature components of s can then be assigned by:
cos[ s ]  cos[  s ]

 sin[  s ], if  ow  0 .
sin[  s ]  

sin[  s ], if  ow  0
From this construction the signal phase s can be extracted by taking the arctangent and

unwrapping. The ghost and signal contributions to the reflectivity are determined from a,
Is  Io /(1 a)
.
Ig  aIo /(1 a)

II. Drude-semiconductor model
The Drude-semiconductor model we use has been applied successfully to estimate the
Fresnel reflectivity of shock fronts in several other shock compressed molecular systems, such as
Al2O3, LiF[3] and H2O[4].
For the shock front in the pre-compressed sample the Fresnel
reflectivity is given by R = |(n-n0)/(n+n0)|2, where n0 is the index of the pre-compressed sample in
its initial state. The Drude model for the conductivity, σ(ω)=(nee2τ/2meff)(1-iωτ)-1 is used to
determine the complex index of refraction n = [b+iσ(ω)/ω]1/2, where b is the contribution of the
bound electrons, estimated as b  n 0 from experimental refractive index data[5]. Combining
these expressions the reflectivity is straightforwardly expressed as a function of the electron
relaxation time, τ, andthe electron concentration, ne; here meff is the electron effective mass. For
disordered materials undergoing insulating-conducting transitions the electron relaxation time τ is
generally found to be close to the Ioffe-Regel limit[6] at which the scattering length is equal to
the interatomic spacing R0; therefore, in our fit we assume τ = R0/Ve where Ve the electron
velocity. The temperature activation is described by the expression for the carrier concentration
of
an
intrinsic
semiconductor[7],
ne  2(meff kT /2 2 )3 / 2 f1/ 2 [E g /(2kT)] ,

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where
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P.M. Celliers et al.
f m [x]  2/ 


0
y m /(1 e yx )dy is the Fermi-Dirac integral and Eg is the gap energy. The
corresponding carrier velocity is given by Ve  (2kT/meff )( f 3 / 2[E g /(2kT)]/ f1/ 2[E g /(2kT)]) .

Since meff is unknown we introduce an additional parameter,  = meff/me where me is the electron

vacuum mass, and allow the fit to determine typically  ~ 1). The resulting reflectivity model
is a function of three other observables: n0 (which depends on the initial density), final density
and final temperature, i.e. R = RM(n0, , T). The data is collected over a range of densities and
temperatures, and we find that the best fit is obtained when the gap energy Eg is assumed to vary
linearly with density and/or temperature.
This semiconducting Drude model with density
dependent gap has been applied successfully by us to fit optical data of shocked water[4], Al2O3
and LiF[3], and by others in the case of compressed Xe[8] and shock-compressed H2[9]. In the
case of shock reflectivity in LiF[3] a subsequent independent theoretical investigation[10] found
that this model provided a good representation of the density functional calculations. In fitting
the data to the model the fit parameters include  and the parameters describing Eg as a function
of density and temperature. Since significant experimental uncertainties are associated with all of
the variables we fit the model to the data set using orthogonal distance regression[11].
III. Impedance matching and quartz equation of state
The densities and pressures we used to analyze the current data set were reported
originally in Eggert et al.[12]; those parameters were inferred through an impedance matching
construction that depends on accurate knowledge of the principal shock Hugoniot of quartz.
Following submission of this work to the journal an extensive and detailed study of the alphaquartz principal Hugoniot was published by Knudson and Desjarlais[13] with improved accuracy
over the data set that had existed up to that point. An improved fit to quartz Hugoniot including
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P.M. Celliers et al.
the new data leads to a slightly non-linear fit that is also shifted to a slightly stiffer (lower
density) path: the new fit can be regarded as a recalibration of the EOS standard used for the
measurements reported in Eggert et al. While Knudson and Desjarlais reported that the density
shift for He could be as large as 23% this is a worst case example; our own preliminary estimates
using this new calibration suggest that the inferred He shock densities will be shifted lower by an
average of 10%, implying a similar shift of the energy gap function extracted from our fit to the
reflectivity data.
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