Tantalum sound velocity under shock compression

J. Appl. Phys. 

125

, 145903 (2019); 

https://doi.org/10.1063/1.5054332

125

, 145903

© 2019 Author(s).

Tantalum sound velocity under shock

compression

Cite as: J. Appl. Phys. 

125

, 145903 (2019); 

https://doi.org/10.1063/1.5054332

Submitted: 30 August 2018 . Accepted: 21 March 2019 . Published Online: 11 April 2019

Minta C. Akin

, Jeffrey H. Nguyen

, Martha A. Beckwith

, Ricky Chau

, W. Patrick Ambrose

, Oleg V.

Fat’yanov

, Paul D. Asimow

, and 

Neil C. Holmes

ARTICLES YOU MAY BE INTERESTED IN

Application of Al-Cu-W-Ta graded density impactors in dynamic ramp compression

experiments

Journal of Applied Physics 

125

, 145902 (2019); 

https://doi.org/10.1063/1.5055398

Inferring the high-pressure strength of copper by measurement of longitudinal sound speed

in a symmetric impact and release experiment

Journal of Applied Physics 

125

, 145901 (2019); 

https://doi.org/10.1063/1.5068730

Pressure-induced reentrant structural transition and equation of state of indium

Journal of Applied Physics 

125

, 075901 (2019); 

https://doi.org/10.1063/1.5086318

Tantalum sound velocity under shock

compression

Cite as: J. Appl. Phys.

125

, 145903 (2019);

doi: 10.1063/1.5054332

View Online

Export Citation

CrossMar

Submitted: 30 August 2018 · Accepted: 21 March 2019 ·

Published Online: 11 April 2019

Minta C. Akin,

Jeffrey H. Nguyen,

Martha A. Beckwith,

Ricky Chau,

W. Patrick Ambrose,

Oleg V. Fat

’

yanov,

Paul D. Asimow,

and Neil C. Holmes

AFFILIATIONS

Lawrence Livermore National Laboratory, Livermore, California 94550, USA

California Institute of Technology, Pasadena, California 91125, USA

akin1@llnl.gov

Present address:

Structural Biology Initiative, CUNY Advanced Science Research Center, New York, New York 10031, USA.

ABSTRACT

We used several variations of the shock compression method to measure the longitudinal sound velocity of shocked tantalum over the

pressure range 37

–

363 GPa with a typical uncertainty of 1.0%. These data are consistent with Ta remaining in the bcc phase along the prin-

cipal Hugoniot from ambient pressure to

300 GPa, at which pressure melting occurs. These data also do not support the putative melting

phenomena reported below 100 GPa in some static compression experiments.

Published under license by AIP Publishing.

https://doi.org/10.1063/1.5054332

I. INTRODUCTION

Tantalum, for reasons of its simple and stable crystalline struc-

ture, high density, and chemical stability, is one of the candidates

for a high pressure standard. There are nevertheless, disagreements

on its physical properties such as high pressure sound velocity,

the melt line, existence of the

-phase,

and low pressure phase

transitions. Many of these questions can be directly or indirectly

addressed by high-pressure measurements of the longitudinal

sound velocity on the principal Hugoniot.

The bcc transition metals, including Ta, Mo, W, and V, have

been observed to have

“

”

(that is, nearly isothermal above a certain

pressure) melting curves in diamond anvil cell (DAC) studies.

These studies include data collected from X-ray di

raction (XRD).

These results are controversial and are inconsistent with melting

curves that one would expect using the Lindemann melt criterion.

Other DAC experiments using similar techniques

give quite di

erent

results and appear consistent with Lindemann-like behavior.

Dynamic experiments such as shock compression in two-stage

light gas guns can produce stress

–

density data at better than 0.5%

uncertainty.

For this reason, Hugoniot data for tantalum and plati-

num

have been used to derive pressure standards for

100 GPa

experiments for the past few decades. For use as a pressure stan-

dard, though, it is essential to know the phase to which the

obtained equation of state data pertains. Unfortunately, dynamic

phase transitions such as melting in metals, including Fe, Mo, Sn,

and Ta, are di

ffi

cult to discern through changes in slope or discon-

tinuities in either stress

–

density (P

–

) or shock velocity

–

particle

velocity (

–

) along the Hugoniot.

–

Certainly, no such changes

are resolved in the currently available Ta

–

data, which are

remarkably linear.

A closer examination of

vs time

may

yield subtle signs of phase transitions.

Sound speed experiments can yield a much clearer signature

of a phase transition, especially for melting. The discontinuous

changes in longitudinal sound speed (

) in such experiments can

occur either because of shear-softening solid

–

solid transitions or

because the shear modulus

approaches zero upon melting.

Structurally determined adiabatic bulk (

) and shear moduli like-

wise manifest a discontinuity in sound speed in a solid

–

solid phase

transition, as the sound speed is related to these moduli through

(1)

where

is the density.

Many sound speed experiments, examin-

ing a variety of materials, have been carried out on the Hugoniot

for this purpose.

–

However, there have been discrepancies

among Hugoniot sound speed studies of iron and molybdenum

concerning the existence of high pressure phases and the quanti

cation of melting pressures,

–

some of which may be attributable

Journal of

Applied Physics

ARTICLE

scitation.org/journal/jap

J. Appl. Phys.

125,

145903 (2019); doi: 10.1063/1.5054332

125,

145903-1

Published under license by AIP Publishing.

to the use of Ta as a reference material in some studies. To recon-

cile both the Mo phase transition issue and the discrepancies

among reported melting pressures of Ta, a new high-precision

campaign of Ta sound speed measurements covering the span of

reported melting pressure is needed.

At lower pressures on the Hugoniot, well before any claimed

melting points, there are discrepancies in the existence of a phase

transition near 60 GPa between the

ndings of Hu

et al.,

and Xi

et al.

To complicate the picture even further, Rigg

et al.

demonstrate that an apparent transition at

51 GPa, the elastic

–

plastic overtake pressure of Ta, can be made to appear through a

choice of the analytical method and selection of the correction factor

for the shocked window.

These studies

–

rely heavily on the

accurate measurement of particle velocity in front-surface overtake

(FSO) experiments. A more accurate sound velocity measurement

method and analysis is needed to resolve these di

erences.

II. METHODS

A. Experimental methods

The experiments described in this study rely upon the

rear-surface overtake (RSO) method, which has been described

extensively elsewhere.

–

Our focus here is to get the most accu-

rate sound velocity data with minimum uncertainty. This requires a

very precise measurement of the sample and impactor thicknesses.

Flatness and parallelism on the surfaces are kept to

m (see

Table I

). In our experiments, 0.8-mm thick Ta impactors were

launched into Ta targets using the propellant and light gas guns at

velocities ranging from 1.1 to 6.03 km/s. We used the same target

design and analysis as described in Nguyen

et al.

We used sym-

metric impact in all experiments, where both the target and the

impactor are made of the same material, to minimize the overall

experimental errors and dependence on properties of materials

other than Ta. Ta samples and impactors were 99.98% pure poly-

crystalline materials and were sourced from legacy LLNL material

and ESPI Metals, with individual sample densities ranging from

16.6 to 16.7 g/cm

. Impactor velocities were determined in each

experiment through dual x-ray

ash imaging and redundantly by

the laser beam breaks on the propellant gun and by magnetic loops

on the light gas gun. The Ta targets incorporated six di

erent step

heights, allowing the overtake distance to be more carefully deter-

mined. Step thicknesses were created by machining countersunk

“

pockets

”

into a single uniform Ta plate (

Fig. 1

). Two-dimensional

simulations were done

a priori

to maximize the number of pockets

and to avoid pocket to pocket wave interactions that might increase

error in the measurements.

TABLE I.

Data from this study. Uncertainty in the measurement of

fl

yer velocity is 0.005 km/s; uncertainty in

fl

yer thickness is 2

m. Calculated shock pressures and densities

use an initial Ta density of 16.67 g/cm

Shot

flyer

(km/s)

Flyer thickness

(mm)

Flyer density

(g/cm

)

(km/s)

Pressure

(GPa)

Density (

ρ

)

(g/cm

)

Catchup

distance (mm)

(km/s)

1082 1.100

0.785

16.69

4.01(0.02) 0.550(0.003) 36.8(0.3) 19.32(0.03)

5.38(0.13)

4.64(0.04)

1085 1.460

0.757

16.64

4.25(0.02) 0.730(0.004) 51.7(0.4) 20.13(0.03)

4.84(0.18)

4.82(0.06)

1081 1.490

0.800

16.68

4.27(0.02) 0.745(0.004) 53.0(0.4) 20.20(0.03)

5.2(1.5)

4.8(0.4)

1083 1.540

0.782

16.67

4.30(0.02) 0.770(0.004) 55.2(0.4) 20.31(0.03)

5.09(0.12)

4.81(0.04)

1094 1.900

0.780

16.61

4.535(0.017) 0.950(0.004) 71.8(0.4) 21.09(0.03)

4.72(0.15)

5.00(0.06)

1092 2.180

0.782

16.64

4.718(0.015) 1.090(0.004) 85.7(0.4) 21.68(0.03)

4.22(0.07)

5.28(0.04)

1086 2.600

0.770

16.64

4.992(0.014) 1.300(0.004) 108.2(0.4) 22.54(0.03)

3.81(0.12)

5.56(0.08)

446

3.533

0.800

16.69

5.60(0.011) 1.766(0.004) 165.0(0.5) 24.35(0.03)

3.37(0.06)

6.22(0.06)

445

3.870

0.800

16.67

5.822(0.011) 1.935(0.004) 187.8(0.5) 24.97(0.03)

3.15(0.05)

6.53(0.06)

444

4.449

0.800

16.68

6.200(0.010) 2.224(0.004) 229.9(0.5) 26.00(0.03)

3.07(0.04)

6.78(0.06)

447

5.017

0.800

16.68

6.572(0.009) 2.509(0.004) 274.8(0.5) 26.96(0.03)

3.26(0.02)

6.71(0.03)

452

5.100

0.800

16.67

6.626(0.009) 2.550(0.004) 281.7(0.6) 27.10(0.03)

3.26(0.06)

6.73(0.07)

448

5.386

0.800

16.64

6.813(0.009) 2.693(0.004) 305.8(0.6) 27.57(0.03)

3.67(0.07)

6.41(0.06)

449

6.027

0.800

16.67

7.232(0.008) 3.014(0.004) 363.3(0.6) 28.58(0.03)

3.52(0.08)

6.70(0.07)

FIG. 1.

Target design for the Ta baseplates used in this series. Each

“

pocket

”

cut in the baseplate creates a uniform step thickness. This allows six thick-

nesses to be studied per shot and improves uncertainties by correcting for

fl

ections at the downrange surface. Reproduced with permission from Nguyen

etal

., Phys. Rev. B

174109,

Journal of

Applied Physics

ARTICLE

scitation.org/journal/jap

J. Appl. Phys.

125,

145903 (2019); doi: 10.1063/1.5054332

125,

145903-2

Published under license by AIP Publishing.

Two detection methods were used. The

rst method uses

bromoform (CHBr

) as an analyzer. This system has been

described in detail in previous papers,

so we will include only a

short description here. The target is enclosed in a vacuum-tight

chamber, which is

lled with degassed bromoform. Upon impact,

twin shocks are launched, downrange into the target, and uprange

into the impactor. When the downrange shock front enters the

bromoform, it emits brightly, with the response scaling as

The emitted light is optically relayed to photomultiplier tubes with

a large linear dynamic range, which serve as the detector system.

The voltage output from the photomultipliers is recorded on

fast oscilloscopes. When the uprange shock reaches the rear

surface of the impactor, now moving at

, it launches a release

wave that moves downrange in the impactor at

, a velocity

which always exceeds

. This release wave moves through the

compressed sample to eventually catch up to the shock front in

the bromoform. When the release and shock fronts interact, the

shock pressure and

decrease, leading to a decrease in emission.

We identify this catchup point as the intercept of two linear seg-

ments through a Monte Carlo algorithm described elsewhere.

The goal is to identify, by linear extrapolation, the sample thickness

at which shock and rarefaction would arrive at the sample/bromo-

form interface simultaneously, a result which is independent of

multiple wave interactions in the sample or wave speeds in the

bromoform.

Below 30 GPa in bromoform, the emitted light intensity falls

below the detection threshold of our photomultiplier tubes. For

experiments in this pressure range (

,

110 GPa in Ta), we use our

second method, photon Doppler velocimetry (PDV).

In this vari-

ation, the direct measurement of the particle velocity indicates

when the release wave has arrived at the free surface, indicated by a

drop in

(

Fig. 2

). Catchup point identi

cation and subsequent

analysis is the same as with the analyzer-based methods. While

adequate, it is less sensitive to velocity changes, as it requires a

Fourier Transform analysis of data over a

nite time interval and

because the signal scales only linearly with

. As a result, the

identi

cation of catchup times is less precise, as the slope changes

are less distinct than when using an analyzer. This leads to larger

uncertainties in the catchup time and ultimately in

, though it

does provide additional information about the material

’

s state

through the

measurements.

When using PDV as the detector method, we measured from

a bare surface with no windows. By doing so, we avoid additional

interactions of elastic or release waves in the window, though these

interactions do still occur in the sample. Because our analysis

method depends on time di

erences with step heights, the e

ects

of these interactions, which are linearly dependent on thickness,

can be corrected through extrapolation as they are in analyzer

experiments. We also avoid the issues caused by windows, namely,

the choice of correction factor and variation in the equation of

state, faced by Hu

et al.

and Rigg

et al.

and dis-

cussed in detail by Rigg

et al.

Window-related errors are the main

source of uncertainty in FSO measurements, and so this can be a

signi

cant advantage of the RSO method.

We used the Mitchell and Nellis

Ta Hugoniot to determine

pressure, density, and uncertainties

293

307

(2)

This is the same Hugoniot used by Brown and Shaner and Rigg

et al.

to enable easy comparison of the data sets.

The di

erence

between that Hugoniot and the later measurements by Holmes,

et al.

is negligible. Alternate Hugoniots (Lalle

) were also tested

and found to shift the measured value of the sound speed by up to

2.7%, while also changing the position in

space. Xi

et al.

used a slightly di

erent Hugoniot (

310

296

) based

on a largely unpublished data set.

We recalculated their data

using Eq.

(2)

and found it had a negligible e

ect on their results,

and so we have included their as-published data in

gures for

comparison.

B. Methods to estimate bulk and shear moduli

To further explore a change in material response near 60 GPa,

we examined the bulk and shear moduli. In the solid phase, the

measurement of

does not directly determine either the bulk or

shear modulus; one must assume a model for one to infer the

other, and so those inferred values depend directly on model

choices and assumptions. In this study, we applied two methods to

calculate the bulk modulus, and determined the corresponding

shear modulus from our calculations of

and measurement of

using Eq.

(1)

The

rst functional form we considered for

estimating K

(3)

where the ambient bulk modulus

200 GPa, the ambient

density

67 g/cm

, and

is a

tting parameter. Fitting to the

solid Ta bulk sound speed data of Hu

et al.

,we

though we note that this model fails to match the liquid sound

speed data above the melting pressure.

The second model we used to calculate

and the bulk sound

speed

is that of McQueen

et al.

rearranged to give (as in

FIG. 2.

Observed

at selected step heights for shot 1082, which at 37 GPa is

below the elastic

–

plastic overtake pressure. The time between breakout and

release used to calculate sound speed for the 1 mm step is indicated. Traces

are aligned on the onset time of the rarefaction.

Journal of

Applied Physics

ARTICLE

scitation.org/journal/jap

J. Appl. Phys.

125,

145903 (2019); doi: 10.1063/1.5054332

125,

145903-3

Published under license by AIP Publishing.

y and Ahrens

)

γρ

(4)

where

is the pressure, the subscript

indicates the Hugoniot

condition, and

is the Grüneisen parameter. Calculations

–

show that

varies with density, though how it varies can di

er sig-

cantly between calculations. An exhaustive comparison of these

models to select one to use to calculate shear and bulk moduli is left

for future work. Instead, we constrained our choices by

tting to

in the melt region and at ambient; this requires

to be about 1 near

300 GPa and 1.6

–

1.7 near ambient, which the model of Cohen and

Gülseren agrees with. This model has weak thermal dependence,

which is another advantage as the shock temperatures are unknown.

Therefore, we approximated their model with a cubic function

(

7019

85002

0444

17558

where

is the pressure on the Hugoniot) up to 300 GPa, after

which

1, and used that to calculate

and

III. RESULTS AND DISCUSSION

We plot the new

data for Ta with those of previous

studies

–

Fig. 3

. New data are listed in

Table I

. These new

data agree with previously existing data by Brown and Shaner,

et al.

, and Yu

et al.

at pressures above 60 GPa, including an

observed melt transition at just under 300 GPa.

We observed

that the sound speed ceased increasing at 230 GPa. This result is

similar to that seen in Mo

and consistent with predictions by

et al.

which suggests that softening of the metal prior to

melting leads to a decrease in the shear modulus.

Existing controversy regarding Ta phases on the Hugoniot

focuses on pressures at or below 60 GPa, where Hu

et al.

indicated

a transition based on changes in the longitudinal sound speed.

Their data are consistent with those of previous works in the small

regime that overlapped, and were the

rst Ta sound speed data to

be reported at pressures less than 100 GPa. However, Rigg

et al.

recently demonstrated that these results

–

including both the direc-

tion and magnitude of the claimed change in sound speed depend

on the mathematical analysis method used, and demonstrate that

this change, when it is observed, is seen at the elastic

–

plastic over-

take pressure of 51 GPa, remarkably close to the phase transition

claimed by Hu

et al.

That work raised the basic question of

whether the claimed transition was real, or an artifact of the

method used.

Additional e

ort was spent to obtain data above the overtake

pressure of Ta, but below the claimed transition of 60 GPa. Three

shots at 51

–

55 GPa show excellent reproducibility in the deter-

mined values of

(inset,

Fig. 3

), although shot 1081 su

ered

FIG. 3.

The sound speed of Ta on the Hugoniot, as measured. No sharp discontinuity in sound speed is observed near 60 GPa; instead, a gradual change in the sound

speed between

51 and 72 GPa is seen. Inset: Calculated sound speeds near the elastic

–

plastic overtake pressure. Error bars for other authors shown where available.

Journal of

Applied Physics

ARTICLE

scitation.org/journal/jap

J. Appl. Phys.

125,

145903 (2019); doi: 10.1063/1.5054332

125,

145903-4

Published under license by AIP Publishing.

signi

cant data loss (two channels returned no data and a third

was incomplete) resulting in larger error bars. As reproducibility

studies are rarely done on sound speed measurements, we chose to

include the data from shot 1081 as an illustration of both the repro-

ducibility of the calculated sound speed, and the sensitivity of the

measurement to the number and quality of data channels. There is

no evidence of an elastic precursor wave in the PDV signal on any

of these shots, indicating that all shots were above the elastic

–

plastic overtake pressure. Though data are sparse, there is no dis-

continuity in the sound speed data between 51 and 72 GPa compa-

rable to that seen at 300 GPa. In fact, extrapolated

ts of

from

–

55 GPa, shown as a black line segment in

Fig. 3

, intersect

ambient pressure (4.19 km/s) and at 72 GPa (4.72 km/s) within 2

Similarly,

ts from 72

–

230 GPa extrapolated to 51

–

55 GPa (dashed

red line segment) intersect all three of the measured sound speeds

in this range within 2

.(Athird

tfrom51

–

230 GPa, shown as a

solid red line segment, is included for easy comparison by the reader.)

This supports the idea that the claimed transition at 60 GPa is actually

a convolution of nonideal analysis methods, window e

ects, and the

elastic

–

plastic response of the material, consistent with the conclu-

sions of Rigg

et al.

and Xi

et al.

However, extrapolation of

from 51 GPa to ambient condi-

tions substantially underestimates the ambient sound speed, indi-

cating an in

ection point must be present. This in

ection point

need not be due to a phase transition. A known change in elastic

–

plastic material response occurs near this pressure: above about

51 GPa, the plastic deformation wave overtakes the slower elastic

precursor wave, combining the waves such that only a single shock

is seen in the velocimetry. At pressures less than about 51 GPa,

these two waves remain distinct. A single shot (1082) was obtained

below the elastic

–

plastic overtake pressure, which shows clearly

both elastic and plastic waves in the untransformed PDV traces.

The emergence of the plastic wave was determined through fringe

tting, rather than the onset of any motion, which corresponded to

the elastic wave. The calculated

is based upon the onset of the

plastic shock front, not the elastic front. This point falls directly on

the extrapolation between the points at

51 GPa and the ambient

sound speed. We observed small thickness-correlated changes in

the mean

of each step on this shot, as shown in

Fig. 2

. In calcu-

lating

, we used the impactor velocity to determine the

state

of the sample, not the observed

values; this is the same proce-

dure as used on the other shots.

To further explore the behavior in this region, we calculated

two models for the bulk moduli and inferred the corresponding

shear moduli. The results are shown in

Figs. 4

and

. Uncertainties

shown are solely due to uncertainty in the measurement of

and

do not include uncertainties associated with the models. As can be

seen by comparing the Grüneisen and exponential models in

Fig. 4

the true uncertainties are much larger. Within these uncertainties, we

nd that the shear modulus is approximately constant up to 72 GPa,

after which it increases until softening around 250 GPa. The marked

notch in

at 60 GPa observed by Hu is not seen.

at shear modulus is unusual, but not unheard of.

Flattening or softening shear moduli has been predicted for V, Ni,

and Ta,

and was observed in the diamond anvil cell compressed

from about 50

–

100 GPa, though at room temperature.

erences between observed and predicted pressures for these

shear regions is likely due to uncertainties in the equilibrium

volume of the DFT models used and can be substantial. Such a

modulus (within error bars) was then observed in shock com-

pressed V

as well from ambient conditions to about 90 GPa; as

with Ta, there is an in

ection in

in this region.

There are several reasons why one must be cautious to avoid

over-interpreting this result. The change in shear modulus is

heavily dependent on the model choice for bulk modulus, and this

apparent response in shear modulus can be altered substantially by

changing the model for bulk modulus. For example, this response

can be explained by altering the Grüneisen parameter model; ours

was chosen only to

t the bulk data at ambient and in the melt

region, as no experimental data were available in the relevant phase

space. We note that neither of the chosen models for the bulk

modulus match all of the

data from Hu

et al.

which appears

to have an in

ection point between 80 and 100 GPa. If this point

exists, then the bulk modulus must change at this condition, as it

does in the V data.

Ta is thought to be remain bcc until melting,

so one possibility could be an isostructural transition analogous to

that of Zr,

which again is unusual but not unheard of.

Because one can markedly alter the inferred behavior of the

shear modulus by changing assumptions about the bulk modulus,

we conclude that more data are needed on bulk sound speeds to

improve models of the bulk modulus. Measurements of bulk

moduli and sound speeds in the shock-compressed solid are

FIG. 4.

Calculated bulk and shear moduli for Ta using the exponential and

Grüneisen models. The shear modulus is

fl

at within uncertainties up to about

21 g/cm

(72 GPa), after which it begins to rapidly increase until shear softening

is seen near 26 g/cm

. Uncertainties shown are due only to uncertainties in

measurement of

; additional uncertainties arise from the choice of model,

which are not included but can be substantial.

Journal of

Applied Physics

ARTICLE

scitation.org/journal/jap

J. Appl. Phys.

125,

145903 (2019); doi: 10.1063/1.5054332

125,

145903-5

Published under license by AIP Publishing.

notoriously di

ffi

cult, so it is reasonable to expect substantial uncer-

tainties in these models that leave enough ambiguity in the data for

either the bulk or shear moduli to change.

In light of the inferred apparently

at shear modulus, one may

be tempted to dismiss these results in favor of previous work, so it

is worth discussing how the data sets compare. In the studied pres-

sure range, our data agree with the results of Rigg

’

s Method 1.

Above 51 GPa, our results also agree well with Hu

et al.

’

s results,

while our results are systematically lower below 51 GPa.

We note

a systematic 6%

–

8% discrepancy between the

calculated in this

study and those published by Xi

et al.

and Rigg

et al.

when

using their Method 3. As Rigg

et al.

demonstrated, changing the

method used to analyze their FSO data (to Method 1) eliminates

this discrepancy. It is reasonable to expect that similar analysis

changes to the Xi

et al.

data set would lead to similar changes.

We did not have enough information to reanalyze all of

et al.

’

s data using Rigg

’

s Method 1, so instead, we reanalyzed

et al.

’

s data to test if another source of this discrepancy could be

found. We note that impactor velocity was measured on only one

shot in their study. However, reanalyzing their data using the

impactor velocity instead of the measured

had little e

ect. FSO

is sensitive to changes in the particle velocity, including the e

ect

of window corrections. Replacing their linear correction for particle

velocity with a constant correction as described in Jensen

et al.

leads to an

1% decrease in pressure and up to a 0.5% decrease in

sound speed; to approximate the di

erences in results observed the

correction factor would have to be closer to 1.2

–

1.23, which is

extremely unlikely. Changing the Ta Hugoniot has even less impact

(under 0.2%) and so can be neglected. After ruling out these possi-

bilities, three others remain: (1) sensitivities to the analytical

method, as described in Rigg

et al.

; (2) di

erences arising from the

technique applied, such as unaccounted-for wave interactions; and

(3) di

erences in the material properties of the Ta used in each

study. Material property di

erences a

ecting strength or shear

modulus could include the grain size or the metallurgical methods

used

—

that is, whether the samples were cast, wrought, rolled, etc.

While it may be possible to test and correct for the

rst two

reasons, if the systematic discrepancy observed is due to di

erences

in the sample material

’

s origin, the solution is more di

ffi

cult.

Unfortunately, if this is the case, measurement of a universal Ta

sound speed response would not be possible, but would have to be

tailored for the material at hand.

IV. CONCLUSIONS

We measured the longitudinal sound speed of shock com-

pressed high-purity Ta from 37 to 363 GPa using the rear-surface

overtake method. We observe a decrease in the shear modulus

leading to a decreased rate of sound speed rise, consistent with pre-

vious observations, beginning near 230 GPa. A discontinuity in

sound speed consistent with melting near 300 GPa, congruous with

previous work, was also seen.

nd no strong evidence for the phase transition at 60 GPa

as claimed by Hu

et al.

Speci

cally, we do not observe a sharp

drop in

in this pressure range. Instead, the data suggest the pres-

ence of an in

ection point associated with the change in the mate-

rial response near 50

–

70 GPa; this is approximately near the

elastic

–

plastic overtake pressure of 51 GPa that was previously pro-

posed by Rigg

et al.

as a possible source of the apparent transition

seen by Hu

et al

. Three data points were collected at 51

–

55 GPa.

No evidence of an elastic wave was seen, and the high agreement

demonstrates an excellent reproducibility in the method.

A single point measured at 37 GPa did show an elastic response;

the calculated

falls directly between

at the overtake pressure

and at ambient conditions, giving additional con

dence in the

presence of the in

ection point. These results are not necessarily

inconsistent with changes in the sound speed due to a single stable

bcc phase up to 300 GPa, where the metal melts.

To narrow down the origin of this change in material

response, the bulk and shear moduli were calculated using two

erent models for the bulk modulus. Doing so reveals the possi-

ble appearance of a

at shear modulus below

72 GPa, in contrast

to a sharply increasing modulus as a function of pressure between

72 GPa and 230 GPa. This

at shear modulus persists well beyond

the elastic

–

plastic overtake condition, suggesting that some other

mechanism should also be considered. As we used a Grüneisen

model, an obvious possibility is some change in the thermal response

or the function of

. However, we also note that the low-pressure

response is very sensitive to the model chosen for bulk modulus; a

erent model can easily change the apparent slope, and one must

avoid assigning too much signi

cance to the shear response until

more data are available. More work should be done to experimentally

extract the bulk sound speed independent of a bulk modulus

FIG. 5.

Calculated bulk and shear sound speeds for Ta using the exponential

and Grüneisen models. Uncertainties shown are due only to uncertainties in

measurement of

; additional uncertainties arise from the choice of model,

which are not included.

Journal of

Applied Physics

ARTICLE

scitation.org/journal/jap

J. Appl. Phys.

125,

145903 (2019); doi: 10.1063/1.5054332

125,

145903-6

Published under license by AIP Publishing.

model and the crystal phase of shock compressed Ta, especially

below

70 GPa.

A systematic discrepancy was observed between the results cal-

culated using the front-surface overtake method of Xi

et al.

and the

rear-surface overtake method used in this study. Additional work is

required to determine if these di

erences are due to the methods,

the analysis, or the metallurgical history of the samples studied.

ACKNOWLEDGMENTS

We thank P. Rigg, P. Söderlind, and J. Klepeis for useful dis-

cussions, and the editors and reviewers for their work to improve

this paper. We thank Papo Gelle, Mike Long, Russ Oliver, Mike

Burns, Toni Bulai, Bob Nafzinger, Paul Benevento, Sam Weaver,

and Cory McLean for their excellent and dedicated work. Lawrence

Livermore National Laboratory is operated by Lawrence Livermore

National Security, LLC, for the U.S. Department of Energy,

National Nuclear Security Administration under Contract No.

DE-AC52-07NA27344.

REFERENCES

L. Burakovsky, S. P. Chen, D. L. Preston, A. B. Belonoshko, A. Rosengren,

A. S. Mikhaylushkin, S. I. Simak, and J. A. Moriarty,

Phys. Rev. Lett.

104

255702 (2010).

F. Lindemann, Z. Phys.

, 609 (1910).

D. Errandonea, B. Schwager, R. Ditz, C. Gessman, R. Boehler, and M. Ross,

Phys. Rev. B

, 132104 (2001).

D. Santamaria-Perez, M. Ross, D. Errandonea, G. D. Mukherjee, M. Mezouar,

and R. Boehler,

J. Chem. Phys.

130

, 124509 (2009).

A. Dewaele, M. Mezouar, N. Guignot, and P. Loubeyre,

Phys. Rev. Lett.

104

255701 (2010).

A. C. Mitchell and W. J. Nellis,

J. Appl. Phys.

, 3363 (1981).

N. C. Holmes, J. A. Moriarty, G. R. Gathers, and W. J. Nellis,

J. Appl. Phys.

2962 (1989).

S. P. Marsh,

LASL Shock Hugoniot Data

(University of California Press, 1980).

P. Song, L. Cai, T. T. Tao, S. Yuan, H. Chen, J. Huang, X. Zhao, and X. Wang,

J. Appl. Phys.

120

, 195101 (2016).

P. A. Rigg, C. W. Gree

, M. D. Knudson, G. T. Gray III, and R. S. Hixson,

J. Appl. Phys.

106

, 123532 (2009).

J. Yu, W. Wang, and Q. Wu,

Phys. Rev. Lett.

109

, 115701 (2012).

T. S. Du

y and T. J. Ahrens,

J. Geophys. Res.

(B4), 4503

–

4520 (1992).

R. S. Hixson, D. A. Boness, J. W. Shaner, and J. A. Moriarty,

Phys. Rev. Lett.

(6), 637 (1989).

J. H. Nguyen, M. C. Akin, R. Chau, D. E. Fratanduono, W. P. Ambrose,

O. V. Fat

’

yanov, P. D. Asimow, and N. C. Holmes,

Phys. Rev. B

, 174109

(2014).

J. H. Nguyen and N. C. Holmes,

Nature

427

, 339 (2004).

J. M. Brown and R. G. McQueen,

Geophys. Res. Lett.

(7), 533 (1980).

J. Hu, C. Dai, Y. Yu, Z. Liu, Y. Tan, X. Zhou, H. Tan, L. Cai, and Q. Wu,

J. Appl. Phys.

111

, 033511 (2012).

F. Xi, K. Jin, L. Cai, H. Geng, Y. Tan, and J. Li,

J. Appl. Phys.

117

, 185901

(2015).

P. A. Rigg, R. J. Schar

, and R. S. Hixson,

J. Phys. Conf. Ser.

500

, 032018

(2014).

M. C. Akin and J. H. Nguyen,

Rev. Sci. Instrum.

(4), 043903 (2015).

R. G. McQueen and D. G. Isassk, in

Shock Waves in Condensed Matter

–

1989

edited by S. C. Schmidt, l. N. Johnson, and L. W. Davison (North-Holland,

Amsterdam, 1990), pp. 125

–

128.

O. Strand, D. Goosman, C. Martinez, T. Whitworth, and W. Kuhlow,

Rev. Sci.

Instrum.

, 083108 (2006).

J. M. Brown and J. W. Shaner, in

Shock Waves and Condensed Matter

–

1983

edited by J. R. Asay, R. A. Graham, and G. K. Straub (Elsevier, New York, 1984),

pp. 91

–

94.

P. Lalle and R. Courchinoux,

AIP Conf. Proc.

370

, 207 (1996).

Y. Yu, H. Tan, J. Hu, C. Dai, and D. Chen, Explosion Shock Waves

, 486

(2006).

C. J. Wu, P. Söderlind, J. N. Glosli, and J. E. Klepeis,

Nat. Mater.

, 223

(2009).

R. G. McQueen, S. P. Marsh, J. W. Taylor, J. N. Fritz, and W. J. Carter, in

High-Velocity Impact Phenomena

, edited by R. Kinslow (Academic Press, San

Diego, CA, 1970), pp. 294

–

419.

C. Bercegeay and S. Bernard,

Phys. Rev. B

, 214101 (2005).

J. A. Moriarty and J. B. Haskins,

Phys. Rev. B

, 054113 (2014).

S. Taioli, C. Cazorla, M. J. Gillan, and D. Alfè,

Phys. Rev. B

, 214103 (2007).

R. E. Cohen and O. Gülseren,

Phys. Rev. B

, 224101 (2001).

Z.-L. Liu, L.-C. Cai, X.-R. Chen, Q. Wu, and F.-Q. Jing,

J. Phys. Condens.

Matter

, 095408 (2009).

B. J. Jensen, D. B. Holtkamp, P. A. Rigg, and D. H. Dolan,

J. Appl. Phys.

101

13523 (2007).

A. Landa, P. Söderlind, A. V. Ruban, O. E. Peil, and L. Vitos,

Phys. Rev. Lett.

103

, 235501 (2009).

D. Antonangeli, D. L. Farber, A. H. Said, L. R. Benedetti, C. M. Aracne,

A. Landa, P. Söderlind, and J. E. Klepeis,

Phys. Rev. B

, 132101 (2010).

Y. Yu, Y. Tan, C. Dai, X. Li, Y. Li, Q. Wu, and H. Tan,

Appl. Phys. Lett.

105

201910 (2014).

E. Stavrou, L. H. Yang, P. Söderlind, D. Aberg, H. R. Radousky,

M. R. Armstrong, J. L. Belof, M. Kunz, E. Greenberg, V. B. Prakapenka, and

D. A. Young,

Phys. Rev. B

, 220101(R) (2018).

Journal of

Applied Physics

ARTICLE

scitation.org/journal/jap

J. Appl. Phys.

125,

145903 (2019); doi: 10.1063/1.5054332

125,

145903-7

Published under license by AIP Publishing.