Mo elasticity-density

JOURNAL OF APPLIED PHYSICS 106, 043506 ͑2009͒

Experimental and theoretical studies on the elasticity of molybdenum
to 12 GPa
Wei Liu,1,a͒ Qiong Liu,1 Matthew L. Whitaker,1,2 Yusheng Zhao,3 and Baosheng Li1
1
Mineral Physics Institute, Stony Brook University, Stony Brook, New York 11794, USA
2
Department of Geosciences, Stony Brook University, Stony Brook, New York 11794, USA
3
LANSCE, Los Alamos National Lab, Los Alamos, New Mexico 87545, USA
͑Received 26 May 2009; accepted 1 July 2009; published online 19 August 2009͒
Experiments have been conducted to measure compressional ͑V P͒ and shear wave ͑VS͒ velocities as
well as unit-cell volumes ͑densities͒ of molybdenum to 12.0 GPa at room temperature using
ultrasonic interferometry in conjunction with synchrotron x-radiation. Both V P and VS as well as the
adiabatic bulk ͑KS͒ and shear ͑G͒ moduli exhibit monotonic increase with increasing pressure. A
finite strain equation of state analysis of the directly measured velocities and densities yields KS0
Ј Ј
= 260.7͑5͒ GPa, G0 = 125.1͑2͒ GPa, KS0 = 4.7͑1͒, and G0 = 1.5͑1͒ for the elastic bulk and shear
moduli and their pressure derivatives at ambient conditions. Complimentary to the experimental
data, V P and VS as well as the elastic bulk and shear moduli were also computed using density
functional theory ͑DFT͒ at pressures comparable to the current experiment. Comparing with
experimental results, the velocities and elastic moduli from DFT calculations exhibit close
agreement with the current experimental data both in their values as well as in their pressure
dependence. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3197135͔

I. INTRODUCTION II. EXPERIMENT AND COMPUTATION

Molybdenum is a body-centered-cubic ͑bcc͒ 4d transi- The polycrystalline molybdenum rod ͑Alfa, 99.95%͒
tion metal with extensive application in modern technology with a diameter of 2.0 mm was used in this investigation.
for its extreme stability and refractory properties. The equa- Using the Archimedes’ method, a bulk density of
tion of state ͑EOS͒ of molybdenum ͑Mo͒ at high pressures 9.942͑10͒ g cm−3 was obtained, which is 97.3% of the the-
was used to calibrate the ruby fluorescence pressure scale, oretical x-ray density of 10.219 g cm−3. To minimize the
which is widely used as pressure calibrant in diamond-anvil acoustic energy loss, all surfaces along the acoustic travel
cell experiments.1 Numerous investigations have been per- path, including the tungsten carbide ͑WC͒ anvil on which the
formed on molybdenum, including ultrasonic measurements transducer was mounted, and both sides of the buffer rod and
of elastic constants at high temperature or at high sample, were polished using 1 ␮m diamond paste in final
pressure,2–6 static compression experiments,7–12 shock wave finish. The flatness and parallelism of those surfaces are
experiments,13–15 and theoretical calculations.16–18 within 0.5 and 2.5 ␮m.
Elastic bulk ͑KS͒ and shear ͑G͒ moduli and their pres- Compressional ͑P͒ and shear ͑S͒ wave velocities at high
sure derivatives are important parameters in understanding pressure were measured using ultrasonic interferometry tech-
the structural behavior and physical properties of materials niques in conjunction with in situ x-ray diffraction and radi-
under compression. The elastic properties of Mo have been ography in a DIA-type, multianvil, high pressure apparatus
͑SAM85͒ installed on the superconducting wiggler beamline
measured only up to 0.5 GPa.6 Recently, the state-of-the-art
͑X17B2͒ of the National Synchrotron Light Source at the
techniques developed in our laboratory have enabled simul-
Brookhaven National Laboratory. Details of this experimen-
taneous measurements of elastic compressional ͑P͒ and
tal setup and the ultrasonic interferometry have been de-
shear ͑S͒ wave velocities, and hence the elastic properties of
scribed elsewhere.19–21 The sample length at high pressure
materials at high pressure and high temperature using com-
was directly obtained by the x-radiographic imaging method,
bined ultrasonic interferometry, x-ray diffraction, and
the precision of this direct measurement of sample length
x-radiography in large-volume high-pressure apparatus.19 was reported to be 0.2%–0.4%.19
In this paper, we present the results of the compressional X-ray diffraction patterns for the sample were recorded
and shear wave velocities to 12 GPa from both experiments in the energy dispersive mode using a solid state Ge detector.
and density functional theory ͑DFT͒ calculations. The elastic The incident x-ray beam was collimated to 0.2 mm by 0.1
bulk and shear moduli and their pressure derivatives are de- mm and the diffraction angle was set at 2␪ = 6.52°. The
rived from the measured V P-VS volume data using third- ͑110͒, ͑211͒, ͑220͒, and ͑310͒ diffraction lines were used in
order finite strain equations of state; the results are compared the refinement of unit-cell volumes for Mo, with a relative
with those from DFT calculations as well as those from pre- standard deviation less than 0.1%.
vious experimental studies. Theoretical calculations were performed within the
framework of DFT using all-electron full potential linear
a͒
Electronic mail: weiliu3@notes.cc.sunysb.edu. augmented plane wave plus local orbitals method as em-

0021-8979/2009/106͑4͒/043506/4/$25.00 106, 043506-1 © 2009 American Institute of Physics

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043506-2 Liu et al. J. Appl. Phys. 106, 043506 ͑2009͒

͑1958͒ and Neutron diffraction result of 31.181͑3͒ Å3 from
Zhao et al.10 ͑2000͒, and within mutual uncertainties with the
x-ray diffraction result of 31.150 Å3 ͑3͒ from Zhao et al.
͑2000͒.10 Our DFT calculation at 0 K yields a value of
31.63 Å3 which is in agreement with experimental data
within 1.5%. The unit-cell volumes of Mo as a function of
pressure are compared with previous studies in Fig. 1. The
300 K compression curve from the current experiment is in
excellent agreement with previous results,7,9,10 although
some previous data show minor scattering at the pressure
range from 5 to 9 GPa. The current experimental data also
show an excellent agreement with those from DFT calcula-
tions at ground state ͑Fig. 1͒.
Compressional and shear wave velocities as well as the
adiabatic bulk ͑KS = L − 4G / 3 = ␳V2 − 4G / 3͒ and shear
P
FIG. 1. Variation of unit-cell volumes of molybdenum as a function of ͑G = ␳V2͒ moduli at high pressures are obtained from the
S
pressure. Symbols are from experimental studies, dotted line represents the measured sample lengths, travel times, and densities ͑Table
results from current DFT calculations. I͒. As shown in Fig. 2, both V P and VS exhibit linear increase
with pressure at room temperature. The pressure dependence
ployed in WIEN2K code.22,23 The unit cell is divided into non- is in excellent agreement with those from DFT calculations,
overlapping atomic spheres ͑muffin-tin, radius RMT͒ and an although the values of the velocities from DFT calculations
interstitial region where Kohn–Sham wave functions,24 ͑at 0 K͒ are slightly higher than the experimental data at
charge density and potential are expanded in different sets of temperature of 300 K ͑Fig. 2͒.
basis functions. The charge density and potential are ex- To obtain the zero-pressure adiabatic bulk and shear
panded into lattice harmonics inside muffin-tin spheres and moduli as well as their respective pressure derivatives, the
as a Fourier series in the remaining space. A maximum quan- velocity and density data can be fitted simultaneously to the
tum number of ten was used for the atomic wave functions finite strain Eqs. ͑1͒ and ͑2͒ without the input of pressure,27
inside the muffin-tin sphere. The generalized gradient ap-
proximation of Perdew et al.25 was used for the exchange- ␳V2 = ͑1 – 2␧͒5/2͑L1 + L2␧͒,
P ͑1͒
correlation functionals. A sphere radius of 2.4 a.u. was used
in the current calculation, and the Brillouin zone was ␳V2 = ͑1 – 2␧͒5/2͑M1 + M2␧͒, ͑2͒
S
sampled by 10 000 k-points ͑268 in the irreducible wedges͒.
‫ء‬
The plane wave cutoff was set by RMTKmax = 8 where Kmax is Ј
in which M1 = G0, M2 = 5G0 − 3KS0G0L1 = KS0 + 4G0 / 3, and
the maximum k vector in the basis set. The largest vector in Ј Ј
L2 = 5L1 − 3KS0͑KS0 + 4G0 / 3͒. The strain ␧ is defined as
the charge density Fourier expansion was Gmax = 14 bohr−1. ͓1 − ͑V0 / V͒2/3͔ / 2, The fitted coefficients, L1, L2, M1, and
The criterion for self-consistency energy convergence was M2, are used for the calculation of the zero-pressure adia-
1 ␮Ry. Convergence tests were conducted by using different batic bulk and shear moduli ͑KS0 and G0͒, as well as their
‫ء‬
values of RMTKmax and k-points to make sure the self- Ј
pressure derivatives ͑KS0 and G0͒. Ј
consistent force calculations is well converged with those The velocity and density data from our experiments are
used in the current calculations. The elastic constants were fitted to Eqs. ͑1͒ and ͑2͒ by minimizing the difference be-
calculated using the procedures described in Ref. 23, in tween the calculated and the observed compressional and
which a series of energy minimizations were performed after shear wave velocities, yielding KS0 = 260.7͑5͒ GPa, G0
applying small strains to the Mo unit cells and the elastic Ј Ј
= 125.1͑2͒ GPa, KS0 = 4.7͑1͒, and G0 = 1.5͑1͒. These param-
constants were then derived from the second derivatives of eters reproduce the observed compressional and shear wave
the energy with respect to the applied strains. The bulk and velocities remarkably well in the entire pressure range, with
shear moduli corresponding to polycrystalline sample were a RMS ͑root mean squares͒ misfit of 0.006 km s−1 for V P
obtained using a Voigt–Reuss–Hill average method. The and 0.002 km s−1 for VS, respectively. These results, to-
‫ء‬
same values of RMTKmax were used for the current calcula- gether with those from the current DFT calculations, are
tions at all pressures. compared with previous results in Table II.
The ambient pressure adiabatic bulk modulus KS0, 260.7
III. RESULTS AND DISCUSSION
GPa, agrees very well with previous values ͑259.3–263.8
GPa͒ from acoustic measurements on single crystal,3–6
The x-ray powder diffraction patterns show that the Mo within their respective estimates of error ͑ϳ1% – 2%͒. One
sample remains in the bcc phase up to the peak pressure of exception is that the KS0 value of Bolef and de Klerk2 ͑1962͒,
the current study ͑12.0 GPa͒. The refined unit-cell volumes which is only in marginal agreement with the upper bound of
show a smooth decrease with increasing pressure ͑Fig. 1͒. the current data as well as other single crystal acoustic mea-
We obtain a value of 31.175͑25͒ Å3 from our experiment for surements. The zero-pressure adiabatic shear modulus G0
the unit-cell volume at ambient conditions, which is in good from our measurement is in excellent agreement with the
agreement with the value of 31.161 Å3 from Pearson26 Voigt-Reuss-Hill ͑VRH͒ averages of shear moduli from pre-



TABLE I. Experimental ultrasonic data and unit-cell parameters of molybdenum.

Pa L 2tp 2ts VP VS V ␳ KS G
͑GPa͒ ͑mm͒ ͑␮s͒ ͑␮s͒ ͑km s−1͒ ͑km s−1͒ ͑Å3͒ ͑g cm−3͒ ͑GPa͒ ͑GPa͒

1.7 0.7631 0.2333 0.4326 6.542 3.528 30.972 10.286 269.5 128.0
2.4 0.7616 0.2323 0.431 6.557 3.534 30.893 10.312 271.4 128.8
3.1 0.7602 0.2309 0.4293 6.585 3.542 30.820 10.337 275.3 129.7
4.1 0.7597 0.2299 0.4276 6.609 3.553 30.708 10.374 278.5 131.0
4.6 0.7587 0.2287 0.426 6.635 3.562 30.656 10.392 281.7 131.9
5.3 0.7587 0.2279 0.425 6.658 3.570 30.577 10.419 284.8 132.8
5.6 0.7582 0.2271 0.4239 6.677 3.577 30.544 10.43 287.1 133.5
6.1 0.7577 0.2263 0.4228 6.696 3.584 30.493 10.448 289.5 134.2
6.7 0.7577 0.2257 0.422 6.714 3.591 30.429 10.47 292.0 135.0
7.1 0.7582 0.2253 0.4212 6.726 3.598 30.387 10.484 293.3 135.7
8.4 0.7592 0.2245 0.4202 6.763 3.614 30.251 10.531 298.3 137.6
9.1 0.7592 0.2239 0.4192 6.782 3.622 30.187 10.553 300.8 138.4
9.6 0.7592 0.2231 0.4182 6.806 3.631 30.137 10.571 303.8 139.4
10.1 0.7587 0.2223 0.4174 6.826 3.635 30.086 10.589 306.8 139.9
10.8 0.7582 0.2215 0.4162 6.846 3.643 30.018 10.613 309.6 140.9
11.1 0.7587 0.2209 0.4156 6.869 3.651 29.985 10.625 312.5 141.6
12.0 0.7582 0.2199 0.4144 6.896 3.659 29.903 10.654 316.5 142.6
a
P = −3KS0␧͑1 – 2␧͒5/2͓1 + 3͑4 − KЈ ͒␧ / 2͔ ͑Ref. 27͒. Values in parentheses are 1␴ error in the last digits. The uncertainty of the unit-cell volumes are less than
S0
0.15%. Travel times have 1␴ of ϳ0.4 ns for S wave, and ϳ0.2 ns for P wave. The uncertainties are less than 0.5% in velocities and less than 1.5% in the
derived elastic moduli. The densities at pressures are calculated using the unit-cell volumes and the theoretical density at room temperature ͑␳ = ␳‫ء‬V / V0͒.
0

vious single crystal elastic constant measurements,3–6 except
that of Bolef and de Klerk2 ͑1962͒, with which the agreement
is within ϳ2%.
Our experimental results for the pressure derivatives of
Ј Ј
the bulk and shear moduli, KS0 = 4.7͑1͒ and G0 = 1.5͑1͒, ap-
pear to be slightly higher than the respective value of 4.44
and 1.43 derived using experimental data up to 0.5 GPa from
Katahara et al. ͑1979͒. It should be noted that the current
results are derived from a finite strain fit instead of a linear fit
to pressure. If we fit the current elastic moduli data with a
linear expression of pressure as in Katahara et al. ͑1979͒, we
Ј
obtain KS0 = 261.0 GPa, G0 = 125.3͑2͒ GPa, KS0 = 4.54͑1͒,
Ј
and G0 = 1.45͑1͒ ͑see Table II͒, which is in excellent agree-
FIG. 2. ͑Color online͒ Elastic compressional ͑V P͒ and shear ͑VS͒ wave ment with those from Katahara et al. ͑1979͒. The close
velocities for molybdenum as a function of pressure from the current ultra- Ј Ј
agreement of KS0, G0, KS0, and G0 derived from the current
sonic and x-ray measurements to 12.0 GPa ͑symbols͒ and from DFT calcu- measurements with the results of Katahara et al. ͑1979͒ sug-
lations ͑lines͒.

TABLE II. Elastic moduli and their pressure derivatives of Mo.

References KS0 GV GR GVRH KЈ
S0 GЈ KT Ј
KT Notes

2 268.3 124.5 121.0 122.7 Single crystal, ultrasonic, 77–500 K
3 261.9 126.7 126.4 126.6 Single crystal, ultrasonic, 4.2–300 K
4 259.3 126.2 122.9 124.5 Single crystal, pulse echo, Ϫ198–650 C
4 261.6 125.8 122.6 124.2 Single crystal, thin rod resonance, Ϫ198–650 C
5 263.8 126.4 123.2 124.8 Single crystal, ultrasonic, Ϫ190–100 C
6 262.6 126.0 122.7 124.4 4.44 1.43 single crystal, ultrasonic, to 0.5 GPa
7 267͑11͒ ͓4.46͔ DAC, to 10 GPa
13 267 3.9 Shock wave experiment
8 262.8 3.95 DAC, to 272 GPa
10 268 ͑1͒ 3.81 ͑6͒ EoS, x-ray, to 10 GPa and 1475 K, combine shock data
10 264 ͑1͒ 4.05 ͑2͒ EoS, x-ray, to 10 GPa and 1475 K, combine shock data
10 266͑9͒ 4.1͑9͒ EoS, x-ray, to 10 GPa and 1475 K
This study 260.7͑5͒ 125.1͑2͒ 4.7͑1͒ 1.5͑1͒ Ultrasonic, 3rd FS, to 12 GPa
This study 261.0͑5͒ 125.3͑2͒ 4.54͑1͒ 1.45͑1͒ Ultrasonic, linear fit to 12 GPa
This study 261.9 4.5 DFT



ACKNOWLEDGMENTS
This research is supported by DoE/NNSA ͑Contract No.
DEFG5206NA2621 to B.L.͒. We thank Michael Vaughan
and Liping Wang for technical support at the X17B2 beam-
line. These experiments were carried out at the National Syn-
chrotron Light Source ͑NSLS͒, which is supported by the
U.S. Department of Energy, Division of Materials Sciences
and Division of Chemical Sciences under Contract No. DE-
AC02-76CH00016. X-17B2 is supported by COMPRES.
Mineral Physics Institute Publication No. 478.
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Mo elasticity-density

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