Experimental and Computational Studies of Compression and Deformation Behavior of Hafnium Diboride to 208 GPa

The compression behavior of the hexagonal AlB2 phase of Hafnium Diboride (HfB2) was studied in a diamond anvil cell to a pressure of 208 GPa by axial X-ray diffraction employing platinum as an internal pressure standard. The deformation behavior of HfB2 was studied by radial X-ray diffraction technique to 50 GPa, which allows for measurement of maximum differential stress or compressive yield strength at high pressures. The hydrostatic compression curve deduced from radial X-ray diffraction measurements yielded an ambient-pressure volume V0 = 29.73 Å3/atom and a bulk modulus K0 = 282 GPa. Density functional theory calculations showed ambient-pressure volume V0 = 29.84 Å3/atom and bulk modulus K0 = 262 GPa, which are in good agreement with the hydrostatic experimental values. The measured compressive yield strength approaches 3% of the shear modulus at a pressure of 50 GPa. The theoretical strain-stress calculation shows a maximum shear stress τmax~39 GPa along the (1−10) [110] direction of the hexagonal lattice of HfB2, which thereby can be an incompressible high strength material for extreme-environment applications.


Introduction
Hafnium diboride (HfB 2 ) belongs to a class of AlB 2 -type transition metal borides that have exhibited superb physical and thermal properties. The high bulk modulus (K 0 = 260-350 GPa [1,2]), low thermal expansion coefficient (7.49 × 10 −6 K −1 ) [3], and high shear to bulk modulus ratio (G 0~2 70 GPa) have brought much attention to HfB 2 for applications as a material exposed to extreme environments. The combination of strength and temperature resilience has garnered attention from the field of aerospace engineering for hypersonic and re-entry vehicles into Earth's atmosphere. HfB 2 is special among the refractory metal borides for the applications listed as it has a lower density (10.5 g/cm 3 ) [4] than other ultra-incompressible metal borides such as ReB 2 (12.7 g/cm 3 ) and WB 2 (10.8 g/cm 3 ) while still retaining similar elastic moduli values [4]. The incorporation of boron into the hexagonal crystal lattice (symmetry group P6/mmm) also enhances neutron absorption capabilities for protection from damaging radiation that can embrittle materials. Relatively recent theoretical studies have shown that these refractory transition metal borides are as efficient as water at moderating neutrons and suggest applications in reactor and future fusion devices [5]. The broad range of application of HfB 2 for industrial tool use and other extreme environments has led researchers to study its compression behavior, though only to relatively modest pressures 30-50 GPa compared to the conventional limits of commonly used diamond anvil cells (DAC) that can reach >200 GPa. The compression behavior or equation of state for HfB 2 is currently not available in the ultrahigh-pressure region despite the propensity for structural and electronic transitions of Group IV metals with narrow d-bands and broad sp-bands to occur with high pressure [6]. There is also an interest in studying how different pressure states (whether nonhydrostatic or hydrostatic) affect compression behavior. Some previous studies have used pressure mediums such as noble gases to make the sample compression in a DAC more hydrostatic, but this is best classified as quasi-hydrostatic due to crystallization of the gas or fluid with pressure and limited in the applied pressure range (~30 GPa) [7,8]. Theoretically, ab initio evolutionary algorithms have indicated that the AlB 2 structure with the transition metal and boron atoms sitting at the origin and the (1/3, 2/3, 1/2) Wyckoff sites can persist to a few hundred GPa in certain diborides such as ZrB 2 [9]. However, whether the structure in ZrB 2 or HfB 2 can be stable at such as high pressure remains experimentally unknown.
In this study, HfB 2 is compressed using beveled diamond anvils to 208 GPa to probe volumetric compression in a nonhydrostatic pressure environment, accomplished by an axial X-ray diffraction technique. This is also combined with a separate study using DAC with radial X-ray diffraction (R-XRD) and lattice strain theory (LST) to determine the hydrostatic pressure volume relation, as well as an estimation of its compressive yield strength. Density functional theory (DFT) calculations are also performed, and the theoretical results show good agreement with the experimental results on compression and deformation behavior.

Materials and Methods
The nonhydrostatic compression study on HfB 2 was conducted using a pair of beveled diamond anvils with a central flat of 30-microns, 8-degree bevel to 300-micron outer diameter. A spring steel gasket was indented to 30-micron thickness and laser drilled for a sample hole of 5-microns for sample placement. The HfB 2 sample was in powder form (Alfa Aesar, 99.5% metals basis) and mixed in a 3:1 ratio with platinum powder (Alfa Aesar, 99.97% purity) that was used for pressure calibration. Platinum was chosen as the pressure calibrant as its XRD peaks will not overlap with the sample, while the 3:1 ratio was chosen to ensure the platinum intensity does not overwhelm that of HfB 2 for later structural refinement. Given that platinum has a similar shear modulus to the base metal in HfB2, it is not expected that platinum will alter shear strength measurements significantly. This is later addressed in the Discussion section. The axial X-ray diffraction was conducted at the High-Pressure Collaborative Access Team (HPCAT) at Beamline 16-ID-B, Advanced Photon Source, Argonne National Laboratory.
For determination of hydrostatic components of compression, radial X-ray diffraction at beamline 16-BM-D, APS, Argonne National Laboratory was conducted. A pair of 300-micron flat diamond anvils were used for HfB 2 sample compression. The same HfB 2 and Pt sample mixture from the nonhydrostatic compression study was used for the R-XRD study and was placed in a beryllium gasket on the diamond culet with a sample hole of 70 microns in diameter. The X-ray beam was focused through Double Multilayer Monochromator into a collimating pinhole to a beam size of 3.4 µm (vertical) × 4.7 µm (horizontal) at 30 keV.
The pressure state at the center of a sample in a DAC is defined as nonhydrostatic due to differential stress between the axial and radial directions. The maximum stress lies along the diamond compression axis, while the minimum stress is exerted radially by the gasket material. This differential stress is a measure of compressive yield strength of the sample giving rise to a yielding of the crystallites that can be observed experimentally via R-XRD. To ensure the lattice strain is observed in the R-XRD profiles, a nonhydrostatic pressure state is necessitated, and no pressure medium is used in the sample hole. The lattice strain can be shown using Lattice Strain Theory (LST) produced by Singh et al. [10], where lattice strain from the nonhydrostatic environment will deform the measured d-spacing d m (hkl) by: where d p (hkl) is the hydrostatic component of the d-spacing, ψ is the angle between the cell compression axis and normal to the diffracting plane, and Q(hkl) is the lattice strain. Experimentally, the angle ψ can be determined by the relation ψ = cos(δ)cos(θ), where δ is the azimuthal angle around the XRD image and θ is the diffracting angle. The angle ψ can then be chosen such that 1 − 3cos 2 (ψ) = 0, in which case the d-spacing measured at this angle is the hydrostatic d-spacing d m (hkl) = d p (hkl). This provides a unique method for determining hydrostatic components of compression while the sample is under nonhydrostatic stress. The lattice strain term Q(hkl) is also very useful as it can be utilized to derive the differential stress and shear strength of a sample. This is shown in Equation (2): where t is the differential stress, α is a weighted factor between 0 and 1, and G R (hkl) and G V are the shear moduli approximations for the Reuss (iso-stress) and Voigt (iso-strain) conditions. By assuming the Voigt approximation, Equation (2) can be simplified to form a lower bound to compute the differential stress (t) and shear strength (τ): The measured d-spacing d m (hkl) from Equation (1) and the lattice strain term Q(hkl) were determined in this study by using the Rietveld refinement software MAUD [11]. The collected R-XRD images were integrated into azimuthal sections of δ = 5 degrees to determine the angle ψ, and each section was sequentially refined in MAUD to determine the variation of 1 − 3cos 2 (ψ) with d m (hkl), which allowed for the calculation of d p (hkl) for both HfB 2 sample and Pt pressure marker. The hydrostatic d-spacing d p (hkl) for Pt as well as the d-spacing for the nonhydrostatic experiment to 208 GPa were used in the Birch-Murnaghan equation of state (BM EoS) to determine pressure using Pt equation of state parameters derived by Yokoo et al. [12]: The first-principles calculations were performed using VASP (the Vienna Ab initio Simulation Package, version 5.4.4) [13,14], which is a plane-wave pseudopotential density functional theory (DFT) code [15,16]. The Perdew-Burke-Ernzerhof generalized gradient approximation (GGA-PBE) [17] functional and the projector augmented wave (PAW) method [18,19] were employed in our calculations. A plane-wave basis set with an energy cut-off of 600 eV and a fine 33 × 33 × 33 (resolution = 0.01 × 2π/Å) Monkhorst-Pack k-point grid were chosen. Structural relaxation calculations were performed with convergence criteria of 10 −3 eV/Å and 10 −6 eV/cell for the atomic force and self-consistent electronic steps, respectively. After the structure optimization, we computed the elastic tensor by the strain-stress method [13,14,20,21] in VASP. The bulk modulus K and shear modulus G are derived from the elastic constants using the Voigt-Reuss-Hill approximation [22][23][24], which averages the upper and lower bounds of the moduli.
We also computed the tensile and shear strengths of HfB 2 by using the QE (Quantum ESPRESSO, version 6.3) [25][26][27] DFT package. The PAW method and GGA-PBE functional were utilized as well. The same calculation settings and convergence criteria as those in the VASP calculations were considered. We used a fully relaxed crystal structure to construct a 2 × 2 × 2 supercell for computing the strain-stress curves. A k-grid of 11 × 9 × 9 points was used in the supercell calculations. Tensile strains were applied along the [001], [100], [110], and [111] directions, respectively, with a strain value up to 0.4 in a strain step of 0.01. In each step, the lattice constants and atomic positions of the tensile axis were fixed, while the lattice constants and atomic positions of the other two axes were allowed to relax fully. The shear strain-stress curves were obtained in a similar way by considering shear deformation in different directions on various planes. Figure 1 shows an example of shear deformation calculation on the (001) plane along the [1−10] direction with different strain values. In each step of the shear deformation calculation, we fix the lattice constant and atomic positions of the axis perpendicular to the shear plane (a-axis in Figure 1) and completely relax the lattice constants and atomic positions of the other two axes parallel to the shear plane (b and c axes in Figure 1). The structures were visualized by the VESTA software (version 3.4.8) [28].
FOR PEER REVIEW 4 of 11 ent strain values. In each step of the shear deformation calculation, we fix the lattice constant and atomic positions of the axis perpendicular to the shear plane (a-axis in Figure 1) and completely relax the lattice constants and atomic positions of the other two axes parallel to the shear plane (b and c axes in Figure 1). The structures were visualized by the VESTA software (version 3.4.8) [28]. Since the atoms are located at high-symmetry Wyckoff positions in HfB2, the actual atomic positions along the strain direction will be directly determined by the lattice parameters. We have benchmarked our QE calculations by comparing to results in the literature for ReB2 [29] and Mg2Si [30] using VASP, in which cases the applied strain is fixed, while the other five independent components of the strain tensors and all the atom positions were simultaneously relaxed. The critical strain and maximal stress values determined from these different approaches and software agree at a quantitative level.

Results
The hexagonal AlB2 phase of the HfB2 sample was found to be stable to the highest pressure of 208 GPa (V/V0 = 0.7). Figure 2a shows the sample HfB2 and Pt mixture at 0.63 GPa and 208 GPa, while Figure 2b shows the nonhydrostatic compression of HfB2 reaching a maximum pressure of 208 GPa fitted with the 3rd Order BM EoS. The least-square fitting of HfB2 pressure-volume data shows a bulk modulus K0 = 353 GPa with first pressure derivative K0′ = 3.08 and an initial volume V0 = 29.65 Å. The hydrostatic pressure-volume curve derived from LST and R-XRD is displayed in Figure 3 with the corresponding 3rd Order BM EoS fit and derived elastic parameters. The Since the atoms are located at high-symmetry Wyckoff positions in HfB 2 , the actual atomic positions along the strain direction will be directly determined by the lattice parameters. We have benchmarked our QE calculations by comparing to results in the literature for ReB 2 [29] and Mg 2 Si [30] using VASP, in which cases the applied strain is fixed, while the other five independent components of the strain tensors and all the atom positions were simultaneously relaxed. The critical strain and maximal stress values determined from these different approaches and software agree at a quantitative level.

Results
The hexagonal AlB 2 phase of the HfB 2 sample was found to be stable to the highest pressure of 208 GPa (V/V 0 = 0.7). Figure 2a shows the sample HfB2 and Pt mixture at 0.63 GPa and 208 GPa, while Figure 2b shows the nonhydrostatic compression of HfB 2 reaching a maximum pressure of 208 GPa fitted with the 3rd Order BM EoS. The leastsquare fitting of HfB 2 pressure-volume data shows a bulk modulus K 0 = 353 GPa with first pressure derivative K 0 = 3.08 and an initial volume V 0 = 29.65 Å.
allel to the shear plane (b and c axes in Figure 1). The structures were visualized by the VESTA software (version 3.4.8) [28]. Since the atoms are located at high-symmetry Wyckoff positions in HfB2, the actual atomic positions along the strain direction will be directly determined by the lattice parameters. We have benchmarked our QE calculations by comparing to results in the literature for ReB2 [29] and Mg2Si [30] using VASP, in which cases the applied strain is fixed, while the other five independent components of the strain tensors and all the atom positions were simultaneously relaxed. The critical strain and maximal stress values determined from these different approaches and software agree at a quantitative level.

Results
The hexagonal AlB2 phase of the HfB2 sample was found to be stable to the highest pressure of 208 GPa (V/V0 = 0.7). Figure 2a shows the sample HfB2 and Pt mixture at 0.63 GPa and 208 GPa, while Figure 2b shows the nonhydrostatic compression of HfB2 reaching a maximum pressure of 208 GPa fitted with the 3rd Order BM EoS. The least-square fitting of HfB2 pressure-volume data shows a bulk modulus K0 = 353 GPa with first pressure derivative K0′ = 3.08 and an initial volume V0 = 29.65 Å. The hydrostatic pressure-volume curve derived from LST and R-XRD is displayed in Figure 3 with the corresponding 3rd Order BM EoS fit and derived elastic parameters. The initial volume from R-XRD hydrostatic parameters gives a slightly higher volume V0 = 29.74 Å with a lower bulk modulus K0 = 282 GPa (with first pressure derivative K0′ = 3.38)  The hydrostatic pressure-volume curve derived from LST and R-XRD is displayed in Figure 3 with the corresponding 3rd Order BM EoS fit and derived elastic parameters. The initial volume from R-XRD hydrostatic parameters gives a slightly higher volume V 0 = 29.74 Å with a lower bulk modulus K 0 = 282 GPa (with first pressure derivative K 0 = 3.38) compared to the nonhydrostatic case. This decrease in bulk modulus for more hydrostatic environments is also noted in other literature for HfB 2 with similar moduli values [1,2]. This is attributed to lattice strain being underestimated under nonhydrostatic stress in axial diffraction geometry [10], i.e., the measured volume for nonhydrostatic cases can be larger with increasing pressure (and strain), rendering a seemingly higher bulk modulus. For HfB 2 , this effect seems to be profound as the difference in nonhydrostatic and hydrostatic bulk moduli are~82 GPa for this study and~79 GPa in Laing et al. for quasihydrostatic bulk modulus [1]. This will be discussed further later in the Discussion section.
OR PEER REVIEW 5 of 11 compared to the nonhydrostatic case. This decrease in bulk modulus for more hydrostatic environments is also noted in other literature for HfB2 with similar moduli values [1,2]. This is attributed to lattice strain being underestimated under nonhydrostatic stress in axial diffraction geometry [10], i.e., the measured volume for nonhydrostatic cases can be larger with increasing pressure (and strain), rendering a seemingly higher bulk modulus. For HfB2, this effect seems to be profound as the difference in nonhydrostatic and hydrostatic bulk moduli are ~82 GPa for this study and ~79 GPa in Laing et al. for quasi-hydrostatic bulk modulus [1]. This will be discussed further later in the Discussion section.  Figure 4a shows the calculated hydrostatic axial compression ratios a/a0 and c/c0 up to 210 GPa. The results indicate a slight anisotropic compression behavior of HfB2. The caxis compresses more easily than the a-axis, and the compression anisotropy decreases gradually with increasing pressure up to 125 GPa. Figure 4b shows the variation of bulk and shear moduli with pressure. The high bulk and shear modulus values under compression suggest highly incompressible and deformation-resistant behavior of HfB2. We next present DFT-GGA calculation results, which can be compared to HfB 2 experiments under hydrostatic pressure. The calculated lattice constants, volume, bulk, and shear moduli under ambient conditions are a 0 = 3.14 Å, c 0 = 3.49 Å, V 0 = 29.84 Å 3 /atom, K 0 = 262 GPa, and G 0 = 246 GPa, respectively. These calculated values are in good agreement with our experiments (V 0 = 29.73 Å 3 /atom, K 0 = 282 GPa) as well as other theoretical and experimental values (V 0 = 29.78-29.90 Å 3 /atom, K 0 = 261-395 GPa, G 0 = 227-248 GPa) reported in the literature [1,2,[31][32][33][34]. Figure 4a shows the calculated hydrostatic axial compression ratios a/a 0 and c/c 0 up to 210 GPa. The results indicate a slight anisotropic compression behavior of HfB 2 . The c-axis compresses more easily than the a-axis, and the compression anisotropy decreases gradually with increasing pressure up to 125 GPa. Figure 4b shows the variation of bulk and shear moduli with pressure. The high bulk and shear modulus values under compression suggest highly incompressible and deformation-resistant behavior of HfB 2 .
ported in the literature [1,2,[31][32][33][34]. Figure 4a shows the calculated hydrostatic axial compression ratios a/a0 and c/c0 up to 210 GPa. The results indicate a slight anisotropic compression behavior of HfB2. The caxis compresses more easily than the a-axis, and the compression anisotropy decreases gradually with increasing pressure up to 125 GPa. Figure 4b shows the variation of bulk and shear moduli with pressure. The high bulk and shear modulus values under compression suggest highly incompressible and deformation-resistant behavior of HfB2. The calculated c/a ratio of HfB 2 under pressure is plotted in Figure 5a along with the experimental data, including both nonhydrostatic and hydrostatic results. The measured c/a ratio decreases with increasing pressure. The anisotropy between the c and a-axes is noted for the pressure region 0-50 GPa, with the c-axis being the most compressible. Figure 5b shows the calculated volume compression V/V 0 under pressure along with the experimental values. Overall, the calculated results are in good agreement with the experiments up to 208 GPa. It is noted that the c/a ratio appears to decrease slightly with pressure in the low-pressure regime, and it starts to increase with pressure in the higher-pressure range.  The calculated c/a ratio of HfB2 under pressure is plotted in Figure 5a along with the experimental data, including both nonhydrostatic and hydrostatic results. The measured c/a ratio decreases with increasing pressure. The anisotropy between the c and a-axes is noted for the pressure region 0-50 GPa, with the c-axis being the most compressible. Figure 5b shows the calculated volume compression V/V0 under pressure along with the experimental values. Overall, the calculated results are in good agreement with the experiments up to 208 GPa. It is noted that the c/a ratio appears to decrease slightly with pressure in the low-pressure regime, and it starts to increase with pressure in the higher-pressure range.  Figure 6 shows the calculated shear and tensile stress-strain relationships for HfB2. The strain-stress curves show a linear behavior at small strain values (Hooke's law). At larger strains, a nonlinear behavior follows. The critical value is reached when the stress starts to decrease with strain. When the critical point is exceeded, the structure can become  Figure 6 shows the calculated shear and tensile stress-strain relationships for HfB 2 . The strain-stress curves show a linear behavior at small strain values (Hooke's law). At larger strains, a nonlinear behavior follows. The critical value is reached when the stress starts to decrease with strain. When the critical point is exceeded, the structure can become unstable, which leads to deformation or fracture. The maximum stress determines the upper limit of a material's mechanical strength.   Figure 6b shows that the ideal shear strength of HfB2 is in the (1−10) plane along the [110] direction. The ideal shear stress is ~39.16 GPa at a critical strain value of 0.2. It is noted that the theoretical shear strength of a perfect crystal is approximately equal to the shear modulus G0 divided by 2π. Our calculation results are consistent with this theoretical estimation. Table 1 lists the calculated data for the maximum tensile and shear stresses in each direction and their corresponding critical strain values.

Discussion
HfB2 is similar in crystal structure to other AlB2-type transition-metal borides (TMBs) such as TiB2 and ZrB2, and has certain similarities to hexagonal WB2-type TMBs. These   Figure 6b shows that the ideal shear strength of HfB 2 is in the (1−10) plane along the [110] direction. The ideal shear stress is~39.16 GPa at a critical strain value of 0.2. It is noted that the theoretical shear strength of a perfect crystal is approximately equal to the shear modulus G 0 divided by 2π. Our calculation results are consistent with this theoretical estimation. Table 1 lists the calculated data for the maximum tensile and shear stresses in each direction and their corresponding critical strain values. Table 1. Maximum tensile σ max and shear stresses τ max (in GPa) and their corresponding critical strain ε max values along different directions. The results are based on density functional theory (DFT) calculations using the generalized gradient approximation (GGA) functional.

Discussion
HfB 2 is similar in crystal structure to other AlB 2 -type transition-metal borides (TMBs) such as TiB 2 and ZrB 2 , and has certain similarities to hexagonal WB 2 -type TMBs. These materials typically share similar elastic properties, with nonhydrostatic bulk modulus values in the range of 260-400 GPa (e.g., K 0 = 263-395 GPa for HfB 2 [1,2], K 0 = 350-366 GPa for ReB 2 [35][36][37][38], and K 0 = 358 GPa for Os 2 B 3 [39,40]). However, it is interesting to note the behavior of HfB 2 with respect to differing pressure (hydrostatic versus nonhydrostatic) environments as compared to other TMBs. ReB 2 has displayed a nonhydrostatic bulk modulus of 360 GPa and a hydrostatic bulk modulus of 366 GPa. This is further replicated by DFT calculations (K 0 = 360 GPa) and quasi-hydrostatic compression to 30 GPa (K 0 = 360 GPa) [35][36][37][38]. HfB 2 , however, has shown a~80 GPa difference in bulk modulus between our nonhydrostatic and hydrostatic experiments, as well as in nonhydrostatic and quasi-hydrostatic studies in the literature [1,2]. This suggests the impact of high compressive yield strength of HfB 2 on the nonhydrostatic X-ray diffraction measurements. To further illustrate this point, we have analyzed the maximum differential stress (t) obtained from the radial X-ray diffraction data using Equation (3) and average measured lattice strains <Q(hkl)>. Figure 7 shows that the maximum differential stress (t) or compressive yield strength normalized to shear modulus (G) increases with increasing pressure and approaches a limiting value of 3% of the shear modulus (0.03 G).

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behavior of HfB2 with respect to differing pressure (hydrostatic versus nonhydrostatic) environments as compared to other TMBs. ReB2 has displayed a nonhydrostatic bulk modulus of 360 GPa and a hydrostatic bulk modulus of 366 GPa. This is further replicated by DFT calculations (K0 = 360 GPa) and quasi-hydrostatic compression to 30 GPa (K0 = 360 GPa) [35][36][37][38]. HfB2, however, has shown a ~ 80 GPa difference in bulk modulus between our nonhydrostatic and hydrostatic experiments, as well as in nonhydrostatic and quasihydrostatic studies in the literature [1,2]. This suggests the impact of high compressive yield strength of HfB2 on the nonhydrostatic X-ray diffraction measurements. To further illustrate this point, we have analyzed the maximum differential stress (t) obtained from the radial X-ray diffraction data using Equation (3) and average measured lattice strains <Q(hkl)>. Figure 7 shows that the maximum differential stress (t) or compressive yield strength normalized to shear modulus (G) increases with increasing pressure and approaches a limiting value of 3% of the shear modulus (0.03 G). The lattice strains Q(hkl) for (001), (101), and (002) planes given by Equation (2) for HfB2 as a function of pressure are also shown in Figure 7. The (001) plane shows a considerably higher strain Q and thus higher compressive yield strength than the other planes. This is an interesting result, as the axis of most incompressibility is along the a lattice parameter while maximum strength is along the (001) plane. This is actually opposite to what is seen in WB2-type TMBs, which display higher incompressibility along the c-axis and maximum strength in the (100) plane [36,37]. The WB2-type differs from the AlB2-type in that the interstitial boron layers are puckered, whereas the AlB2 displays a flat plane of boron atoms [41,42]. In a puckered boron layer, the distance between TM and B atoms is shorter than that of the flat boron layer. Therefore, the puckered boron layer has strong covalent hybridization between TM and B atoms. It has been shown in some WB2-type structures that electron density congregates along the c-axis, giving them their characteristic incompressibility [40]. Based on results in the c/a ratio and the lattice-dependent strain Q(hkl) in this study, it is inferred that the orientation of boron atoms in the AlB2type structure of HfB2 plays a large role in the selection of electronic density along the a- The lattice strains Q(hkl) for (001), (101), and (002) planes given by Equation (2) for HfB 2 as a function of pressure are also shown in Figure 7. The (001) plane shows a considerably higher strain Q and thus higher compressive yield strength than the other planes. This is an interesting result, as the axis of most incompressibility is along the a lattice parameter while maximum strength is along the (001) plane. This is actually opposite to what is seen in WB 2 -type TMBs, which display higher incompressibility along the c-axis and maximum strength in the (100) plane [36,37]. The WB 2 -type differs from the AlB 2 -type in that the interstitial boron layers are puckered, whereas the AlB 2 displays a flat plane of boron atoms [41,42]. In a puckered boron layer, the distance between TM and B atoms is shorter than that of the flat boron layer. Therefore, the puckered boron layer has strong covalent hybridization between TM and B atoms. It has been shown in some WB 2 -type structures that electron density congregates along the c-axis, giving them their characteristic incompressibility [40]. Based on results in the c/a ratio and the lattice-dependent strain Q(hkl) in this study, it is inferred that the orientation of boron atoms in the AlB 2 -type structure of HfB 2 plays a large role in the selection of electronic density along the a-axis as compared to the WB 2 -type structure. When the applied pressure increases, the Hf-Hf bond along the c-axis becomes shorter, which leads to a more repulsive electron-electron interaction and a stronger hybridization between Hf and B atoms; therefore, the compression of the c-axis can then become comparable to that of the a-axis at higher pressure.
Finally, our DFT calculations show that the weakest tensile strength is along the [100] direction, with a peak tensile stress value of σ~41.3 GPa at critical strain ε~0.13. Moreover, the ideal shear strength of HfB 2 occurs along the (1)(2)(3)(4)(5)(6)(7)(8)(9)(10) [110] direction, with a peak shear stress value of τ~39.2 GPa at critical strain ε~0.2. Based on the Frenkel model [43], the shear stress τ can be approximated as a sinusoidal stress-strain relationship: where G is the shear modulus, a is the spacing between atoms in the direction of shear stress, h is the atomic row spacing, and x is the shear translation. The maximum shear stress value τ max can be obtained by letting sin(2πx/a) = 1. Since h~a, τ max~G /2π~0.16 G, our DFT-calculated shear modulus value of 246.5 GPa will lead to τ max = 246.5 GPa/2π ≈ 39 GPa, which is consistent with the ideal shear strength from the strain-stress curves. Experimentally, the compressive yield strength of 0.03 G is smaller, potentially due to crystal imperfections. The presence of platinum (an elastically weaker material than HfB 2 ) within the sample hole of the DAC poses a natural question to what degree the mechanical properties of HfB 2 are altered. Given the excellent agreement in shear modulus, bulk modulus, and lattice parameter in this study with other literature values, it is not expected that the platinum pressure calibrant would cause the large difference in theoretical and experimental shear strength values of HfB 2 . A further study into fully sintered HfB 2 without the presence of platinum would have to be conducted to understand this effect fully.

Conclusions
The equation of state of HfB 2 was measured up to 208 GPa in a diamond anvil cell employing platinum as an internal pressure standard. The hexagonal AlB 2 was found to be stable up to 30% volume compression or V/V 0 = 0.7. The c-axis was observed to be more compressible relative to the a-axis below 50 GPa, in agreement with theoretical calculations. The hydrostatic compression curve deduced from radial X-ray diffraction measurements yielded an ambient-pressure volume V 0 = 29.73 Å 3 /atom and a bulk modulus K 0 = 282 GPa. Density functional theory calculations showed ambient-pressure volume V 0 = 29.84 Å 3 /atom and bulk modulus K 0 = 262 GPa, which are in good agreement with the hydrostatic experimental values. The theoretical strain-stress calculation shows that the weakest deformation direction is along the (1-10) [110] shear direction, and the maximum shear stress is τ max~3 9 GPa~G 0 /2π. Our comprehensive experimental and computational studies of the compression and deformation behavior of HfB 2 provide new results and understanding of this incompressible high strength material, which has potential applications in extreme environments.