Pressure-induced isostructural phase transformation in gamma-B(28)

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gamma-B(28)

Evgeniya Zarechnaya, Natalia Dubrovinskaia, Razvan Caracas, Marco Merlini,

Michael Hanfland, Yaroslav Filinchuk, Dmitry Chernyshov, Vladimir

Dmitriev, Leonid Dubrovinsky

To cite this version:

Evgeniya Zarechnaya, Natalia Dubrovinskaia, Razvan Caracas, Marco Merlini, Michael Hanfland,

et al.. Pressure-induced isostructural phase transformation in gamma-B(28). Physical Review B:

Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2010, 82 (18),

pp.184111. �10.1103/PhysRevB.82.184111�. �hal-00676894�

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Pressure-induced isostructural phase transformation in

␥

-B

₂₈

Evgeniya Zarechnaya,1 _{Natalia Dubrovinskaia,}2,3,

_*

_{Razvan Caracas,}4 _{Marco Merlini,}5 _{Michael Hanfland,}6

Yaroslav Filinchuk,6_{Dmitry Chernyshov,}6_{Vladimir Dmitriev,}6_{and Leonid Dubrovinsky}1

1_{Bayerisches Geoinstitut, Universität Bayreuth, 95440 Bayreuth, Germany}

2_{Mineralphysik, Institut für Geowissenschaften, Universität Heidelberg, 69120 Heidelberg, Germany} 3_{Lehrstuhl für Kristallographie, Physikalisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany}

4_{Ecole Normale Supérieure de Lyon, Laboratoire de Sciences de la Terre, Centre National de la Recherche Scientifique, 46, allée d’Italie,}

69364 Lyon Cedex 07, France

5_{Dipartimento di Scienze della Terra, Universita degli Studi di Milano, Via Botticelli 23, 20133 Milano, Italy} 6_{ESRF, Boîte Postale 220, 38043 Grenoble, France}

共Received 15 October 2010; revised manuscript received 24 October 2010; published 18 November 2010兲

Isostructural phase transformations共ISPhTs兲 so far were reported only for heavy elements and their

com-pounds with complex electronic and/or magnetic structures. Studies of one of the lightest elemental material, high-pressure boron␥-B28, by means of in situ single-crystal x-ray diffraction and Raman spectroscopy reveal

abrupt changes in the compressional behavior and Raman spectra at 40 GPa. Combined experimental and ab

initio theoretical analysis of the structural and vibrational properties of␥-B₂₈suggests that the ISPhT under compression is due to the changes in the polarity of the covalent bonds between the boron atoms in the complex quasimolecular structure of␥-B28.

DOI:10.1103/PhysRevB.82.184111 PACS number共s兲: 61.50.Ks

I. INTRODUCTION

Isostructural phase transformations are particular cases of isosymmetric transformations and include discontinuous iso-structural transitions and crossovers. They are rare, intrigu-ing phenomena in solids often associated with a significant volume collapse, giant magnetoelastic coupling, or negative thermal expansion.1–5_{Elemental boron and boron-rich}

com-pounds have been given a particular scientific and techno-logical attention because of their specific physical properties: high melting temperatures, a wide energy band gap, high hardness, strong absorbance of neutrons, etc. The structures of these materials are based on icosahedral units as a com-mon feature. Even high pressure does not destroy the rigid units so that the very recently described high-pressure boron phase6–8_␥_-B

28 consists of B12 icosahedra typical for

rhom-bohedral ␣- and ␤-boron. Consisting of only B12 units, ␣-boron does not show any phase transitions on compression to at least 80 GPa at ambient temperature. The arrangement of B12 icosahedra in ␤-boron is complex and its structure

contains partially occupied sites.9,10 _The _␤_{-boron is more}

compressible 共the bulk modulus K₃₀₀= 185– 210 GPa兲 in comparison with␣-boron共K₃₀₀= 213– 224 GPa兲.11,12_The

re-sistivity measurements13 _{indicate a transformation from}

␤-boron into yet unknown superconducting phase at about 160 GPa. At pressures above 8.5 GPa and temperatures higher than 1500 ° C pure boron crystallizes as a␥-B₂₈phase with a Pnnm space group.6–8It contains 28 atoms in the unit cell. The structure consists of B12 icosahedra and B2

dumb-bells linked covalently6,7 _共Fig. ₁_{兲. The} _␥_-B

28 phase was

demonstrated7 _{to be stable at least to 30 GPa and 2000 K.}

However, experimental information about the high-pressure behavior of the␥-B28phase is so far limited. Here we present

the results of the high-pressure Raman spectroscopy study of

␥-B28 up to 105 GPa and its single-crystal x-ray diffraction

over 65 GPa.

II. METHODS

Single crystals and polycrystalline aggregates of ␥-B₂₈ were grown as described in our previous publications 共see Refs. 6and7兲.

A. Diamond-anvil cell experiments

In different runs the sample was clamped between dia-mond anvils with culets of 300, 250, or 120 ␮m in diameter. Rhenium 共in the Raman spectroscopy experiments兲 or steel 共in single-crystal x-ray diffraction experiments兲 gaskets were indented to the thickness of about 50 ␮m and holes with a diameter of 100 or 80 ␮m were drilled in the center. Small isometric pieces of the high-pressure boron phase synthe-sized in a multianvil apparatus were loaded into the holes along with ruby balls served as pressure markers. As a pres-sure transmitting medium Ne or He were loaded at 1.4 kbar

FIG. 1. 共Color online兲 Structure of ␥-B28 共atom labels are

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共all single-crystal x-ray diffraction experiments were per-formed in He pressure medium兲. One- and double-side laser annealing of samples compressed in Ne pressure medium at pressures above 20 GPa at temperatures up to 2000 K was conducted at BGI or at ESRF.

B. Raman spectroscopy

Raman spectroscopy studies were done on single crystals 共with characteristic dimensions of ⬃5⫻5⫻25 ␮m3_{兲 and}

polycrystalline aggregates of the B₂₈ phase. Measurements were performed with LabRam and Dilor XY 共DILOR GmbH兲 systems with a resolution better than 2 cm−1_.

The 632.8 nm line of a He-Ne laser and 514 nm of an Ar laser were used for an excitation with a power at the sample position of 15–50 mW. The positions of Raman peaks were determined by processing experimental data using PEAKFIT© v4.12 software.

C. Single-crystal diffraction

Crystals with a size of about 7 ␮m⫻7 ␮m⫻30 ␮m were selected and measured at ID09a at the European Syn-chrotron Radiation Facility 共ESRF兲. Diffraction data were collected at 293 K using the MAR555 image plate detector, radiation with a wavelength of 0.4143 Å and the crystal-to-detector distance of 399 mm. 120 frames in the omega scan-ning range of −30° to +30° were collected 共0.5° scanning step size兲 with exposure time of 1 s. The data were processed using the CRYSALIS software 关Oxford Diffraction 共2006兲

CRYSALIS RED, version 1.171.31.8. Oxford Diffraction Ltd., Abingdon, Oxfordshire兴. Crystal-structure refinements on in-tegrated intensities were carried out withJANA2006software.

D. Ab initio calculation

We performed first-principles calculations using the density-functional theory and density-functional perturbation theory formalisms as implemented in theABINITcode.14We employed a 6⫻6⫻6 regular grid of high special k points15

and a kinetic-energy cutoff of 30 Ha 共1 Ha=27.2116 eV兲. These ensured a precision on the order of 1 mHa in energy and better than 1 GPa in pressure. The calculations were performed on the JADE machine of CINES.

III. RESULTS

A. Raman spectroscopy at high pressure.

The␥-B28gives very distinct and strong Raman spectra at

ambient conditions6,7,16 _{so that the Raman spectroscopy}

could be the most convenient and useful technique for inves-tigation of the high-pressure behavior of this phase. Polar-ized Raman spectroscopy studies16_{allowed assigning}

vibra-tion modes of ␥-B28 thus creating a basis for the present

study. Some examples of Raman spectra of polycrystalline

␥-B₂₈ collected at high pressure are presented in Fig. 2共a兲. The pressure dependence of Raman modes关Fig.2共b兲兴 can be shown in terms of the mode-Grüneisen parameter, defined as

␥␻i= −⳵ln␻i/⳵ln V = −共B0/␻i兲⳵␻i/⳵P, where␻iis the pho-non frequency, B₀is the bulk modulus共227 GPa at ambient

conditions7兲, V is the molar volume, and P is the pressure. According to the values of the mode-Grüneisen parameters, all observed Raman peaks could be divided into three groups 共Table I兲. To the first group belongs the mode B2g at

319共3兲 cm−1_{for which the Grüneisen parameter}_{共2.8兲 is}

dis-tinctly high. According to theoretical analysis共TableII兲 this

mode is associated with tilts共rotations兲 around the a axis of the B12 icosahedra. The second group of mode consists of

weakly varying共␥= 0.03– 0.2兲 narrow bands in the frequency Raman shift, cm-1 200 400 600 800 1000 1200 In ten sit y, ar b.un its 0 2000 4000 6000 8000 10000 12000 3.8(1) GPa 31(1) GPa 40(1) GPa 44(1) GPa 65(2) GPa 105(3) GPa (a) Pressure (GPa) 0 20 40 60 80 100 Wa ve num be r (c m -1) 200 400 600 800 1000 1200 1400 Ag B1g Ag Ag B B AgB AgB Ag Ag BB3g B1gAg Ag B1g B B1g B1g (b)

FIG. 2. 共Color online兲共a兲 Examples of Raman spectra of␥-B28

compressed in a neon pressure transmitting medium in a diamond-anvil cell. Arrows highlight some parts of the spectra which

un-dergo rapid changes at pressure above 40 GPa.共b兲 Variation in the

positions of Raman lines of␥-B28as a function of pressure. Mode

assignment at ambient conditions is given according to Ref.16关for some modes we could not assign their type共B2gor B3g兲 so that they

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range 380– 470 cm−1_{. All these modes are assigned to}

vibrations of B12 icosahedra 共380, 388, and 460 cm−1兲 or

both B12icosahedra and B2 dumbbells 共444 and 470 cm−1兲

along the unit-cell axes共TableII兲. Note that similar behavior

of icosahedron and icosahedron-dumbbell vibrations were observed in ␣-boron,17 _in _highly _ordered _pnictides

共B12As2and B12P2兲,18and boron carbide 共B4C兲.19 The

Grü-neisen parameters of modes observed between 507 and 1200 cm−1 _{共the third group兲 are in the range of 0.42–1.05.}

These bands arise mainly from intraicosahedral vibrations, such as breathing or rocking-stretching modes. Note that ap-parent changes in the shape of spectra in the 800– 1000 cm−1 region on compression to about 39 GPa

共Fig. 2兲 are entirely due to differences in mode-Grüneisen

parameters of the bands.

The observed high-pressure behavior of Raman modes can be qualitatively related to the response of the ␥-B28

structure on compression. Indeed, rotations of B12

icosahe-dra, or of the B12icosahedra and B2dumbbells around b and c axes 共Fig.1, TableII兲 require changes in the length of the

shortest 共i.e., strongest兲 B-B contacts between icosahedra and/or between an icosahedron and a dumbbell7_{; they}

corre-spond to the modes with the smallest mode-Grüneisen pa-rameters. The mode which is most sensitive to the volume changes is associated with the rotation of B12 icosahedra

TABLE I. Assignment of experimental共Ref.16兲 and theoretical phonon frequencies at ambient conditions

with experimentally determined mode-Grüneisen parameters共␥_exp兲.

Experiment Theory

␻exp

共cm−1_兲 _Assignment _␥

expbelow 39 GPa ␥expabove 39 GPa

␻th 共cm−1_兲 _Assignment 217共2兲 B2g, B3g 222共2兲 A_g 319共3兲 B2g, B3ga 2.76 1.04 303 B2g 380共1兲 A_g 0.07 −0.03 387 A_g 388共2兲 B1g 0.17 389 B3g 440共2兲 B_2g, B_3g 0.20 −0.02 454 B_3g 460共2兲 B1ga 464/475 B3g/B1g 470共2兲 B_2g, B_3g 0.03 −0.03 464/475 B_3g/B_1g 507共2兲 Ag 0.65 0.34 536 Ag 531共2兲 B_2g, B_3g 0.68 537 B_2g 593共3兲 Ag 0.67 0.41 611 Ag 632共2兲 B2g, B3g 0.53 0.42 585/575 B2g/B3g 654共2兲 A_g 0.84 0.47 687 A_g 686共3兲 B1g, B2g, B3ga 658 B1g 702共3兲 A_ga _0.85 _0.50 ₇₃₂ _A g 712共3兲 B2g, B3g 714/719 B3g/B2g 729共2兲 B_1g 0.78 0.40 726 B_1g 735共2兲 B1g, B2g, B3ga 0.79 0.42 735 B2g 758共2兲 B_3g 0.82 0.45 765 B_3g 786共2兲 B1g 0.87 0.54 781, 784 B1g, B1g 857共3兲 B_1g 0.81 0.52 862 B_1g 878共2兲 B1g, B3ga 884 B3g 895共2兲 Ag 1.05 0.55 887 Ag 912共2兲 Ag 1.26 0.66 920 Ag 929共2兲 Ag 970?b Ag 952共2兲 B_2g, B_3ga ₉₅₉ _B 2g 1095共2兲 B1g, B2g, B3ga 1118共2兲 B_1g 0.98 0.40 1118 B_1g 1138共2兲 B1g 1138 B1g 1161共2兲 A_g 0.42 1189 A_g 1189共2兲 Ag 1194 Ag 1218共2兲 A_g

a_{Peaks which are visible also in forbidden scattering geometries}_共Ref.₁₆_兲.

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TABLE II. Calculated Raman-active modes of␥-B28at 0 GPa.

Assignment ωcalc(cm-1₎

Description

Ag 387

The B6half-caps move up and down

and the whole B12 icosahedron tilts

around the c axis

Ag 536 B2 dumbells tilt around the c axis; slight participation of the B12

icosahedra that tilt much less

Ag 611 _{Breathing mode of the B12}_icosahedra

Ag 732 Tilt of B2_B12 _{icosahedra, looking like a dangling bond; the B atoms from the B12}associated with a movement of the B atoms from the sides of the

that move are the furthest apart from B2

Ag 811

Breathing mode of the B12

icosahedra; here the B atoms from the side, the ones that form a bond

perpendicular to the B2-B12 plane

participate in the motion

Ag 920 Breathing mode of the B12

icosahedra

Ag 1119 Tilt of the B2_{from the B12}dumbell associated with a displacement of the two B atoms_{icosahedra situated closest to the B2}_{dumbells along the b}

axis

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TABLE II. 共Continued.兲

the apex atoms (these are the only ones that really move)

B1g 475 _{Tilts of the B12}_{icosahedra around the c axis}

B1g 554 _{Tilts of the B2}_dumbells

B1g 658

Vibration corresponding to half the

B12 icosahedra vibrating against the

other half along the b axis

B1g 726 _{Breathing mode of the B12}_icosahedra

B1g 781

Breathing mode of the B12

icosahedra; each half-cap breathes independently

B1g 784 Breathing_{indepedently; the movement has an important a component, the sizes of}mode of the B12 icosahedra; each half-cap breathes

the half-caps rotating around the z axis

B1g 793 Breathing mode of the B12icosahedra; dominated by a tilt of the B-B bond

between the B12icosahedra, in the (010) plane

B1g 862 _{Breathing mode of the B12}_{icosahedra; B2}_{do not participate}

B1g 906 Dominated by B2 extension mode with some smaller participation from

the breathing of B12icosahedra from the B2side bonds parallel to the c axis

B1g 979

Breathing mode of the B12 icosahedra,

with lateral extension in the (100) planes

B1g 1118 Tilt of the B2_{from the B12}dumbell associated with a displacement of the two B atoms_{icosahedra situated closest to the B2}_{dumbbells along the b}

axis

B1g 1189 Breathing-like mode of the B12icosahedra dominated by the movement of

the apex atoms (these are the only ones that really move)

B2g 303 _{Tilts (=rotations) of the B12}_{icosahedra along the x axis}

B2g 454 Tilts (rotations) of the B12_icosahedra _{a axis, while the B}icosahedra and of the B2dumbells. The B12

2 component along the b axis

dumbbells have a around the

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TABLE II. 共Continued.兲

axis

B2g 585 _{Tilts of the B12}_icosahedra_{around the a axis}

B2g 719

Shear-like mode of the B12 icosahedra;

such that each half of the cage tilts around the b axis producing a

sheared final B12cage

B2g 735

Shear-like mode of the B12icosahedra;

the tilt system is slightly different than from the previous mode; there are two subunits within each half that tilt in anti-phase

B2g 771 The apex of the B12atoms along the X axis and the B2dumbbells vibrate

around the b axis

B2g 880 _{Dominated by a tilt system of the B12}_{apex against the B2}_dumbbells

B2g 959 Shear mode of the B12icosahedra, associated with a tilt of the half-caps

the b axis

B3g 389 _{Tilt mode of the B12}_icosahedra_{around the b axis}

B3g 403 Tilt of the B2dumbbells with a sight participation from the B12unit that tilt

as well

B3g 464 _{Another tilt of both B12}_{and B2}_{around the a axis}

B3g 575 Shear mode of the B12icosahedra; half-caps vibrate against the other half

a tilt around the b axis

B3g 714 Double shear of the B12 icosahedra; the mode is dominated by the B2

bond parallel to the c axis

B3g 765 Breathing mode of the B12icosahedra

B3g 810 Breathing mode of the B12_{differential displacement in the (100) plane soft the different}associated with a shear

12 icosahedra

B3g 884 Breathing mode of the B12_{differential displacement along the a axis of the different B of B12}associated with a shear

icosahedra; dominated by lateral displacement of the apex atoms

B3g 929 _{Complex breathing mode of the B12}_icosahedra

B2g 537 Tilts of the B12icosahedra around the

b along with B of B due to icosahedra, due to icosahedra,

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around the a axis and does not involve tension of the shortest interatomic contacts.

Most the Raman modes monotonically shift to higher wave numbers with increasing pressure up to about 35 GPa. At pressures higher than 40 GPa in all three independent experiments we observed an inflection in the modes behavior 关Fig.2共b兲, Table I兴. Under further compression to pressures

about 40–45 GPa several broaden peaks split 共Fig. 2共a兲兲. Although above⬃40 GPa we detect up to 32 Raman modes in the spectra, the total number of modes was always less than 42 allowed in orthorhombic␥-B₂₈.16_{At the same}

pres-sure two modes共Agat 380 cm−1and B3gat 470 cm−1兲

dras-tically change their behavior and become soft with Grü-neisen parameters of about −0.03 共Fig.2; TableI, Fig.3兲.

Phonon determination based on first-principles calcula-tions show that the lowest six Raman modes 共one A_g, one B1g, two B2g, and two B3g兲 present pressure softening in

perfect agreement with the experimental observations. They are lattice modes corresponding mainly to tilts of the B12

icosahedra. In particular, the behavior of the lowest B_2g mode is relevant as it hardens up to about 50 GPa and it softens beyond that pressure.

Laser annealing at 1500–1800 K at pressures above 40 GPa does not affect the Raman spectra and the changes are fully reversible on decompression suggesting that the ortho-rhombic␥-B28may undergo a phase transition at high

pres-sures. In order to test this hypothesis we have undertaken an intensive x-ray diffraction investigation of␥-B28under

com-pression.

B. Single-crystal x-ray diffraction at high pressure

Being the fifth element in the Periodic Table, boron is a very weak x-ray scatterer. X-ray powder-diffraction experi-ments on boron phases above 30 GPa are difficult7,19 _and

could hardly provide even accurate lattice parameters. At the same time, single-crystal studies allowed investigating the behavior of lattice parameters of ␤-boron at pressure up to 100 GPa.12 _{In three independent diamond-anvil cell} _共DAC兲

experiments we compressed single crystals of␥-B28共grown

in a multianvil apparatus in metallic fluxes20_{兲 to pressures}

over 65 GPa using He as a quasihydrostatic pressure trans-mitting medium. The collected x-ray diffraction data were sufficient in quality to refine both lattice parameters and atomic coordinates共TableIII兲. Remarkably, the quality of the

structural refinement based on the data collected in DACs was not worse than that based on the data obtained at ambi-ent conditions共TableIII兲, and it also proves that the crystals

were in a good quasihydrostatic environment to highest pres-sures reached.

Up to the highest pressure reached in this study共65 GPa兲 all observed reflections perfectly match the ␥-B28 共space

group Pnnm兲 structure 共Table III兲. However, the

compres-sional behavior of the material at pressures below and above about 40 GPa is considerably different共Fig.4兲. It manifests

in an abrupt increase in the stiffness of the structure 共more than 20% increase in the bulk modulus兲 and in the relative softening along the a axis in comparison with the c and especially the b axes. There is a clear change in the behavior of the b/a ratio 共it decreases monotonously to about 35 GPa and increases above 40 GPa兲, and there is a discontinuity in “F-f” plot共normalized stress vs Eulerian strain, see insets in Fig. 4兲. An extrapolation of molar volumes of low-pressure

and high-pressure materials to ambient conditions gives a difference of about 3%共Fig.4兲.

On compression up to 40 GPa the shortest bond lengths as well as the average distances between any atom and its neighbors 共“atom’s sphere radius” as defined in IVTON

software22_{兲 homogeneously decrease; their compressibilities}

are linear with values in the 0.019– 0.024 GPa−1 _range

共Fig. 5兲. The volume of the B12 icosahedra also decreases

smoothly to 40 GPa共Fig.6兲. However the bulk modulus of

this polyhedron 共285 GPa兲 is significantly higher than the bulk modulus of the bulk material 共227 GPa兲 confirming a molecular-solidlike behavior with intericosahedral bonding weaker than intraicosahedral ones. At pressures above 40 GPa the compressibility of the spheres around the B1 and B4 atoms 共Fig. 1 for designation of atoms兲 does not change

共these two atoms are not involved into the shortest interi-cosahedral contacts or into the contacts of the B2dumbbells

and the B12 icosahedra兲. Contrary, the atoms involved in

in-traicosahedral bonding 共labeled B3兲 or in bonding between the B12icosahedra and the B2dumbbells共labeled B2 and B5兲

show pronounced changes in the compressional behavior. In parallel the B12 icosahedra become much more

incompress-ible, exhibiting a 1/3 increase in the bulk modulus共Fig.6兲.

IV. DISCUSSIONS

Both the Raman spectroscopy and the single-crystal x-ray diffraction studies reveal the existence of two states of␥-B₂₈ with a pronounced difference in their vibrational and compressional properties. The abrupt change in the properties occurs at pressures ⬃40 GPa. However, neither the symmetry nor the structure change was detected in this pressure range. We conclude, therefore, that B₂₈ undergoes an isostructural phase transformation.

Isostructural phase transformations are relatively exotic phenomena. They were reported for a few elements, in par-ticular, in Ce 共a well-known example of the isostructural Pressure, GPa 0 20 40 60 80 100 W ave num ber (cm -1) 374 376 378 380 382 384 386 388 390 A_g

FIG. 3. 共Color online兲 Example of the measured 共dotted curve兲

and calculated 共solid curve兲 Ag Raman mode as a function of

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TABLE III. Structural parameters of the␥-B28phase共Pnnm, Z=28兲 as revealed by single-crystal diffraction experiments in

diamond-anvil cells at high pressures.

Run Pressure

共GPa兲

Lattice parameters共Å兲 Atomic coordinates R1共%兲

wR₂共%兲 a b c B1 8h B2 4g B3 4g B4 8h B5 4g B4_1 16.1共2兲 4.9554共3兲 5.5097共8兲 6.8011共6兲 0.1540共5兲 0.5927共7兲 0.7902共5兲 0.6498共7兲 0.7264共11兲 0.5 0.3358共7兲 0.5082共9兲 0.0 0.8395共5兲 0.7188共7兲 0.8731共5兲 0.6698共7兲 0.9803共10兲 0.0 7.57 8.07 B4_2 22.6共2兲 4.9231共4兲 5.4720共14兲 6.7388共10兲 0.1545共5兲 0.5920共7兲 0.7898共5兲 0.6513共8兲 0.7242共11兲 0.5 0.3355共7兲 0.5071共10兲 0.0 0.8406共6兲 0.7181共6兲 0.8731共7兲 0.6698共7兲 0.9793共10兲 0.0 6.39 7.12 B4_3 30.2共3兲 4.8767共3兲 5.4260共12兲 6.6606共9兲 0.1538共6兲 0.5914共8兲 0.7895共6兲 0.6518共9兲 0.7234共13兲 0.5 0.3359共9兲 0.5066共11兲 0.0 0.8411共7兲 0.7178共9兲 0.8733共6兲 0.6698共8兲 0.9769共12兲 0.0 6.92 8.18 B4_4 35.5共3兲 4.8535共4兲 5.4025共13兲 6.6187共10兲 0.1534共6兲 0.5902共7兲 0.7885共5兲 0.6514共7兲 0.7231共10兲 0.5 0.3371共7兲 0.5082共9兲 0.0 0.8404共6兲 0.7184共7兲 0.8723共5兲 0.6691共7兲 0.9776共9兲 0.0 4.68 5.21 B4_5 37.8共3兲 4.8346共3兲 5.3831共9兲 6.5920共6兲 0.1541共6兲 0.5910共8兲 0.7884共5兲 0.6519共8兲 0.7229共12兲 0.5 0.3358共8兲 0.5072共10兲 0.0 0.8410共6兲 0.7188共8兲 0.8724共5兲 0.6703共8兲 0.9766共11兲 0.0 6.02 6.92 B4_6 45.1共3兲 4.8078共5兲 5.3532共17兲 6.5465共13兲 0.1537共6兲 0.5902共8兲 0.7878共5兲 0.6513共8兲 0.7230共11兲 0.5 0.3367共7兲 0.5072共9兲 0.0 0.8406共6兲 0.7187共7兲 0.8719共5兲 0.6701共7兲 0.9765共10兲 0.0 5.46 5.62 B5_1 0.001 5.0508共2兲 5.6181共4兲 6.9793共5兲 0.1543共4兲 0.5928共5兲 0.7930共4兲 0.6481共6兲 0.7279共7兲 0.5 0.3357共5兲 0.5072共6兲 0.0 0.8386共4兲 0.7195共5兲 0.8737共4兲 0.6695共5兲 0.9834共7兲 0.0 6.61 7.43 B5_2 5.8共1兲 5.0115共5兲 5.5731共10兲 6.9041共12兲 0.1542共4兲 0.5921共5兲 0.7917共3兲 0.6485共6兲 0.7267共7兲 0.5 0.3355共5兲 0.5059共6兲 0.0 0.8390共4兲 0.7195共4兲 0.8733共4兲 0.6701共5兲 0.9820共7兲 0.0 5.58 6.74 B5_3 9.6共2兲 4.9810共2兲 5.5392共3兲 6.8535共4兲 0.1542共5兲 0.5924共6兲 0.7914共4兲 0.6483共6兲 0.7277共8兲 0.5 0.3364共6兲 0.5079共7兲 0.0 0.8392共4兲 0.7198共5兲 0.8731共4兲 0.6699共6兲 0.9813共7兲 0.0 6.28 7.59 B5_4 14.7共2兲 4.9636共2兲 5.5192共4兲 6.8160共5兲 0.1542共6兲 0.5921共7兲 0.7907共5兲 0.6487共8兲 0.7264共9兲 0.5 0.3357共7兲 0.5082共8兲 0.0 0.8392共5兲 0.7193共6兲 0.8728共5兲 0.6710共7兲 0.9798共8兲 0.0 7.86 8.58 B5_5 15.1共2兲 4.9267共2兲 5.4751共4兲 6.7439共6兲 0.1538共6兲 0.5907共6兲 0.7893共4兲 0.6499共9兲 0.7253共8兲 0.5 0.3353共8兲 0.5064共7兲 0.0 0.8403共6兲 0.7184共5兲 0.8722共5兲 0.6718共8兲 0.9779共7兲 0.0 5.42 5.95 B5_6 21.5共2兲 4.8986共4兲 5.4462共9兲 6.6990共11兲 0.1544共4兲 0.5910共4兲 0.7887共3兲 0.6494共7兲 0.7244共7兲 0.5 0.3357共6兲 0.5076共6兲 0.0 0.8401共4兲 0.7188共4兲 0.8717共4兲 0.6712共6兲 0.9779共6兲 0.0 5.65 6.26 B5_7 26.5共2兲 4.8715共3兲 5.4189共6兲 6.6651共7兲 0.1550共11兲 0.5930共13兲 0.7891共8兲 0.6481共18兲 0.7187共16兲 0.5 0.3352共14兲 0.5092共13兲 0.0 0.8389共10兲 0.7146共10兲 0.8732共9兲 0.6707共16兲 0.9818共16兲 0.0 10.17 15.55 B5_8 31.7共3兲 4.8492共5兲 5.3910共10兲 6.6116共13兲 0.1542共5兲 0.5912共7兲 0.7873共5兲 0.6494共7兲 0.7239共9兲 0.5 0.3356共7兲 0.5098共9兲 0.0 0.8402共5兲 0.7194共7兲 0.8714共5兲 0.6712共7兲 0.9756共9兲 0.0 6.79 7.56 B5_9 36.2共3兲 4.8246共6兲 5.3679共12兲 6.5713共11兲 0.1539共4兲 0.5906共5兲 0.7871共3兲 0.6495共6兲 0.7240共7兲 0.5 0.3358共6兲 0.5093共7兲 0.0 0.8409共4兲 0.7189共5兲 0.8713共4兲 0.6710共6兲 0.9758共7兲 0.0 4.63 5.72 B5_10 41.2共3兲 4.8100共2兲 5.3531共5兲 6.5486共5兲 0.1542共5兲 0.5905共6兲 0.7867共3兲 0.6494共7兲 0.7235共8兲 0.5 0.3357共7兲 0.5089共7兲 0.0 0.8408共5兲 0.7182共6兲 0.8711共4兲 0.6716共7兲 0.9754共7兲 0.0 5.13 6.08 B5_11 45.0共3兲 4.7857共3兲 5.3291共7兲 6.5091共7兲 0.1544共6兲 0.5909共6兲 0.7862共4兲 0.6498共8兲 0.7227共8兲 0.5 0.3366共7兲 0.5086共7兲 0.0 0.8407共5兲 0.7180共6兲 0.8709共5兲 0.6710共7兲 0.9749共7兲 0.0 5.02 5.98

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transition,1 _{see Ref.} ₂₃ _{for review兲 at high pressures, Zr}

and24 _Os.25 _{However, established cases of the ISPhTs are}

related to the elements or compounds containing elements with the variable chemical valence due to fully or partially free d- or/and f-electronic shells 共for example, SmS,26

EuCo2P2,27rare-earth chalcogenides28and pnictides, see Ref.

29for review兲, or/and materials which undergo magnetic or spin transitions, for example, MnO共Ref. 30兲 or YCo5.2

Bo-ron is an element of the second row of the Periodic Table and the conventional mechanisms of the ISPhT in its crystal structure are highly unlikely 共indeed, theoretical calculations7,8 _{do not indicate any significant changes in}

band structure of ␥-B28 at least to 100 GPa兲. At the same

time␥-B28is a quasimolecular material. The detailed

single-crystal x-ray diffraction study conducted at ambient pressure7_{revealed the strong polar-covalent bonding with an}

electron density excess at B2 and B4 atoms共they belong to the B12 icosahedra兲 and an electron density deficit at B5

at-oms 共which form B2 dumbbells兲. This suggests that the

ISPhT in ␥-B₂₈ can be a result of changes in the electron density distribution between boron atoms and/or bonds or, in other words, changes in the character of chemical bonding. For compounds with the polar-covalent bonding, the splitting of the IR longitudinal- and transversal-optical 共LO-TO兲 phonon modes can be used as a measure of the degree of the charge transfer,8,9,31 _{which can be characterized}

by the generalized Lyddane-Sachs-Teller 共LST兲 parameter

␰=兿i n_共_␻

i

LO_/_␻

i

TO_兲2_{. Using calculated values of frequencies for}

all IR-active modes at different pressures共Fig.7兲 we found

that above 50 GPa, in strict similarity with experimental data

TABLE III. 共Continued.兲

Run Pressure

共GPa兲

Lattice parameters共Å兲 Atomic coordinates R₁共%兲

wR2共%兲 a b c B1 8h B2 4g B3 4g B4 8h B5 4g B5_12 51.2共4兲 4.7208共5兲 5.2508共12兲 6.4015共14兲 0.1543共11兲 0.5912共12兲 0.7858共9兲 0.6529共17兲 0.7214共14兲 0.5 0.3340共13兲 0.5076共13兲 0.0 0.8401共10兲 0.7168共11兲 0.8703共8兲 0.6731共14兲 0.9729共12兲 0.0 7.54 8.70 B5_p1 65.3共5兲 4.731共3兲 5.280共5兲 6.4285共39兲 0.1533共12兲 0.5891共9兲 0.7844共6兲 0.6521共16兲 0.7204共10兲 0.5 0.3341共15兲 0.5082共9兲 0.0 0.8425共9兲 0.7190共7兲 0.8729共8兲 0.6705共15兲 0.9763共10兲 0.0 6.94 6.91 B5_p2 55.6共5兲 4.772共3兲 5.318共5兲 6.486共7兲 0.1541共2兲 0.5901共6兲 0.7859共4兲 0.6506共9兲 0.7221共7兲 0.5 0.3369共7兲 0.5079共8兲 0.0 0.8415共6兲 0.7190共1兲 0.8712共4兲 0.6723共8兲 0.9760共7兲 0.0 5.55 5.60 B28关7兴 0.001 5.0576共4兲 5.6245共8兲 6.9884共10兲 0.1539共3兲 0.5938共2兲 0.7924共2兲 0.6469共4兲 0.7284共4兲 0.5 0.3362共4兲 0.5076共4兲 0.0 0.8391共3兲 0.7189共3兲 0.8737共2兲 0.6690共4兲, 0.9823共4兲 0.0 3.73% 11.5% Pressure, GPa 0 20 40 60 V/V 0 0.80 0.85 0.90 0.95 1.00

Zarechnaya et al. (2009),γ-B powder in Ne

This study, single crystals ofγ-B in He

Le Godec et al. (2009), K300=237(5) GPa, K'=2.7(3), V0=197.6(1) A3

K300=227(3) GPa, K'=2.5(2), V0=198.1(3) A

3_{(single crystals, P<40 GPa)}

K₃₀₀=281(6) GPa, K'=2.8(9), V₀=192.6(3) A3_{(single crystals, P>45 GPa)}

Pressure, GPa 0 20 40 60 b/a 1.110 1.113 1.116 1.119 f 0.02 0.03 0.04 0.05 0.06 0.07 0.08 F 195 200 205 210 215 220 225 230 235

FIG. 4. 共Color online兲 The relative unit-cell volume of␥-B₂₈as

a function of pressure共red diamonds—single-crystal data from this

study; blue circles—powder-diffraction data共Ref.7兲; lines are fits

with the third-order Birch-Murnaghan 共BM3兲 equation of state:

dark red for single-crystal data below 40 GPa, dark purple for single-crystal data above 45 GPa, and dashed line as reported by Le Godec et al.共Ref.21兲 based on powder x-ray diffraction data to 70 GPa. Insets show the variation in the ratio of the lattice parameters

b/a as a function of pressure and the F-f plot 共normalized stress vs

Eulerian strain兲. Pressure, GPa 0 10 20 30 40 50 60 70 Atom 's Sphere Radius, Å 1.65 1.70 1.75 1.80 B1_B2 B3 B4 B5

FIG. 5.共Color online兲 The average distance of the atom from its neighbors关“sphere radius,” as defined inIVTONsoftware共Ref.22兲兴 as a function of pressure. Continuous lines for B3 and B4 atoms are guide for eyes.

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on changes in Raman spectra and compressional behavior, the LST parameter changes its pressure dependence. This result strengthens our assignment of the isostructural phase transformation in ␥-B28 as due to alteration the polarity of

the covalent bonding between the boron atoms in the com-plex quasimolecular structure.

Boron-rich carbides, nitrides, oxides, and borides of light elements with polar covalent and/or partially ionic bonding represent a large group of materials with prospective appli-cations due to a number of unusual and potentially useful properties.9,17 _{Our finding of the ISPhT in} _␥_-B

28 suggests

that the same or similar mechanism, involving rapid pressure- or chemically induced changes in chemical

bond-ing, could drive isostructural transformations in other boron-rich phases as well.

ACKNOWLEDGMENTS

We thank S. van Smaalen, S. Mondal, A. S. Mikhaylush-kin, S. I. Simak, and I. A. Abrikosov for very useful discus-sions. E.Z., N.D., and L.D. acknowledge financial support of German Science Foundation共DFG兲 and Elite Network Bay-ern 共ENB兲. R.C. acknowledges support from CINES under Grant No. stl2816.

*Author to whom correspondence should be addressed;

natalia.dubrovinskaia@geow.uni-heidelberg.de

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