Submitted Manuscript: Confidential Title: Quantum and isotope effects in lithium metal Authors: Graeme J. Ackland 1, Mihindra Dunuwille 2 Miguel Martinez- Canales 1 , Ingo Loa 1, Rong Zhang 2 , Stanislav Sinogeikin 3 , Weizhao Cai 2 and Shanti Deemyad 2* Affiliations: 1 SUPA, School of Physics and Astronomy and Centre for Science at Extreme Conditions, The University of Edinburgh, Edinburgh, EH9 3FD, UK 2 Department of Physics and Astronomy, University of Utah, 115S 1400E, Salt Lake City, Utah 84112, USA 3 HPCAT, Geophysical Laboratory, Carnegie Institution of Washington, Argonne, Illinois 60439, USA *Correspondence to: [email protected]1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Transcript
Submitted Manuscript: Confidential
Title: Quantum and isotope effects in lithium metal
Authors: Graeme J. Ackland1, Mihindra Dunuwille2 Miguel Martinez-Canales1, Ingo Loa1,
Rong Zhang2, Stanislav Sinogeikin3, Weizhao Cai2 and Shanti Deemyad2*
Affiliations:
1 SUPA, School of Physics and Astronomy and Centre for Science at Extreme Conditions, The University of Edinburgh, Edinburgh, EH9 3FD, UK
2 Department of Physics and Astronomy, University of Utah, 115S 1400E, Salt Lake City, Utah 84112, USA
3 HPCAT, Geophysical Laboratory, Carnegie Institution of Washington, Argonne, Illinois 60439, USA
Abstract: The crystal structure of elements at zero pressure and temperature is the most fundamental information in condensed matter physics. For decades it has been believed that lithium, the simplest metallic element, has a complicated ground-state structure. Using synchrotron X-ray diffraction in diamond-anvil-cells and multiscale simulations with density functional theory and molecular dynamics we show that the previously accepted martensitic ground-state is metastable. The actual ground-state is face-centered-cubic. We find that isotopes of lithium, under identical thermal paths, exhibit a large difference in martensitic transition temperature. Lithium exhibits large quantum mechanical effects, serving as a metallic intermediate between helium, with its quantum-effect-dominated structures, and the higher mass elements. By disentangling the quantum-kinetic complexities, we prove that fcc lithium is the groundstate, and we synthesize it by decompression.
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Introduction
At ambient pressure, lithium is the lightest metal in the periodic table. We might expect that this would lead to simple chemistry of the electrons, and complex quantum behavior of the nuclei. In fact, crystallography shows highly complicated structural behavior, while the isotope differences that are indicative of nuclear quantum effect have remained largely unstudied. Here, we address both these issues presenting a comprehensive picture of how the quantum interplay of the nuclei and electrons leads to surprising and hitherto unsuspected behavior in this simplest of elements.
For light elements, the nuclear zero-point vibrations, a wholly quantum effect arising from the fact that the nuclei are never truly at rest, can be orders of magnitude larger than the very small energy differences between competing crystal structures (1). Quantum ground states are responsible for superconductivity and superfluid behavior, which allow flow of charge or matter without resistance. Quantum effects are especially pronounced in metallic systems of low mass at high densities, leading to extraordinary properties as well as exotic states of matter(2-5). The effects manifest as differences between isotopes. For example 4He is also almost twice as dense as 3He at the same pressure and temperature(6), and has a hexagonal rather than cubic crystal structure.
Away from ambient conditions of pressure and temperature, lithium exhibits unintuitive and complicated behavior including numerous temperature- and pressure-induced phase transitions to low-symmetry structures(7, 8), metal to semiconductor to metal transitions(9, 10), both a maximum and a minimum temperature in its melting line(11, 12), and superconductivity with an anomalous isotope effect(13-16). At room temperature and pressures below 7 GPa, 7Li crystallizes in the bcc structure. When cooled below T 77 K (at P = 0 GPa), bcc 7Li undergoes a martensitic transition to a close-packed structure, identified as the 9R structure which has a nine-layer stacking sequence, previously assumed to be the ground-state structure. It is odd that a simple metal would adopt a complicated atomic arrangement at zero pressure, and despite tremendous theoretical efforts to understand the multifaceted physics underlying the small differences between the energies of various close-packed structures, there is no clear explanation in the literature; e.g. (17-19).
The free-energy landscape of materials often contains multiple local minima, and the critical role of quantum effects in controlling the kinetics of changing between them has only recently been recognized(20). Consequently, care in identifying the P-T paths, along which a phase diagram is constructed is required for a proper determination of thermodynamic states. However, experimentally, the high-pressure and low-temperature structural phase diagram of lithium has previously been constructed based on limited studies and only using 7Li (7, 21, 22). This is mainly because lithium is an extremely challenging material for high-pressure studies. Lithium reacts chemically with most
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materials, and diffraction experiments are challenging due to low X-ray and neutron scattering cross sections.
Structural transitions — experiment
With only three electrons, lithium is a very weak scatterer of X-rays. The low temperature structures of lithium are mainly studied by neutron scattering. However, because of the high neutron absorption cross section of 6Li (940 barn) these studies have been only performed on 7Li (0.0454 barn);e.g.(8, 21, 23, 24). Hence, the P-T structural phase diagram of 6Li was not reported. Moreover, the structural boundaries of 7Li below 80K were unknown. This includes the boundary between the martensite and the fcc phase. Advances in synchrotron beam quality allow exploration of the low temperature and high pressure regions of the lithium phase diagram using diamond anvil pressure cells. (25)
We observed a large difference between the phase transitions in the 6Li and 7Li samples under similar conditions (Fig. 1). This contrasts early ambient-pressure studies (25-27). We found that the 6Li samples do not show any evidence of a martensitic transition or any other structural phase transitions during isobaric cooling between 0.2 and 2.0 GPa (Fig 1A, 2A). They remained in the pure bcc structure down to the lowest temperature we measured (~16 K), regardless of whether we used helium or mineral oil as the pressure medium. Cooling 7Li at pressures below 3.3 ± 0.3 GPa always resulted in the appearance of martensite peaks mixed with bcc peaks below 75 K. For 6Li we observed the first evidence of a transformation from the bcc phase to the martensite at ~20 K and ~2 GPa during both isobaric cooling and isothermal compression. With further pressurization of the 6Li sample to ≥4.5 GPa at T = 20 K, we observed the appearance of additional fcc peaks (Fig. S2).
Under hydrostatic conditions during isobaric cooling to low temperature, bcc 7Li transforms to fcc for pressures higher than 3.3 ± 0.3 GPa. This removes an ambiguity in the previous boundaries of the fcc and martensite states(21). For 6Li the point at which bcc transforms to fcc was above 4.0 GPa (Fig. 1A). The boundaries are based on isobaric cooling, which do not always match up with the pressures from isothermal compression. The fcc structure always remains stable during isobaric cooling (Fig. 1 A, B) and unlike the bcc structure it does not undergo a martensitic phase transition. We found the critical pressure for bccfcc at room temperature was independent of the isotopic mass.
We found that the pressure onset of phase transitions that include low-temperature paths depended strongly on the specific temperature-pressure path taken (25). For example, reaching ~3.5 GPa and ~20 K during isothermal compression of the 7Li sample leads to mixed bcc and martensite phases (data point 3 Fig. 1B), whereas
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during isobaric cooling to the same P-T point, no evidence of the martensite was observed, and instead the sample crystallized in fcc plus bcc (data point 7 Fig. 1B). We found (Fig. 1, S2 and S3) that the martensite phase was only ever obtained from bccmartensite transformations. Although transitions from martensite to fcc were seen, the reverse process never occurred.
Of particular interest is the P-T path shown in Fig. 1D, where we pressurized the bcc 7Li sample at ambient temperature to above 8 GPa to produce fcc and then cycled back across the phase boundary to demonstrate low hysteresis. We then cooled at ~10 GPa to 20 K and depressurized to 2 GPa, accessing the region previously ascribed to the 9R phase. We then warmed the sample to 120 K, which is above the reported 9R phase boundary (21) K and re-pressurized until the martensitic phase appeared above 3 GPa. A similar result was found for 6Li where we synthesized the fcc ground state by pressurizing to 9 GPa, cooling to 20K, then depressurizing to ambient pressure (Fig. 1C).
Explaining the 9R/martensite transition
The 9R structure was believed to be the ground state of lithium based on reproducibility of diffraction patterns from ambient pressure isobaric cooling studies (25, 28-33). The martensite is a well-defined crystal structure and it cannot be dismissed as a highly defective version of something simpler(34). The bcc-9R transition provides evidence that contradicts the 9R structure as ground state as the transition is not reversible. On heating, the martensite transforms to fcc before returning to bcc (Fig. 3A)(35).
We performed well converged free energy calculations using all-electron density functional theory (DFT) and the quasiharmonic approximation for phonons (QHA)(25). Additional calculations using the SCAILD(36) method showed that anharmonic effects are negligible at these temperatures (Fig. S9). The DFT calculations reveal a fcc ground state. The bcc structure is stable at high temperature and low pressure as it has higher entropy and volume. The fcc-bcc transition temperature we calculated increases with pressure (Fig. 3) and the value of 166 ± 10 K at zero pressure is in good agreement with the reported values for the retransformation on heating (8, 19, 23, 34, 35, 37).
The 9R phase is always unstable with respect to fcc and other close-packed stackings such as hcp. Our calculations show that 9R becomes stable relative to bcc at 80 K and 0 GPa, with the phase boundary rising with pressure. To reconcile these DFT results with the experimental observation of 9R, we directly simulated the martensitic transformation. This required modeling very large numbers of atoms with molecular dynamics (MD) which is currently unfeasible using DFT. We therefore derived a bespoke many-body interatomic potential, fitted directly to the relevant DFT properties
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(25). The new potential reproduces the DFT phase behavior (with fcc being the thermodynamically stable phase at low temperature and pressure) as well as the lattice parameters and elastic properties of relevant phases. As with DFT, the potential model shows that the bcc phase is entropically stabilized over fcc above ~150 K.
We used the large-scale MD simulations to simulate the cooling transition in the NPT ensemble. We used 128,000 atoms, starting with either a bcc single-crystal or bcc nanocrystals, and cooled into the region of fcc stability. We observed transformations to a twinned, close-packed structure that is neither fcc nor pure 9R (Fig. 4A). The apparent contradiction between the known fcc ground state of the potential and the observed martensite means that the bcc–fcc transition is kinetically hindered, and that a transition can only occur once all close-packed stackings are favored relative to bcc. Experimentally, the bcc9R transition is always incomplete, so the remnant bcc material in the sample provides nucleation sites for retransformation. Analyzing the stacking sequences of the largest twins leads to the surprising result that the stacking is non-random and has significantly more hcp-like h-layers than fcc-like k-layers. The 9R-like, three-layer hhk motif is prominent (Fig. 4A).
Using the Debye method, we simulated the powder neutron diffraction patterns for the three main twins generated in the MD simulation (Fig. 5). . We compare these simulations with the best ambient-pressure neutron diffraction experimental data (38). Our simulated diffraction patterns for the martensite twins show all of the features observed in the experiment without any fitting. The simulated martensite patterns yield a better approximation of the experimental data than the 9R structure itself. This includes the shift and broadening of the (1 0 4), (0 1 5), (1 0 13) and (0 1 14) reflections compared to the ideal 9R prediction, and the suppression of peak intensities in the 34-48° region. As the stacking sequences extracted from the MD simulation comprise only 21 to 47 layers, the simulated patterns from single twins show signs of finite-size effects, most notably, the (1 0 13) and (0 1 14) reflections in the largest twin pattern are less broadened and shifted more with respect to the ideal 9R pattern than observed in the experiment. However, the three-twin average shows already a much better agreement with the experiment. Fits to the experiment of similar quality to ours can only be achieved by postulating an appropriate concentration of stacking faults within the 9R structure.
The excellent agreement between the simulated and experimental patterns shows that the martensite obtained in the MD simulation correctly describes the experimentally observed structure. The discrepancies between the ideal 9R diffraction pattern and the experimental data have been noted before, and can be explained by postulating an appropriate concentration of stacking faults within the 9R structure. However, the key point is that our patterns are generated ab initio, without adjustable parameters or any reference to 9R. Our structure arises from the transformation kinetics, not the
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thermodynamic stability. Absence of reverse transformation from fcc to bcc or a martensite structure during the isothermal decompression at low temperature is precisely what we theoretically predict to occur (Figs. 1C and 1D). We conclude that the diffusive bcc→fcc transition has no low-energy path, so instead bcc undergoes a martensitic transition with large hysteresis on both isobaric and isothermal paths. In other words, not only is the previously reported martensite not thermodynamically stable, it is does not have the 9R crystal structure either.
Isotope and quantum effects
In classical thermodynamics, free-energy differences between phases are independent of the nuclear mass, but at low temperatures, quantum effects can lead to differences in behavior for different isotopes since both vibrational and zero point effects are mass dependent. In lithium, the zero-point energy is large at ~40 meV/atom, equivalent to
500 K. The zero-point energy is larger in 6Li than 7Li by a factor of
7 6 . Early ambient-pressure measurements showed that at low temperature, 6Li has a slightly larger bcc lattice constant (27). Also, isotope effects in the shear modulus in lithium under pressure are evidence of the quantum contribution (3, 39). Although the large zero-point energy of lithium has a large contribution to the vibrational energies and may influence its equilibrium structures (3, 33), the difference in the martensite transition line during isobaric cooling paths of the two isotopes is unexpected.
Entropy differences between phases ultimately drive thermal phase transitions, but the kinetics of the transition can prevent some transformations happening. The MD simulations demonstrated that the martensitic transformation path is complicated by the need to generate many different stackings. We postulate therefore that the transformation will begin only when all close-packed stackings are stable against bcc. Because 9R is the least favorable stacking we have found, we associate the observed martensitic transition with the P-T conditions where 9R becomes more stable than bcc (Fig 3B).
We approached this issue with DFT calculations and the QHA for the vibrational properties. The fcc equation of state (Fig. 6) allows a direct comparison between measurements and calculations uncompromised by hysteresis, where we find that 6Li is at a larger pressure for a given lattice parameter at low temperature (Fig. S11), consistent with ambient-pressure experiments (27), and that it has a lower compressibility than 7Li. Zero-point effects contribute to the pressure, so that a higher pressure is required to compress 6Li to the same volume as 7Li. Our calculations show this implies a difference in the equation of state of around 25 MPa. The 9R structure is always thermodynamically unstable with respect to fcc, while bcc is stabilized at higher temperatures by vibrational effects. However, the isotope effect only shifts the phase
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boundary by a few degrees, so even with quantum corrections thermodynamics alone cannot explain the survival of bcc to zero temperature in 6Li. Martensitic transitions were reported to be sensitive to impurities in several systems (40, 41). While our samples show very similar impurity levels(25), contribution from slight differences between the impurity levels of the samples, below our resolution, cannot be entirely excluded. However, the similarity in the martensitic transition of 7Li and natural lithium samples, regardless of their source and purity levels, reported by many groups, indicates that the observed phenomenon here is mass-related.
To understand the extent that mass can impact the stability of lithium, we have looked at the extreme, unphysical case of 3Li and 14Li (neither exists naturally). While they do not describe a physical system, they provide a remarkable insight. Whereas the bcc–fcc transition shows only a very small mass dependence, the bcc–9R transition shows a very strong mass dependence, and the 9R phase would be entirely suppressed for 3Li up to 1.2 GPa (Fig. S14). These results qualitatively resemble our experimental observations for stable isotopes of lithium, albeit for unphysical masses. The stability line obeys the third law of thermodynamics, which requires that the phase boundary must become vertical if it crosses the T = 0 K axis. This makes it extremely sensitive to small changes in the energy difference between bcc and martensite: If the real martensite has a higher enthalpy than 9R, then the ZPE may be enough to destabilize it in 6Li (20).
Metastability Regions
Although 9R is not a stable phase of lithium, and can only be obtained starting from bcc, the conditions where this occurs are reproducible and isotope-dependent. Under hydrostatic conditions, the transition line bcc→9R is suppressed to lower temperature and higher pressure in the 6Li samples. By contrast, the transition line 9Rfcc appears to be at higher pressure in 6Li: As discussed above, the 9R→fcc crossover pressure for isobaric cooling is 3.3 ± 0.3 GPa for the 7Li and at least 4.0 GPa for the 6Li material. This suggests that the role of quantum effects is to inhibit the transition rather than to destabilize the 9R phase.
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Combining these observations with the DFT results suggests that there are two important lines on the phase diagram (Fig. 3B). The true thermodynamic phase boundary between bcc and fcc, and a line designating conditions at which bcc is unstable with respect to any close-packed stacking sequence. The second line denotes the onset of the martensitic transformation mechanism, and it is the one observed in experiments. The ground state of lithium can be synthesized if this martensite line is avoided via a high-pressure path.
The Structure of Lithium
The experimental and calculated properties of lithium can be seen in a consistent manner which rewrites the understanding of this simplest of metals. In both 6Li and 7Li, the ground state structure is fcc rather than 9R. A combination of zero-point energy and vibrational entropy stabilized the bcc structure where it exists. Only the transition from bcc creates a metastable martensite as it does not arise from the structure’s thermodynamic stability. A natural question that arises is whether the 9R state observed in other elements (i.e. Na, Sm) and alloys are thermodynamically stable phases.
In contrast to 7Li, we observed that in the 6Li samples, the bcc structure remained stable compared to the martensite to the lowest measured temperatures for pressures up to ~2 GPa. The difference between 6Li and 7Li cannot be explained by quantum stability analysis, and it indicates that quantum effects may play a crucial role in the transition kinetics. By circumventing the quantum-kinetic barriers of the martensitic transformation, after 70 years of trying, we have finally synthesized the ground state of lithium.
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Fig. 1Observed phases of 6Li and 7Li measured along the identified P-T paths. (A) Isobaric cooling paths are connected by gray lines as guides to eye. Data points we collected during isothermal compression or isobaric warming are labeled in numerical order. We used mineral oil (crossed symbols) or He (dotted and solid symbols) as pressure transmitting media. Blue dotted lines show the onset of the bcc to close-packed transitions on cooling. (B) isobaric results for 7Li. Open symbols are data from previous studies measured either using mineral oil or no pressure medium during isothermal compression and isobaric cooling (7, 21, 42). Points 3 and 7 are very close in P and T but were approached via different thermal paths — the resulting structures are 9R + bcc vs. fcc + bcc, respectively. (C) Experimental paths for 6Li in P-T space to examine the possibility of a reverse phase transition from fcc→9R during decompression. Dotted lines are the transition lines from (A) and (B). During decompression, we observed the pure fcc structure deep in what was previously identified as the 9R stability region. (D) Experimental paths for 7Li in P-T space with the same observation of the fcc structure in the 9R stability region. Points 12–14 show the martensitic transition of 7Li during isothermal compression, followed by a transition to fcc.
Fig. 2Synchrotron X-ray diffraction patterns of 6Li at variable pressures and temperatures. The angle-dispersive diffraction measurements were performed using a wavelength of 0.4066 Å. (A) Selected diffraction patterns of 6Li from three different cooling paths. (Points 1→4, 8→10 and 11→13 in Fig. S4.). The reflections from bcc (red) and martensite (blue) phases are labelled by their hkl indices, using the 9R structure for the martensite. Not all 9R peaks are visible because the sample recrystallized to a highly textured quasi-single-crystal. (B) Diffraction patterns of 6Li during cooling to the base temperature 17 K and isothermal decompression to 0.5 GPa (points 18→22 in Fig. S4). Only pure the fcc phase (green) was observed (Fig. 2). For clarity, the Compton scattering of the diamonds and reflections from the cryostat window have been removed in both panels A and B.
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Fig. 3(A) Experimental observations of bcc, fcc and martensitic (9R and disordered) polytypes of 7Li on cooling and warming at zero pressure (8, 19, 34, 35, 37) (B) Calculated thermodynamic phase boundary between bcc and fcc for 7Li, and metastable bcc–hcp and bcc–9R phase boundaries. Small symbols indicate the DFT results, and the corresponding lines are interpolations. White lines indicate the experimental transition lines of 7Li during isobaric cooling from this work and (21). Large symbols show the observed phases on isothermal pressure changes (Fig. 1D). We can relate the calculated phase boundaries to the temperature-driven transitions at zero pressure (gray bars connecting (A) and (B)).
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Fig. 4(A) Complex, non-9R stacking sequence in the martensite from the MD simulations, viewed along the [111]bcc body diagonal from the initial bcc crystal. The close-packed layers are shown edge-on, exposing the stacking sequence. Atoms with a local hcp (h) or fcc (k) environment are shown in dark purple or cyan, respectively (8). Other colors mark atoms without close-packed coordination at cell and grain boundaries. There are three separate twins, and a region of pure fcc, which is a known finite-size effect (19). Such fcc regions form only in MD cells with orientations incompatible with a two-twin microstructure. MD simulations in supercells rotated 45o about [100]bcc produced microstructures with only two twins, nanocrystals produced ultrafine twins which anneals slowly, and simulations with high cooling rates remained in bcc. (B) Relationship between fcc, hcp, 9R and bcc. (Top) The close-packed structures can be identified by their stacking sequences of hexagonal layers along the z-axis: In the ABC notation, the letters refer to different atomic positions in the xy-plane. The hk notation removes the arbitrary choice of origin and labels a layer as hexagonal (h) if the layers directly above and below the central layer are of the same type, and as cubic (k) otherwise. The sequence ABACACBCB for the 9R structure translates to hhk hhk hhk. Unit cells are indicated with dark dashed lines. (Bottom) Illustration of the martensitic mechanism in which the bcc (110)bcc layers of atoms transform into close-packed (001)hcp layers and shuffle into the appropriate stacking sequence, shown here for the bcc-to-hcp transformation. The bcc-to-fcc transition, requires a substantial shear strain parallel to (110), compared with the transition to 9R.
Fig. 5Comparison of neutron diffraction data (wavelength λ = 1.288 Å) from 7Li at T < 20 K and zero pressure(38) with simulated diffraction patterns from: various candidate phases of lithium; a random sequence of close-packed layers; the stacking sequence of the main twin obtained in the MD simulation; and an unweighted average of all three stacking sequences observed in the simulated martensite shown in Fig. 4A. The neutron diffraction pattern is a combination of bcc and martensite, the main 9R and the bcc peak positions are shown by the tick marks at the bottom. The Miller indices refer to the 9R structure.
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Fig. 6 Atomic volume of the fcc phase of 6Li and 7Li at low temperature as a function showing agreement of experiment and theory. The orange and gray lines indicate the third-order Birch-Murnaghan EOS fits to the data from 7Li and 6Li respectively in the region where data obtained for both isotopes (shaded region). Open symbols show the calculated EOS for the lithium isotopes in fcc phase.
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Acknowledgments
Extensive assistant was provided for data collection by J. K. Hinton and M. C. MacLean and for data analysis by T. Taylor. We would like to acknowledge Dr. S. Tkachev and GSECARS for providing the helium gas loading and C. Kenney-Benson in 16-ID-B for tremendous experimental support. The experimental works were performed at HPCAT (Sector 16), Advanced Photon Source (APS), Argonne National Laboratory. Beam time for these experiments was provided by the Carnegie-DOE Alliance Center, which is supported by DOE-NNSA under grant number DE-NA-0002006. HPCAT operations are supported by DOE-NNSA under Award No. DE-NA0001974 and DOE-BES under Award No. DE-FG02-99ER45775, with partial instrumentation funding by NSF. The APS is a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357. The research in University of Utah was supported by National Science Foundation-Division of Materials Research Award No. 1351986.The research at the University of Edinburgh is supported by EPSRC with computing time on EPCC’s ARCHER supercomputer (UKCP grant K01465X). GJA was supported by an ERC fellowship “Hecate”' and a Royal Society Wolfson fellowship. This work also used resources provided by the Edinburgh Compute and Data Facility (ECDF, www.ecdf.ed.ac.uk; the ECDF is partially supported by the eDIKT initiative (www.edikt.org.uk).
SupplementaryTextMaterials and MethodsFigs. S1 to S15Tables S1 to S5
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Supplementary TextHistory of the 9R phase in LithiumThe low-temperature, ambient-pressure state of lithium has been reported to have
the 9R crystal structure in many previous experiments. In this section we critically review this work, and explain why these studies do not prove that 9R is the ground state.
In 1947, lithium was first observed by Barrett to have a martensitic transformation on cooling from its room-temperature bcc phase to a close-packed structure (8). Barrett noted that the transformation “is accompanied by a series of audible clicks, as in the twinning of tin or magnesium and the formation of martensite.” The structure of that phase, and the explanation for its stability, has proved controversial ever since. The initial fcc and hcp identifications were convincingly disproven in a neutron scattering experiment that showed the absence of the hcp (011) and (012) lines, or the fcc (200) line(19). In 1984, re-examination of these data led Overhauser(43) to conclude that the low-temperature phase has the 9R crystal structure, a belief which has gone unchallenged for many years. The relations between the different structure types and the mechanism of the martensitic transition are illustrated in Fig. 3B of the main paper.
The experiments are complicated by the fact that the transition is martensitic, which gives rise to a coexistence of a residual bcc phase and twinned martensite crystals. The structure anneals slowly: McCarthy(19) reported that some peak intensities nearly doubled over the course of 24 hours after cooling to ~4K. Annealing processes are inevitably slow at such low temperatures, and considerable hysteresis can be expected.
Smith(34) reported again a partial bcc9R structural transformation near 75 K on cooling a single crystal, with the sample returning at 170 K to a bcc crystal in the original orientation. Together with the coexistence and hysteresis, the well-defined orientation relation between the bcc and 9R lattices provides conclusive evidence that the bcc9R transition is martensitic. In 1990, Schwarz(35) also reported observation of the 9R structure at around 80 K upon cooling, coexisting with a ”disordered polytype'” (and bcc). On warming, they found that the 9R component transformed first to fcc at 120 K, before it returned to bcc above 150 K (Fig. 2A).The most detailed analysis of neutron diffraction data(34, 35, 37) shows clear discrepancies between the martensite pattern and that expected from the 9R structure. In order to obtain a satisfactory fit, it is necessary to assume that the stacking sequence is even more complex. The pattern can be made to match the experimental data by using extra parameters describing stacking faults of various types. However, a high stacking fault density is required so that the average distance between faults is shorter than the 9-layer repeat distance of the 9R structure. These analyses focused on trying to explain and fit the observed pattern, assuming that the stable structure is 9R, and the stacking faults were defects in the 9R structure, rather than representing an even more complex stacking sequence.
The existing studies on the properties of the lithium isotopes in the martensitic region of their phase diagram predate Overhauser’s proposal of the 9R structure, and are limited to zero pressure. In 1959/60, Martin reported anomalies in the temperature dependences of the specific heats of 6Li and 7Li samples during heating, which occurred between 90 and 170 K with a pronounced peak near 105 K in both cases(26, 27, 44). These anomalies were attributed to a reverse transition from the martensite. Martin concluded that the details of the transformation appeared to be very similar in both isotopes (26, 27), but that in 6Li a somewhat smaller fraction of the bcc phase appeared to transform to the martensite(45) and that the transformation ended at a slightly higher temperature (26, 27). Later experiments by Schwartz and Blaschko on 7Li(35), as discussed above, showed the appearance and growth of fcc reflections at the expense of the 9R reflections during heating above 80 K. The only experiment on the low temperature crystal structures of 6Li was reported by Kogan in 1963(27), which indicated that diffraction peaks corresponding to a “hexagonal” structure were observed in both isotopes, and diffraction patterns with similar phase fractions were used for a comparison of the bcc lattice parameters. Diffraction data were collected at three temperatures and the lattice parameters for the bcc phase reported, showing that the bcc 6Li sample had slightly larger lattice parameters than 7Li. However, in this study, diffraction patterns of the samples at low temperature were not presented, and no attempt was made to resolve the crystal structure of what was thought to be the martensite.
Despite the lithium system being ideally suited to DFT calculation, numerous previous works have avoided publishing the relative stability of 9R and fcc (29-33, 46, 47). As we show here, such calculations would give fcc as the stable phase.
A possible reason for the stability of such a complex structure was advanced by Ashcroft(1): interactions between the fermi surface electrons and the Brillouin zone could open pseudogaps and lower the energy. The implication was that the more complex structure would have a more complex Fermi Surface, and hence more opportunity for this effect, which is a common cause of complexity in nearly-free electron materials. However, detailed calculations of the electronic structure close to the Fermi surface show that the pseudogap effect is both small and also present in fcc, and the effect on the free energy difference is negligible. (Fig S13)
More recently, transitions between the bcc, fcc, and 9R phases of 7Li were mapped out as a function of both pressure and temperature (7, 21, 22). Below 3 GPa, previous results for the bcc9R transition line were reproduced. However, starting from the fcc phase at 7 GPa and room temperature, no fcc9R transition was observed on cooling down to temperatures as low as 7 K.
Synchrotron Diffraction with Diamond Anvil CellThe crystal structures of both lithium isotopes were mapped out as a function of
pressure and temperature using X-ray angle dispersive diffraction by the High Pressure Collaborative Access Team (HPCAT) at the Advanced Photon Source (APS). In-situ diffraction patterns of samples with 99.99% lithium content were collected using a ~30.5 keV X-ray beam (λ = 0.4066 Å). The DAC was rotated by 20° at a rate of 0.25°/s and the data were integrated in 83 s exposure time. The samples were isotopically enriched 7Li (99.9% 7Li obtained from Oak Ridge National Labs with metallic trace of Ca(<195ppm), K(<75ppm) and Na(<300ppm)) and 6Li-rich (95.6% 6Li and 4.4% 7Li together with metallic trace elements of Ca(<280ppm), K(<100ppm) and Na(<240ppm) ; Sigma-Aldrich) and formed polycrystalline patterns (Fig. S2). The samples were checked independently for contaminations and impurities by several methods. We used x-ray photoelectron spectroscopy (XPS) on freshly cut samples and found the samples to be free of nitrogen and trace metallic impurities. Trace of oxygen and carbon, with similar values was found on the first few nanometer surface layers of all of our samples (which were similar to the stainless steel stub holding the samples). This layer was rapidly removed after sputtering ~15nm off of the surface. In addition we also checked for metallic impurities using inductively coupled plasma mass spectrometry (ICP-MS, Agilent 7500ce) on all of our samples which was found to be very close to those provided by suppliers and similar for all of our samples. We also verified the isotopic enrichment of the samples, consistent with supplier information. All samples showed
excellent metallic conductivity and similar residual resistivity ratio (R (298 K )R(10 K )
>1000 ¿,
indicative of similar metallic purity level. The isotopes of lithium used here have different level of enrichment, however earlier resistivity studies of lithium with different isotopic mixing ratios show that isotopic mixing does not act as defect in lithium but it modify the phonon spectra (48).
Rhenium or stainless steel was used for gasket materials. Conical diamond anvils were used allowing large angular access of ~70° to the sample(49). In some of the experimental runs, the diamonds were coated with Al2O3 to prevent the possible contact between lithium and diamond and inhibiting the diffusion of helium to the diamonds, which would cause damage to the anvils. The alumina film coating was carried out in the Utah Nanofab first by etching the diamonds in argon plasma in Oxford Plasmalab 80 PECVD dry etching machine followed by deposition of 15nm of Al2O3 on the surface using Cambridge Fiji F200 atomic layer deposition instrument.
To prevent contamination all the sample preparation was performed inside high-purity argon glovebox (O2 and H2O <0.1 ppm) and samples never been exposed to the atmosphere even for short period of time. The mineral oil on the samples was removed by multiple steps rinsing in ultra-high purity pentane. The surface layer was then removed using sharp razor blade. The samples with mirror-like appearance (~50 micrometer thickness and 100 micrometer diameter) were cut and loaded in the gasket together with pressure markers (ruby and NaCl). Helium gas loading was done in GSECARS sector of APS. The experiments were conducted during multiple beamtimes in APS beamline 16-ID-B. Samples were pressurized using double membrane DACs cooled in a liquid helium cryostat. Both helium and mineral oil were used as pressure transmitting mediums in different runs. The diffraction pattern had better quality when helium was used as a pressure medium in all cases. However, unless noted, no appreciable differences in the structures of the samples under otherwise similar conditions were detected. In the lowest pressure isobaric cooling path (0.2-0.5GPa) we used a twin chamber design (Fig S1). In this design both 7Li and 6Li are loaded in the same gaskets but in two different chambers. This would allow following identical thermal path. During this run we used mineral oil as pressure medium and noticed that the martensitic transition occurs in 7Li but not in the 6Li down to ~20K. To eliminate the possible effect of non-hydrostatic conditions at higher pressures we performed runs using helium as pressure medium in traditional single-chamber gaskets. We monitored the pressure constantly and kept the rate of cooling very similar (Fig. S2). Unfortunately using twin chamber design is complicated when helium is used as pressure medium since the volume collapse of helium is very large and drilling symmetric holes of proper initial size to both accommodate stability of the gasket and provide sufficient final volume of the sample is difficult and for this reason these measurements were performed using a single hole gasket.An online ruby spectrometer was used to monitor the pressure of the sample at each point. In addition, EOS of NaCl was used to confirm the pressure. A double membrane configuration and periodic monitoring of the pressure and temperature allowed isobaric cooling conditions(50). We explored numerous P-T paths for each sample and carefully monitored the pressure and temperature throughout the experiments. However, the fine tuning of pressure was not always possible and, at some pressures, paucity of data can be observed. Therefore, the analysis is done based on the most similar paths.The rate of temperature change has been kept very slow and comparable for proper isotopic comparison studies. The cooling rate has been kept less than 1.5K/min between room temperature and 150K. Below 150K the rate was decreased to <1K/min. At every reported point below 150K the sample was kept at least for 30 minutes. For isothermal compression/decompression, controlling the exact rate of compression/decompression was more difficult. Overall rate of decompression at base
temperature was <0.5 GPa/hr. The compression rate for isothermal compression of 7Li at 100K was ~1 GPa/hr. Figures 1, S4 and S5 show the paths across which the data has been collected in numerical order. Data collected with different pressure mediums are plotted in separate panels. Despite the high quality of the beam and optimized experimental setup, we could not always observe all the peaks from lithium and refinement of data was not possible. This was due to the weak scattering of lithium, geometric restrictions for rotating the DAC with the sample inside the cryostat, and the presence of a scattering background from the cryostat and the diamond anvil cell. The structures were determined by assigning the lithium diffraction peaks to those of bcc, fcc, hcp or 9R structures. Both fcc and 9R have similar d-spacing (or 2θ) values for most of the strong reflections. The most distinct difference between the reflections from the 9R and fcc phases is a single strong reflection that was only observed in one of the structures. For the fcc structure, it is the (200) reflection that is expected at 10.595° (2θ, when λ = 0.4066 Å, P = 0 GPa), where as in 9R it is the (104) reflection that is expected at 9.604° (2θ, when λ = 0.4066 Å, P = 0 GPa).
Thermal paths and equations of state of different structures of lithium isotopesStructural phase transitions of 6Li and 7Li above 3 GPa were measured using helium
as a pressure medium along comparable thermal paths. Our data suggests that the 9Rfcc phase boundaries of the two isotopes is located at a slightly lower pressure for 7Li. Upon isobaric cooling from 3 GPa (using helium as pressure medium), we start seeing a mixture of bcc and 9R below 100 K for both isotopes. The fraction of 9R increases during cooling, similar to the behavior of 7Li at ambient pressure. Initial isothermal compression at 20 K, to 3.5 and 4 GPa for 7Li and 6Li respectively, results in a mixture of mainly 9R with residual bcc. For 7Li, annealing at 3.5 GPa to 150K leads to the appearance of weak fcc peaks together with bcc and martensite peaks (Point 4 Fig.1B main text). Notably, cooling down from this point leads to disappearance of the 9R peaks and only bcc and fcc peaks can be observed (Fig. 1B of main paper, points 5-7) There were slight differences after this point between the thermal paths of the two samples. However, it is clear that the bcc 7Li sample transformed to the martensite upon cooling at pressures below 3.3 ± 0.3 GPa and to fcc for higher pressures, whereas in the 6Li sample, the transition to the martensite was found to occur up to a higher pressure of at least 4.0 GPa (Fig. 1A). For 7Li, isothermal compression of bcc at 100 K led to a martensitic phase transition of mixed bcc+9R phases at 1.8 GPa. The mixed bcc+9R phase was detected up to ~5GPa. In the pressure range of 1.8-3GPa, the bcc peaks remained stronger than the 9R peaks; while for 3-5GPa, the 9R peaks became stronger. Upon further compression to ~5 GPa, the 9R peaks disappeared and strong fcc peaks mixed with bcc were observed.
Large thermal hysteresis, even under hydrostatic conditions in helium, for the martensitic transition were present and a residual martensite phase could be observed after warming to 200K at 4GPa (Fig. S2).
Theoretical details
Empirical potential generationWe use the Finnis-Sinclair form for the potential
U=∑i
F(¿ ρi)+12 ∑
i , j≠ iV ij (rij)¿ (1)
with
ρi=∑i ≠ j
ϕ ij (r ij ) . (2)
Using cubic splines for the functions, 9 splines for the pairwise part V (r ) and 4 splines for the hopping integralsϕ (r ), we fitted explicitly 13 relevant quantities obtained in the DFT calculations: lattice parameters, elastic constants, and vacancy formation energies of fcc and bcc Li as well as the energy differences between the bcc, fcc, hcp and 9R phases. The vacancy formation energies were included because the transition may create different coordination. The spline points were adjusted to smooth the functions, leaving 13 data points and 13 parameters which can be combined to give a unique linear fit.(51) The potential was tested for transferability by calculating the phonon spectra for the fcc, bcc and hcp phases. This confirmed the stability of all phonon modes, reproduced the correct relation between the Debye frequencies and showed that the bcc phase has a lower Debye temperature than fcc. Quasiharmonic free energy calculations suggest a bcc–fcc transition temperature of 150 K at 0 GPa, increasing with increasing pressure, in agreement with the DFT results (Fig. 3B). The fcc structure is stable at all temperatures and pressures with respect to any alternative close-packed structure.
MD simulationsThe MD simulations were carried out using the MOLDY code (52) in the NVT
ensemble using systems with 128,000 atoms at 50 K. We started with either single crystal or polycrystalline bcc arrangements which had been previously equilibrated to the correct volume using NPT ensemble calculations. The simulations that had been started with single crystals were found to produce the largest twins, which were therefore used in the analysis. The phase transition was observed to proceed by the (110)bcc layers becoming close-packed layers, and the strain being accommodated by
twinning. Despite the system size, the single crystal samples produced twins which spanned the periodic boundary conditions. Analysis of the “local crystal structure” using the BallViewer software (53) revealed, surprisingly, that the twins contained more hcp like ABA (h) stacking than the lower-energy fcc-like ABC(k) stacking. The stacking sequences bear little resemblance with the 9-layer repeat associated with the 9R structure. However, the stacking sequences are far from random (Fig. 4), with hhk and hh motifs dominant.
X-ray diffraction pattern simulation Neutron diffraction patterns were simulated with the Debye method (54) using the
DEBYER code(http://github.com/wojdyr/debyer). In this approach, the structure factor is calculated from the distribution of interatomic distances in a finite arrangement of atoms rather than from the atomic positions in an infinite and perfectly periodic crystal as is common practice. The Debye method allows us to calculate the powder diffraction pattern from the atomic positions in the MD simulation box without imposing periodic boundary conditions and from the finite-length stacking sequences without imposing an artificial long-range periodicity along the stacking direction, which avoids artefacts in the diffraction patterns. The peak widths depend on the size of the simulated region: a good match to the neutron data was found with 128,000 atoms. The experimental widths are limited by instrumental resolution, implying that the twins in the experiment were at least this size.For the simulations shown in Fig. 5, ~128,000 atoms were placed in spheres with a diameter of 173 Å to create nanoparticles with the structures of the various simple structure types and according to the three stacking sequences identified in the MD simulations. All nanocrystals had the same atomic density as the bcc phase with a lattice parameter of a = 3.4769 Å. For the rhombohedral and hexagonal phases, axial ratios corresponding to the experimental value for Li-9R (38) were assumed, i.e., c /a=1.630 for hcp. The effect of thermal motion of the atoms (Debye-Waller factor) was not included in the simulated diffraction patterns.
The powder diffraction pattern from the simulation cell as a whole was also calculated, but found to be significantly broadened relative to the experiment (Fig. S7). This may be due to the small size of the martensitic twins, the disordered grain boundaries or interference between twins.
Density Functional Theory calculationsAll DFT calculations were performed using the planewave pseudopotential method
as implemented in QUANTUM-ESPRESSO(55) version 5.2.1. Lattice dynamics calculations were performed with perturbation theory for all phases. For the
quasiharmonic approximation, we ran 10 calculations per phase between -0.75 GPa and 5 GPa. All DFT calculations were performed using the PBE functional(56).
Convergence parametersWe have used the GBRV 1.4 ultrasoft pseudopotential (57) for lithium, which has a Δ-
factor (58) of 0.014 meV. All calculations have been performed using a 60 Ry cutoff for the wavefunctions and a 480 Ry cutoff for the charge density. This guarantees total energies to be converged to within 1 meV, formation energies converged to within 0.1 meV, and bcc–fcc energy differences converged to 1 μeV, as seen in Fig. S7. The bcc and fcc stresses are converged to better than 0.1 kbar. The formation energy is a worst-case scenario in terms of energy convergence; the chemical environment of 9R, hcp and fcc Li is very similar, and we expect the energy difference to be converged to close to 1 μeV. All the studied structures are metallic, so the electron occupation has been modelled via Marzari-Vanderbilt cold smearing(59) with a 0.02 Ry width. We used suitably dense meshes for both Brillouin zone integration (roughly 0.015 Å-1) and interatomic force constant interpolation. We also performed selected calculations with denser grids, to make sure the phonon free energies are converged to better than 0.05 meV/atom. The meshes can be found in table S2.
Elastic constants
The DFT elastic constants have been used as fitting parameters for our classical potential. We have obtained these via the stress-strain relations of volume-conserving distortions, except for C11, which has been found using the known relationship between
the bulk modulus and the elastic constants in cubic systems: B=C11+2C12
3. Our values
at zero DFT pressure are shown in Table S3.
Vacancy formation energies We have computed the vacancy formation energy for bcc Li in 33, 43 ,53 supercells (53 to 249 atoms), both in a fixed-cell and relaxed-cell environment. The values we obtain are consistent with the literature: E f=0.54 eV and Er=0.52eV . This is roughly one third of our computed cohesive energy, 1.60 eV .
Evolution of phonon dispersion with pressureFor all 4 structures considered, we have computed the phonon dispersions at 10 different volumes. The DFT pressures for bcc Li at the chosen volumes are: -0.75, -0.5, -0.25, 0.0, 0.5,1.0, 2.0, 3.0, 4.0 and 5.0 GPa. This is dense enough to allow both numerical derivatives and fits with Birch–Murnaghan equations of state. The chosen values guarantee data points with negative pressure at 300 K at in all the considered
phases. The evolution of the different phonon dispersions are depicted in Figs. S10 and S11.
Quasiharmonic ApproximationAs stated in the main paper, we have extended the phase diagram of Li to finite temperatures via the quasiharmonic approximation (QHA). From the lattice dynamics calculations above, we have obtained the zero point energy (ZPE) and the phonon free energy at finite temperatures. Within the QHA, after discarding thermal effects on the electrons, the Helmholtz free energy is given by
F (V ,T )=EDFT (V , T )+Fvibr (V ,T )=EDFT+∑k
[ 12ℏωk+K B T ( log (1−e
−ℏωk
K BT ))]=EDFT+KB T ∑k
log [2 sinh( ℏωk
2K BT )] (3)
where log denotes the natural logarithm. The correct thermodynamic potential to use at finite pressure is the Gibbs free energy, for which we need to obtain the pressure given by
P (V )=−∂ F (V , T )∂ V |
T (4)
One can either opt to fit an equation of state to the data produced or compute the derivatives numerically. We have opted for the latter approach. However, as seen in Table S4, opting for an EoS such as Birch–Murnaghan produces essentially the same results. The Gibbs free energy is now given by
G ( p ,T )=F (V ( p ) , T )+ pV (5)
and the transition curves are given by the crossing of the Gibbs free energies of the different phases. We estimate the numerical error in our procedure to be about 0.1 meV. This error corresponds roughly to ±10 K in the transition temperature.
Zero point energy and equation of state
We calculated the isotope effect on the equation of state within the quasiharmonic approximation. The larger zero point energy of 6Li causes it to have a slightly larger volume for a given pressure. Although the zero point energy itself is substantial in Li,
731732733734735736737738739
740
741742743744745746747
748
749750751752753754755
756
757758759760761762763764765766
some 40meV, the difference between the isotopes is only 8% (√ 76−1) of this, which
gives a very rough estimate of the size of the expected effect. In fact, although zero point vibrations make bcc more favorable in 6Li, this is partly cancelled by thermal vibrations which are substantial by 150 K. The isotope effect on the bulk modulus is negligible. Our calculations show an unusually low value of B' 0 ≈ 3 compared with the typical value of 4 adopted for second-order Birch-Murnaghan, though still higher than the free electron gas value of 5/3.
767
768769770
771
772773774
Fermi SurfacesA qualitative argument for the stability of 9R over bcc comes from the opening of pseudogaps at the Fermi Surface, which lowers the energy relative to a free electron system(1, 29) .We calculated these gaps, and show them in Fig. S15. The bcc Fermi surface does not touch the Brillouin Zone boundary, but both close packed structures do, eight times. The gaps in 9R are larger than in fcc, but because 9R has a 3-atom primitive cell there are only 8 gaps per three bands, not per band as in fcc. The mean gap opening per band is then 15 meV in fcc, and 17 meV in 9R. The distortion from the sphere is slight, showing that only a small fraction of electrons, less than 0.1%, are located in these regions, so the effect in stabilizing 9R over fcc is in the hundredths of meV. This is insignificant.
Dependence on Exchange-correlation Functional
Our calculations use the PBE generalized gradient functional, which has become a de facto standard for DFT calculations. We tested two other functionals, the Local density approximation and the PBEsol re-parameterization of PBE, in each case also regenerating the pseudopotential with the same settings.
With all functionals, the same sequence of phases and structural stability is observed, fcc is the ground state, 9R is metastable, and bcc has softer phonons which stabilize it at some finite temperature. Energies agree to within hundredths of meV. The results for free energy differences and Birch-Murnaghan fits to static lattice calculations are given in the table below.
As is well known, LDA has an overbinding effect which makes the LDA “zero pressure” equivalent to about 0.5GPa when considering lattice parameter, bulk modulus and phonon spectrum. Consequently the LDA phase diagram has the fcc-bcc line shifted to lower pressure by about this amount. The PBEsol data is very similar to the PBE used for the main calculations. Our results are robust to choice of functional (Table S5).
Fig. S1. Left image shows a micrograph of the helium loaded gasket with piece of lithium and ruby inside the helium. On the right 2-D x-ray image of the lithium isotope samples loaded inside the twin chamber design gasket (red spots).
809810811
812
Fig. S2. Image plates of 6Li together with pressure monometers at different pressure and temperatures. The diffraction lines of NaCl (bcc) (and solid helium at low temperature (fcc)) together with ruby fluorescence used to determine the pressure in the vicinity of the sample. Panel F shows lithium in mixed 9R+bcc phase where diffraction lines of lithium are clearly smeared and split showing highly texture features. Unlabeled patterns are cryostat and DAC background that has no pressure dependence. Diamond reflections are masked.
813814815816817818819
820
Fig. S3. Suggested phase boundaries of lithium isotopes during isobaric cooling. Both isotopes show same bcc-fcc transition pressure at room temperature and we assigned the bcc-fcc
821
823
824825826
Fig. S4.Various thermal paths and the structures of the 6Li sample using helium as pressure medium. Numbers and arrows are guides to eye for following the thermal history of the sample. Open symbols are used for data acquired during warming.
827
828
829830831
Fig. S5Various thermal paths and the structures of the 6Li sample using mineral oil as the pressure medium. Numbers and arrows are guides to eye for following the thermal history of the sample.
832833
834835836837
838
Fig. S6Birch-Murnaghan parameters of 6Li and 7Li phases at various temperatures. (A) Atomic volume (V/Z) of 7Li bcc, fcc and 9R phases as a function of pressure using helium (He) as the pressure-transmitting medium. Due to the limited data, 9R (102 K) and fcc (300 K) phase are fitted by polynomial function. (B) Atomic volume of 6Li fcc and 9R phases compressed in mineral oil (MO) at 22 K together with theoretically calculated values for 9R phase under similar conditions (C) conditions (C) Atomic volume of the 6Li fcc phase isothermally compressed in MO at 22, 26 and 300 K. The second- and third-order Birch–Murnaghan (BM) equation of states are used to fit the atomic volume data, see Table S1.
839
840841842843844845846847848
Fig. S7Figure 4 from the main text, also showing the diffraction pattern from the whole MD simulation box containing 128,000 atoms when treated as a nanocrystal.
849
850
851852
Fig. S8
Convergence of energies with respect to the wavefunction cutoff ( cutρ E. 8E ). The black and red lines show the total energy convergence of bcc and fcc Li and the green line shows the evolution of the convergence of the bcc-fcc energy difference. The blue line shows the evolution of the formation energy difference as computed by the SSSP (http://materialscloud.org/sssp/ ). The reference values were obtained at 200 Ry.
Fig. S9Example of the self -consistent ab initio lattice dynamics calculation (SCAILD) for Li at 3GPa. The left hand panel shows that the phonons evaluated at 10K, 100K and 300K have essentially identical frequencies (i.e. there is negligible anharmonicity). The only noticeable effect is that the softest Gamma-N branch stiffens with increasing temperature. The right hand panel compares natural Li (6.94 mass) and pure 6Li at 100K, showing that the expected harmonic scaling of about sqrt(7/6) is reproduced in the anharmonic calculations.
862
863
864865866867868869870
Fig. S10The calculated phonon spectra and densities of state as a function of pressure of bcc (top) and fcc (bottom) Li.
871
872
873874
Fig. S11The calculated phonon spectra and densities of state as a function of pressure of hcp (top) and 9R (bottom) Li.
875
876
877878
Fig. S12Equation of state for 7Li and 6Li calculated with quasiharmonic density functional theory for fcc at 25K. The numbers shown are fits to the Birch- Murnaghan equation of state. The zero point contribution to the pressure, omitted from the captions in the phonon spectra (S7&S8), is about 0.3GPa and hence the pressure difference between 7Li and 6Li at the same volume is about 25MPa.
879
880
881882883884885
Fig. S13Free energy differences relative to fcc at T=0 for 7Li (solid), 6Li (dashed) showing stability limits of bcc with respect to fcc and 9R.
886
887
888889
890
Fig. S14Quantum-driven isotope effects on the thermodynamic phase boundary (fcc-bcc, black) and the kinetic instability line (9R-bcc, red). The blue line is the experimentally ob-served martensitic transition on cooling for 7Li. We recalculated the vibrational free en-ergy assuming different values for the nuclear mass. Using a mass number of 14 (solid lines) gives a semiclassical quasiharmonic approximation and raises the lines slightly. A nuclear mass number of 3 is sufficient for zero point effects to stabilize bcc to zero pres -sure. According to the third law of thermodynamics, P-T phase boundaries must be ver-tical at T = 0, which explains why there is a dramatic shift in the bcc-9R line, consistent with the data in Fig 1 of the main paper, whereas there is only a modest effect on bcc-fcc. Stabilization of bcc due to zero point energy would be possible for 6Li if the static lattice energy difference were just 1meV closer, which is within the accuracy of the ap-proximations using DFT. However, this would also move the 7Li line away from its cur-rent good agreement with experiment. It is extremely unlikely that there are significant errors in the calculated phonon frequencies, since these are in very good agreement with inelastic neutron scattering data.
891
892
893894895896897898899900901902903904905906907908
Fig. S15Fermi surface for (A) bcc lithium: no contact with the Brillouin Zone boundary. (B) fcc lithium: There are 8 symmetry-related gaps at the L-point in total. The energy at L is 15 meV below the Fermi energy. (C) 9R lithium, showing three sheets in the Jones (backfolded) zone. The Fermi sphere gaps to touch the boundaries six times in the first
909
910
911
912913914915
Brillouin Zone (at meV 65EE:L f ) and twice in the third zone (at meV 17EE:T f
).916917
918
Phase Medium Equations of state fitted
V0 (Å3) B0 (GPa) B′ T (K)
7Li-bcc He 3rd-order BM 21.9(5) 10.4(26) 3.7(7) 3007Li-bcc He 2nd-order BM 21.4(4) 12.0(14) 4 1027Li-fcc He 2nd-order BM 21.9(9) 10.0(23) 4 1027Li-fcc He 2nd-order BM 21.8(5) 9.5(11) 4 227Li-fcc He 3rd-order BM 21.3(2) 12.0(7) 3.5(1) 266Li-fcc He 3rd-order BM 21.0(2) 13.5(6) 3.4(1) 236Li-9R MO 2nd-order BM 20.5(5) 14.1(22) 4 226Li-fcc MO 2nd-order BM 20.6(7) 12.3(18) 4 226Li-fcc MO 3rd-order BM 20.6(5) 14.4(22) 3.6(1) 266Li-fcc MO 2nd-order BM 22.1(9) 9.2(16) 4 300
Table S1.Experimental bulk moduli of 6Li and 7Li phases at several temperatures. All the data were calculated by the EosFit7-GUI software. (60)
Table S4.The various pressures, in GPa, as a function of the conventional lattice parameter, for bcc Li at 300 K. The DFT pressure is provided for the reference. PEoS is computed via a least-squares fit to Birch–Murnaghan equation.
937
938939940
941
942943944945
PBEsol dE (meV) V0 (A3/atom) B (GPa) B’
bcc 1.76 20.247 13.68 3.41
9R 0.16 20.215 13.61 3.40
fcc 0 20.205 13.60 3.40
PBE dE (meV) V0 (A3/atom) B (GPa) B’
bcc 1.74 20.262 13.88 3.42
9R 0.16 20.232 13.81 3.41
fcc 0.0 20.222 13.81 3.41
LDA dE (meV) V0 (A3/atom) B (GPa) B’
bcc 2.24 18.993 15.15 3.39
9R 0.26 18.953 15.07 3.37
fcc 0.0 18.938 15.06 3.39
Table S5.Dependence on Exchange-correlation Functional. The parameters calculated for different structures using PBEsol (Top), PBE (middle) and LDA (bottom) regenerating the pseudopotential with the same settings.
946
947
948949950951952953954
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973974
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