Abstract
Two dimensionless fundamental physical constants, the fine structure constant α and the proton-to-electron mass ratio
INTRODUCTION
Several notable properties of condensed matter phases are defined by fundamental physical constants. The Bohr radius gives a characteristic scale of interatomic distance on the order of the angstrom, in terms of electron mass me, charge e, and Planck constant ħ. These same fundamental constants enter the Rydberg energy, setting the scale of a characteristic bonding energy in condensed phases and chemical compounds (1).
Among the fundamental constants, those that are dimensionless and do not depend on the choice of units play a special role in physics (2). Two important dimensionless constants are the fine structure constant α and the proton-to-electron mass ratio,
We show that a simple combination of α and
Identifying and understanding bounds on physical properties is important from the point of view of fundamental physics, predictions for theory and experiment, and searching for and rationalizing universal behavior [see, e.g., (3–11)]. Properties for which bounds were recently discussed include viscosity and diffusivity. The proposed lower bounds for these two properties feature in a range of areas including, for example, strongly interacting field theories, quark-gluon plasmas, holographic duality, electron diffusion, transport properties in metals and superconductors, and spin transport in Fermi gases (3–11). Recently, two of us found a lower bound for the kinematic viscosity of liquids set by fundamental physical constants (12). Here, we propose a new, upper bound for the speed of sound in condensed matter phases in terms of fundamental constants.
Apart from setting the speed of elastic interactions in solids, v is related to elasticity and hardness and affects important low-temperature thermodynamic properties such as energy, entropy, and heat capacity (13). As discussed below, the upper bound of v sets the smallest possible entropy and heat capacity at a given temperature.
In solids, v depends on elastic properties and density. These strongly depend on the bonding type and structure, which are interdependent (14). As a result, it was not thought that v can be predicted analytically without simulations, contrary to other properties such as energy or heat capacity, which are universal in the classical harmonic approximation (13). In view of this, representing the upper bound of v in terms of fundamental constants is notable.
RESULTS AND DISCUSSION
There are two approaches in which v can be evaluated. The two approaches start with system elasticity and vibrational properties, respectively.
We begin with system elasticity. The longitudinal speed of sound is
We now recall that the bonding energy in condensed phases is given by the Rydberg energy on the order of several electron volts (1) as
ER is used for order-of-magnitude estimations of the bonding energy E (1). Using E = ER from Eq. 3 in Eq. 2 gives
A result similar to Eq. 4 can be obtained in the second approach that starts with the consideration of the vibrational properties of the system. The longitudinal speed of sound, v, can be evaluated as the phase velocity from the longitudinal dispersion curve ω = ω(k) in the Debye approximation:
We recall that the characteristic scale of interatomic separation is given by the Bohr radius aB on the order of the angstrom as
We now use the known ratio between the phonon energy, ħωD, and E. The phonon energy ħωD can be approximated as
Using Eq. 7 in Eq. 5 gives
As compared to the first approach, the second approach to evaluating v involves additional approximations, including evaluating v from the dispersion relation in the Debye model, using a = aB in Eq. 6, and the ratio between the phonon and bonding energies (Eq. 7). We therefore focus on the result from the first approach (Eq. 4).
We now discuss Eq. 4 and its implications. me characterizes electrons, which are responsible for the interactions between atoms. The electronic contribution is further reflected in the factor αc (
We note that αc and v do not depend on c. The reason for writing v in terms of αc in Eq. 4 and the ratio
m in Eq. 4 characterizes atoms involved in sound propagation. Its scale is set by the proton mass mp: m = Amp, where A is the atomic mass. Recall that aB in Eq. 6 and ER in Eq. 3 are characteristic values derived for the H atom. We similarly set A = 1 and m = mp in Eq. 4 to arrive at the upper bound of v in Eq. 4, vu, as
Equation 9 is the extension of Eq. 4 to atomic hydrogen. We will calculate v in atomic H later in the paper.
Combining Eqs. 4 and 9 and m = Amp gives
Before discussing the experimental data in relation to Eq. 4 and its consequences (Eqs. 9 and 10), we note that the speed of sound is governed by the elastic moduli and density, which substantially vary with bonding type: from strong covalent, ionic, or metallic bonding, typically giving a large bonding energy, to intermediate hydrogen bonding, and weak dipole and van der Waals interactions. Elastic moduli and density also vary with the particular structure that a system adopts. Furthermore, the bonding type and structure are themselves interdependent: Covalent and ionic bonding result in open and close-packed structures, respectively (14). As a result, the speed of sound for a particular system cannot be predicted analytically and without the explicit knowledge of structure and interactions (17), similarly to other system-dependent properties such as viscosity or thermal conductivity [but differently to other properties such as the classical energy and specific heat, which are universal in the harmonic approximation (13)]. Nevertheless, the dependence of v on m or A can be studied in a family of elemental solids. Elemental solids do not have confounding features existing in compounds due to mixed bonding between different atomic species (including mixed covalent-ionic bonding between the same atomic pairs as well as different bonding types between different pairs).
To compare Eq. 10 to experiments, we plot the available data of v as a function of A for 36 elemental solids (18–20) in Fig. 1, including semiconductors and metals with large bonding energies. The data are depicted in a log-log plot. Equation 10 is the straight line in Fig. 1 ending in its upper theoretical bound (Eq. 9) for A = 1. The linear Pearson correlation coefficient calculated for the experimental set (log A, log v) is −0.71. Its absolute value is slightly above the boundary notionally separating moderate and strong correlations (21). The ratio of calculated and experimental v is in the range of 0.6 to 2.4, consistent with the range of
The solid line is the plot of Eq. 10:
We also show the fit of the experimental data points to the inverse square root function predicted by Eq. 10 as the dashed line in Fig. 1 and observe that it lies close to Eq. 10. The fitted curve gives the intercept at 37,350
The agreement of Eq. 10 with experimental data supports Eq. 4 and its consequence, the upper limit vu in Eq. 9. We now show that vu agrees with a wider experimental set. In Fig. 2, we show experimental v (18–20) in 133 systems, including compounds together with the elemental solids in Fig. 1. We observe that experimental v are smaller than the upper theoretical bound vu in Eq. 9. vu is about twice as large as v in diamond, the highest speed of sound measured at ambient conditions [the in-plane speed of sound in graphite is slightly above v in diamond (10)].
Solids are as follows: Al, Be, brass, Cu, duralumin, Au, Fe, Pb, Mg, diamond, Ni, Pt, Ag, steel, Sn, Ti, W, Zn, fused silica, Pyrex glass, Lucite, polyethylene, polyesterene, WC, B, Mo, NaCl, RbCl, RbI, Tl, Li, Na, Si, S, K, Mn, Co, Ge, Y, Nb, Mo, Pd, Cd, In, Sb, Ta, Bi, Th, U, LiF, LiCl, BeO, NH4H2PO4, NH4Cl, NH4Br, NaNO3, NaClO3, NaF, NaBr, NaBrO3, NaI, Mg2SiO4, α-Al2O8, AlPO4, AlSb, KH2PO4, KAl(SO4)2, KCl, KBr, KI, CaBaTiO3, CaF2, ZnO, α-ZnS, GaAs, GaSb, RbF, RbBr, Sr(NO3)2, SrSO4, SrTiO3, AgCl, AgBr, CdS, InSb, CsCl, CsBr, CsI, CsF, Ba(NO3)2, BaF2, BaSO4, BaTiO3, TlCl, Pb(NO3)2, PbS, apatite, aragonite, barite, beryl, biotite, galena, hematite, garnet, diopside, calcite, cancrinite, α-quartz, corundum, labradorite, magnetite, microcline, muscovite, nepheline, pyrite, rutile, staurolite, tourmaline, phlogopite, chromite, celestine, zircon, spinel, and aegirite. Liquids are as follows: mercury, water, acetone, ethanol, ethylene, benzene, nitrobenzene, butane, and glycerol. See (18–20) for system specifications, including density and symmetry groups.
Equation 10 can be used to roughly predict the average or characteristic speed of sound v.
We have included the experimental speed of sound of room-temperature liquids in Fig. 2, with typical v in the range of 1000 to 2000
With regard to liquids, we note that an expression similar to Eq. 2 was earlier obtained by evaluating the elastic modulus using the liquid state theory and applied to liquid metals (28). The speed of sound can also be evaluated in the theory of metals using the ionic plasma frequency and subsequently accounting for the conduction electrons screening. This results in the Bohm-Staver relation
We make three further remarks about the calculated v and its bound. First, this derivation involves approximations as mentioned earlier. The approximations may affect the numerical factor in Eqs. 4 and 9. At the same time, the characteristic scale of v in Eq. 4 and its upper bound (Eq. 9) is set by fundamental constants. Second, Eqs. 3, 6, and 7 used in the second derivation of v assume valence electrons directly involved in bonding and hence strongly bonded systems, including metallic, covalent, and ionic solids. Although bonding in weakly bonded solids such as noble, molecular, and hydrogen-bonded solids is also electromagnetic in origin, weak dipole and van der Waals interactions result in smaller E (29) and smaller v as a result. Therefore, the upper bound vu applies to weakly bonded systems, too. We note here that our evaluation does not directly distinguish between bonding types and hence does not consider the trend of v to increase along the rows of the periodic table, from soft metals to hard covalent materials in Fig. 1. This trend can be accounted for by (i) noting that v in Eq. 8 and
Our upper bound in Eq. 9 corresponds to solid hydrogen with strong metallic bonding. Although this phase only exists at megabar pressures (30, 31) and is dynamically unstable at ambient pressure where molecular formation occurs, it is interesting to calculate v in atomic hydrogen to check the validity of our upper bound. In addition, there has been strong interest in the properties of atomic hydrogen at high pressure [see, e.g., (30–32)], although the speed of sound in these phases was not discussed and remains unknown.
We have calculated the speed of sound in atomic hydrogen for the I41/amd structure (33, 34), which is currently the best candidate structure for solid atomic metallic hydrogen. This structure is calculated to become thermodynamically stable in the pressure range of 400 to 500 GPa (35, 36), below which solid hydrogen is a molecular solid. However, we find that I41/amd is dynamically stable at pressures above about 250 GPa, and therefore, we perform calculations in the pressure range of 250 to 1000 GPa. The speed of sound as a function of pressure and density reported in Fig. 3 corresponds to the highest energy acoustic branch and is averaged over stochastically generated directions in q-space.
The dashed line shows the upper bound vu in Eq. 9.
Our upper bound (Eq. 9) does not account for the enthalpic contribution to the system energy as mentioned earlier; including the pressure effect would increase vu considerably at pressures shown in Fig. 3. Despite this, the calculated v remains below vu in a wide pressure range and starts increasing above vu only above very high pressures of about 600 GPa. In this regard, we note that hydrogen is a unique element with no core electrons. This results in the absence of strong repulsive contributions to the interatomic interaction as compared with heavier elements and, consequently, weaker pressure dependence of elastic moduli and the speed of sound (37). We also note that sharper change of v at lower pressure shown in Fig. 3 is related to approaching the limit of dynamical stability of the I41/amd structure around 250 GPa.
We make three remarks related to previous work. It was noted that thermal diffusivity of insulators does not fall below a threshold value given by the product of v2 and the Planckian time (8). Later work linked the upper bound on the speed of sound to the melting velocity related to melting temperature and Lindemann criterion (11). Last, the upper bound of the speed of sound for hadronic matter was conjectured as
As discussed above, v features in several thermodynamic properties of solids. For example, the low-temperature entropy and heat capacity per volume are
We conclude by returning to dimensionless fundamental physical constants. Rewriting Eq. 9 as
MATERIALS AND METHODS
We have performed density functional theory calculations using the castep package (39), with the Perdew-Burke-Ernzerhof exchange-correlation functional (40), an energy cutoff of 1200 eV, and a k-point grid of spacing 2π × 0.025 Å̊−1 to sample the electronic Brillouin zone. We have relaxed the cell parameters and internal coordinates to obtain a pressure to within 10−4 GPa of the target pressure and forces smaller than 10−5 eV/Å̊. We have then calculated the phonon spectrum using the finite difference method (41) in conjunction with nondiagonal supercells (42) with a 4 × 4 × 4 coarse q-point grid to sample the vibrational Brillouin zone. We have used Fourier interpolation to calculate the phonon frequencies at q-vectors close to the Γ-point and then used finite differences to calculate the corresponding speed of sound.
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.
REFERENCES AND NOTES
- Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).
