ISSN 10283358, Doklady Physics, 2014, Vol. 59, No. 10, pp. 446–448. © Pleiades Publishing, Ltd., 2014. Original Russian Text © G.I. Barenblatt, G.S. Golitsyn, N.N. Eremin, V.S. Urusov, 2014, published in Doklady Akademii Nauk, 2014, Vol. 458, No. 5, pp. 528–530.
PHYSICS
Universality of the Linear Nanoscale G. I. Barenblatta, Academician G. S. Golitsynb, N. N. Ereminc, and Academician V. S. Urusovc Received June 16, 2014
Abstract—Based on our analysis of factual material (27 various crystals), we have shown the universality of the linear scale composed of Planck’s constant, the density, and Young’s modulus by the dimensional method. This scale is 1 angstrom, i.e., 10–10 m, with a logarithmic accuracy. The quantum effects become sig nificant starting from this scale. DOI: 10.1134/S1028335814100061
1. As is well known, PierreSimon Laplace during the French Revolution proposed to introduce a new “revolutionary” unit of length, the meter, equal to 1/40 000 000 of the length of the Paris meridian. It is this interpretation of the meter that enabled Laplace to find the necessary means for measuring a quantity that was actually of interest to him, the length of the Paris meridian. The measurement result was adopted as a standard of the unit of length; it is kept at the Bureau of Weights and Measures in Paris. Much later, a defini tion of the meter related to a conservative natural pro cess was proposed: 1.65073673 wavelengths of emis sion in a vacuum during the transition from the 2p10 level to the 2d5 level of the krypton86 atom. The introduction of the new unit of length encountered resistance beginning with the unrestrained criticism of N.I. Lobachevsky’s book “Geometry” by the influen tial academician Fuss (see [1]), who accused the great geometer of indulgence in the “rage of the nation” for using the metric measures. However, with the intro duction of the SI system, using the meter as a unit of length became obligatory; in our country, it entered into the State Standard (GOST). Naturally, various derivatives of the unit of length arose: 1 km, 1 cm, 1 μm, 1 nm, etc. In optics and atomic and molecular physics, as well as when measuring processes in crystal structures, the unit of length 1 Å equal to 10–10 m = 10–8 cm, the angstrom, most commensurable with the sizes of atoms and the lengths of interatomic distances
a
Shirshov Institute of Oceanology, Russian Academy of Sciences, Nakhimovskii pr. 36, Moscow, 117997 Russia b Oboukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Pyzhevskii per. 3, Moscow, 109017 Russia c Moscow State University, Moscow, 119991 Russia email:
[email protected],
[email protected],
[email protected],
[email protected]
is used. Thus, the angstrom is a pixel (1/10) of the nanoscale. 2. It was shown in [2] that 1 Å is a natural unit of length associated with a more indepth analysis of the deformation of a crystal structure. Indeed, during the deformation of a crystal, even an arbitrarily slow one, the breakage of bonds leading to the generation and propagation of elastic waves occurs in the crystal struc ture. The density ρ and the microscopic Young’s mod ulus E are the control parameters of the phenomenon of elastic waves in a crystal lattice; these two quantities define the propagation speeds of longitudinal and 1/ 2
⎛ ⎞ transverse elastic waves proportional to ⎜ E ⎟ . If we ⎝ρ⎠ take into account the fact that the quantum effects become significant on the scales of a crystal lattice, then Planck’s constant h should be added to the con trol parameters ρ and Е. A single quantity with the dimensions of length can be composed of the three quantities ρ, Е, and h = 6.626 ⋅ 10–27 erg s by the method of dimensional analysis [3, 4]: h ⎞ 1/4 = ⎛ h⎞ 1/4 , λ = ⎛ ⎝ ρE⎠ ⎝ ρc⎠
(1)
where с, for definiteness, is the propagation speed of longitudinal waves. 3. Let us turn to quantitative estimations. The table presented here was compiled on the basis of data from the ICSD database [5] and the review [6]. This table gives the density ρ, Young’s modulus Е, the propaga tion speed of longitudinal waves с1, the smallest inter atomic distance in the crystal lattice r0, the linear scale λ calculated from Eq. (1), and the dimensionless scal r ing parameter β = 0 for 27 crystals belonging to differ λ ent chemical and structural types from simple mona tomic (metals nos. 3, 4, 5, 27, diamond no. 2) to the
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1/ 2
⎛ ⎞ Density ρ, Young’s modulus E, longitudinal wave speed c1 = ⎜ E ⎟ , parameter λ, interatomic distance r0, and similarity ⎝ρ⎠ r0 parameter β = λ No.
Compound
ρ, g/cm3
E, 10–10, g cm–1 s–2
c1, km/s
r0, Å
λ, Å
β
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Si, silicon C, diamond Cu, copper αFe, ferrite Au, gold ZnS, sphalerite FeS2, pyrite CaF2, fluorite NaCl, halite KCl, sylvine CaCO3, aragonite MgSiO3, enstatite Mg2SiO4, forsterite CaO, lime MgO, periclase Fe3O4, magnetite MgAl2O4, spinel PbS, galenite TiO2, rutile SnO2, cassiterite CaWO4, wulfenite Mg3Al2Si3O12, pyrope UO2, uraninite Te, tellurium FeCr2O4, chromite H2O, ice (1H) Pt, platinum
2.3 3.12 8.3 7.7 19.8 4.9 5.2 3.8 2.6 1.9 2.3 3.0 3.2 3.8 3.5 5.1 3.8 7.0 4.26 6.98 6.12 3.57 10.97 6.24 5.09 0.917 21.45
109 1146 126 211 79 84 292 108 38 23 92 185 201 197 307 230 276 81 286 264 96 234 220 35 269 295 174
6.83 18.06 3.76 5.18 2.02 4.53 7.63 5.83 4.14 3.40 5.60 7.60 7.90 7.42 9.57 6.64 8.78 3.26 8.19 6.15 3.96 8.10 4.48 2.37 5.28 3.22 2.85
2.35 1.54 2.56 2.48 2.88 2.30 2.26 2.34 2.81 3.15 2.44 1.78 2.05 2.40 2.09 1.93 1.92 2.96 1.99 2.05 1.77 1.63 2.37 2.82–3.35 2.05 1.00–2.76* 2.79
0.80 0.56 0.67 0.67 0.75 0.78 0.65 0.78 0.76 1.00 0.80 0.72 0.71 0.71 0.67 0.66 0.68 0.72 0.66 0.63 0.72 0.69 0.61 0.82 0.65 0.80 0.54
2.96 2.77 3.82 3.69 4.48 2.97 3.50 3.02 3.68 3.16 3.06 2.47 2.86 2.29 2.10 2.92 2.83 4.12 3.01 3.27 2.45 2.36 3.90 3.35–4.09 3.50 1.25–3.46 5.14
* The range of distances from 1.0 Å (short arm of O–H bond) to 2.76 Å (distance between the centers of adjacent water molecules) is shown for 1H ice.
very complex such as pyrope belonging to the class of orthosilicates–garnets (no. 22). As can be seen, λ lies within the range from 0.54 (platinum, no. 27) to 1.00 Å for potassium chloride and has a mean value of 0.71 ± 0.16 Å. The scaling parameter β has the lowest value of 2.10 for periclase no. 15 (if the value of 1.25 for 1H ice obtained by tak ing into account the short hydrogen bonds in the water molecule is disregarded). For diamond, β is also small, β = 2.77, while the value of β for platinum (no. 27) is the highest: β = 5.14. The value of β averaged over all 27 materials is 3.13 ± 0.48. Similar results can also be obtained by the method of dimensional analysis if the atomic weight is taken instead of the density ρ. DOKLADY PHYSICS
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These estimates lead us to the following conclu sion: the parameter λ defined by Eq. (1) (see also [2]) is universal with the logarithmic accuracy adopted in physics; its value is the same for all crystal structures and is equal to 1 Å. The scale λ is the threshold size starting from which the quantum effects become sig nificant. 4. Comparison of the parameter λ determined from the material properties with the mean interatomic dis r tance in crystal structures shows that the ratio β = 0 is λ close to unity for all of the investigated materials. Thus, when the deformation parameters and, conse quently, the strength of structures made from crystal
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line and polycrystalline materials are considered, a dimensionless scaling parameter appears, G = Λ , λ where Λ is the characteristic scale of the structure. Clearly, the scaling parameter G is very large in macro scopic phenomena. However, this does not mean that its influence may be neglected. Indeed, this can be done only if there is complete selfsimilarity in G. This means that the scaling relations for the dimen sionless deformation and destruction characteristics tend to finite limiting relations that may be considered the scaling laws as G → ∞ . In general, this is not the case (see [4]), and the dimensionless parameter G remains a significant control parameter at any of its values. However, this means that the microscopic quantum effects continue to influence the behavior of the structure and, in particular, its strength even at very large values of this parameter. Neglecting the scaling parameter, the Reynolds number, in the problem of a turbulent shear flow at large Reynolds numbers (see [7]) can be an illustration of the incorrectness of the “naive” approach in which the scaling parameter is neglected if it is large. ACKNOWLEDGMENTS We wish to thank Prof. P.J.M. Monteiro (Univer sity of California, Berkeley) and Dr. V.M. Prostokishin
(Institute of Oceanology, Russian Academy of Sci ences) for their valuable discussion of the work. The study was partially supported by the Program of the Presidium of RAS (No. 4); G.S. Golitsyn—by the Program of the Presidium of RAS (No. 19). The work of N.N. Eremin and V.S. Urusov was financially sup ported by RFBR project no. 1205983a. REFERENCES 1. V. F. Kagan and N.I. Lobachevsky, in Essays on Geometry (Moscow State University, Moscow, 1963) [in Russian]. 2. G. I. Barenblatt and P. J. M. Monteiro, Fiz. Mego mekhanika 13 (5), 41 (2010). 3. P. W. Bridgman, Dimensional Analysis (Yale University Press, New Haven, 1932). 4. G. I. Barenblatt, SelfSimilar Phenomena. Dimensional Analysis and Scaling (Intellekt, Dolgoprudnyi, 2009) [in Russian]. 5. Inorganic Crystal Structure Database. http://icsd.fiz karlsruhe.de/ 6. Mineral Physics and Crystallography: A Handbook of Physical Constants, Ed. by T. J. Abreas (AGU, 1995). 7. G. I. Barenblatt, A. J. Chorin, and V. M. Prostokishin, Usp. Fiz. Nauk 184 (3), 265 (2014).
Translated by V. Astakhov
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