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Open Access Journal of Mathematical and Theoretical

Letter to Editor Open Access

4 2 He and zero−point

Abstract Volume 1 Issue 2 - 2018 4 At very low , 2 He becomes crystalline under high pressure of order of 65 Atmospheres or more. The solid structure is hexagonal close packed (HCP) and the Namwetako JS,1 Tonui JK,1 Tanui PK,1 Khanna KM1 4 1 of the crystal increases. 2 He , being a solid, is intrinsically restless with a large Department of Physics, University of Eldoret, Kenya

zero – point energy∈o , which is of the order of 98 percent of the total energy, E0 , in the Correspondence: Tanui PK, Department of Physics, University crystalline state. The balance two percent will be the energy due to interparticle interaction. of Eldoret, P.O Box 1125−30100, Kenya, Tel +2547 2080 0446, The total energy in the crystalline state can be calculated using the Gaussian form of Email [email protected] the Lennard – Jones potential and the t – matrix formalism. The zero – point energy is Received: February 26, 2018 | Published: March 26, 2018 calculated using Heisenberg’s . If ∈o is to be of the order of 98 percent

of E0 , the interparticle distance X turns out to be of the order of 1.5 Å, which is less than the hard sphere radius.

Keywords: zero – point energy, λ – transition, Hexagonal close pack, bose condensate

Introduction The degree of the quantumness in a solid is sometimes characterized by

4 It is found that cooling 2 He liquid from Tλ towards absolute

4 1 λ h zero does not result in the solidification of 2 He liquid. Λ= = πσ 1 2 ………… (1) 4 σ 2m∈ However, 2 He becomes solid under large external pressure of ( ) 2 the order of 25 Atmospheres or more. Helium was first solidified Where λ , is the de Boer parameter, which is the ratio of the de in by W. H. Keesom.1 on specific and 2 thermal conductivity of solid Helium are given by H. R. Glyde. To hh Broglie , λ= = to the minimum separation of 4 p 2m∈ study the properties of any interacting system, such as solid 2 He , the pair potential between the has to be precisely known. Between the atoms in the crystal which isσ . If λ is comparable toσ , the solid is highly quantum. Moreover, the de Boer parameter Λ and the the Helium atoms, the pair potential,3,4 Vr( ) , is weakly attractive Lindmann ratio δ are practically identical, and thus a large Λ means at large separation, r≥3 Å, with a minimum well depth ∈=10.95K . quantum and very anharmonic vibration. At close approach, r≤=σ 2.63 Å, where hard – core radius isσ . For In physics, a quantum solid is the type of solid that is intrinsically  V (σ )= 090 , Vr()becomes steeply repulsive. The volume of solid restless, in the sense that atoms continuously vibrate about their 4 3 and exchange places even at absolute zero temperature. The 2 He is 21.1cm /mole. Since Helium is , its thermal wavelength typical quantum solid, both in low density and high density, is the λ , is long. At TK= 1.0 , λ ≅ 10Å for 4He . Helium is thus difficult T T 2 4 to localize. Attempts to localize it leads to a high kinetic or zero – solid 2 He which is crystalline and is hexagonal – close – packed system. The atoms in a quantum crystalline solid are arranged in a point energy. SinceVr( ) , is weak, Helium does not solidify under regular array that may be characterized by a lattice of points. The atoms are held in the lattice by the inter – atomic bonds, meaning attraction viaVr( ) . Rather, it solidifies only under large external thereby that the atoms interact with each other, and there may be some pressure as a Consequence of the existence of the hard core of Vr( ) , specific form of interaction potential between the atoms. We now turn much as billiard balls form a lattice under pressure. At higher pressure, to the basic ideas and information that lead us to write the present

4 manuscript. The properties of 4 He are dominated by the quantum – 2 He gets compressed into close−packed (FCC and HCP) phases. At 2 mechanical effect called zero – point which is the dynamical 4.9 k-bars, FCC 4He has a volume of 9.03cm3/mole andσ =2.11A ° . 2 4 property of 2 He . Due to the low atomic and weak inter–atomic

Submit Manuscript | http://medcraveonline.com Open Acc J Math Theor Phy. 2018;1(2):46–47. 46 © 2018 Namwetako et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Copyright: ©2018 Namwetako et al. 47 Solid 4 and zero point energy 2 He −

forces E , atoms do not freeze even in the limit of zero temperature. shows that the value of zero–point–energy ∈ changes as ∆X , varies O o o 4 and the value of ∆X depend on the value of X as shown in Eq. (4). Liquid 2 He can be solidified only on the application of external o pressure of 25bars or more, in addition to cooling. The effect allows the presence of delocalized vacancies in the crystalline solid that can Here we have quoted two extreme values for condense into a at low temperatures.5 Even when it X ( X =°=° 2.11 A and X 3.67 A ) . It is simple to calculate the value becomes solid, the liquid – solid transition is not well understood. ∈ of X that will lead to the value of o that will be 98 percent of the In this manuscript, we have concentrated on the physical total energy EO . Again there are two values for EO in the literature;7 4 4 EK= 25 and EK= 14 . In order that∈ be 98 percent of E characteristics of solid 2 He , one is that 2 He crystals will be treated as o o o O quantum crystals, and the second is the calculation of the interaction ,the value of X will be the order of 1.7 A° which means that in the crystalline state the inter− distance between the atoms 4 energy between the 2 He atoms. Since a quantum solid is intrinsically should be of the order of 1.7 A° . However, this needs to be verified restless, the in the values of position r and the experimentally, may be by using the scattering method. 4 p should be finite, and they are large in 2 He because of its low mass. Conclusion 6,7 Hence, ∆r and ∆p should be large. At absolute zero temperature, the Our calculations bring out that if the zero−point−energy , is to (∈o ) 4 zero – point fluctuations of atoms in solid 2 He are about 30 percent of be 98 percent of the total energy (E ) , then the inter-particle distance the inter−atomic spacing and as a consequence the zero – point energy O between 4 He atoms should be of the order of 1.5 A° which is less than represents roughly 98 percent of the total energy of the crystal. 2 the hard−core radius. More experimental and theoretical work needs This means that in the crystalline state (T →0) , the interaction to be done to determine the exact value of the fluctuation in position energy between the helium atoms will be roughly 2 percent of the (∆X ) since this will determine∈ , and we then calculate whether o o total energy of the system. Thus, if E is the total energy and∈ is ∈ is really 98 percent of E . In our next attempt, many−body o 1 o o interaction energy, and∈ is the zero – point energy, then, will be used to calculate∈ . o o 2 Acknowledgements ∈=1 EO ……………………… (2) 100 None 98 ∈= E oO100 ……………………… (3) Conflict of interest Author declares that there is no conflict of interest. Zero – Point Energy (ZPE) References If X is the average inter–particle distance or inter−atomic spacing of the linear lattice, then the zero – point fluctuationof the atoms in 1. Keesom WH. Comm. Kamerlingh Onnes. 1926. solid helium is about 30 percent of. Hence, 2. Glyde HR. Excitations in Liquid and Solid Helium. Clarendon Press, Oxford. 1994. 30 ∆=XX 0 100 …………………… (4) 3. Aziz RA. In Inert Gases. Springer −Verlag, Berlin: Heidelberg; 1984. 7 In the literature, X has two values, one is when XÅ = 2.11 , 4. Korona T, Wiilliams HL, Bukowski R, et al. Helium dimer potential and second when XÅ = 3.67 . Both of these values will be used from −adapted perturbation theory calculations using large to get the numerical value for the zero – point – energy ∈ . Now, o Gaussian geminal and orbital sets. Journal of . if ∆X is the uncertainty in the value of the position, and ∆p is 1977;106:5109 o 0 the uncertainty in the value of the momentum, then according to 5. Khanna KM, Ayodo YK, Sakwa TW, et al. Pair distribution function for Heisenberg’s , interacting and the ground−state energy of solid helium−4. Indian Journal of Pure and . 2009;47(5):325−331.  ∆=po ……………………… (5) 6. Khanna KM, Das BK. Excitation spectrum for interacting bosons. ∆XO Physica. 1973;69(2):611−622. The zero−point−energy is then given by 7. Nosanov HL. Phys Rev. 1966;18:120 4 2 2 8. Glyde HR. Private Communication on Solid 2 He .2010. (∆po )  ∈=0 = 2m 2( mX∆ O )……… (6)

−24 4 Where m= 6.46 xg 10 is the mass of the 2 He . Eq. (6)

4 Citation: Namwetako JS, Tonui JK, Tanui PK. Solid 2 He and zero−point energy. Open Acc J Math Theor Phy. 2018;1(2):46–47. DOI: 10.15406/oajmtp.2018.01.00008