Calculating the optimum pressure and temperature for vacancy minimization from theory; Niobium is an example. Jozsef Garai

Size: px

Start display at page:

Download "Calculating the optimum pressure and temperature for vacancy minimization from theory; Niobium is an example. Jozsef Garai"

Silas Smith
5 years ago
Views:

1 Calculating the ptimum pressure and temperature fr vacancy minimizatin frm thery; Nibium is an example Jzsef Garai Department f Mechanical and Materials Engineering, Flrida Internatinal University, Miami, USA Self-resnance in the atmic vibratin ccurs when the average wavelength f the phnn thermal vibratin is equivalent r harmnic f the diameters f the atms. It is suggested that applying pressure at temperature crrespnding t the self-resnance shuld effectively reduce the number f vacancies. his theretical predictin is tested n Nibium by measuring the magnetic susceptibility f the untreated and treated samples. he applied pressuretemperature treatment increased the critical temperature f Nibium by abut 30 percent which was als accmpanied with vlume increase. he reductin f the number f vacancy in a substance imprves the physical prperties f the materials. raditinally pressure and temperature treatments are emplyed t achieve this gal. he ptimum treatments are usually determined by try and errr methd. Fllwing a theretical apprach and using cnventinal thermdynamic relatinships equatins allw t calculate the ptimum pressure and temperature fr vacancy minimizatin are derived here. When the wavelength f the average thermal phnn vibratin is equal t r harmnic with the atmic diameter then self resnance ccurs. It is suggested that the applicatin f pressure at the temperature crrespnding t self-resnance shuld result in the reductin f the number f vacancies. he same prcedure can als be used t enhance diffusin. Using fundamental thermdynamic relatinships the pressure-temperature curve fr self-resnance is derived here. he average wavelength [ λ] f the phnn frequency at a given temperature can be calculated as: hυ λ = () k where h is the Planck cnstant, k is the ltzmann cnstant, and is the temperature. he bulk seismic velcity is calculated as: - - υ is the bulk seismic velcity

2 V υ = =, () ρ N M A where is the isthermal bulk mdulus, ρ is the density, V is the vlume, N is the A Avgadr s number and M is the mass f the atm. In Eq. () the isthermal bulk mdulus is used instead f the adiabatic. he difference between the tw bulk mdulus is usually under % which is ignred in this study. he effect f temperature n the bulk mdulus at bar pressure is calculated as: δ d (3) = 0 0 = e where is the bulk mdulus at zer pressure and temperature, is the vlume cefficient f thermal expansin and δ is the Andersn- Grüneisen parameter, which defined as: ln ln δ = =. ln V p V p p V p p Assuming that the temperature and pressure effect n the vlume cefficient f thermal expansin and bulk mdulus is linear respectively then Eq. (3) can be written as: p, where ' ( + ) ( p) e δ + = (5) ' is the linear term fr the pressure dependence f the bulk mdulus, - - (4) is the prjected value f the vlume cefficient f thermal expansin at zer pressure and temperature and is the linear term fr the vlume cefficient f thermal expansin. Eventhugh, the temperature dependence f the cefficient belw the Debye temperature is nt linear 3, the linear apprximatin and Eq. (5) can be used fr ambient cnditins and at higher temperatures 4 because the intrduced errr is minr. he mlar vlume f the slid at the temperature f interest is calculated by using the ES f Garai 4, which is given as: a P P P P P P P P+ P P V = nv e (6) where, a is a linear, is a quadratic term fr the pressure dependence f the bulk P mdulus, is a linear and is a quadratic term fr the pressure dependence f the vlume P cefficient f thermal expansin, is a linear term fr the temperature dependence f the vlume cefficient f thermal expansin and a is cnstant, characteristic f the substance. he theretical explanatins fr Eq. (6) and the physics f the parameters are discussed in detail 4. he atmic diameter [d] crrespnding t the size f the vacancy is apprximated as: ( p) V, d = 3. (7) N A Self resnance can ccur when

3 n (,p) λ = d where n ù *. (8) If the thermdynamic parameters are available then the p- curve fr the atmic self-vibratin can be calculated by using Eq. ()-(8). he vibratinal mtin f crystals is very cmplex and idealized apprach is valid nly fr mnatmic highly symmetrical atmic arrangement. Nibium, which satisfies this criterin, has been selected t test the prpsed hypthesis because f its imprtance in supercnductivity. he thermdynamic parameters required fr Eqs. ()-(8) are determined by unrestricted fitting using the available experimental data 5-7. he 77 experiments cver the temperature and pressure range and 0-34 GPa respectively. he determined parameters are given in able. he temperatures fr the fundamental and the secnd harmnics at atmspheric pressure and at 0 GPa are calculated by using Eqs. ()-(8). Using atmic radius.46 A fr Nibium 8 the temperatures fr the first harmnic at atmspheric pressure and 0 GPa are 340 and 360 respectively while fr the secnd harmnic the temperatures are 640 and 68. he ¼ x 0.5 mm Nb (99.95%) samples were bught frm Smart Elements. he apprximately 0. GPa pressure was achieved by screws in a pressure vessel. he sample under pressure was heated up frm rm temperature t the targeted temperature (640 ) in 5 hurs. he sample was annealed at 640 fr 8 hurs and cl dwn t rm temperature in 5 hurs. he sample was taken ut frm the pressure vessel right befre the experiment. he effectiveness f the treatment was tested by measuring the critical temperature f the untreated and the treated samples. he temperature dependence f the ac susceptibility frm the treated and untreated Nb sample, perfrmed by using a mutual inductance technique at an applied field f H = 0 Oe and frequency f f = khz is shwn in Figure. he real part, χ, reveals a large diamagnetic signal belw 6.8 and 8.8 marking the supercnducting transitin fr the untreated and treated sample respectively. elw 6.0 and 8.0, χ is flat, indicating that the supercnducting transitin is cmplete fr bth the untreated and the treated samples 9. he critical temperature f Nb reprted in the literature fr untreated sample is lwer 0 then 6.8 measured here. he mst likely explanatin fr this lwer critical temperature is the presence f irn in the Nb sample. Cmparing the untreated and treated samples it is evident that the treatment increased bth the critical temperature and the vlume by abut 30% (Fig. ). It is cncluded that the presented theretical apprach can successfully applied t calculate the ptimum pressure and temperature cnditins fr vacancy minimizatin

4 Acknwledgement: I wuld like t thank Rngying Jin fr measuring the magnetic susceptibility and Andriy Durygin fr helping in the design and the manufacturing f the pressure vessel. References: C. ittel, Intrductin t Slid State Physics, 8 th Editin (Wiley, New Yrk, 004). J. Garai, and A. Laugier, J. Appl. Phys., 0, 0354 (007). 3 J. Garai, CALPHAD, 30, 354 (006). 4 J. Garai, J. Appl. Phys., 0, 3506 (007). 5 L-C Ming and M.H. Manghani, J. Appl. Phys (978). 6. enichi and A.. Singh, Phys. Rev. 73, 49 (006). 7 J.W. Edwards, R. Speiser, and H.L. Jhnstn, J. App. Phys., 44 (95). 8 J.A. Dean, Lange s Handbk f Chemistry, Fifteenth ed., (McGraw Hill, Inc, New Yrk, 999). 9 he magnetic measurements were cnducted by Rngying Jin at the Cndensed Matter Sciences Divisin f Oak Ridge Natinal Labratry. 0 J. Eisenstein, Mdern Phys. 6, 77 (954)

5 FIG. Magnetic measurements f the treated and untreated samples

6 ALE. hermdynamic parameters fr the Equatin f States GPa [N=334] m V [cm 3 ] [GPa] 0 [0-5 - ] [0-9 - ] P [ 3 0 GPa ] 7 [ 0 GPa ] a δ P 0 [ 0 GPa ] RMS misfit V(p,) (G) cm 3 P(V,) (-M) GPa -M = Universal irch-murnaghan ES G = ES f Garai (007) 6

Theoretical study of third virial coefficient with Kihara potential

Theoretical study of third virial coefficient with Kihara potential Theretical study f third virial cefficient with Kihara ptential Jurnal: Manuscript ID cjp-017-0705.r Manuscript Type: Article Date Submitted by the Authr: 6-Dec-017 Cmplete List f Authrs: Smuncu E.; Giresun