European Journal of Basic and Applied Sciences Vol. 3 No. 1, 2016 ISSN

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1 uopean Jounal of asic and Applied Sciences ol. No., 6 ISSN DMIN MOS ONIAL, HMAL XANSION COFFICIN AND DSCI ASYMMICAL COMONNS HOUGH DY-WALL FACO Y ANHAMONIC COLAD INSIN MODL Assoc. of. Nguen a Duc hd. anao Univesit, uenquang o. I NAM ASAC ffective potential in anhamonic coelated instein model had detemined on to base analtics calculation Mose potential between absobe and bacscatte atoms with neaest neighbo atoms, this wo was epesented the expession of themal expansion coefficient at high tempeatues and expessions descibe asmmet components (he fist cumulants o themal expansion coefficient, the second cumulant o mean squae elative displacement o Debe-Walle facto, the thid cumulant and the fouth cumulant chaacteie fo asmmet popeties of potential) and themodnamic quantities though Debe-Walle facto. xpessions of coelative function between the cumulants and between cumulants and themal expansion coefficient fo cubic stuctual cstals also was detemined. he expessions had obtained inclusion the classical theo at high tempeatue and quantum effects at low tempeatue. Kewod: Anhamonic; coelate; themodnamic; asmmet; cumulant. INODUCION Anhamonic coelated instein model was used the calculation cumulants, fequenc and tempeatue instein and themodnamic paametes of the cubic stuctual cstals, esults obtained agee well with expeimental values [6]. In the instein model, atomic inteaction potential is Mose pais potential, howeve Mose potential usuall deduced fom expeiment [], so analtics calculation the phsics quantit when to need Mose potential be ve had, theefoe if themodnamic paametes of Mose potential ae calculated in advance will educe the numbe calculations. In this studing scope, we ae will analtics calculation in advance Mose inteactive potential in anhamonic coelated instein model and application to detemine the expessions of themal expansion coefficient, build the expessions themodnamic paametes and cumulants though Debe-Walle facto, conside coelative functions and themodnamics paametes in classical appoximation at high tempeatue and quantum effects at low tempeatue. FOMALISM Anhamonic coelated instein model is descibed b effective inteaction potential as fom [, 9]: U x eff x x () In which x is deviation of the instantaneous bond length of two atoms fom thei equilibium distance o the location of the minimum potential inteaction, eff is effective sping constant, because it include all contibutions of neighbo atoms, is anhamonicit paamete and descibing an asmmet in inteactive potential. Anhamonic coelated ogessive Academic ublishing, UK age

2 uopean Jounal of asic and Applied Sciences ol. No., 6 ISSN instein model is detemined b vibation of single pais atoms with M and M mass of absobe and bacscatte atoms. ibation of atoms affected b neighbo atoms so inteactive potential in expession () is witten as fom [6]: U x Ux U x ˆ i ˆ ij () ji Mi MM with is educed mass, ˆ is the unit bond length vecto, U(x) chaacteie to the M M single pais potential between absobe and bacscatte atoms, the second tem in equation () chaacteie fo contibution of neaest neighbos atoms and calculation b sum i which is ove absobe ( i ) and bacscatte ( i ), and the sum j which is ove all thei neaest neighbos, excludes the absobe and bacscatte themselves because the contibute in the U x. he atomic vibation is calculated on based quantum statistical pocedue with appoximate quasi - hamonic vibation [], in which the Hamiltonian of the sstem is witten as hamonic tem with espect to the equilibium at a given tempeatue plus an anhamonic petubation. aing account fom that we have: H U x eff eff x a a... x... a a a a a eff... a a a... eff a a eff eff () Setup H is sum of fist tem and fouth tem, U (a) is second tem and U () is sum of thid tem and fifth tem, we have expessions: H eff a () U a eff a a (5) a a a U eff (6) xpession () will become: H H U a U (7) in which a is themal expansion coefficient with: x, x a, a x x a Fom equation (7) deduced inteactive potential accoding to anhamonic coelated instein model can wite as fom: U x U a eff U (8) In anhamonic coelated instein model, inteactive potential is Mose pais anhamonic potential [5], conside appoximation fo cubic stuctual cstals, Mose anhamonic potential as fom: D e e U (9) ogessive Academic ublishing, UK age

3 uopean Jounal of asic and Applied Sciences ol. No., 6 ISSN in which (Å - ) is themal expansion coefficient, D(e) is the dissociation eneg b U D. We can wite expession of Mose potential accoding to fom of x: x x De e U () xpand the equation () accoding to x, we have: x x x x U x D...!!!! x x x x... D x x!!!! x x... x x x x 6 aing appoximate to the thid-ode tem, we can wite eduction: Ux D x x x x x x... hus, expession of Mose potential accoding to deviation of the instantaneous bond length of two atoms x will wite become: D x x... U x () he inteaction between pais atoms in anhamonic coelated instein model is descibed b expession effective inteaction potential of Mose pais anhamonic potential in eq. (). Fom equations () and () we have: U x D x x... U x ˆ ˆ iij () ji Mi With cubic stuctual cstals and pue, mass of absobe and bacscatte atoms is equal, so can M tae appoximation M M M, simultaneousl expand second tem of eq.() and calculation, we deduced themodnamic paametes, eff, U() [, ]. eff D c a D c a c c, cd () U () in which c, c, c ae stuctual paametes with values coesponding has detemined [5]. Anhamonic coelated instein model have been used to analtics calculation cumulants [6], the expand cumulants accoding to the expession: n i i n e exp i ; n,,... (5) n n! with n ae cumulants and x is themal expansion coefficient and a() ; x a,. xpand cumulants fom the fist-ode to the sixthode, we have: ; ; ogessive Academic ublishing, UK age 5

4 uopean Jounal of asic and Applied Sciences ol. No., 6 ISSN ogessive Academic ublishing, UK age 6 ; (6) ; ; In above expessions of cumulants, the second cumulant o the mean squae elative displacement (MSD) othewise nown as Debe-Walle facto (DWF). xpessions annaltics calculation cumulants fo cubic stuctue cstals has detemined fom the fist-ode to the thid-ode cumulants [, 5], as fom: he fist cumulant o net expansion coefficient D c c a (7) he second cumulant o Debe-Walle facto: D c (8) he thid cumulant chaacteie to the anhamonicit: c D c (9) Next, we calculate themal expansion coefficient due to effect of anhamonicit when high aise tempeatue b fomula [, ]: () in which is volume coesponding the change of absolute tempeatue unde pessue. Use equation state of themal sstem: () Fom expessions () and (), we have: () Setup K is elastic modulus detemination the change of volume due to inteaction of pessue. Ignoe lins between vibations of atoms and assume feedom eneg Helmholt as fom q F q U F with U is sum potential eneg, q F is fee eneg and was ceated fom vibation of lattice with wave vecto q, then pessue dependence to volume accoding to expession [,5]:

5 uopean Jounal of asic and Applied Sciences ol. No., 6 ISSN F du dfq d d q du q () d q q exp When appeaance anhamonic effect, the sstem equilibium at new location and volume expanded so impotant phenomena of anhamonic effect is dependence of fequenc net vibation to volume, this dependence descibed though b second tem in expession (). o simple, assume dependence to volume of all fequencies net vibation the same and wite though Guneisen facto as fom: ln ln a G ~ G () ln a ln Facto G chaacteie fo anhamonic effect with net themal coefficient: a a Simultaneousl we have: a a da d Deduce themal expansion coefficient: da (5) d Substitute (7) into (5) we get: with eplace c c D / d c d e d / D d c D e / / / / e e e e / e c c c c c D c D c c ogessive Academic ublishing, UK age 7 D c c we have: c D ln, we obtained themal expansion coefficient: c D ln c. (6) c D o educe calculated and measue, to need simplification the desciption expessions of themodnamic paametes, thus we can desciption themodnamic paametes though DWF [6,7,8] b: (7) ;

6 uopean Jounal of asic and Applied Sciences ol. No., 6 ISSN Substitute fomula (7) into equations (7, 8, 9, 6), we have: () () c () c ; (8) c c ; c D (9) () () () c ; () c c D c ; () c D Simultaneousl we deduce coelative expessions between cumulants togethe and between cumulants with themal expansion coefficient, distance between atoms and absolute tempeatue accoding to stuctual paametes and Debe-Walle facto: () cd () () ; () () () () () whee, and ae contibutions eo-point into, and, stuctual paametes was descibed in [5]. Accoding to the desciption above, outside the Mose () () potential paametes analtics calculation, to calculate cumulants,, and themal expansion coefficient, we onl need to calculate DWF, theefoe has educe analtics calculation and pogammable calculato fo themodnamic paametes. he expessions is detemined fom quantum theo, theefoe can appling fo an tempeatue, at high tempeatue it include appoximate esults of classical theo and at low tempeatue limit it alwas shows quantum effects though contibutions of eo-point eneg. In high tempeatue limit (H) we use appoximate fomula change e with We get: (5), deduced: e x () x, and appoximate () ogessive Academic ublishing, UK age 8

7 uopean Jounal of asic and Applied Sciences ol. No., 6 ISSN so we educe expessions of cumulants and themodnamic paametes at high tempeatue (see in able statistic). In low tempeatue limit (L) we use appoximate fomulas: ; (6) because in low tempeatue limit, thus, we can ignoe and highe powes, we educe expessions of cumulants and themodnamic paametes at low tempeatue limit (see in able statistic). () () Note cumulants,, include contibutions eo-point eneg, appoaching the value constant at high tempeatues but the destuctivel accoding to exponential of at low tempeatue and both coelative expessions () and () appoximatel with classical esults and expeiment as / at high tempeatues and ight eflection with esults of classical theo and expeimentalist. able : xpessions of cumulants, themal expansion coefficient, coelative expessions at low tempeatue limit ( ) and appoximation at high tempeatue ( ) Quantities () () () () c /c D /cd /cd 6c ln () / ln(/ ) () () / / / / / Figue : Anhamonic effective inteatomic Figue : Dependence tempeatue and net expansion x of anhamonic petubation ogessive Academic ublishing, UK age 9

8 uopean Jounal of asic and Applied Sciences ol. No., 6 ISSN CONCLUSION he effective inteaction potential in anhamonic coelated instein model was detemined on base analtics calculation Mose inteactive potential between pais absobe and bacscatte atoms with neaest neighbo atoms, this wo was educed the calculation themodnamic paametes and measues, because eplace the calculation b complex matices thee dimensions we onl need to solve poblem one dimension with the inteaction of cluste neaest neighbo atoms and esults obtained agee well with expeimental data. Figue desciption anhamonic effective potential inteatomic and compaed with expeimental data fo FeMo cstal and gaph show the shifts between pesent theo and expeimental data small than shifts between hamonic tem with expeimentalist, this esults to see pesent pocedue can to use good fo the stud anhamonic vibation of atoms. Figue desciption the dependence tempeatue and net expansion of anhamonic petubation facto in anhamonic coelated instein model and fom the gaph appoximate classical at high tempeatue and quantum effects at low tempeatue. Figue descibe the dependence tempeatue of the themal expansion coefficient, we see appoaching the value constant at high tempeatue but has destucted accoding to exponential at low tempeatues. he themal expansion coefficient, cumulants and themodnamic paametes was pesented though Debe-Walle facto and stuctue paametes and has educed the calculations also measue and pogammable calculato. he expession of coelative function between cumulants, coelative function between cumulants and themal expansion coefficient fo cubic stuctual cstals was detemined and inclusion both classical theo at high tempeatue and quantum effects at low tempeatue limit. FNCS Figue : Dependence tempeatue of the themal expansion coefficient [] Nguen a Duc et al. (5) Anhamonic coelated instein model in XAFS theo and application, LAM Academic ublishing, Geman; [] Nguen a Duc (5) Statistical phsics, ublishe hai Nguen Univesit; [] Nguen an Hung (999) Solid state theo, ublishe National Univesit, Ha Noi; [] Giifalco, L. A., Weie,. G. (959), Application of the Mose potential Function to cubic Metals, hs. ev. (), pp [5] Nguen a Duc, (), using the anhamonic coelated instein model to define the expessions of cumulants and themodnamic paametes in the cubic cstals with new stuctue factos, Jounal of hsics and Astonom eseach (AUS). (), pp.-6. [6] Hung, N.. and Duc, N.., (), Anhamonic-Coelated instein model hemal xpansion and XAFS Cumulants of Cubic Cstals: Compaison with xpeiment and othe heoies, J. Commun. in hs., (), pp. 5-. ogessive Academic ublishing, UK age

9 uopean Jounal of asic and Applied Sciences ol. No., 6 ISSN [7] Hung N.. and N.. Duc, (999), Stud of hemodnamic opeties of Cubic Sstems in XAFS, oceedings of the hid Intenational Woshop on Mateial Science (IWOM' 99), Hanoi. pp. 95, 98. [8] Hung, N.., Duc, N.., Fahm,.., (), A New Anhamonic Facto and XAFS including anhamonic contibutions, accepted fo publications in J. hs. Soc., Japan, ol. 7, N o.. [9] Hung, N.. and eh, J. J., (997) Anhamonic coelated instein-model Debe-Walle factos hs. ev. (56), pp.. [] Hung, N.. and Duc, N.., and Dinh Quoc uong, (), heo of themal expansion and cumulants in XAFS technique, J. Commun. in hs () pp. -9. ogessive Academic ublishing, UK age