Determine morse potential, thermal expansion coefficient and describe asymmetrical components through debye -Waller factor by anharmonic correlated einstein model - PGS.TS. Nguyễn Bá Đức

TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO SỐ 02 – THÁNG 3 NĂM 2016 14 DETERMINE MORSE POTENTIAL, THERMAL EXPANSION COEFFICIENT AND DESCRIBE ASYMMETRICAL COMPONENTS THROUGH DEBYE -WALLER FACTOR BY ANHARMONIC CORRELATED EINSTEIN MODEL Xác định thế Morse, hệ số dãn nở nhiệt và mô tả các thành phần bất đối xứng qua hệ số Debye - Waller bằng mô hình Einstein tương quan phi điều hòa PGS.TS. Nguyễn Bá Đức* TÓM TẮT Thế tương tác hiệu dụng trong mô hình Einstein tương quan phi điều hòa đã được xây dựng dựa trên cơ sở tính giải tích thế tương tác Morse giữa cặp nguyên tử hấp thụ và tán xạ với các nguyên tử lân cận gần nhất, nghiên cứu đã biểu diễn hệ thức hệ số dãn nở nhiệt tại nhiệt độ cao và các biểu thức mô tả thành phần bất đối xứng (cumulant) và các đại lượng nhiệt động qua hệ số Debye-Waller. Hệ thức hàm tương quan giữa các cumulant, hàm tương quan giữa các cumulant và hệ số dãn nở nhiệt đối với các tinh thể có cấu trúc lập phương cũng đã được xác định. Các hệ thức nhận được bao chứa cả lý thuyết cổ điển tại nhiệt độ cao và hiệu ứng lượng tử tại nhiệt độ thấp. Từ khóa: Phi điều hòa; tương quan; nhiệt động; bất đối xứng; cumulant. ABSTRACT Effective potential in anharmonic correlated Einstein model was determined on to base analytics calculation Morse potential between absorber and backscatter atoms with nearest neighbor atoms, this work was represented the expression of thermal expansion coefficient at high temperatures and expressions was described ansymmetry components (cumulants) and thermodynamic quantity through Debye-Waller factor. Expressions of correlative function between the cumulants and between cumulants and thermal expansion coefficient for cubic structural crystals also was determined. The expressions obtained include classical theory at high temperature and quantum effects at low temperature. Keyword: Anharmonic; correlate; thermodynamic; ansymmetry; cumulant. *Trường Đại học Tân Trào TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO SỐ 02 – THÁNG 3 NĂM 2016 15 1. Introduction Anharmonic correlated Einstein model was used the calculation cumulants, frequency and temperature Einstein and thermodynamic parameters of the cubic structural crystals, results obtained agree well with experimental values [6]. In the Einstein model, atomic interaction potential is Morse pairs potential, however Morse potential usually deduced from experiment [4], so analytics calculation the physics quantity when to need Morse potential be very hard, therefore if thermodynamic parameters of Morse potential are calculated in advance will reduce the number calculations. In this studying scope, we are will analytics calculation in advance Morse interactive potential in anharmonic correlated Einstein model and application to determine the expressions of thermal expansion coefficient, build the expressions thermodynamic parameters and cumulants through Debye-Waller factor, consider correlative functions and thermodynamics parameters in classical approximation at high temperature and quantum effects at low temperature. 2. Formalism Anharmonic correlated Einstein model is described by effective interaction potential as form [1, 9]: ( ) ++≈ 332effE xkxk2 1 xU (1) In which 0rrx −= is deviation of the instantaneous bond length of two atoms from their equilibrium distance or the location of the minimum potential interaction, effk is effective spring constant, because it include all contributions of neighbor atoms, 3k is anharmonicity parameter and describing an asymmetry in interactive potential. Anharmonic correlated Einstein model is determined by vibration of single pairs atoms with 1M and 2M mass of absober and backscatter atoms. Vibration of atoms affected by neighbor atoms so interactive potential in expression (1) is written as form [6]: ( ) ( ) ∑ ≠       µ += ij iji0 i E ˆˆx M UxUxU RR (2) with 21 21 MM MM + =µ is reduced mass, Rˆ is the unit bond length vector, )x(U characterize to the single pairs potential between absorber and backscatter atoms, the second term in equation (2) characterize for contribution of nearest neighbors atoms and calculation by sum i which is over absorber )1i( = and backscatter )2i( = , and the sum j which is over all their nearest neighbors, excludes the absorber and backscatter themselves because they contribute in the ( )xU . The atomic vibration is calculated on based quantum statistical procedure with approximate quasi - hamonic vibration [1], in which the Hamiltonian of the system is written as harmonic term with respect to the equilibrium at a given temperature plus an anharmonic perturbation. Taking account from that we have: ( ) ( ) ( ) ...aykayk 2 1 2 P ...xkxk 2 1 2 P xU 2 PH 3 3 2 eff 2 3 3 2 eff 2 E 2 +++++ µ = =+++ µ =+ µ = ( ) ( ) =++++++++= ...332 2 1 2 2233 3 22 2 ayyaaykaayykP effµ 2 2 3 3 1 2 2 e f f P k a k a µ   = + + +    ( )2 2 33 3 313 3 ...2eff effy k a k a y k k a k y + + + + + +   (3) Setup 0H is sum of first term and fourth term, )a(U E is second term and )y(U Eδ is sum of TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO SỐ 02 – THÁNG 3 NĂM 2016 16 third term and fifth term, we have expressions: 2 2 0 22 ykPH += µ , ak kk eff 3322 += (4) ( ) 332effE akak2 1 aU += (5) ( ) ( ) 3323effE ykyak3akaU ++=δ (6) Expression (3) will become: ( ) ( )yUaUHH EE0 δ++= (7) in which a is thermal expansion coefficient with: 0rrx −= , axy −= , →= xa 0rrrraxy 00 =+−−=−= From equation (7) deduced interactive potential according to anharmonic correlated Einstein model can write as form: ( ) ( ) ( )yUyk 2 1 aUxU E 2 effEE δ++= (8) In anharmonic correlated Einstein model, interactive potential is Morse pairs anharmonic potential [5], consider approximation for cubic structural crystals, Morse anharmonic potential as form: ( ) ( ) ( )( )00 rrrr2 e2eDrU −α−−α− −= (9) in which α (Å-1) is thermal expansion coefficient, D(eV) is the dissociation energy by ( ) DrU 0 −= . We can write expression of Morse potential according to form of x: ( ) ( )xx2 e2eDrU α−α− −= (10) Expand the equation (10) according to x, we have: ( ) ( ) ( ) ( )         + α− + α− + α− + α− += ... !4 x2 !3 x2 !2 x2 !1 x21DxU 432 ( ) ( ) ( ) ( ) [ −α+α−=                 + α− + α− + α− + α− +− 22 432 x2x21D... !4 x !3 x !2 x !1 x12          α+α−α+α−−+α+α− 4433224433 x 12 1 x 6 1 x 2 1 x12...x 3 2 x 3 4 Taking approximate to the third-order term, we can write reduction: ( ) [ ]...x 3 1 xx22x 3 4 x2x21DxU 33223322 +α+α−α+−α−α+α−≈ Thus, expression of Morse potential according to deviation of the instantaneous bond length of two atoms x will write become: ( ) ( )...xx1DxU 3322 +α−α+−= (11) The interaction between pairs atoms in anharmonic correlated Einstein model is described by expression effective interaction potential of Morse pairs anharmonic potential in eq. (11). From equations (2) and (11) we have: ( ) ( )2 2 3 3E 0i ij j i i ˆ ˆU x D 1 x x ... U xR R M≠  µ = − + α − α + +     ∑ (12) With cubic structural crystals and pure, mass of absober and bacscatter atoms is equal, so can take approximation 2 MMMM 21 =µ→≈≈ , simultaneously expand second term of eq.(12) and calculation, we deduced thermodynamic parameters )E(U,k,k eff3 δ [1, 10]. ( ) 2E3223eff akcDck µω=+α= , 313 Dck α−= (13) ( ) ( )3132E ycaycDyU α−α=δ (14) in which 321 c,c,c are structural parameters with values corresponding has determined [5]. Anharmonic correlated Einstein model have been used to analytics calculation cumulants [6], the expand cumulants according to the expression: ( ) ( )         σ+= ∑ n n n 0 ikr2 !n i2ikr2expe ; ...3,2,1n= (15) with ( )nσ are cumulants and 0rrx −= is TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO SỐ 02 – THÁNG 3 NĂM 2016 17 thermal expansion coefficient and ( )1 0rr)T(a σ=−= ; axy −= , 0y = . Expand cumulants from the first-order to the sixth-order, we have: ( ) rR1 −=σ ; 0y = ( ) ( ) 2222 yRr =−=σ=σ ; ( ) ( ) 333 yRr =−=σ ; (16) ( ) ( ) ( ) ( )2242244 3yRr3Rr σ−=−−−=σ ( ) ( ) ( ) ( ) ( )2352355 y10yRrRr10Rr σ−=−−−−=σ ; ( ) ( ) ( ) ( ) ( ) ( ) =−+−−−−−−=σ 32232466 Rr30Rr10RrRr15Rr ( ) ( )3223246 30y10y15y σ+−σ−= . In above expressions of cumulants, the second cumulant ( ) 22 σ=σ or the mean square relative displacement (MSRD) otherwise known as Debye-Waller factor (DWF). Expressions annalytics calculation cumulants for cubic structure crystals has determined from the first-order to the third- order cumulants [10, 5], as form: The first cumulant or net expansion coefficient ( ) ( ) ( )z1 z1 Dc2 c3 a 2 1 E31 − + α ω ==σ ℏ (17) The second cumulant or Debye-Waller factor: ( ) ( ) ( )z1 z1 Dc2 y 2 1 E22 − + α ω ==σ ℏ (18) The third cumulant characterize to the anharmonicity: ( ) ( ) ( ) ( )2 2 3 1 32 2 E33 z1 zz101 cD2 c3 − ++ α ω =σ ℏ (19) Next, we calculate thermal expansion coefficient due to effect of anharmonicity when high raise temperature by fomula [1, 3]: P T T V V 1       ∂ ∂ =α (20) in which V is volume corresponding the change of absolute temperature T under pressure P. Use equation state of thermal system: 1 P V V T T P TPV −=      ∂ ∂       ∂ ∂       ∂ ∂ TV P P V T P 1 V T       ∂ ∂       ∂ ∂−=     ∂ ∂ ⇒ (21) From expressions (20) and (21), we have: VT T T P P V V 1       ∂ ∂       ∂ ∂ =α (22) Setup TV PVK       ∂ ∂ −= is elastic modulus determination the change of volume due to interaction of pressure. Ignore links between vibrations of atoms and assume freedom energy Helmholtz as form ∑+= q qFUF with U is sum potential energy, qF is free energy and was created from vibration of lattice with wave vector q, then pressure dependence to volume according to expression [2,5]: q q q q qT B dFF dU dU 1 1P V dV dV dV V 2 exp 1 k T    ∂ω∂    =− =− − =− − +   ω∂ ∂    −       ∑ ∑ℏ ℏ (23) When appearance anharmonic effect, the system equilibrium at new location and volume expanded so important phenomena of anharmonic effect is dependence of frequency net vibration to volume, this dependence described through by second term in expression (23). To simple, assume dependence to volume of all frequencies net vibration the same and write through Gruneisen factor as TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO SỐ 02 – THÁNG 3 NĂM 2016 18 form: ( ) ( ) ( ) ( )Vln r r a a ln Vln lnV~ GG ∂ ∂ ∂ ∂ ∂ ω∂ −= ∂ ω∂ −=γ⇒ω γ− (24) Factor Gγ characterize for anharmonic effect with net thermal coefficient: 1 r a rra 0 =∂ ∂ →−= Simultaneously we have: ( ) ( ) rdTdaTaTa T0 α==− Deduce thermal expansion coefficient: dT da r 1 T =α (25) Substitute (17) into (25) we get: ( ) ( ) ( )( ) =      − + α ω =      − + α ω =α θ− θ− T/ T/ 2 1 E3 2 1 E3 T E E e1 e1 dT d rDc2 c3 z1 z1 dT d rDc2 c3 ℏℏ ( ) ( )( ) ( ) =− θ −−− θ − α ω = θ− θ−θ−θ−θ− 2T/ 2 ET/T/ 2 ET/T/ 2 1 E3 E EEEE e1 T ee1 T ee1 rDc2 c3 ℏ ( ) ( ) ( ) ( ) =− θ α ω = − θ ++ θ − α ω = 2 2 E 2 1 E3 2 2 E 2 E 2 1 E3 z1 T z2 rDc2 c3 z1 T zz1 T zz1 rDc2 c3 ℏℏ ( )22 E 2 1 E3 T z1 z TrDc c3 − θ α ω =α⇒ ℏ with B E k ω =θ ℏ we have: ( ) ( ) ( )2 2 B E 2 1 B3 22 B 2 E 2 1 3 T z1 z TkrDc kc3 z1 z TkrDc c3 −       ω α = − ω α =α ℏℏ ; replace zln Tk B E = ωℏ , we obtained thermal expansion coefficient: ( ) ( )2 2 2 1 B3 T z1 zlnz rDc kc3 −α =α . (26) To reduce calculated and measure, to need simplification the description expressions of thermodynamic parameters, thus we can description thermodynamic parameters through DWF 2σ [6,7,8] by: 2 0 2 2 0 2 z σ+σ σ−σ = (27) Substitute formula (27) into equations (17, 18, 19, 26), we have: 2 0 1 3)1( 0 2 1 3)1( 0 )1( c c3 ; c c3 z1 z1 σ α =σσ α = − + σ=σ (28) 2 1 E2 0 2 0 2 Dc2 ; z1 z1 α ω =σ − + σ=σ ℏ (29) ( ) ( ) ( ) ( ) 22 0 1 3)3( 022 0 22 0 22 )3( 0 )3( c c3 ; 23 σ α =σ σ σ−σ σ=σ (30) rDc kc3 ;1 Tk Dc 2 1 B30 T 2 2 2 0 2 B 22 10 TT α =α                 σ σ −         σα α=α (31) Simultaneously we deduce correlative expressions between cumulants together and between cumulants with thermal expansion coefficient Tα , distance between atoms r and absolute temperature T according to structural parameters and Debye-Waller factor: 22 0 22 2 2 T 1 (3) 22B 0 2 1 rT c D 2k T 21 3  σ −   σα σ α σ   = σ  σ −   σ  ; (32) 2 2 2 0 )3( 2)1( 3 42 1         σ σ − = σ σσ (33) where 20)1(0 , σσ and )3(0σ are contributions zero- point into 2)1( , σσ and )3(σ , structural parameters was described in [5]. According to the description above, outside the Morse potential parameters analytics calculation, to calculate cumulants 2)1( , σσ , )3(σ and thermal expansion coefficient α T, we only need to calculate DWF 2σ , therefore has reduce analytics calculation and programmable calculator for thermodynamic parameters. The expressions is determined from quantum theory, therefore TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO SỐ 02 – THÁNG 3 NĂM 2016 19 can applying for any temperature, at high temperature it include approximate results of classical theory and at low temperature limit it always shows quantum effects through contributions of zero-point energy. In high temperature limit (HT) we use approximate formula x1e x +≈ , and approximate change T E ez θ − = with B E E k ω =θ ℏ , deduced: Tk 1z B Eω −≈ ℏ (34) We get: ω ω− = ω ω − ≈ − + ℏ ℏ ℏ ℏ Tk2 Tk Tk 2 z1 z1 B B B (35) so we reduce expressions of cumulants and thermodynamic parameters at high temperature (see in Table statistic) In low temperature limit (LT) we use approximate formulas: z1 z1 1 +≈ − ; z21zz1 z1 z1 +=++≈ − + (36) because in low temperature limit 0z → , thus, we can ignore 2z and higher powers, we reduce expressions of cumulants and thermodynamic parameters at low temperature limit (see in Table statistic). Note cumulants )1(σ , 2σ , )3(σ include contributions zero-point energy, Tα approaching the value constant 0Tα at high temperatures but they destructively according to exponential of T Eθ at low temperature and both correlative expressions (32) and (33) approximately with classical results and experiment as 2/1 at high temperatures and right reflection with results of classical theory and experimentalist. Table 1: Expressions of cumulants, thermal expansion coefficient, correlative expressions at low temperature limit ( 0T→ ) and approximation at high temperature ( ∞→T ) Quantities 0T → ∞→T )1(σ ( )z21)1(0 +σ αDc/Tkc3 21B3 2σ ( )z2120 +σ 21B Dc/Tk α )3(σ ( )z121)3(0 +σ ( ) 32312B3 Dc/Tkc6 α Tα ( ) ( )z21zlnz 20T +α T0α )3(2 T /rT σσα )z/1ln(z3 → 0 2/1 )3(2)1( / σσσ ( ) ( )z1212/z213 2 ++ → 3/2 2/1 TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO SỐ 02 – THÁNG 3 NĂM 2016 20 3. Conclusion The effective interaction potential in anharmonic correlated Einstein model was determined on base analytics calculation Morse interactive potential between pairs absorber and backscatter atoms with nearest neighbor atoms, this work was reduced the calculation thermodynamic parameters and measures, because replace the calculation by complex matrices three dimensions we only need to solve problem one dimension with the interaction of cluster nearest neighbor atoms and results obtained agree well with experimental data. Figure 1 description anharmonic effective potential interatomic and compared with experimental data for FeMo crystal and graph show the shifts between present theory and experimental data small than shifts between harmonic term with experimentalist, this results to see present procedure can to use good for the study anharmonic vibration of atoms. Figure 2 description the dependence temperature and net expansion of anharmonic perturbation factor in anharmonic correlated Einstein model and form the graph approximate classical at high temperature and quantum effects at low temperature. Figure 3 describe the dependence temperature T of the thermal expansion coefficient, we see Tα approaching the value constant 0Tα at high temperature but has destructed according to exponential at low temperatures. The thermal expansion coefficient, cumulants and thermodynamic parameters was presented through Debye-Waller factor and structure parameters and has reduced the calculations also measure and programmable calculator. The expression of correlative function between cumulants, correlative function between cumulants and thermal expansion coefficient for cubic structural crystals was determined and inclusion both classical theory at high temperature and quantum effects at low temperature limit. Figure 1: Anharmonic effective interatomic Figure 2: Dependence temperature and net expansion x of anharmonic pertubation factor Figure 3: Dependence temperature T of the thermal expansioncoefficient Tα TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO SỐ 02 – THÁNG 3 NĂM 2016 21 REFERENCES 1. Nguyen Ba Duc (2015) Anharmonic correlated Einstein model in XAFS theory and application, LAMBERT Academic Publishing, Germany; 2. Nguyen Ba Duc (2015) Statistical physics, Publisher Thai Nguyen University; 3. Nguyen Van Hung (1999) Solid state theory, Publisher National University, Ha Noi; 4. Girifalco, L. A., Weizer, V. G. (1959), “Application of the Morse potential Function to cubic Metals”, Phys. Rev. (114), pp. 687. 5. Nguyen Ba Duc, (2014), “By using the anharmonic correlated Einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new structure factors”, Journal of Physics and Astronomy Research (AUS). (1), pp.02-06. 6. Hung, N. V. and Duc, N. 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