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

Chia sẻ: Hân Hân | Ngày: | Loại File: PDF | Số trang:8

Thêm vào BST

Báo xấu

76
lượt xem 1
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

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.

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: 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

TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO 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 14 SỐ 02 – THÁNG 3 NĂM 2016 TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO 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]: U E (x ) ≈ Anharmonic correlated Einstein model is determined by vibration of single pairs atoms with M 1 and M 2 mass of absober and backscatter atoms. Vibration of atoms affected by neighbor atoms so interactive potential in expression (1) is written as form [6]: with µ = 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 (i = 1) and backscatter ( i = 2 ) , and the sum j which is over all their nearest neighbors, excludes the absorber and backscatter themselves because they contribute in the U (x ) . 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: H= is deviation of the = instantaneous bond length of two atoms from their equilibrium distance or the location of the minimum potential interaction, k eff is = effective spring constant, because it include all contributions of neighbor atoms, k 3 is anharmonicity parameter and describing an asymmetry in interactive potential. P2 P2 1 + U E (x ) = + k eff x 2 + k 3 x 3 + ... = 2µ 2µ 2 = (1) x = r − r0 M1M 2 is reduced mass, Rˆ is the M1 + M 2 unit bond length vector, U ( x ) characterize to 1 k eff x 2 + k 3 x 3 + … 2 In which (2)  µ  U E (x ) = U (x ) + ∑ U  x Rˆ 0 i Rˆ ij  j≠ i  M i  P2 1 + k eff (y + a )2 + k 3 (y + a )3 + ... 2µ 2 P2 1 + keff y 2 + 2ay + a 2 + k3 y 3 + a 3 + 3a 2 y + 3ay 2 + ... = 2µ 2 ( P 2 2 µ ) (  1 +  k  2 e ff a 2 ) + k 3 a 3   +  1  +y keff a +3k3a2 + y2  keff +3k3a + k3 y3 +... (3) 2  ( ) Setup H 0 is sum of first term and fourth term, U E ( a ) is second term and δ U E ( y ) is sum of SỐ 02 – THÁNG 3 NĂM 2016 15 TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO third term and fifth term, we have expressions:  (−αx) (−αx)2 (−αx)3 (−αx)4  −21+ + + + +... = D1−2αx + 2α2x2 −   1! 2! 3! 4!   [ k keff P2 k + 3k3a H0 = + y2 , = 2µ 2 2 2 (4) 1 UE (a ) = k eff a 2 + k 3a 3 2 4 2 1 1 1   − α3x3 + α4x4 +...−21−αx+ α2x2 − α3x3 + α4x4  3 3 2 6 12   (5) Taking approximate to the third-order term, we can write reduction: (6) 4 1 U(x) ≈ D1− 2αx + 2α2x2 − α3x3 − 2 + 2αx − α2x2 + α3x3 +... 3 3 ( ) δ U E (a ) = k eff a + 3k 3 a 2 y + k 3 y 3 Expression (3) will become: H = H0 + U E (a ) + δU (7) (y ) E in which a is thermal expansion coefficient with: y = x −a , x = r − r0 , a= x → y = x − a = r − r0 − r + r0 = 0 From equation (7) deduced interactive potential according to anharmonic correlated Einstein model can write as form: 1 U E (x ) = U E (a ) + k eff y 2 + δU E (y) 2 (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: U(r ) = D e −2α(r −r0 ) − 2e −α(r −r0 ) ) (9) in which α (Å-1) is thermal expansion coefficient, D(eV) is the dissociation energy by U (r0 ) = − D . We can write expression of Morse potential according to form of x: ( U(r ) = D e−2αx − 2e−αx ) (10) Expand the equation (10) according to x, we have:  − 2αx (− 2αx)2 (− 2αx)3 (− 2αx)4 U(x) = D1+ + + + + ... 1! 2! 3! 4!  [ ] Thus, expression of Morse potential according to deviation of the instantaneous bond length of two atoms x will write become: ( ) U(x) = D −1+ α2x2 − α3x3 +... (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:  µ ˆ ˆ  U E ( x ) = D −1 + α 2 x 2 − α 3 x 3 + ... + ∑ U  xR 0i R ij  j≠ i  Mi  ( ) (12) With cubic structural crystals and pure, mass of absober and bacscatter atoms is equal, so can take approximation M1 ≈ M 2 ≈ M → µ = M , 2 simultaneously expand second term of eq.(12) and calculation, we deduced thermodynamic parameters k 3 , k eff , δ U ( E ) [1, 10]. ( ) k eff = c 3 Dα 2 + c 2 ak 3 = µω2E , k 3 = − c1 D α 3 ( δUE (y) = Dα2 c3ay − c1αy3 ) (13) (14) in which c 1 , c 2 , c 3 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: e 2ikr  (2i)n σ(n)  ; n=1,2,3... = exp2ikr0 + ∑  n n!   (15) with σ (n ) are cumulants and x = r − r 0 is 16 SỐ 02 – THÁNG 3 NĂM 2016 TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO thermal expansion coefficient and y = x −a , y = 0. a (T) = r − r0 = σ (1) ; Expand cumulants from the first-order to the sixth-order, we have: σ(1) = R − r ; y =0 σ(2) = σ2 = (r − R)2 = y2 ; σ(3) = (r − R)3 = y3 ; 2 2 ( ) = y4 − 3 σ2 ( ); ( ) = y6 − 15 y 4 σ2 − 10 y (r − R)2 3 2 −10 (r − R)3 2 + 30 (r − R)2 3 = ( ). + 30 σ (20) in which V is volume corresponding the change of absolute temperature T under pressure P. Use equation state of thermal system: ⇒ σ(5) = (r − R)5 −10 (r − R)3 (r − R)2 = y5 −10 y3 σ2 σ(6) = (r − R)6 −15 (r − R)4 1  ∂V    V  ∂T  P  ∂P   ∂T   ∂V        = −1  ∂T  V  ∂V  P  ∂P  T (16) σ (4 ) = (r − R )4 − 3 (r − R )2 αT = 1  ∂T    =−  ∂P   ∂V   ∂V  P      ∂T  V  ∂P  T From expressions (20) and (21), we have: 2 3 In above expressions of cumulants, the second cumulant σ (2 ) = σ 2 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 thirdorder cumulants [10, 5], as form: (21) αT = 1  ∂V   ∂P      V  ∂P  T  ∂T  V (22)  ∂P  K = − V  is elastic modulus  ∂V  T determination the change of volume due to interaction of pressure. Ignore links between vibrations of atoms and assume freedom energy Helmholtz as form F = U + ∑ Fq with U Setup q The first cumulant or net expansion coefficient is sum potential energy, Fq is free energy and 3c ℏω (1 + z) σ(1) = a = 32 E 2c1 Dα (1− z) was created from vibration of lattice with wave vector q, then pressure dependence to volume according to expression [2,5]: (17) The second cumulant or Debye-Waller factor: ℏωE (1+ z) σ(2) = y2 = 2c1Dα2 (1− z) (18) The third cumulant characterize to the anharmonicity: ( 3c 3 (ℏω E )2 1 + 10z + z 2 ( 3) σ = 2D 2 α 3c13 (1 − z )2 ) (19) Next, we calculate thermal expansion coefficient due to effect of anharmonicity when high raise temperature by fomula [1, 3]:     dFq ∂ωq  1 dU dU 1  ∂F   P = −  = − − ∑ = − − ∑ℏ + dV q dV dV q ∂V  2  ℏωq    ∂V T −1   exp   kBT   (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 SỐ 02 – THÁNG 3 NĂM 2016 17 TẠP CHÍ KHOA HỌC ĐẠI HỌC TÂN TRÀO form: ω ~ V−γG ⇒ γG = − ∂(ln ω) ∂(ln ω) ∂a ∂r (24) =− ∂(ln V) ∂a ∂r ∂(ln V) Factor γ G characterize for anharmonic effect with net thermal coefficient: Substitute formula (27) into equations (17, 18, 19, 26), we have: σ(1) = σ0(1) Simultaneously we have: ℏωE 1+ z ; σ02 = 1− z 2c1Dα2 σ ( 3) = σ 0( 3) thermal expansion 1 da = r dT coefficient: α T (25) 2 2 0 = = ( ( (1− e )e T ( T (1 − z )2 2 c1 2 D α r 3c 3ℏωE θ E )( T ) −θE / T 2 = 3c 3 ℏω E 2z )Tθ E 2 = θE T2 = 2c1 2 Dα r (1 − z )2 z z c1 2 D α r k B T 2 (1 − z )2 3c k  ℏ ω E  = 23 B   c1 D αr  k B T  c12 Dαr (1 − z)2 2 z ; z= 18 σ2  3c3kB c12Dαr 1 = (3) 4 2 − 3 (1 − z )2  σ   σ  2 0 2     2 (33) where σ 0 (1) , σ 02 point into σ and (1 ) σ 0 ( 3) ,σ 2 are contributions zeroand σ (3) , structural parameters was described in [5]. . (26) To reduce calculated and measure, to need simplification the description expressions of thermodynamic parameters, thus we can description thermodynamic parameters through DWF ; α0T = (30) absolute temperature T according to structural parameters and Debye-Waller factor: σ expansion coefficient: αT = 2 ( ) Simultaneously we deduce correlative expressions between cumulants together and between cumulants with thermal expansion coefficient αT , distance between atoms r and (32) σ ( 1 ) σ 2 ℏω E = ln z , we obtained thermal replace k BT 3c3k B z(ln z)2 2 2 2    2 2   0  c1 Dα σ    σ0   1− αT = αT  kBT    σ2       3c 3 α 2 σ0 c1 2 ℏω we have: kB (ℏω E )2 σ (03) = ;  σ20  1 −  2 αT rTσ2 c1Dα 2 σ2 σ  ; = (3) 2 2k BT σ 2  σ2  1 −  02  3σ  c12 Dαr T 2 (1 − z )2 with θ E = 3c 3 ) = ) − 1− e−θE / T − e−θE / T (1 − z )z θ E2 + (1 + z )z θ E2 3c 3 ℏω E αT = 2 (1− e 2c12Dαr ⇒ αT = θE −θE / T −θE / T (29) (31) 3c 3ℏω E d  (1 + z )  3c 3 ℏω E d  1 + e −θE / T   = 2c12 Dαr dT  (1 − z )  2c12 Dαr dT  1 − e −θE / T 3c3ℏωE 2 2 0  Substitute (17) into (25) we get: αT = 2 ( ) − 2(σ ) (σ ) 3 σ2 a (T ) − a (T 0 ) = da = α T rdT Deduce (28) 1 σ2 = σ02 ∂a a = r − r0 → =1 ∂r 3c α 1 + z 3c3α 2 = σ ; σ0(1) = 3 σ02 1− z c1 c [6,7,8] by: σ2 − σ02 σ2 + σ02 SỐ 02 – THÁNG 3 NĂM 2016 (27) According to the description above, outside the Morse potential parameters analytics calculation, to calculate cumulants σ (1 ) , σ 2 coefficient 2 , σ (3) and thermal expansion α T, we only need to calculate DWF σ , therefore has reduce analytics calculation and programmable calculator for thermodynamic parameters. The expressions is determined from quantum theory, therefore