Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM

II-P-1.19 POSITRON ANNIHILATION IN PERFECT AND DEFECTIVE ZRO2 MONOCLINIC CRYSTAL WITH SINGLE PARTICLE WAVE FUNCTION: SLATER TYPE ORBITAL AND MODIFIED JASTROW FUNCTIONS Trinh Hoa Lang, Chau Van Tao, Le Hoang Chien Faculty of Physics and Engineering Physics, University of Science – HCM City, Vietnam Email: [email protected] ABSTRACT The positron annihilation in ZrO2 monoclinic crystal is studied by assumption that a positron binds with the valance electrons of zirconium and oxygen to form the pseudo ZrO 2 – positron molecule before it annihilates with these. The modifications of explicit electron-positron and electron-electron correlations of the electrons and positron wave functions are carried out by Pade and Yakawa approximations. It is applied these wave functions and VQMC to estimate the positron lifetime in the unmitigated and the defective ZrO2 crystals Keywords: positron, annihilation, Jastrow, lifetime, VQMC, ZrO 2. INTRODUCTION In this work, the study of the positron annihilation with the valence electrons of the material is assumed that positron will bind with the unit structure to form the new ground - state before it destroys with the electrons of this unit. By this assumption, the orthonormalized Slater type orbital will be used for describing the electrons and the positron wave functions in the irreducible element and VQMC[8] will be used to find their ground-state wave functions. When a positron is going into the material, the positron will be slowed down to thermal energy and then bound with the atoms in the unit structure. In this state, the positron wave function is approximated by the hybrid wave function of the valence electrons. The ground state of many-body problem is determined by minimizing the energy of each particle rather than the total energy. The problem of total wave function, Slater determinant representing for exchange effects, is avoided by solving the individual particle equation with the Hamiltonian of exchange potential. One particle energy is derived by one particle equation, which is constructed from the Kohn-Sham method and a single particle wave function. This wave function consists of an atomic wave function and a correlation function. By choosing a trial single wave function, the VQMC method is used to find the ground state of positron-valence electron of a unit element. THEORY The theoretical model [14] of the positron annihilation in a specific structure is derived from the two – component density functional theory [3] and quantum Monte Carlo. The electron and the positron wave functions are used with some modifications of the correlation functions and the exchange – correlation potential [14]. The Slater type wave functions are used for the single electron wave function and the linear combination of them is used for the single positron wave function. The correlation function has the form J(r) = exp[-u(r)] and u(r) is the Jastrow function. This function is suggested in the new modified Jastrow function which is a combination of long correlation Padé [8] and short correlation Yukawa [5]. The Jastrow function for the electron – electron correlation is given by

u ee r  

e A 1  e  Fe r   e 1  er r

and for electron – positron correlation is given by

u ep r  

 ep r 1   ep r





A ep 1  e r

 Fepr

(1)



(2)

where e, e, Ae, Fe, ep, ep, Aep, and Fep are the variational parameters. These parameters depend on the electronic structure of material. Then the single wave function of the electron and positron can be given by

i r   i r Jr 

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210

Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM where i is atomic wave function for single electron or positron wave function ith. The electron and positron densities are calculated by summing over the occupied states (n- and n+ are the number of electrons and positrons respectively).

    i r  ;     i r  n

2

i

n

2

(4)

i

Basing on the theoretical model of tight – binding of the positron and the unit structure[14], the pair correlation function, the positron annihilation rate and the positron lifetime can be determined in the ground – state of this system. Positron annihilation rate depending on the overlap of electron and positron densities and enhancement factor is directly given by the value of the pair-correlation function at the origin via the relation [7][9]





re2 c N  i r   r g 0;   ,   dr    i



where g 0;   ,   is the enhancement factor which is the value of the pair – correlation function g at r = 0 for given electron density. This function is fitted in Chebyshev polynomial [11] as given by N  2r  L  gr;   ,     gr    c i Ti   i 0  L 

(5)

r;  ,   



(6)

L is maximum interaction range of electron and positron, radius of spherical cell, which is used for collecting electron distribution data in space around a positron. This model is applied to study the positron annihilation in the unit cell of zirconium dioxide of monoclinic structure consisting of two zirconiums and four oxygens as shown in figure 1. The single-particle wave function in ZrO2 is constructed by LCAO [10] approximation and the atomic wave function of oxygen and zirconium.

Figure 1. The schematic of the zirconium dioxide – positron system and the configuration of the zirconium dioxide in unit cell of monoclinic crystal structure.The big spheres are zirconiums and the smaller spheres are oxygens. The single – particle wave functions for the valance electrons of zirconium and oxygen atoms in 4d, 5s and 2p shells are obtained from the Slater Type Orbital[6] and are constructed by LCAO approximation as shown.

 i4dz2 Z Zr , Z O1 , Z O 2 ; r   C1i4dz2 Z Zr , r  R Zr   C 2 i4dz2 Z O1 , r  R O1  C 3  i4dz2 Z O 2 , r  R O 2 

(7)

 i4dxz Z Zr , Z O1 , Z O 2 ; r   C1 i4dxz Z Zr , r  R Zri   C 2 i4dxz Z O1 , r  R O1  C 3  i4dxz Z O 2 , r  R O 2 

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Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM i4dyz Z Zr , Z O1 , Z O 2 ; r   C1i4dyz Z Zr , r  R Zr   C 2 i4dyz Z O1 , r  R O1  C 3 i4dyz Z O 2 , r  R O 2 

Z Zr , Z O1 , Z O2 ; r   C1

(9)

Z Zr , r  R Zr   C 2  Z O1 , r  R O1  C 3  Z O 2 , r  R O 2  4 dx 2  y 2 Z Zr , Z O1 , Z O2 ; r   C1i4dx2 y 2 Z Zr , r  R Zr   C 2 i4dx2 y 2 Z O1 , r  R O1  i C 3  i4dx2 y 2 Z O 2 , r  R O 2   5is Z Zr , Z O1 , Z O 2 ; r   C1 5is Z Zr , r  R Ti   C 2  5is Z O1 , r  R O1  C 3  5is Z O 2 , r  R O 2   i2 px Z Zr , Z O1 , Z O 2 ; r   C1 i2 px Z O1 , r  R O1   C 2  i2 px Z O 2 , r  R O 2  C 3  i2 px Z Zr , r  R Zri   i2 py Z Zr , Z O1 , Z O 2 ; r   C1 i2 py Z O1 , r  R O1   C 2  i2 py Z O 2 , r  R O 2  C 3  i2 pz Z Zr , r  R Zr   i2 pz Z Zr , Z O1 , Z O 2 ; r   C1 i2 pz Z O1 , r  R O1   C 2  i2 pz Z O 2 , r  R O 2  C 3  i2 pz Z Zr , r  R Zr  

4 dxy i

4 dxy i

4 dxy i

4 dxy i

(10)

(11) (12)

(13)

(14)

(15)

i4dz2(Z,r), i4dyz(Z,r), i4dxz(Z,r), i4dxy(Z,r), i4dx2-y2(Z,r), i5s(Z,r), i2px(Z,r), i2py(Z,r), and i2pz(Z,r) are Slatertype orbitals of atomic wave functions. r – RZr , r – RO1 and r – RO2 are respectively distances of ith electron to zirconium and two oxygens. The single – particle wave function for the positron in the bound state with valance electrons of oxygen and zirconium is hybrid wave function of these electron wave functions. According to the principle of linear superposition, the single – particle wave function for positron is supposed to take the form as

ip Z pZr , ZOp 1 , ZOp 2 ; r   c1i4dz2 Z pZr , ZOp 1 , ZOp 2 ; r   c 2 i4 dxz Z pZr , ZOp 1 , ZOp 2 ; r   c 3i4dyz Z pZr , ZOp 1 , ZOp 2 ; r 

c 4 i4dx2 y 2 Z pZr , ZOp 1 , ZOp 2 ; r   c 5 i4 dxy Z pZr , ZOp 1 , ZOp 2 ; r   c 6 5is Z pZr , ZOp 1 , ZOp 2 ; r  (16) c 7 i2 px Z pZr , ZOp 1 , ZOp 2 ; r   c8 i2 py Z pZr , ZOp 1 , ZOp 2 ; r   c9 i2 pz Z pZr , ZOp 1 , ZOp 2 ; r 

where ZZr, ZO1, ZO2, ZpZr, ZpO1 and ZpO2 are the variational parameters in the atomic wave function of electron and positron in ZrO2; C1, C2 and C3 are the weighted coefficients of electron wave function with C1 = C2 = C3 =

1 / 3 ; and c1, c2, c3, c4, c5, c6, c7, c8 and c9 are the weighted coefficients of positron wave function with c1 = c2 = c3= c4= c5= c6= c7= c8= c9 = 1/3. Using equations from (1) to (3) and (7) to (16), the minimizing energy is calculated by VQMC and then the ground-state wave function of this system can be obtained. Assumption that these ground-state wave function is valid for ZrO2 monoclinic, the properties of electron – positron annihilation in ZrO2 monoclinic, therefore, are determined. RESULTS AND DISCUSSION. In calculation of ground-state wave function of electrons and positron in ZrO2 molecule, the fourteen sets of energy data which correspond to the variation of the fourteen parameters ZTi, ZO1 , ZO2 , e, e, Ae, Fe, ZpTi, ZpO1 , ZpO2 , ep, ep, Aep, and Fep are generated. Each result is calculated by varying one of these parameters, and the other thirteen parameters are kept as constants. The set of optimized parameters, which make the trial wave function well approach to exact wave function, is determined from these sets by minimization of energy. These optimized parameters are given in table 1. Table 1. The values of optimized parameters of electron and positron wave function ZTi

ZO1

ZO2

e

e

Ae

Fe

3.7

1.1

22.1

5.5

2.2

4.5

2.1

Zp

Zp

Zp

ep

ep

Aep

Fep

0.4

0.5

0.3

0.2

0.9

1.4

0.6

Ti

O1

O2

The enhancement factor and the positron lifetime in perfect ZrO 2 monoclinic crystal are determined by using these parameters. The electron – positron pair correlation distribution is collected by Monte – Carlo’s simulation as shown in figure 2a. The enhancement factor is extrapolated by the pair correlation function which ISBN: 978-604-82-1375-6

212

Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM is fitted by Chebyshev polynomial in equation (6). The number of coefficient N in equation (6) is determined by the minimization of chi square of fitting goodness of pair correlation function as shown in figure 2b. Minimal chi square is corresponding to N = 38 with the value of chi square 2 =0.219442 and the values of the coefficients are given in the table 2. After fitted analytical form of the pair correlation function of electron – positron g(r, -, +) is constructed, the enhancement factor is calculated with g(r=0, -, +) = 1126.65 as shown in figure 2c and 2d. This enhancement factor can be considered as effective number electron in the paper[2]. So this value of the enhancement factor is resonable value in comparing to the result in this paper[2]. By applying the formular (5), the annihilation rate is estimated by overlap of electron and positron densities, which is determined from Monte – Carlo simulation, and the enhancement factor  = (5.225  0.005)×109 (s-1). The positron lifetime is inverse of the annihilation rate  = (191  2) ps. This result is a little greater than the result in the work[1]. The fitting parameters of Chebyshev polynomial are shown on table 2.

Figure 2. a) The electron – positron pair distribution obtained by Monte – Carlo calculation with the optimized wave function of electron and positron in ZrO2 perfect monoclinic crystal;b) The chi square of goodness of fit in the term of the number of Chebyshev coefficients;c), d) The graph of fitted pair correlation fucntion of electron – positron. The black dot is Monte – Carlo simulation and the solid line is its fitting curve. Table 2. The fitting parameters of Chebyshev polynomial of electron – positron correlation function in ZrO2 perfect monoclinic crystal. c0

c1

c2

c3

c4

c5

c6

4.10434×108

4.81244×108

-2.56131×108

-7.80682×108

-6.54615×108

3.76495×107

7.65425×108

c7

c8

c9

c10

c11

c12

c13

9.61306×108

4.2027×108

-6.00696×108

-1.60198×109

-2.16855×109

-2.17012×109

-1.7474×109

c14

c15

c16

c17

c18

c19

c20

-1.1601×109

-6.30282×108

-2.66306×108

-7.18435×107

3.87314×106

1.85853×107

1.24801×107

c21

c22

c23

c24

c25

c26

c27

4.78327×106

720345.

-435030.

-407027.

-191380.

-67305.4

-21279.8

c28

c29

c30

c31

c32

c33

c34

-5577.02

129.659

1310.91

650.631

-24.1979

-220.413

-141.774

c35

c36

c37

c38

-42.7249

2.32369

6.68755

2.40294

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Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM The enhancement factor and positron lifetime in ZrO2 monoclinic crystal of an oxygen defect, created by removing one oxygen atom from unit cell of ZrO 2 perfect monoclinic crystal, are also done by these parameters in table 1, which is assumed that these parameters is still valid for describing the electron and the positron wave functions in the ZrO2 monoclinic crystal of an oxygen defect. The distribution of electron – positron pair correlation in the ZrO2 of an oxygen defect is obtained from Monte-Carlo’s simulation as shown in figure 3a. The number of coefficient N in this case is identified by the minimization of chi square corresponding to N = 57 with the value of chi square 2 = 0.0190853 as shown in figure 3b and the values of the coefficients are given in the table 3. The enhancement factor in ZrO2 of an oxygen defect is obtained by the extrapolation of pair correlation function at r = 0 with g(r=0, -, +) = 374.099 as shown in figure 3c and 3d. The annihilation rate and the lifetime of positron,which are determined from this enhancement factor and the overlap of electron and positron densities, are  = (3,860  0,005)×109 (s-1) and  = (259  3) ps respectively. This positron lifetime is greater than one in the case of perfect crystal of ZrO2. It showed that this result is suitable to predict the positron lifetime in monovacancy defect.

Figure 3. a) The electron – positron pair distribution obtained by Monte – Carlo calculation with the optimized wavefunction of electron and positron in ZrO2 monoclinic crystal of an oxygen defect;b) The chi square of goodness of fit in the term of the number of Chebyshev coefficients; c),d) The graph of fitted pair correlation function of electron – positron. The black dot is Monte – Carlo simulation and the solid line is its fitting curve. Table 3. The fitting parameters of Chebyshev polynomial of electron – positron correlation function in ZrO2 monoclinic crystal of an oxygen defect. c0

c1

c2

c3

c4

c5

c6

8.15083×107

9.59066×107

-5.07185×107

-1.56796×108

-1.33118×108

7.68232×106

1.59929×108

c7

c8

c9

c10

c11

c12

c13

2.02804×108

8.48488×107

-1.45945×108

-3.80463×108

-5.19148×108

-5.23778×108

-4.22709×108

c14

c15

c16

c17

c18

c19

c20

-2.78634×108

-1.48599×108

-6.11782×107

-1.67121×107

-842527.

1.79588×106

828211.

c21

c22

c23

c24

c25

c26

c27

-4281.16

-186759.

-102855.

-24256.1

160.68

692.06

-930.967

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Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM c28

c29

c30

c31

c32

c33

c34

-810.87

-359.432

-181.732

-126.745

-67.4853

-15.0278

9.08281

c35

c36

c37

c38

c39

c40

c41

9.24742

2.38405

-0.947838

-0.74746

0.126992

0.133303

0.0511814

c42

c43

c44

c45

c46

c47

c48

-0.0321264

0.0155505

-0.00851802

0.00633494

-0.000669446

0.000742321

-0.00025831

c49

c50

c51

c52

c53

c54

c55

-0.0000481428

0.0000232338

0.0000241346

-5.11878×10-6

3.0873×10-6

3.31341×10-7

-1.08377×10-7

c56

c57

2.7218×10-7

-5.1034×10-8

The enhancement factor and positron lifetime in ZrO 2 monoclinic crystal of a zirconium defect, created by removing one zirconium atom from unit cell of ZrO 2 perfect monoclinic crystal, are familiarly determined in the oxygen defect case. These parameters of wave function of electron and positron in the ZrO 2 monoclinic crystal of a zirconium defect is still valid, and the distribution of electron – positron pair correlation is obtained from Monte - Carlo’s simulation as shown in figure 4a. The number of coefficient N is determined by minimization of chi square of fitting goodness of pair correlation function as shown in figure 4b. Minimal chi square is corresponding to N = 35 with the value of chi square 2 = 2.3126 and the values of the coefficients are given in the table 4. By this pair correlation function, the enhancement factor is calculated with g(r=0, -, +) = 621.632 as shown in the figure 4c and 4d. The results of annihilation rate and life time of positronare  = (4,024  0,003)×109 (s-1) and  = (248  2) ps respectively. This result is as well greater than the positron lifetime in the perfect ZrO2 crystal but it is smaller than the positron lifetime in the ZrO2 of oxygen defect. The results of the positron lifetime in oxygen and zirconium defects in ZrO2 monoclinic crystal are greater than the shorter lifetime component of positron in the vacancies of ZrO2 monoclinic with impurities[12][13]. Table 4. The fitting parameters of Chebyshev polynomial of electron – positron correlation function in ZrO2 monoclinic crystal of a zirconium defect. c0

c1

c2

c3

c4

c5

c6

1.92204×1010

2.15438×1010

-1.41187×1010

-3.69786×1010

-2.69704×1010

7.35944×109

3.70847×1010

c7

c8

c9

c10

c11

c12

c13

3.75041×1010

6.95324×109

-3.46087×1010

-6.2828×1010

-6.58483×1010

-4.86054×1010

-2.49157×1010

c14

c15

c16

c17

c18

c19

c20

-6.5702×109

2.2694×109

3.81818×109

2.31116×109

7.14815×108

-2.34901×107

-1.34714×108

c21

c22

c23

c24

c25

c26

c27

-5.396×107

1.80813×106

9.45851×106

2.28975×106

-1.26514×106

-865454.

1930.12

c28

c29

c30

c31

c32

c33

c34

199272.

94105.5

21483.8

9161.1

8449.59

4574.24

1293.56

c35 158.223

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Báo cáo toàn văn Kỷ yếu hội nghị khoa học lần IX Trường Đại học Khoa học Tự nhiên, ĐHQG-HCM

Figure 4. a) The electron – positron pair distribution obtained by Monte – Carlo calculation with the optimized wavefunction of electron and positron in ZrO2 monoclinic crystal of a zirconium defect; b) The chi square of goodness of fit in the term of the number of Chebyshev coefficients;c),d) The graph of fitted pair correlation fucntion of electron – positron. The black dot is Monte – Carlo simulation and the solid line is its fitting curve. CONCLUSION On the basic of the principle of linear superposition, Kohn – Sham approximation, the Slater-type orbital, the modification of Jastrow and VQMC method, we have managed to derive a theoretical model to determine the positron lifetime in some element – specific structure of materials. In this scenario of pure theoretical calculations basing on real space electron and positron distributions, it is applied to estimate the value of the positron lifetime in ZrO2 monoclinic crystal structure. The calculated results showed that the lifetime of positron in perfect crystal  = 191ps is shorter than in oxygen defect crystal  = 259ps and zirconium defect crystal  = 248ps. In comparison with the experimental result, the positron lifetimes of this calculation is a little greater than one of ZrO2 monoclinic in the works[1][12][13]. These deviation come from the experimental results performing in the compound of ZrO 2 containing impurities.Therefore, this result can be used to predict for lifetime component of positron in the perfect and defects ZrO2 monoclinic crystal.

NGHIÊN CỨU SỰ HỦY POSITRON TRONG TINH THỂ ZRO2 MONOCLINIC SỬ DỤNG XẤP XĨ HÀM SÓNG ĐƠN HẠT SLATER VÀ HÀM JASTROW HIỆU CHỈNH TÓM TẮT Khảo sát sự hủy positron trong tinh thể ZrO2 monoclinic với giải thuyết positron sẽ liên kết với các electron hóa trị của oxy và zircon để hình thành nên trạng thái giả bền của hệ ZrO2 – positron trước khi positron hủy với các electron. Trong mô hình này các hàm sóng đơn hạt của electron và positron được xây dựng từ các xấp xĩ hàm sóng đơn hạt trong nguyên tử dạng Slater, xấp xĩ LCAO và hàm tương quan Jastrow đã được hiệu chỉnh. Các thời gian sống của positron trong tinh thể ZrO2 monoclinic hoàn hảo và trong sai hỏng điểm oxy và zircon được xác định từ các hàm sóng này là 191ps, 248ps và 259ps. REFERENCES [1]. I. Proch azkaa, J. · C ³· zeka, J. Kuriplacha, O. Melikhovaa,T.E. Konstantinovaband I.A. Danilenkob(2008), “Positron Lifetimes in Zirconia-Based Nanomaterials”, Acta Physica Polonica A Vol.113. 1945 - 1949

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