臺灣博碩士論文加值系統

莫來石由於具有優異的熱機械性質與耐久性，因此廣泛地被應用於高溫結構陶瓷。從晶體結構觀點來探討莫來石的性質，發現其結構具有大量的氧空缺以及鋁原子會取代特定矽原子，到目前為止試驗結果的解釋也都圍繞著點缺陷的影響來探討莫來石的性質，然而有時不同性質的解釋，會提出不同的模型來分析，往往造成矛盾，因此本研究利用電腦模擬來研究單一點缺陷與缺陷組合對基本結構性質與熱學性質的影響。
由於莫來石的晶體結構相當複雜，我們首先提出一個合適的原子模型，進而使用傳統分子動力學來研究不同比例的點缺陷對晶格常數的影響以及在高溫下莫來石的晶格常數變化，藉以推算熱膨脹係數，另外也計算各原子受到高溫驅動下的自體擴散係數。為了釐清單一缺陷與特定缺陷組合對常溫與高溫下的晶格常數影響，我們使用第一原理全能量計算，另外我們也利用該計算方法來計算產生單一缺陷與特定缺陷組合所需的能量，以及透過施加外壓力於原子模型，然後利用狀態方程式擬合，藉以計算體模數係數來明瞭壓力對特定缺陷結構的影響。另外我們也使用了第一原理分子動力學來探討高溫下莫來石的結構變化，最後為了釐清氧擴散的機制，我們利用微彈性帶(Climb-image nudged elastic band)方法來計算兩種氧擴散路徑所需要的能量。
研究結果顯示莫來石的平均結構模型是非常適合用來電腦模擬，晶格常數受不同氧缺陷含量與鋁取代矽的交互影響造成b軸會隨著缺陷量增加而往下掉，而單一氧缺陷會造成晶格縮小，而鋁取代矽造成晶格膨脹，莫來石特有的缺陷組合(氧缺陷加上兩個鋁取代矽)會造成a軸增加，b軸減少。外加等向壓力於晶格上，無論何種缺陷結構，晶格都會縮小。高溫下莫來石的晶格常數是隨著溫度增加而增加，然而到攝氏1200度後，a軸與b軸斜率突然大增，而c軸變的比較小。在常溫下，AlO6的鍵長是影響莫來石的晶格變化最主要的因素，然而在高溫下，以莫來石而言最主要的是SiO4的鍵長變化，透過能量分析(Energy landscape)證實鍵長變化跟玻璃相轉換的形成有關。
氧空缺在O(C)位置以及鋁取代矽所需要的缺陷能量最低，因此這兩種缺陷最容易產生，而以最小平方法來估算原子的自體擴散係數似乎不適合用在莫來石上，計算的自體擴散係數與試驗值有很大的誤差。莫來石的氧擴散是氧O(C)位置移動，如果是完全的氧原子躍遷到已存在的結構缺陷，所需的能量非常大，約是15電子伏特，然而從試驗的活化能與微彈性帶計算的活化能顯示，氧原子並沒有完全跑到結構缺陷中，只是稍微往該缺陷方向動一下而已，而從第一原理分子動力學計算也觀察到這個現象。
本研究所建立的研究方法與流程適合應用在陶瓷材料的模擬與分析，而我們已經探討了莫來石的缺陷對晶體結構、晶格常數、熱膨脹、體模數、缺陷能、氧擴散的影響，接下來可以朝向以下幾點深入研究，包含研究熱狀態方程式、找出最適當的方法來計算擴散係數、提出更好的分子勢能模型、研究莫來石在高溫下的相變化與力學性質。

Mullite has extraordinary thermo-mechanical properties and durability; therefore, it is a strong candidate for high-temperature structural applications. From the structural viewpoints, such properties are strongly interwoven with the oxygen vacancies and Si atoms replaced by Al atoms. Based on experimental observations, several mechanisms have been proposed to explain mullite properties. However, some of them are controversial in which further insights are much needed. The objective of this work is to fulfill the needs through atomistic simulations. We applied classical molecular dynamics and first-principles simulations to study the effect of point defects on static and thermal properties of mullite.
Due to the existence of partial atom occupancy of oxygen, aluminum, and silicon atoms in mullite, we first proposed suitable atomistic models of mullite for computer simulations. We then utilized classical molecular dynamics to calculate the lattice constants with different mullite compositions, the change of lattice constants with temperatures, and self-diffusivity of each species at high-temperature. In order to clarify the effects of the single point defect and defect-pair, the ab initio total-energy calculation was used to compute the lattice constants of different defective structures and defect formation energy for each single point defect and the defect pair. We also applied the third Birch-Murnaghan equation-of-state fitting to calculate bulk modulus of mullite with different external pressures. Furthermore, ab- initio molecular dynamics simulation was applied to study the structural change of mullite at high temperature. Finally, we used the climb-image nudged elastic band method to evaluate the activation energies of two migration paths for oxygen atoms.
The simulation results show that the average structure model of mullite is suitable for the computer simulations. The increase of oxygen vacancies and Si atoms replaced by Al atoms in mullite causes the decrease of lattice constant b. The single oxygen vacancy reduces the lattice constants, and the single Si atom replaced by Al atom expands the lattices. The special defect pair (one oxygen vacancy with two Si atoms replaced by Al atoms) for mullite causes the increase of lattice constant a but decrease of lattice constant b. External pressure on all defective structures shortens all lattice constants. The lattice constants of mullite expand while temperature is increasing from 0℃ to 1200℃. At 1200℃-1400℃, lattice constants a and b increase rapidly, but lattice constant c decreases slightly. The bond lengths in AlO6 units affect lattice constants at room temperature, but at high temperature the bond lengths of SiO4 mainly influence lattice constants of mullite. We assert that such phenomenological change is related to the glass transition. This assertion is further reinforced by analysis of energy landscape. Oxygen vacancy at O(C) site and the Si atom replaced by Al atom have the lowest formation energy; therefore, these might explain why the two defects occur normally in mullite. The calculated self-diffusivity values for each species using the mean-square-displacement method do not match the experimental ones. This indicates that the mean-square-displacement method might not appropriate to study the diffusion of mullite. From nudged elastic band analysis, oxygen diffusion in mullite is the migration of oxygen atom at O(C) site. The activation energy for the oxygen to be fully hopping into the existed vacancy is around 15 eV. The value is quite high and thus such migration path is not likely happen during the diffusion. According to the experimental values and the nudged elastic band calculation, the oxygen at O(C) site moves slightly toward the existed structural vacancy only. We also confirmed such small movement in the ab-initio molecular dynamics simulation.
We have created a research process for conducting the atomistic simulation of ceramics. The anomalies of lattice constants and thermal expansion and oxygen diffusion have been revealed in this study. Moreover, the effects of point defects within the structure and equation of state of mullite with different pressure have also been studied. Despite significant progress has been made in this study on mullite simulation, much work remains to be done in the future. For example, it is stall need to study the thermal equation-of-state, to find a proper methodology for calculating self-diffusivity, to propose a better potential model for mullite and sillimanite, to calculate mechanical properties at high temperature, and to analyze the structural transformation at high temperature.

Chinese abstract……………………………………………………………………….VII
English abstract………………………………………………………………………...IX
Table of contents……………………………………………………………………...XIII
List of tables…………………………………………………………………………...XV
List of figures………………………………………………………………………..XVII
CHAPTER 1 INTRODUCTION………………………………………………………...1
1.1 Motivation…………………………………………………………………….1
1.2 Objectives……………………………………………………………………14
1.3 Methodology…………………………………………………………………15
1.4 Outlines of the dissertation…………………………………………………..16
CHAPTER 2 LITERATURE REVIEW………………………………………………..18
2.1 Computer simulation of mullite……………………………………………...18
2.2 Classical molecular dynamics ……………………………………………….20
2.3 Density functional theory…………………………………………………22
2.4 Nudged elastic band method………………………………………………....31
CHAPTER 3 ATOMISTIC MODELS FOR SIMULATION OF MULLITE………34
3.1 Introduction………………………………………………………………….34
3.2 Crystal structure of mullite…………………………………………………..36
3.3 Angel and Prewitts’ model…………………………………………………..42
3.4 The proposed model…………………………………………………………44
3.5 Concluding remarks………………………………………………………….48
CHAPTER 4 CLASSICAL MOLECULAR DYNAMICS OF MULLITE……………49
4.1 Introduction………………………………………………………………….49
4.2 Simulation details……………………………………………………………50
4.3 Results and discussion…………………………………………………….…54
4.4 Concluding remarks………………………………………………………….79
CHAPTER 5 AB INITIO TOTAL-ENERGY CALCULATION OF MULLITE……..80
5.1 Introduction………………………………………………………………….80
5.2 Simulation details……………………………………………………………81
5.3 Results and discussion……………………………………………………….86
5.4 Concluding remarks………………………………………………………….99
CHAPTER 6 AB INITIO MOLECULAR DYNAMICS AND NUDGED ELASTIC BAND CALCULATION OF MULLITE……………………………………………..101
6.1 Introduction………………………………………………………………...101
6.2 Simulation details…………………………………………………………..101
6.3 Results and discussion……………………………………………………...106
6.4 Concluding remarks………………………………………………………...117
CHAPTER 7 CONCLUSIONS AND SUGGESTIONS……………………………...119
7.1 Conclusions………………………………………………………………...119
7.2 Suggestions…………………………………………………………………123
REFERENCES………………………………………………………………………..125
APPENDIX A REVIEW OF SIMULATION PROGRAMS…………………………134
APPENDIX B INPUT FILES FOR THE VASP PROGRAM………………………..142
APPENDIX C INPUT FILES FOR THE DL_POLY PROGRAM………………..…149

Aksay I.A., Dabbs D.M. and Sarikaya M. (1991), Mullite for structural, electronic, and optical applications, J. Am. Ceram. Soc. 74, 2343-2358.
Allen M.P. and Tildesley D.J. (1987), Computer simulation of liquids, Oxford university press, New York, USA.
Angel R.J. and Prewitt C.T. (1986), Crystal structure of mullite: a re-examination of the average structure, Am. Mineral. 71, 1476-1482.
Angel R.J., McMullan R.K. and Prewitt C.T. (1991), Substructure and superstructure of mullite by neutron diffraction, Am. Mineral. 76, 332-342.
Angel R.J. (2001a), EosFit5.2 users guide, http://www.geol.vt.edu/rja/ .
Angel R.J. (2001b), Equation of state. In: Hazen R.M. and Downs R.T. (eds) High pressure, high temperature crystal chemistry: Rev. Mineral. Geochem. 41, 35-60.
Born M. and Oppenheimer J.R. (1927), Zur Quantentheorie der Molekeln, Ann. Physik 389(20), 457-484.
Burns G. and Glazer A.M. (1978), Space groups for solid state scientists, Academic press, New York, USA.
Burt J.B., Ross N.L., Angel R.J. and Koch M. (2006), Equations of state and structures of andalusite to 9.8Gpa and sillimanite to 8.5Gpa, Am. Mineral. 91, 319-326.
Carrasco J., Lopez N. and Lllas F. (2005), On the convergence of isolated neutral oxygen vacancy and divacancy properties in metal oxides using supercell models, J. Chem. Phys. 122, 224075-1224075-14.
Catlow C.R.A. and Mackrodt W.C. (1982), Computer simulation of solids (Lecture notes in physics:166), Spring-Verlag, Berlin, Germany.
Chiang Y.M., Birnie D.P. and Kingery W.D. (1997), Physical ceramics: principles for ceramic science and engineering, 1st ed., John Wiley & Sons, Indianapolis, USA, 146-147.
Cook R.D., Malkus D.S., Plesha M.E. and Witt R.J. (2001), Concepts and applications of finite element analysis, 4th ed., John Wiley & Sons, New York, USA.
Fielitz P., Borchardt G., Schmücker M., Schneider H., Wiedenbeck M., Rhede D., Weber S. and Scherrer S. (2001a), Secondary ion mass spectroscopy study of oxygen-18 tracer diffusion in 2/1-mullite single crystals, J. Am. Ceram. Soc. 84, 2845-2848.
Fielitz P., Borchardt G., Schneider H., Schmücker M., Wiedenbeck M. and Rhede D. (2001b), Self-diffusion of oxygen in mullite, J. Eur. Ceram. Soc. 21, 2577-2582.
Fielitz P., Borchardt G., Schmücker M. and Schneider H. (2007), A diffusion-controlled mullite formation reaction model based on tracer diffusivity data for aluminum, silicon and oxygen, Phil. Mag. 87, 111-127.
Fischer R.X., Schneider H. and Schmücker M. (1994), Crystal structure of Al-rich mullite. Am. Mineral. 79, 983-990.
Fischer R.X., Schneider H. and Voll D. (1996), Formation of aluminum rich 9:1 mullite and its transformation to low alumina mullite upon heating. J. Eur. Ceram. Soc. 16, 109-113.
Fischer R.X. and Schneider H. (2005), Crystal chemistry of mullite and related phases. In: Schneider H. and Komarneni S. (eds) Mullite, Wiley-VCH, Weinheim, 1-46.
Freimann S. and Rahman S. (2001), Refinement of the real structures of 2:1 and 3:2 mullite, J. Eur. Ceram. Soc. 21, 2453-2461.
Grimes R.W., Catlow C.R.A. and Stoneham A.M. (1989), A comparison of defect energies in MgO using Mott-Littleton and quantum mechanical procedures, J. Phys.: Condens. Matter 1, 7367-7384.
Guse W., Saalfeld H. and Tjandra J. (1979), Thermal transformation of sillimanite single crystals, N. Jb. Miner. Mh. 4, 175-181.
Henkelman G., Uberuaga B.P. and Jónsson H. (2000), A climbing image nudged elastic band method for finding saddle points and minimum energy paths, J. Chem. Phys. 113(22), 9901-9904.
Heyes D.M., Baxter J., Tuzun U. and Qin R.S. (2004), Discrete-element method simulations: from micro to macro scales, Phil. Trans. A Math. Phys. Eng. Sci. 362, 1853-1865.
Hildmann B. and Schneider H. (2004), Heat capacity of mullite: new data and evidence for a high-temperature phase transformation, J. Am. Ceram. Soc. 87, 227-234.
Hohenberg P. and Kohn W. (1964), Inhomogeneous electron gas, Phys. Rev. 136, B864-B871.
Hoover W.G. (1985), Canonical dynamics: equilibrium phase-space distributions, Phys. Rev. A 31(3), 1695-1697.
Hülsmans A., Schmücker M., Mader W. and Schneider H. (2000a), The transformation of andalusite to mullite and silica: Part I. Transfromation mechanism in [001]A direction, Am. Mineral. 85, 980-986.
Hülsmans A., Schmücker M., Mader W. and Schneider H. (2000b), The transformation of andalusite to mullite and silica: Part II. Transfromation mechanism in [100]A and [010]A directions, Am. Mineral. 85, 987-992.
Jónsson H., Mills G. and Jacobsen K.W. (1998), Nudged elastic band method for finding minimum energy paths of transitions. In: Berne B.J., Ciccotti G. and Coker D.F. (eds) Classical and quantum dynamics in condensed phase simulations, World Scientific, Singapore, 385-404.
Kilo M., Argirusis C., Borchardt G. and Jackson R.A. (2003), Oxygen diffusion in yttria stabilized zirconia-experimental results and molecular dynamics calculations, Phys. Chem. Chem. Phys. 5, 2219-2224.
Kittel C. (1996), Introduction to solid state physics, 7th ed., John Wiley & Sons, New York, USA.
Kohn W. and Sham L.J. (1965), Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133-A1138.
Kresse G. and Furthmüller J. (1996), Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54(16), 11169-11186.
Kresse G. and Joubert D. (1999), From ultrasoft pesudopotentials to the projector augmented-wave method. Phys Rev B 59,1758-1775.
Kriven W.M., Siah L.F., Schmücker M. and Schneider H. (2004), High temperature microhardness of single crystal mullite, J. Am. Ceram. Soc. 87, 970-972.
Lacks D.J., Hildmann B. and Schneider H. (2005), Effects of disorder in mullite: molecular dynamics simulation and energy landscape analysis, Phys. Rev. B 72, 214305:1-5.
Leslie M. and Gillan M.J. (1985), The energy and elastic dipole tensor of defects in ionic crystals calculated by the supercell method, J. Phys. C: Solid State Phys. 18, 973-982.
Lessing P.A., Gordon R.S. and Mazdiyasni K.S. (1975), Creep of polycrystalline mullite, J. Am. Ceram. Soc. 58, 149.
Liu C.Z.-W. and Oppenheim I. (1996), Enhanced diffusion upon approaching the kinetic glass transition, Phys Rev. E 53, 799-802.
Martin R.M. (2004), Electronic structure: basic theory and practical methods, Cambridge university press, Cambridge, UK.
Martinón-Torres M, Rehren T. and Freestone I.C. (2006), Mullite and the mystery of Hessian wares, Nature, 444 (23), 437-438.
Matsui M. (1994), A transferable interatomic potential model for crystals and melts in the system CaO-MgO-Al2O3-SiO2, Mineralo. Mag. 58A, 571-572.
Matsui M. (1996), Molecular dynamics study of the structures and bulk moduli of crystals in the system CaO-MgO-Al2O3-SiO2, Phys. Chem. Minerals 23, 345-353.
Melchionna S., Ciccotti G. and Holian B.L. (1993), Hoover NPT dynamics for systems varying in shape and size. Molec. Phys. 78, 533-544.
Monkhorst H.J. and Pack J.D. (1976), Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188-5192.
Nosé S. (1984), A unified formulation of the constant temperature molecular dynamics methods, J. Chem.. Phys. 81, 511-519.
Oganov A.R., Price G.D. and Brodholt J.P. (2001), Theoretical investigation of metastable Al2SiO5 polymorphs, Acta Cryst. Sec. A A57, 548-557.
Oganov A.R. and Dorogokupets P.I. (2003), All-electron and pseudopotential study of MgO: equation of state, anharmonicity, and stability, Phys. Rev. B 67, 224110:1-11.
Padlewski S., Heine V. and Price G.D. (1992), The energetic of interaction between oxygen vacancies in sillimanite: a model for the mullite structure, Phys. Chem. Minerals 19, 196-202.
Padlewski S., Heine V. and Price G.D. (1993), A microscopic model for a very stable incommensurate modulated mineral: mullite, J. Phys.: Condens. Matter 5, 3417-3430.
Parrinello M. and Rahman A. (1981), Polymorphic transitions in single crystals: a new molecular dynamics method. J. Apply. Phys. 52, 7182-7190.
Payne M.C., Teter M.P., Allan D.C., Arias T.A. and Joannopoulos J.D. (1992), Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients, Rev. Mod. Phys. 64(4), 1045-1097.
Perdew J.P., Burke K. and Ernzerhof M. (1996), Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865-3868.
Perdew J.P., Chevary J.A., Vosko S.H., Jackson K.A., Pederson M.R. and Singh D.J. (1992), Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation, Phys. Rev. B 46, 6671-6687.
Phillips R. (2001), Crystals, defects and microstructures: modeling across scales, Cambridge university press, Cambridge, UK.
Probert M.I.J. (2003), Improved algorithm for geometry optimization using damped molecular dynamics, J. Comput. Phys. 191, 130-146.
Raabe D. (1998), Computational materials science: the simulation of materials microstructures and properties, Wiley-VCH, Weinheim, Germany.
Rahman S.H. (1993), The videographic method: a new procedure for the simulation and reconstruction of real structures, Acta. Cryst. A49, 56-68.
Rahman S.H., Strothenk S., Paulmann C. and Feustel U. (1996), Interpretation of mullite real structure via intervacancy correlation vectors, J. Eur. Ceram. Soc. 16, 177-186.
Rehak P., Kunath-Fandrei G., Losso P., Hildmann B., Schneider H. and Jäger C. (1998), Study of the Al coordination in mullites with varying Al:Si ratio by 27Al NMR spectroscopy and X-ray diffraction, Am Mineral. 83, 1266-1276.
Sadanaga R., Tokonami M. and Takéuchi Y. (1962), The structure of mullite, 2Al2O3.SiO2, and relationships with the structures of sillimanite and andalusite, Acta Cryst. 15, 65-68.
Schmücker M. and Schneider H. (2002), New evidence for tetrahedral triclusters in aluminosilicate glasses, J. Non-Crystal. Solids 311, 211-215.
Schmücker M., Hildmann B. and Schneider H. (2002), Mechanism of 2/1- to 3/2-mullite transformation at 1650°C, Am. Mineral. 87, 1190-1193.
Schmücker M., Schneider H., MacKenzie K.J.D, Smith, M.E. and Carroll D.L. (2005), AlO4/SiO4 distribution in thtrahedral double chains of mullite, J. Am. Ceram. Soc. 88, 2935-2937.
Schneider H. and Eberhard E. (1990), Thermal expansion of mullite, J. Am. Ceram. Soc. 73, 2073-2076.
Schneider H., Rodewald K. and Eberhard E. (1993), Thermal expansion discontinuities of mullite, J. Am. Ceram. Soc. 76, 2896-2898.
Schneider H., Okada K. and Pask J.A. (1994), Mullite and mullite ceramics, John Wiley & Sons Ltd, West Sussex, UK.
Schneider H. and Komarneni S. (2005), Mullite, Wiley-VCH, Weinheim, Germany.
Schneider H. (2005), Basic properties of mullite. In: Schneider H. and Komarneni S. (eds) Mullite, Wiley-VCH, Weinheim, 141-188.
Schneider H., Schreuer J. and Hildmann B. (2007), Structure and properties of mullite—A review, J. Eur. Ceram. Soc. (in press), doi:10.1016/j.jeurceramsoc 2007.03.017.
Schreuer J., Hildmann B. and Schneider H. (2006), Elastic properties of mullite single crystals up to 1400℃, J. Am. Ceram. Soc. 89, 1624-1631.
Smith W. and Forester T.R. (1996), DL_POLY_2.0: A general-purpose parallel molecular dynamics simulation package. J. Molec. Graphics 14, 136-141.
Smith W., Forester T.R., Todorov I.T. and Leslie, M. (2006), The DL_POLY_2 user manual (version 2.17), Daresbury Laboratory, UK.
Sosman R.B. (1956), A pilgrimage to mull, Am. Ceram. Soc. Bull. 35 (3), 130-131.
Sastry S., Debenedetti P.G. and Stillinger F.H. (1998), Signatures of distinct dynamical regimes in the energy landscape of a glass-forming liquid, Nature 393(11), 554-557.
Tannehill J.C., Anderson D.A. and Pletcher R.H. (1997), Computational fluid mechanics and heat transfer, 2nd ed., Taylor & Francis, Philadelphia, USA.
Truhlar D.G., Garrett B.C. and Klippenstein S.J. (1996), Current status of transition-state theory, J. Phys. Chem. 100, 12771-12800.
Voter A.F., Montalenti F. and Germann T.C. (2002), Extending the time scale in atomistic simulation of materials, Annu. Rev. Mater. Res. 32, 321-346.
Wang Y. and Perdew J.P. (1991), Correlation hole of the spin-polarized electron gas, with exact small-wave-vector and high-density scaling. Phys. Rev. B 44,13298-13307.
Welberry T.R. and Withers R.L. (1990), An optical transform and Monte Carlo study of the diffuse X-ray scattering in mullite, Al2(Al2+2xSi2-2x)O10-x, Phys. Chem. Minerals 17, 117-124.
Winkler A., Horbach J., Kob W. and Binder K. (2004), Structure and diffusion in amorphous aluminum silicate: a molecular dynamics computer simulation, J. Chem. Phys. 120, 384-393.
Winkler B., Hytha M., Warren M.C., Milman V., Gale J.D. and Schreuer J. (2001), Calulcation of the elastic constants of the Al2SiO5 polymorphs andalusite, sillimanite and kyanite. Z Kristallogr. 216, 67-70.
Winkler B., Milman V., Hennion B., Payne M.C., Lee M.-H. and Lin J.S. (1995), Ab initio total energy study of brucite, diaspore and hypothetical hydrus wadsleyite, Phys Chem Minerals 22, 461-467.
Winter J.K. and Ghose S. (1979), Thermal expansion and high-temperature crystal chemistry of the Al2SiO5 polymorphs, Am. Mineral. 64, 573-586.
Wondraczek L., Heide G., Kilo M., Nedeljkovic N., Borchardt G. and Jackson R.A. (2002), computer simulation of defect structure in sillimanite and mullite, Phys Chem. Minerals 29, 341-345.
Zettili N. (2001), Quantum mechanics: concepts and applications, John Wiley & Sons Ltd, West Sussex, UK.