chapter 1 晶體的形成

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1.1 Quantum Mechanics and atomic structure

SUMMARY OF 1.1

1.1.1 Old Quantum Theory

problems of planetary model:

atom would be unstable
radiate EM wave of continuous frequency

to solve the prablom of planetary model:

Bohr: Quantum atomic structure
Planck: Quantum

Old Quantum Theory: Planck, Einstein, Bohr, de Broglie

Planck’s theory: Each atomic oscillator can have only discrete values of energy. $\nu, n=0, 1, 2, \dots$
Einstein: photon, $E=h\nu=\hbar \omega$ , $\frac{E}{c}n=\frac{h\nu}{c}n=\frac{h}{\lambda}n=\hbar k$
Bohr: H atom model
de Broglie: Matter wave, $E=h\nu=\hbar \omega$ , $E_k=\frac{1}{2}m\nu^2=\frac{p^2}{2m}=\frac{(\hbar k)^2}{2m}$

from de Broglie’s Hypothesis, the motion of a particle is governed by the wave propagation properties of matter wave, which means wave function.

1.1.2 Method of Quantum Mechanics

In method of Quantum Mechanics, we should get the Schrodinger Equation and solve it, then find the wave function $\psi$ .

Schrodinger Equation (very important):
$i\hbar \frac{\partial \Psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2 \Psi+U\Psi$

a. Schrodinger Equation of free particle

$KaTeX parse error: Undefined control sequence: \pPsi at position 24: …\frac{\partial \?p?P?s?i?}{\partial t}= …$

wave function of free particle: $\psi (\vec r, t)=A e^{-\frac{i}{\hbar} (Et-p \cdot r)}$

$\longrightarrow i \hbar \frac{\partial}{\partial t}$

$\mathbf{p} \longrightarrow - i\hbar \nabla$

b. Schrodinger Equation of particle in a force field

$\hbar \frac{\partial \Psi}{\partial t}= - \frac{\hbar^2}{2m}\nabla^2\Psi + U \Psi$

We consider time-independent Schrodinger Equation:

$U(\mathbf{r},t) \longrightarrow U(\mathbf{r})$

then separation of variables: $KaTeX parse error: Undefined control sequence: \math at position 28: …f{r} ,t)= \psi(\?m?a?t?h?{r})f(t)$

Halmiton operator: $\hat H = -\frac{\hbar^2}{2m} \nabla^2+U$

so the Schrodinger Equ becomes a new style:

$\hat H \psi = E \psi$

$\hat H \Psi = i\hbar \frac{\partial }{\partial t} \Psi$

c. Infinite Potential Well

$-(\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+U(x))\psi (x) = E \psi(x)$

$\psi (x) = \sqrt{\frac{2}{a}}sin{\frac{n \pi}{a}x}$

$E=E_n = \frac{\pi^2 \hbar^2}{2ma^2}n^2, n = 1, 2, 3, \dots$

d. Harmonic Oscillator 1D

$-(\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac{1}{2}m\omega^2 x^2)\psi (x) = E \psi(x)$

$E_n = (n + \frac{1}{2})\hbar \omega = (n+\frac{1}{2})h \nu, n = 0, 1, 2, 3, \dots$

$E_{min}= \frac{1}{2}h\nu(\neq 0)$ , which is different from Planck’s blackbody theory ( $E=nh\nu, E_{min}=0$ )
In classical mechanics, the particle can bot exceed x(max), but in quantum mechanics, the particle may exceed x(max) (qith low probabilities)

e. Finite Potential Well

$(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+U(x))\psi(x) = E\psi(x)$

Quantum Tunneling

f. Atomic Structure, Schrodinger Equ. for H Atom

$(-\frac{\hbar^2}{2m}\nabla^2 + U)\psi = E\psi, \nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$

Schrodinger equ. becomes:

$\frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial \psi}{\partial r}) +\frac{1}{r^2 \sin{\theta}} \frac{\partial}{\partial \theta}(\sin{\theta} \frac{\partial \psi}{\partial \theta}) + \frac{1}{r^2\sin^2{\theta}}\frac{\partial^2\theta}{\partial \phi^2} +\frac{2m}{\hbar}(E-U)\psi = 0$

use seperation of variables: $\psi(r, \theta, \phi) = R(r) \Theta(\theta) \Phi(\phi)$

Schrodinger equ. becomes:

$\frac{-\sin^2{\theta}}{R} \frac{d}{dr}(r^2\frac{dR}{dr}) -\frac{2m}{\hbar^2} r^2 \sin^2{\theta} (E-U) -\frac{\sin{\theta}}{\Theta} = 0$

Both Equal to a constant

$\begin{cases} \frac{1}{\Phi} \frac{d^2\Phi}{d\phi^2} = -m_l^2 \\ \frac{m_l^2}{\sin^2{\theta}} -\frac{1}{\Theta} \frac{1}{\sin{\theta}} \frac{d}{d\theta} (\sin{\theta} \frac{d\Theta}{dtheta}) =l(l+1) \\ \frac{1}{R} \frac{d}{dr}(r^2 \frac{dR}{dr})+\frac{2m}{\hbar^2} r^2(E-U) = l(l+1) \end{cases}$

(1) $\phi$ must be single-valued: $m_l = 0, \pm 1, \pm2, \dots$

(2) $\Theta$ must be finite: $\dots and \ \ l \ge |m_l|$

(3) R must be finite: $E=E_n = -\frac{Z^2e^4 m}{8 \epsilon_0^2 h^2}\frac{1}{n^2}, \ n= 1, 2, 3, \dots \ \ and \ \ l<n$

$\begin{cases} 主量子數 \ \ n:\ principle\ quantum\ number\ \longrightarrow\ decide\ E_n\\ 角量子數 \ \ l:\ orbital\ quantum\ number\ \longrightarrow\ 0, 1, 2, \dots , n-1 \\ 磁量子數 \ \ m_l: \ magnetic\ quantum\ number\ \longrightarrow\ 0, \pm 1, \pm2, \pm 3, \dots, \pm l \end{cases}$

不考慮自旋，量子數=波函數個數=量子態數=軌道數

Pauli’s Exclusion Principle: not 2 electrons in a system ( an atom or a solid ) can be in the same quantum state ( have the same n, l, $m_l$ , $m_s$ )

1.1.3 Distributing functions of micro-particles

A system with N identical micro-particles, without either generation of new particles or vanishing of existed particles, without energy exchange—an isolated system

energy level: $E_1, E_2, E_3, \dots,E_l, \dots$

degeneracy: $\omega_1, \omega_2,\omega_3,\dots.\omega_l,\dots$

particle number: $a_1, a_2, a_3, \dots,a_l,\dots$

全同性原理給量子統計和經典統計帶來重要差別；
泡利不相容原理又給費米子和玻色子的統計帶來重要差別。

自旋為 $\pm\frac{1}{2}$ 的粒子服從泡利不相容原理。

a. Boltzmann system

Every particle is identified, the number of particles in an quantum state is unlimited.

標號可分辨，能級上粒子數無限制

$a_l = \frac{\omega_l}{e^{\alpha+\beta E_l}}$

Boltzmann statistics: $f_l = \frac{a_l}{\omega_l} = \frac{1}{e^{\alpha+\beta E_l}} = \frac{1}{e^{ \frac{E_l-\mu}{k_B T} }}$

b. Bose system

Every particle is unidentified, the number of particles in an quantum state is unlimited.

(photon,phonon…) - Boson

玻色子：聲子、光子

不可分辨，能級上粒子無限

$a_l = \frac{\omega_l}{e^{\alpha+\beta E_l} -1}$

Bose-Einstein statistics: $f_l = \frac{a_l}{\omega_l} = \frac{1}{e^{\alpha+\beta E_l}-1} = \frac{1}{e^{ \frac{E_l-\mu}{k_B T} } -1}$

c.Femi system

Every particle is unidentified, the number of particles in an quantum state is limited by Pauli repulsive principle.

(electron, proton…) - Fermion

費米子：電子、質子

不可分辨，能級上粒子個數有限

$a_l = \frac{\omega_l}{e^{\alpha+\beta E_l} +1}$

Fermi-Dirac statistics: $f_l = \frac{a_l}{\omega_l} = \frac{1}{e^{\alpha+\beta E_l}+1} = \frac{1}{e^{ \frac{E_l-\mu}{k_B T} } +1}$

$\alpha = - \frac{\mu}{k_B T},\ \ \ \beta = \frac{1}{k_B t}$

統計力學

1.2 binding

1.2.1 interatomic bonding

potential between two atoms: $U(R)=\frac{-a}{R^m}+\frac{b}{R^n}$

attraction and repulsion A higher binding energy means a higher melting point!

1.2.2 ionic bond

Ionic bond is formed between atoms with large differences in electronegativity. (電負性相差較大)

binding energy: 150~370 kcal/mol

Cohesive Energy in Ionic Crystal

$\frac{Naq^2}{4\pi \epsilon_0^2 r} +\frac{NB'}{r^n}$

Madelung constant: $\sum_{j=1}^{2N-1} \frac{b}{l_j^n}, \ \ \ \ \alpha = \sum_{j=1}^{2N-1} \frac{\delta_j}{l_j}$

The bigger the cell, the more exactness the Madelung constant is.

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1.2.3Van der Waals bond

1.2.4 Hydrogen bond

1.2.5 Covalent bond

1.2.6 Metallic bond

1.3 crystal structure and typical crystals

1.3.1 crystal structure

basic concept:

無定形晶體 Amorphous (Non-crystalline) Solid: All atoms have order only within a few atomic or molecular dimensions. — random arrangement in a bigger size
長程有序 Crystal: All atoms or molecules in the solid have a regular geometric arrangement or periodicity. — highly ordered
平移對稱性 Periodicity: The quality of recurring at regular intervals.
基元 Basis: Repeatable structure units.
格點 Latice site: The dot representing a basis.
晶格 Lattice (Crystal lattice): Geometric pattern of crystal structure

Crystal Structure = Lattice + Basis

primitive vectors 基矢

position vectors 格矢

primitive unit cell 原胞

conventional unit cell 晶胞

Bravais Lattice 布拉伐點陣：The geometric pattern of basis’ arrangement; all points of the lattice is identical.

Bravais lattice only summarizes the geometry of crystals, regardless of what the actual units may be.

The basis consists of the atoms, their spaces and bond angles.

Bravais lattice:

Cubic 立方
Hexahonal 六方
Tetragonal 四方
Trigonal 三方
Monoclinic 單斜
Orthorhomic 正交
Triclinic 三斜

7種bravais晶系，14種bravais點陣，32點群
Bravais Lattice
Catalog of the 14 Bravais lattices classified according to their lattice system Lattice System Point Group Primitive Base-Centered Body-Centered Face-Centered

1.3.2 typical crystal structure

a. important parameters in crystal structure

number of atoms per unit cell: n

the number of nearest neighbors, or Coordination Number: CN 配位數

Atomic Packing Factor: APF 原子堆積因數

$\frac{volume\ \ of \ \ atoms \ \ in \ \ unit \ \ cell}{volume \ \ of \ \ unit\ \ cell}$

Atomic Radius: 原子半徑

b. typical cubic structure of metal

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c. typical crystal structure of semiconductor

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d. typical crystal structure of Ionic Crystal

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e. typical crystal structure and the Bravais Lattice

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1.4 Reciprocal Lattice and Brillouin Zone

1.4.1 Reciprocal Lattice 倒易點陣

晶體衍射得到的圖象（衍射斑點）是倒易點陣的二維投影空間放大。

Fourier series: 傅里葉級數
$f(x+2\pi) = f(x)$
$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n \cos {nx} +b_n \sin{nx})$
$=\sum_n c_n e^{inx},\ \ \ c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x)e^{-inx}dx$