第十九讲金属自由气体模型一、固体物理中的主要模型（理论）：Atoms in the solid matter= ion cores (离子实)+ valence electrons（价电子）= nuclei + core electrons + valence electrons1.最简单的模型—金属自由电子气体模型a)认为离子实静止不动；b)通过“自由电子近似（凝胶模型--离子实系统产生的势场是均匀的）”和“独立电子近似（忽略电子与电子之间的作用）”形成一类最简单的“单电子近似”模型：i.Drude Model (1900)ii.Sommerfeld Model (1928)2.次简单模型Ⅰ—晶格模型和能带理论a)认为离子实仍然静止不动；b)离子实系统产生的势场随空间是周期变化，不再是均匀的。

3.次简单模型Ⅱ—晶格振动理论和声子模型a)不考虑电子的运动；b)离子以简正模式运动。

4.最复杂的模型—电子与声子相互作用理论，光子与声子相互作用理论，光子与电子（固体、半导体中的电子，）相互作用理论，…总结：学习这种将复杂的大问题（真实的物理体系）化成可以局部求解的小问题（简化的物理体系）；通过不断对简单模型的修正，来处理复杂的体系。

在学会这种思维方式的同时，保持头脑清醒，牢记各种模型的成立前提（或条件，或可忽略的物理内容），才能正确使用模型，得到合理的有价值的结论。

二、Sommerfeld量子金属自由电子气体模型通过三个近似，将一块体积为V的金属简单地看成一堆价电子在体积为V的“空盒子”中运动的单纯由电子组成的体系。

1.自由电子近似——对金属来说是个比较好的近似。

a)忽略价电子与离子实之间的作用，认为离子实系统产生的势场对处在其中的价电子来说是均匀的。

b)将离子实系统看成是保持体系电中性的均匀正电荷背景。

c)价电子的自由运动范围仅限于金属块的体积V内，由金属的表面势垒将价电子限制在样品内部。

2.独立电子近似——对其它晶体（包括半导体和绝缘体）来说也是一个比较好的近似。

a)忽略价电子与电子之间的作用，把其它电子对某一个价电子的作用看成是平均场；b) 认为某一个价电子的运动不影响其他电子的运动；c) 把多价电子问题转换成单电子问题求解单电子能量本征态。

d) 最后让多个价电子按照一些规则（能量最小原理（T=0），费米分布（T>0）,泡利不相容原理）来填充单电子能量本征态。

e) 多个价电子填充单电子能量本征态的规则：1. 泡利不相容原理2. 能量最小原理（T=0）3. 费米分布（T>0）3. 弛豫电子近似——在考虑电子输运过程中，不能忽略电子与电子之间的作用，加入一个唯象的近似假设a) 在有外场（电场，磁场，电磁波场和光）作用时对价电子体系采取的一种近似；b) 认为每一个价电子会受到散射和碰撞（由于其他电子的存在与运动）。

三、金属自由电子气体密度3:n n cm ⎡⎤⎢⎥⎣⎦单质金属晶体原子密度3:N N cm ⎡⎤⎢⎥⎣⎦，比重/质量密度（density ）3g cm ρ⎡⎤⎢⎥⎣⎦，原子量(atomic mass)g A mol ⎡⎤⎢⎥⎣⎦, 阿佛加德罗常数236.02210A N mol =⨯，Z 每个原子的价电子个数。

3A AZ N n ZN Zcm A A N ρρ⎡⎤===⎢⎥⎣⎦ 金属自由电子气体密度n 典型值为222331010cm :，比理想气体密度大1000倍。

Condensed gas四、金属自由电子模型中单电子能量本征态和能量本征值目的：求金属自由电子气体的能量密度3:J cm ε⎡⎤⎢⎥⎣⎦。

由独立电子近似，金属中每一个价电子是相互独立的，有相同的运动规律；每个电子都有自己的能量本征态，只要求出单个电子的能量本征态，再复制n 套，在把价电子填充在这些能量本征态上，就可以得到n 个电子的总能量，即金属自由电子气体的能量密度3:J cm ε⎡⎤⎢⎥⎣⎦。

T. YANG Page 5 of 24 6/30/2019波函数归一化和k r矢量意义：T. YANG Page 6 of 24 6/30/2019周期边界条件和k r矢量取值：Periodic boundary conditions1) Surface is not important.2) Useful for large N (bulk solids N ≈ 1023/cm 3). 3) Similar procedure; slightly different results.--------…………--T. YANG Page 7 of 24 6/30/2019T. YANG Page 8 of 24 6/30/2019k r 空间（倒空间）：把波矢k r 看成空间矢量，在直角坐标系中用k r 矢量的末端的位置表示每一个允许的k r 值。

这个直角坐标系所在的空间叫做k r空间（或倒空间）k r 空间（或倒空间）中一个点占据的k r空间（或倒空间）的体积为()()3333222L L V πππ⎛⎫==⎪⎝⎭。

定义态密度：单位k r 空间体积内包含电子态的个数（k r 点的个数）称为k r空间的态密度()()33122Vf Vππ==。

单电子本征能量及单电子态在k r空间（或倒空间）中的能级图1. 等能面是球面2.抛物面（抛物线）能带五、金属自由电子气体基态能量本征态和基态能量本征值T= 0 K，N个价电子所处的状态为基态。

The ground state (T=0) of a system of fermions is governed by the Pauli Exclusion Principle: all the low-energy states are filled to a certain maximum energy E F which depends on the density of the fermionic “gas” (the so-called Fermi energy).N free electrons at T=0 K,()()2222/321/321/3323322432383FF F F F FV N N k k n V k k E n m mv mπππππ⎛⎫=⋅⋅⇒== ⎪⎝=⎭==h h hThe average thermal velocity for ideal gas depend on temperature, T ,k B T ~ 0.026 eV21322e B m v k TWhen we consider ensembles of identical particles , we need to distinguish two regimes: the low-density limit (the inter-particle distance is much greater than the de Broglie wavelength of these particles) and the high-density limit (the opposite is true). In the low-density limit, bosonic and fermionic ensembles behave in a similar way (Boltzmann statistics). In the high-density limit, fermions and bosons are quite different!high-density limitOccupancy (= the distribution function): the mean number of fermions in a particular quantum staten=N/V – the average density of particles()31C dB n n λ≥∝()01,0,F FD T FE f E εεε=≤⎧=⎨>⎩()()f n εε≡()if E n=∑3D Cubic Infinite Potential Well•1-D Well•3-D “Cubic” Well (with sides length L)三维各向同性“立方盒子”的能级简并度d(Degeneracy for an Energy Level)，34 Electrons in 3D Infinite Well34 Electrons in 3D Infinite Well•In this configuration, Array–What is the probability at T =0 that a level with energy 14E0 or less will be occupied?–It is 1!–What is the probability that the level withenergy above 14E0 will be occupied?–It is 0!Pauli Exclusion Principle●How do we extend the quantum theory to systemsbeyond the hydrogen atom?●For systems of 2 electrons, we simply have a ψ thatdepends on time and the coordinates of each of the two electrons:●ψ(x1,y1,z1,x2,y2,z2,t)●and the Schrodinger’s equation has two kineticenergies instead of one.●It turns out that the Schodinger’s Equation can beseparated:T. YANG Page 17 of 24 6/30/2019●ψ = X a(x1,y1,z1) * X b(x2,y2,z2) * T(t) .●This is like having electron one in state a, and having electron two in state b. Note that eachstate has its own particular set of quantum numbers.●However, from the Heisenberg Uncertainty Principle (from wave/particle duality), we are notreally sure which electron is electron number #1 and which is number #2. This means that the wavefunction must also reflect this uncertainty.●There are two ways of making the wavefunction reflect the indistinguishability of the twoelectrons:ψsym = [X a(r1)*X b(r2) + X b(r1)*X a(r2) ]* T(t)andψanti = [X a(r1)*X b(r2) - X b(r1)*X a(r2) ]* T(t) .●Which (if either) possibility agrees with experiment?●It turns out that some particles are explained nicely by the symmetric, and some are explainedby the antisymmetric.●Those particles that work with the symmetric form are called BOSONS. All of theseparticles have integer spin as well. Note that if boson 1 and boson 2 both have the same state, ψ > 0. This means that both particles CAN be in the same state at the same location at the same time.●Those particles that work with the anti-symmetric wavefunction are called FERMIONS. Allof these particles have half-integer spin. Note that if fermion 1 and fermion 2 both have the same state, ψ= 0. This means that both particles can NOT be in the same state at the same location at the same time.T. YANG Page 18 of 24 6/30/2019●FERMIONS. Electrons, protons and neutrons are fermions. These particles can NOT bein the same location with the same energy state at the same time.●This means that two electrons going around the same nucleus can NOT both be in theexact same state at the same time!This is known as the Pauli Exclusion Principle!●BOSONS. Photons and alpha particles(2 neutrons + 2 protons) are bosons. Theseparticles can be in the same location with the same energy state at the same time.●This occurs in a laser beam, where all the photons are at the same energy (monochromatic). Electron SpinT. YANG Page 19 of 24 6/30/2019T. YANGPage 20 of 246/30/20192Im()sin 11sin nlm nlm nlm r m j e r e e e r r r ϕθϕϕμμθθθϕ*=ψ∇ψ=ψ∂∂∂∇=++∂∂∂r rhh r r r 222222200222000sin (sin )sin (sin )sin sin 22e nlm e L Z Z nlm ZL Z Z nlm Znlm L Z Zem j e r d e rd dr e dI j d e em d e S dI e r rd dr e r em e e r rd drr em e d r d drem e e ϕϕϕϕπππμθσθσμπθθμθμπθθμθϕθθμμμ∞∞=-ψ=⋅=⋅=⋅=-ψ⋅=-ψ⋅=-ψ⋅=-⎰⎰⎰⎰⎰r r h r r r r r r r h r r h r h r r h2222Z L Z Zz L Z Z L zL em e e eL e e e L e Lμμμμμμμμ=-=-=-=-r r h r r v vT. YANG,,2(,)()()()alkai atom n l l s l R E E E hcT n l r L S hc r L S n ξξ-=+∆=-+⋅=-+⋅-∆r r v v),,,,(),,,,(j s l m j s l n m s m l nThe other kinds of micro-particles:The properties of proton : s=1/2, c=+e, m=m p The properties of neutron : s=1/2, c=0, m=m n The properties of photon : s=1, c=0, m=0The properties of electron : s=1/2, c=-e, m=m eT>0 K 自由电子气体能量的计算The DoS for a 3D free electron gas (m is the electron mass) : n, l, m l , s, m ss=1/2 :spinm s =+1/2,- 1/2 :spin magneticm l =0,±1, ±2, …,±l :magneticl=0,1, 2, …,n-1 :orbitaln=1, 2, … :principal()3/231/222122D m g εεπ⎛⎫= ⎪⎝⎭h。