固体物理往年试题

卷 A 学期： 2011 至 2012 学年度 第 1 学期

一、Fill in the blanks with the proper concepts and formula for the contents of Chapter I.

The volume 体积 of a parallelepiped 平行六面体 with axes 轴

is defined 定义 by ： ；

Please write out the five 2D Bravais lattices 布拉维格子 as : 正方晶格、六角晶格、长方晶格、有心长方晶格和斜方晶格

;

The possible 14 primitive cells 原胞 are : 简单三斜晶格、简单立方晶格、体心立方晶格、面心立方晶格、三角晶格、六角晶格、简单单斜晶格、底心单斜晶格、简单正交晶格、底心正交晶格、体

心正交晶格、面心正交晶格、简单四角晶格和体心四角晶格

;

For the plane whose intercepts are 4,2,3, the reciprocals 倒数 are 1/4、1/2、1/3 ,the smallest three integers 整数 having the same ratio 比率 are 3、6、4 .

The cube faces of a cubic crystal 立方晶体的立方体面are

二、 Expression and the calculation for the contents of Chapter II.

1）Please write out three vectors 向量 of the reciprocal lattice 倒格子: 321,,b b b

.by using vectors 321,,a a a

。

b1=(2π)·(a2*a3)/(a1·(a2*a3)） b2=(2

π)·(a3*a1)/(a1·(a2*a3)） b3=(2π)·(a1*a2)/(a1·(a2*a3））

2) Calculate 计算 the volume of the primitive cell of fcc lattice 面心立方晶格：

晶格基矢k a c ,j a b ,i a a

体积V=3

).(a c b a

原胞基矢)(2

),(2),(2a 321j i a a i k a a k j a

体积4

a )a a .(a 3321

三、 Derivation for the contents of the contents of Chapter III.

Please derive out the van der Waals-London Interaction 范德瓦尔斯伦敦相互作用 from the linear harmonic oscillators model.线性谐振子模型

解：作为一个模型，考虑两个值距为R 的全同线性谐振子1和2，每个振子带有一个正电荷（+e ）和一个负电荷（-e ），正负电荷之间的距离分别为X1和X2，粒子沿X 轴振动，动量分别用R1和R2表示，力常量为C 。在未受个数扰作用时，该系统的哈密顿量为：

3

21,,a a a

2

222212102

1212121y CX p m CX p m l

令1y l 表示两个振子之间的库伦相互作用能，核间坐标为R ，于是有

2

2

1221221y X R e X R e X X R e R e l

在R X X 《21的近似下，将上式展开，使得到最低级近似表达式为3

2

1212y R X X e l

通过简正模变换：)(21);(212121s X X X X X X a

并解出X1和X2：)(2

1);2121a s a s X X X X X X

（ 同时取1y l 的近似形式，是系统的中哈密顿量对角化，可以得出这两种模式相联系的动量Ps 和Pa,P1)(2

1

2),21Pa Ps P Pa Ps

（ 则总哈密顿量可以写成])2(21m 21])2c (2121[y 2322

232210Xa R

e c Pa Xs R e Ps m yl yl l 【 可得来。。振子的两个频率为W=...]2(81)2(211[]/)2[(2

323202

132 ）R

e R e W m R e c

其中，W0=（c/m)^(1/2)

该系统的零点能量为)s 2

1

Wa W （ 由于存在相互作用，这个值比未。。的值2-1/2V W 低△0

△V=))2(81.0)a s 216232R

A R e W W W △（△

四、Expression and the explanation f o

r the contents of Chapter IV.

1) Please write out the dispersion relation 色散关系of ω(q) for two atoms 原子 Per Primitive Basis 每个原始依据 , and explain the physical meaning of the formula 公式..

五、 Concepts and the derivation for the contents of Chpater V.

1) What is the Debye model 德拜模型 and Debye T 3

lawT3法? What is the concept 概念 of Debye temperature ?

2) Please derive the Density of State in Three Dimension 三维状态密度.

六、Derivations for the contents of Chapter VI.

1) Please derive the formula 公式 of energy levels of free electrons 自由电子的能量水平 in one dimension 维.

2) Please derive the the Hall coefficient 霍尔系数of Hall effect.

七、Explanation and derivation for the contents of Chapter VII.

Please explain the origin of the energy gap , and write out the free electron bands for [110] direction of wavevector space.

Solution:olthe origin of the energy gap is the two standing waves and pile up electors at different regions

and therefore the two waves have different values of the potential energy ,Ihtsis the origin

of the energy gap.

2)the free electron bands for [110] direction of wavevetor space is Energy band Ga/2π （000） （y x k k 0) 1 000 0 2

2k y x k

2,3 100,100

2)/2a π（ y x 22

k 2k π）（ 4,5,6,7 010,010,001,001

2)/2a π（ 8,9,10,11 110,101,110,101

12,13,14,15 110,101011011，，

16,17,18,19 110,101,110,011

八、Concepts and the explanation for the contents of Chapter VIII.

1) A hole acts in applied electric and magnetic fields as if it has a positive charge +e. The possible reasons in five steps are:

Solution ：1）,k e h k the electrons in the full band the total wave vector is zero:0k

2）.()k e h h

）（k e let the valerve band energy zero point in the conduction band above

3）,h e V V the velocity of the hole is equal to the velocity of the missing electron. 4）,m e h m the effective mass is inversely propertional to the crrvature

,/d 22dk and for the hde band ,

this has the opposite sum to that for an electron in the valence band.

5)),1(dk XB V c E e dt h h this come from the equation of motion ),1

(-dk e XB V c

E e dt h

2) Please explian the physical meaning of energy-k relation of following three semiconductor materials 半导体材料 .

卷 B

学期： 2011 至 2012 学年度 第 1 学期

一、Fill in the blanks with the proper data or concepts in Chapter I.

Solid state physics largely concerned 主要关注: (1）crystals 晶体 (2) electrons in crystals ;

Atoms

density

密

度

：

3

23/10cm atoms n a

3

29~28/10cm electrons n e ；

Translation vector 平移矢量： 3 translation vector vs a1、a2、a3 /

332211u a u a u a T

；

The volume of a parallelepiped 平行六面体 with axes is ： )a a .(a 321

；

The posibble five 2D Bravais lattice are : 正方晶格、六角晶格、长方晶格、有心长方晶格和斜方晶格 ;

Seven lattice system are : 三斜、单斜、正交、立方、四角、六角和三角晶系 ;

For the plane whose intercepts are 3,1,2, the reciprocals are 1/3、1/1、1/2 , ,the smallest three

integers having the same ratio are （ 263 ） .

321,,a a a

The cube faces of a cubic crystal are （100）（010）( 001) (100)( 010)和（001）

二、Calculations for the contents of Chapter II.

1） Please write out three vector of the reciprocal lattice: 321,,b b b

.

Explain:）

π，）π，）π

3212133211323213

21(2(2b (2b a a a a

a b a a a a a a a a a a

2)Please verify 验证 the relation: .

3) Calculate the volume 体积of the primitive cell of bcc lattice ：

三、Calculations and the concept explanation for the contents of Chapter III.

Please calculate the Madelung constant 马德龙常数 for the infinite 无限的 line of ions 离子 of alternating sign 交替的迹象 for the one-dimensional chain 一维链to be : :

四、Expression and exlanations for the contents of Chapter IV.

1) Please write out the 1D dispersion relation of ω(q), and explain the physical meaning of the formula.. ω(q)=,2

1

sin

)/c 4(2

1qa m 其中C 是最近邻平面之间的力常量，M 是一个原子的质量。 The special signifcance of phonon wavevetors that lie on the zone.

boundary is developed from the formula ,we can obtain when q=0,w(q)=0,when q=2

1)/4()(,a

m c q w

2) What is the long wave limit 长波极限 and what result 结果 we can get from this limit?

一维单原子链、一维双原子链中，q 的取值都只在一定范围之内。（一维单 原子链：

, 一维双原子链： ），长波极限就是q 取值趋向于范 围边界

时ω的情况。

研究的意义在于了解极限情况下格波振动频率的情况。

Or

.2ln 2 ij

j i a b 2 ?

当qa<<1时，将cosqa 展开并取得近似，可得cosqa 1-2)qa 2

1

（.

由此色散系度为222q /(w a m c ） 表明在长波极限下，频率与波长成正比。

五、 Explanations for the contents of Chapter V.

1) What is the Debye model and Debye T 3

law? What is the concept of Debye temperature?

Solution:1)Debye model is the low of the Max planck blackbody radiation solid equivalents.in the Debye model

,the allow model vectors smaller than the K

2) Debye T 3 law is when T 《 ，U=3435/3 T NK B ，that can obtain

34v )(234)(512

T Nk T Nk C B B

3) Debye temperature can define 3

1

2)6(h V

N k v B

2) Please explain the physics menaning of Umklapp Processes:UP 过程：

Solution:To the thermal nesistivity of electrons,which have more important effect three phonon processes isn ’t 321k k k ,it is G k 321k k ,G is reriprocal lattice vectors called

Umklapp processes .In the processes the energy is constant .The phonon vector ,in the first ,Brillouin zones

has physics menaning .The umklapp processes can let the phonon vector back to the first Brillouin zones.

六、 Derivation for the contents of Chapter VI

1) Please derive the formula of energy levels of free electrons in one dimension.请导出一维自由电子能级的公式

Solution:For

schrodinger equation H ,we can obtain

H H H

dx

d H 2

22m 2- ,where H is the electron orbital energy . To infintle potential boundary conditions

,0,0)()0(n l n

We can obtain ,2

1

),2sin(n L x A n n

where A is a constant,so that we can obtain energy n

2

2n (2）L

m

3) Please derive the the Hall coefficient of Hall effect.请导出霍尔效应的霍尔系数

Solution:To the state electric field steady state ,the time derivative is

zero ,then Vx=y c x V w E m

e

, x c y y V w E m e V

,z z E m e V

,where c

c m eB w is eyctotion frequency ,when 0 y V , We can get x x c y E m eB E w E

.And

the Hall coffinient defined is B

j E R x y H ,we can used m E ne nqV j x x x /2 to get ne

R H 1

. 七、Explanation and the derivation for the contents of Chapter VII.

Please explain the origin of the energy gap, and write out the free electron bands for [111] direction of wavevector space.请解释能隙的起源，并写了[ 111 ]方向的波矢空间的自由电子带。

八、Deravation and the calculation for the contents of Chapter VIII 第八章.

1) Starting from the definition of group velocity v g , please give the effect mass m *

described by the energy band vs wavevector k . 从群速Vg 的定义，请把影响质量m *的能带与波矢k 描述P135

3) Based on the concept of the effective mass 质量, please write out the energy of an electron 电子 near the low edge 边缘 of the conduction 传导 band and that of an electron near the top of the valance band,respectively.

P139基于有效质量概念，请写出能靠近导带低边和一个电子，在价带顶的电子，分别。

一、

1、【马德隆常数的物理意义】在一个晶体内，其中一个离子的总电势能，可表示为一个与它距离最近的另一个离子电势能的M倍，E=ME0，其中E0为两个离子的系统的电势能，M称为马德隆常数（Madelung constant），其值与晶体结构有关。

2、【德拜温度】1912 年德拜提出以连续介质的弹性波来代表格波，将布喇菲晶格看作是各向同性的连续介质。有1个纵波和 2 个独立的横波。温度愈低，德拜模型近似计算结果愈好；温度很低，主要的只有长波格波的激发。

3、费米面：如果固体中有N个自由电子，按照泡利原理它们基态是由N个电子由低到高填充的N个量子态。N个电子在k空间填充半径为kF球，球内包含的状态数恰好等于N。一般称这个球为费米球，kF为费米半径，球的表面为费米面。

二、

1、证明：面心立方的原胞基矢：

体心立方的原胞基矢为：

4、

一、

1、爱因斯坦理论能够反映出Cv在低温时下降的基本趋势。......