Solid State Physics-I

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Chapter#1: Crystal Structure

Team Quanta gladly presents all possible short questions of BS Physics book Solid State Physics – I’s Chapter#1: Crystal Structure for the ease of students.

Q.1 Differentiate between unit cell and primitive cell by explaining all terms used.

Answer.  Unit cell:

  • Unit cell is the smallest building block which repeats itself in three dimensions to form a crystal.
  • Volume of unit cell is;V=\ \vec{a}.\vec{b}\times\vec{c}. .
  • Unit cell may or may not be primitive cell.

Primitive cell:

  • The smallest unit cell that can be defined for a given crystal is called primitive cell.
  • Primitive cell is minimum volume unit cell.
  • A primitive cell is always a unit cell.

Q.2 Differentiate between crystalline and amorphous solids by explaining all the terms used.

Answer.  Crystalline solids:

  • A solid of definite shape with its atoms, ions or molecules arranged in some regular repetitious three-dimensional pattern is called a crystal.
  • The crystals have generally sharp melting points and get converted into liquid very soon.
  • The various properties like thermal conductivity, compressibility, tensile strength etc. depend on structure of crystal or arrangement of particles in it.

Amorphous solids:

  • The substances like plastic, silicate glass etc. in which the molecules do not form regular lattice and more or less chaotic and randomly distributed throughout the solid, are known as amorphous solids.
  • Amorphous solids can be regarded to be highly viscous, super cooled liquids without any sharp melting points.
  • The various properties like conductivity, elasticity, tensile strength etc. remains same, no matter in which direction these are measured with in the substance.

Q.3 Define the following:

Answer.  Non-primitive cell: A cell which is not primitive is called non primitive cell. Example, some unit cell are also non primitive.

Lattice parameters of unit cell: In order to specify a lattice in three dimensions, six parameters are needed; three are basis vectors  and three are angles . These are called lattice parameters.

Bravais lattice: In a Bravais lattice, all the lattice points are equivalent and hence by the necessity all the atoms in the crystal are of same kind.

Non –Bravais lattice: In non-Bravais lattice some of lattice points are not equivalent. A non-Bravais lattice is referred to as a lattice with a basis, the referring to a set of atoms stationed near each side of Bravais lattice.

Packing fraction (packing density): It is defined as the ratio of atoms occupying the unit cell to the volume of unit cell relating to structure of crystal.

Q.4 What do you know about point group and space group symmetries? Give characteristics of space group.

Answer.  There are mainly four types of symmetry operations:

  • Rotation
  • Reflection
  • Inversion
  • Translation

The first three are point operations and give certain symmetry elements which collectively determine the symmetry of space around a point. The group of such symmetry operations at a point is called point group symmetry. We define a lattice group as the collection of symmetry operations which when applied about a lattice point leaves the lattice invariant. The group of all the symmetry elements of a crystal structure is called space group. It determines the symmetry of crystal structure as a whole. There are seventeen distinct space groups possible in two dimension and two hundred thirty in three dimensions. Some characteristics of space group symmetry are listed below:

  • The symmetry of a crystal structure is specified completely when space group is known.
  • The space is characterized by Bravais lattice and by location of point grow and other symmetry elements in a unit cell.
  • The space group through its symmetry elements determines the position of equivalent points with in the unit cell.

Q.5 Define coordination number and write its value for  bcc and  .

Answer.  In a crystal the number of nearest neighbors in a given structure is called coordination number. It gives idea of closeness of packing of the atoms. In simple cubic cell, coordination number is six in body centered lattice is twelve.

Q.6 In two-dimensional lattice, the lattice parameters are \left|\vec{a}\right|=\left|\vec{b}\right|\ ,\ \varphi={90}^0 . Name the lattice. Under what rotation the lattice is invariant?

Answer.  The lattice is a square lattice. The lattice is invariant under rotation of . Thus the lattice has five four symmetry.

Q.7 Which type of unit cell has maximum packing fraction in cubic lattice? Write value of its packing fraction.

Answer.  In cubic lattice,  has maximum packing fraction, which is 74%.

Q.8 In a crystal, a plane cuts intercepts of  \mathbf{2\vec{a}\ ,\ 3\vec{b}\ ,6\vec{c} } along three crystallographic axes, Determines Miller indices of the plane.

Answer.  Intercepts are 2, 3, 6. Reciprocal are \frac{1}{2}\ ,\ \frac{1}{3}\ ,\ \frac{1}{6}  . Reductions to smallest integers gives 3, 2, 1. So miller indices are (321).

Q.9 Calculate atomic radii of bcc and fcc.

Answer.  Atomic radius of  In this type of unit cell, in addition to lattice points at corners, there is one atom at center of cube. Hence unit cell is non-primitive. Number of atoms per unit cell are 2.

    \[\left(AD\right)^2=\ \left(AB\right)^2+\ \left(BC\right)^2+\left(CD\right)^2\]

    \[\left(4r\right)^2=\ a^2+a^2+a^2\]

    \[\left(4r\right)^2=3a^2\]

    \[\sqrt{\left(4r\right)^2}=\ \sqrt{3a^2}\]

    \[4r=\ \sqrt{3a}\]

    \[r=\frac{\sqrt{3a}}{4}\]

The atomic radius of bcc is \frac{\sqrt3}{4}

Atomic radius of In this type of unit cell, in addition to lattice points at corners, there is one atom at center of each face. Hence unit cell is non-primitive. Number of atoms per unit cell is 4.

    \[\left(AC\right)^\mathbf{2}=\left(AB\right)^\mathbf{2}+\left(BC\right)^2\]

    \[\left(4r\right)^2=a^2+a^2\]

    \[\left(4r\right)^2=2a^2\]

    \[16r^2=2a^2\]

    \[r^2=\frac{2a^2}{16}\]

    \[r=\frac{a}{\sqrt[2]{2}}\]

The atomic radius of fcc is \frac{a}{\sqrt[2]{2}}.

Q.10 Determine miller indices of a plane which is parallel to x-axis and cuts intercepts of 2 and 1/2 respectively along y and z axes.

Answer.  Intercepts are \infty,\ 2,\ \frac{1}{2}. Reciprocal are \frac{1}{\infty},\ \frac{1}{2},\ \frac{2}{1}. Reduction to smallest integers gives 0, 1, 4. So miller indices are (014).

Q.11 Calculate the intercepts spacing for (321) plane in simple cubic lattice with inter-atomic spacing 4.21Ao .

Answer.  For simple cubic lattice,

    \[d_{hkl}=\frac{a}{\sqrt{h^2+k^2+l^2}}\]

    \[d_{hkl}=\frac{4.21A^0}{\sqrt{3^2+2^2+1^2}}=1.01A^0\]

Q.12 Give miller indices of family of close packed planes and directions in

Answer.  Miller indices of family of close packed planes and directions are given below:

Family of close packed planes:

  • sc,\ \left{100\right}\ ;bcc\ \left{110\right}\ ;fcc\left{111\right}

Family of close packed directions:

  • c,\ <100>\ ;bcc\ <\overline{1}11>\ ;fcc<1\overline{1}0>

Q.13 Is actual lattice of diamond

Answer.  No, actual lattice is not  The diamond structure is obtained by inserting one lattice into another  lattice displaced along the space diagonal to quarter of its length.

Q.14 A crystal has a basis of one atom per lattice point and set of primitive translation vectors (in Ao) are;

    \[\mathbf{\vec{a}=3\hat{i},\ \ \ \ \ \ \ \ \vec{b}=3\hat{j},\ \ \ \ \ \ \ \ \vec{c}=1.5\left(\hat{i}+\hat{j}+\hat{k}\right)}\]

What is lattice type of this crystal? Calculate volume of conventional unit cell and primitive unit cell?

Answer.  Let us write given vectors as,

    \[\vec{c}=1.5\left(\vec{a}+\vec{b}+\vec{c}\right)\]

Where \vec{c^\prime}=3\hat{k}. It is clear that lattice is body centered cubic with unit cell defined by primitive translation vectors   .

Volume of conventional unit cell  =\ a^3=27A^{0\ 3\ }

Volume of primitive unit cell =\ \frac{a^3}{2}=13.5A^{0\ \ 3}

Q.15   The primitive translation vectors of hexagonal space lattice are;

    \[\vec{a}=\left(\frac{\sqrt{3a}}{2}\right)\hat{i}+\left(\frac{a}{2}\right)\hat{j},\ \ \ \ \ \ \ \ \ \ \ \ \vec{b}=-\left(\frac{\sqrt{3a}}{2}\right)\hat{i}+\left(\frac{a}{2}\right)\hat{j},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vec{c}=c\hat{k}\]

Find volume of primitive cell.

Answer.  Volume of primitive cell is;

    \[V=\vec{a}.\vec{b}\times\vec{c}=\left|\begin{matrix}\frac{\sqrt{3a}}{2}&\frac{a}{2}&0\-\frac{\sqrt{3a}}{2}&\frac{a}{2}&0\0&0&c\\end{matrix}\right|=\frac{\sqrt3}{2}\ a^2c\]

Q.16   How many atoms are there in primitive cell of diamond? What is length of a primitive translation vector if cube edge is

Answer.  Diamond has Bravais lattice with two carbon atoms in primitive cell. The number of atoms in a conventional cube is eight.

Length of primitive translation vector

Q.17 An orthorhombic crystal whose primitive translation vectors have magnitudes a=1.21A^0,\ \ b=1.84A^0,\ \ \ c=1.97A^0. If plane with miller indices (231) cuts an intercept of 1.21  along x-axis, find length of intercepts along y and z-axes.

Answer.  Miller indices are k:k:l=2:3:-1; 

Intercepts are; \frac{1}{2},\ \frac{1}{3},-\frac{1}{1} 

Ratios of actual lengths of intercepts are;

    \[\frac{1.21}{2},\ \frac{1.84}{3},\ -\frac{1.97}{1}=1.21:\frac{3.68}{3}:-3.94=1.21:1.23:-3.94\]

.

So lengths of y and z-intercepts are 1.23A^0and-3.94A^0. 

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