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GNDU Question Paper-2024

Bachelor of Computer Application (BCA) (Hons.)

Semester (Batch 2024-28) (CBGS)

PHYSICS: Paper-A

(Condensed Matter Physics)

Time Allowed: Three Hours Max. Marks:35

Note: Attempt Five questions in all, selecting at least One question from each section. The

Fifth question may be attempted from any section. All questions carry equal marks.

SECTION-A

1. (a) Name and draw the possible Bravais lattices in a cubic crystal system. How many

atoms per unit cell are there in each case?.

(b) Explain the crystal structure of NaCl with suitable diagram.

2. (a) What do you mean by Miller indices of a crystal plane? Draw the planes having

Miller indices (101) and (002).

(b) Differentiate between a primitive and a non-primitive unit cell.

SECTION-B

3. (a) Construct the first Brillouin zone for an fcc lattice.

(b) Prove the Bragg diffraction condition in reciprocal lattice.

4. (a) Show that the reciprocal lattice of bcc lattice is an fcc lattice.

(b) Determine the wavelength of X-rays incident on a single cubic crystal of lattice

constant 0.4 nm suffering second order Bragg diffraction at an angle of 30°.

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SECTION-C

5. Discuss the Einstein's theory of specific heat of solids and obtain an expression for

specific heat of solids at constant volume. What are the limitations of this theory?

6. (a) Explain the inelastic scattering of photons by phonons.

(b) Determine the Debye temperature if the Debye frequency for a vibration mode in a

solid is 6 × 10

Hz.

SECTION-D

7. Explain the formation of energy bands in solids on the basis of Kronig-Penny model.

8. (a) Explain the free electron gas model of metals.

(b) Draw the Fermi levels in intrinsic and extrinsic semiconductors.

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GNDU Answer Paper-2024

Bachelor of Computer Application (BCA) (Hons.)

Semester (Batch 2024-28) (CBGS)

PHYSICS: Paper-A

(Condensed Matter Physics)

Time Allowed: Three Hours Max. Marks:35

Note: Attempt Five questions in all, selecting at least One question from each section. The

Fifth question may be attempted from any section. All questions carry equal marks.

SECTION-A

1. (a) Name and draw the possible Bravais lattices in a cubic crystal system. How many

atoms per unit cell are there in each case?.

(b) Explain the crystal structure of NaCl with suitable diagram.

Ans: Introduction

To understand this question, first imagine that you are building a huge wall using identical

bricks. Every brick is placed at equal distances and follows the same pattern. This repeated

arrangement of bricks is similar to how atoms are arranged inside a crystal. Scientists call

this repeating arrangement a crystal lattice.

A unit cell is the smallest repeating part of a crystal. By repeating this unit cell again and

again in all directions, the entire crystal is formed.

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In the cubic crystal system, the unit cell has the shape of a cube. All three sides are equal in

length, and all angles are 90°. There are three possible Bravais lattices in the cubic crystal

system.

1. Simple Cubic (SC) Lattice

Structure

In a Simple Cubic (SC) lattice, atoms are present only at the eight corners of the cube.

How Many Atoms Are Present?

Each corner atom is shared by 8 neighbouring cubes, so only 1/8th of each atom belongs to

one unit cell.

Calculation:

Number of corner atoms = 8

Contribution of each corner atom = 1/8

Total atoms

= 8 × 1/8

= 1 atom

Features

Atoms only at the corners.

Simplest cubic structure.

Large empty spaces exist inside the cube.

Rare in nature.

2. Body-Centred Cubic (BCC) Lattice

Structure

In the Body-Centred Cubic (BCC) lattice, atoms are present:

At all 8 corners

1 atom at the centre of the cube

The centre atom belongs completely to that unit cell.

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Number of Atoms

Corner atoms

= 8 × 1/8 = 1 atom

Body centre atom

= 1 × 1 = 1 atom

Total atoms

= 2 atoms

Features

More closely packed than Simple Cubic.

Stronger crystal structure.

Examples include Iron (at room temperature), Chromium and Tungsten.

3. Face-Centred Cubic (FCC) Lattice

Structure

In the Face-Centred Cubic (FCC) lattice, atoms are located:

At all 8 corners

At the centre of all six faces

Each face atom is shared between 2 cubes, so only half belongs to one unit cell.

Number of Atoms

Corner atoms

= 8 × 1/8 = 1 atom

Face atoms

= 6 × 1/2 = 3 atoms

Total atoms

= 4 atoms

Features

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Very closely packed structure.

Highest packing efficiency among cubic lattices.

Strong and dense crystal.

Examples include Copper, Silver, Gold and Aluminium.

Summary Table

Bravais Lattice

Position of Atoms

Total Atoms per Unit Cell

Simple Cubic (SC)

Eight corners

Body-Centred Cubic (BCC)

Eight corners + one body centre

Face-Centred Cubic (FCC)

Eight corners + six face centres

(b) Explain the Crystal Structure of NaCl with Suitable Diagram

Introduction

Common salt (Sodium Chloride, NaCl) is one of the most famous examples of an ionic

crystal. It is formed by the attraction between:

Sodium ion (Na⁺) positively charged

Chloride ion (Cl⁻) negatively charged

The opposite charges attract each other strongly, producing a stable crystal.

Crystal Structure of NaCl

The chloride ions (Cl⁻) form a Face-Centred Cubic (FCC) lattice.

The sodium ions (Na⁺) occupy all the octahedral holes (interstitial spaces) present inside

this FCC lattice.

Thus, Na⁺ ions fit perfectly into the empty spaces between Cl⁻ ions.

Arrangement of Ions

Chloride Ions (Cl⁻)

Located at:

Eight corners

Six face centres

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Total Cl⁻ ions

Corners

= 8 × 1/8 = 1

Face centres

= 6 × 1/2 = 3

Total Cl⁻ ions = 4

Sodium Ions (Na⁺)

Located at:

One body centre

Twelve edge centres

Edge atom contribution

= 12 × 1/4 = 3

Body centre

= 1

Total Na⁺ ions

= 3 + 1

= 4

Total Atoms in One Unit Cell

Na⁺ ions = 4

Cl⁻ ions = 4

Total particles = 8 ions

Therefore,

Formula of the unit cell

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= Na₄Cl₄

Simplifying,

= NaCl

This confirms that the crystal maintains a 1 : 1 ratio of sodium and chloride ions.

Coordination Number

One of the most important properties of NaCl is its coordination number.

Every Na⁺ ion is surrounded by 6 Cl⁻ ions.

Every Cl⁻ ion is surrounded by 6 Na⁺ ions.

Hence,

Coordination Number = 6 : 6

This arrangement makes the crystal highly stable and strong.

Simple Representation of NaCl Structure

Cl -------- Na -------- Cl

| | |

Na -------- Cl -------- Na

| | |

Cl -------- Na -------- Cl

Another way to visualize the unit cell:

Corners = Cl⁻ ions

Face centres = Cl⁻ ions

Cl

●-------●

/| /|

●-------● |

| Na | |

| ● | |

| ●-----|-●

|/ |/

●-------●

(Additional Na⁺ ions are present at the edge centres.)

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Characteristics of NaCl Crystal

It has a Face-Centred Cubic (FCC) crystal structure.

Chloride ions form the FCC lattice.

Sodium ions occupy all octahedral voids.

Each ion is surrounded by six oppositely charged ions, giving a coordination number

of 6.

The structure is held together by strong electrostatic (ionic) forces, making NaCl

hard and stable.

It has a 1 : 1 ratio of Na⁺ and Cl⁻ ions, so the chemical formula remains NaCl.

Conclusion

The cubic crystal system contains three Bravais lattices: Simple Cubic (SC) with 1 atom,

Body-Centred Cubic (BCC) with 2 atoms, and Face-Centred Cubic (FCC) with 4 atoms per

unit cell. These lattices describe different ways atoms can be arranged inside a cubic crystal.

The NaCl crystal is a classic example of an FCC ionic crystal, where Cl⁻ ions form the main

lattice and Na⁺ ions occupy the octahedral spaces. Each sodium ion is surrounded by six

chloride ions and vice versa, creating a strong, stable, and highly ordered crystal structure.

This regular arrangement is responsible for the hardness, stability, and characteristic

properties of common table salt.

2. (a) What do you mean by Miller indices of a crystal plane? Draw the planes having

Miller indices (101) and (002).

(b) Differentiate between a primitive and a non-primitive unit cell.

Ans: Introduction

In solid-state physics, chemistry, and materials science, crystals are made up of atoms

arranged in a very regular and repeating pattern. This arrangement forms many flat surfaces

inside the crystal, called crystal planes. Scientists needed a simple and universal way to

identify these planes, just like we use house numbers to identify houses. This numbering

system is called Miller Indices.

Miller indices make it easy to describe the orientation (position and direction) of any crystal

plane. They are written inside brackets, such as (100), (110), (101), (111), (002), etc.

What are Miller Indices?

Definition:

Miller Indices are a set of three integers (h, k, l) that represent the orientation of a crystal

plane with respect to the three crystallographic axes X, Y, and Z.

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They do not tell us the size of the plane, but they tell us how the plane is inclined or

positioned inside the crystal.

Think of a crystal as a building made of thousands of tiny cubes stacked together. Inside this

building, many imaginary flat sheets can pass through the cubes. Miller indices help us

identify exactly which sheet we are talking about.

Steps to Find Miller Indices

To determine Miller indices:

1. Find where the plane cuts the X, Y, and Z axes.

2. Write the intercepts.

3. Take the reciprocal of each intercept.

4. Remove fractions by multiplying with the same number.

5. Write the final integers inside brackets (h k l).

Understanding Plane (101)

The Miller index (101) means:

h = 1

k = 0

l = 1

This tells us:

The plane cuts the X-axis at one unit.

Since k = 0, it does not cut the Y-axis. It remains parallel to the Y-axis.

The plane cuts the Z-axis at one unit.

Diagram of Plane (101)

Z-axis

↑

● | (cuts Z at 1)

/ |

-------------/---|------------→ X-axis

(Plane is parallel to Y-axis)

Here, the plane touches the X-axis and Z-axis but never meets the Y-axis.

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Understanding Plane (002)

The Miller index (002) means:

h = 0

k = 0

l = 2

This means:

The plane is parallel to both X-axis and Y-axis.

It intersects only the Z-axis.

Since the index is 2, the plane is located at half the normal spacing between (001)

planes. A larger Miller index means the crystal planes are more closely spaced.

Diagram of Plane (002)

Z-axis

↑

------------------- (002) Plane

------------------+----------------→ X-axis

Y-axis

The plane is horizontal and parallel to the XY plane.

Why are Miller Indices Important?

Miller indices are very useful because they:

Identify different crystal planes.

Help scientists study crystal structure.

Explain X-ray diffraction patterns.

Describe how crystals grow.

Help engineers understand the strength and properties of materials.

Are widely used in metallurgy, nanotechnology, semiconductor technology, and

mineralogy.

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Without Miller indices, describing crystal planes would be confusing because crystals

contain thousands of different planes.

2(b) Differentiate between a Primitive and a Non-Primitive Unit Cell

What is a Unit Cell?

A unit cell is the smallest repeating building block of a crystal. By repeating this tiny unit in

all directions, the entire crystal is formed.

Imagine making a wall using identical bricks.

One brick = Unit Cell

Complete wall = Crystal

There are two main types of unit cells:

1. Primitive Unit Cell

2. Non-Primitive Unit Cell

Primitive Unit Cell

A Primitive Unit Cell contains only one lattice point.

The lattice points are present only at the eight corners of the cube.

Each corner atom is shared by eight neighboring unit cells.

Contribution of one corner atom:





Total atoms:







 

Therefore,

Primitive Unit Cell contains one effective lattice point.

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Diagram

●---------●

/| /|

●---------● |

| | | |

| ●-------|-●

|/ |/

●---------●

Only corner atoms are present.

Non-Primitive Unit Cell

A Non-Primitive Unit Cell contains more than one lattice point.

Besides the corner atoms, additional atoms are present at:

Face centers

Body center

Base centers

Examples include:

Body-Centered Cubic (BCC)

Face-Centered Cubic (FCC)

Base-Centered Unit Cell

These extra atoms increase the total number of lattice points in the unit cell.

Example: Body-Centered Cubic (BCC)

●---------●

/| /|

●----●----● |

| | ● | |

| ●-------|-●

|/ |/

●---------●

The atom at the center belongs completely to the same unit cell.

Difference Between Primitive and Non-Primitive Unit Cell

Primitive Unit Cell

Non-Primitive Unit Cell

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Contains only one lattice

point.

Contains more than one lattice point.

Atoms are present only at

the corners.

Atoms are present at corners and also at face centers,

body center, or base centers.

Simple crystal structure.

More complex crystal structure.

Lower packing efficiency.

Higher packing efficiency.

Occupies less space

efficiently.

Atoms are packed more closely.

Example: Simple Cubic (SC).

Examples: Body-Centered Cubic (BCC), Face-Centered

Cubic (FCC).

Conclusion

Miller indices provide a simple and standard method to identify the orientation of crystal

planes. For example, the (101) plane intersects the X- and Z-axes while remaining parallel to

the Y-axis, whereas the (002) plane is parallel to the XY plane and intersects only the Z-

axis, indicating closely spaced planes. Understanding Miller indices is essential for studying

crystal structures and diffraction.

A unit cell is the basic repeating block that forms a crystal. A primitive unit cell contains

only one effective lattice point because atoms are located only at the corners. In contrast, a

non-primitive unit cell contains more than one lattice point due to additional atoms at the

body center, face centers, or base centers. These differences influence the packing

efficiency and physical properties of crystalline materials. Together, Miller indices and unit

cells form the foundation for understanding the structure and behavior of crystalline solids.

SECTION-B

3. (a) Construct the first Brillouin zone for an fcc lattice.

(b) Prove the Bragg diffraction condition in reciprocal lattice.

Ans: 3(a) Construct the First Brillouin Zone for an FCC Lattice

The First Brillouin Zone is a very important concept in solid-state physics. Although the

name sounds difficult, the idea is actually simple if we understand it step by step.

Imagine a crystal as a city where atoms are arranged in a perfectly repeating pattern.

Scientists use something called reciprocal space to study how electrons and waves move

inside this crystal. In reciprocal space, the First Brillouin Zone is like the main boundary or

neighborhood around one lattice point.

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What is an FCC Lattice?

FCC (Face-Centered Cubic) means that atoms are placed:

At all 8 corners of a cube.

At the center of each of the 6 faces.

This arrangement is found in metals like Aluminium, Copper, Silver, Gold, and Nickel.

What is the First Brillouin Zone?

The First Brillouin Zone is the smallest region in reciprocal space that is closer to the central

reciprocal lattice point than to any other reciprocal lattice point.

You can think of it like this:

Imagine you are standing exactly in the middle of a field. Around you are many trees placed

at equal distances. If you draw boundaries exactly halfway between you and every nearby

tree, the enclosed area around you is similar to the First Brillouin Zone.

Steps to Construct the First Brillouin Zone of an FCC Lattice

1. Draw the reciprocal lattice of the FCC crystal.

2. Choose the central reciprocal lattice point.

3. Join this point to all the nearest neighboring reciprocal lattice points.

4. Draw perpendicular bisectors of each joining line.

5. The closed region formed by these bisectors is called the First Brillouin Zone.

For an FCC crystal, the reciprocal lattice becomes a Body-Centered Cubic (BCC) lattice.

The shape of the First Brillouin Zone of an FCC crystal is a Truncated Octahedron.

Simple Diagram

________

/ \

| |

| Central |

| Point |

\ /

\________/

(Actual shape is a Truncated Octahedron)

3(b) Prove the Bragg Diffraction Condition in Reciprocal Lattice

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Before understanding the proof, let us first understand what diffraction means.

When X-rays fall on a crystal, they strike many layers of atoms. These waves reflect from

different atomic planes.

If the reflected waves combine and become stronger, this is called Constructive

Interference.

This phenomenon produces a bright diffraction spot.

Bragg's Law

The ordinary Bragg condition is

  

Where

n = Order of diffraction

λ = Wavelength of X-rays

d = Distance between crystal planes

θ = Angle of diffraction

This equation tells us when diffraction occurs.

Reciprocal Lattice Concept

Instead of studying diffraction in real space, scientists study it in reciprocal space because

calculations become much easier.

Let

k = Incident wave vector

k = Diffracted wave vector

G = Reciprocal lattice vector

Diffraction occurs when



󰆒

  

This is called the Laue Condition or Bragg Condition in Reciprocal Space.

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Simple Proof

Suppose an X-ray enters the crystal.

Its wave vector is k.

After scattering, the wave vector becomes k.

The change in wave vector is

  

󰆒



If this change is exactly equal to one reciprocal lattice vector G, then all scattered waves

remain in phase.

Hence,

  



󰆒

  

This proves the Bragg diffraction condition in reciprocal lattice.

Why is this Important?

This condition tells us:

Which crystal planes produce diffraction.

Where diffraction peaks appear.

How scientists determine crystal structure.

Diagram

Incident Wave

------->

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/ |

*-------> k'

Difference

k' - k = G

3(c) Various Methods Used to Study Crystal Structure

Scientists use several experimental methods to determine how atoms are arranged inside a

crystal. Each method works on a different principle and is suitable for different types of

materials.

1. X-ray Diffraction (XRD)

This is the most common and widely used method.

X-rays are passed through the crystal.

They are reflected by atomic planes.

The diffraction pattern helps determine the arrangement of atoms.

Advantages

Very accurate.

Suitable for most crystalline materials.

Gives lattice parameters and crystal symmetry.

2. Electron Diffraction

Instead of X-rays, a beam of electrons is used.

Since electrons have a very small wavelength, they can reveal extremely fine crystal details.

It is mainly used for:

Thin films

Nanomaterials

Very small crystals

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3. Neutron Diffraction

Neutrons are fired onto the crystal.

This method is especially useful because neutrons interact with atomic nuclei rather than

electron clouds.

It is useful for:

Locating light atoms like hydrogen.

Studying magnetic materials.

Investigating magnetic structures.

4. Transmission Electron Microscopy (TEM)

A very thin specimen is examined using high-energy electrons.

TEM provides:

Crystal structure.

Grain boundaries.

Crystal defects.

Atomic-scale images.

5. Scanning Electron Microscopy (SEM)

SEM scans the surface of the crystal using electrons.

It gives:

Surface morphology.

Crystal shape.

Grain size.

Surface defects.

6. Scanning Tunneling Microscopy (STM)

STM uses a very sharp needle-like tip placed extremely close to the crystal surface.

It can produce images of individual atoms and is useful for studying conductive materials.

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Easy Comparison Table

Method

Radiation Used

Main Purpose

X-ray Diffraction

(XRD)

X-rays

Determine crystal structure and lattice

parameters

Electron Diffraction

Electrons

Study thin films and nanocrystals

Neutron Diffraction

Neutrons

Locate light atoms and magnetic structures

TEM

Electron beam

Atomic-scale internal crystal structure

SEM

Electron beam

Surface structure and morphology

STM

Tunneling

current

Imaging individual atoms on conductive

surfaces

Conclusion

The First Brillouin Zone is the smallest region in reciprocal space around a lattice point. For

an FCC crystal, its reciprocal lattice is BCC, and the First Brillouin Zone has the shape of a

truncated octahedron. The Bragg diffraction condition in reciprocal space, 

󰆒

  ,

explains when scattered waves reinforce each other to produce diffraction and is the

foundation of crystal structure analysis. Finally, techniques such as X-ray diffraction,

electron diffraction, neutron diffraction, TEM, SEM, and STM allow scientists to determine

the arrangement of atoms, identify defects, and study the properties of crystalline

materials. Together, these concepts form the basis of understanding how the internal

structure of crystals is investigated in modern physics and materials science.

4. (a) Show that the reciprocal lattice of bcc lattice is an fcc lattice.

(b) Determine the wavelength of X-rays incident on a single cubic crystal of lattice

constant 0.4 nm suffering second order Bragg diffraction at an angle of 30°.

Ans: (a) Show that the Reciprocal Lattice of BCC is FCC

What is a Crystal Lattice?

A crystal is made up of atoms arranged in a regular repeating pattern.

Think of it like bricks in a wall.

Every brick is placed at a fixed position, forming a neat pattern.

Similarly, atoms form a lattice.

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What is a BCC (Body-Centered Cubic) Lattice?

In a Body-Centered Cubic (BCC) lattice,

There is one atom at each of the 8 corners of the cube.

One additional atom is present at the center of the cube.

So,

Corner atoms = 8

One center atom = 1

This arrangement is called Body-Centered Cubic (BCC).

Examples:

Iron (α-Iron)

Chromium

Tungsten

What is a Reciprocal Lattice?

The reciprocal lattice is not a real crystal.

It is an imaginary mathematical lattice used to study diffraction of X-rays, electrons, and

neutrons.

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Instead of describing the positions of atoms, it describes how waves interact with the

crystal.

Scientists use reciprocal lattices because diffraction calculations become much easier.

Relationship Between BCC and FCC

One of the most important results in crystallography is

Reciprocal of BCC = FCC

Reciprocal of FCC = BCC

This is a standard and well-known result.

Why does this happen?

When we mathematically calculate the reciprocal lattice vectors of a BCC crystal, the

resulting points appear at the faces of the cube, exactly matching the arrangement of an

FCC lattice.

Therefore,

Reciprocal Lattice (BCC) = FCC

Similarly,

Reciprocal Lattice (FCC) = BCC

Final Answer for (a)

The reciprocal lattice of a Body-Centered Cubic (BCC) lattice is a Face-Centered Cubic (FCC)

lattice. This result is obtained by applying the reciprocal lattice vector equations. Hence, it

is proved that BCC and FCC are reciprocal to each other.

(b) Determine the Wavelength of X-rays

Given

Lattice constant

   nm

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Second-order diffraction

  

Bragg angle

  



Step 1: Bragg's Law

Where

= spacing between crystal planes

= order of diffraction

= wavelength

= Bragg angle

Step 2: Find the Interplanar Distance

For a simple cubic crystal,

  

Therefore,

   nm

Step 3: Substitute Values





 

Since





 

  

  

   nm

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Final Answer

Wavelength of X-rays = 0.2 nm

Meaning of This Result

The wavelength of the X-ray is 0.2 nanometres, which is nearly the same size as the spacing

between atoms inside a crystal. This is why X-rays are ideal for studying crystal structures

they can be reflected (diffracted) by the atomic planes and reveal how atoms are arranged.

Why Were Laue's Equations Introduced?

When X-rays strike a crystal, every atom scatters the X-rays.

Constructive interference (bright diffraction spots) occurs only if the scattered waves satisfy

certain conditions.

These conditions are called Laue's Equations.

They explain when diffraction is possible.

The Three Laue Equations

If the incident X-ray has direction cosines

󰇛











󰇜

and the diffracted X-ray has direction

cosines

󰇛



󰆒





󰆒





󰆒

󰇜

, then:

󰇛 

󰆒

󰇜  

󰇛

󰆒

󰇜  

󰇛 

󰆒

󰇜  

Where:

= lattice constants along the three crystal axes

= wavelength of X-rays

= integers called Miller indices

= direction cosines of the incident beam



󰆒



󰆒



󰆒

= direction cosines of the diffracted beam

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Physical Meaning of Laue's Equations

These equations tell us that:

Diffraction occurs only when the path difference between scattered X-rays is an

integer multiple of the wavelength.

Only specific directions satisfy these conditions, producing bright diffraction spots.

They help scientists determine the arrangement of atoms in crystals.

Difference Between Bragg's Law and Laue's Equations

Bragg's Law

Laue's Equations

Simple and easy to use

More general and mathematical

Based on reflection from crystal

planes

Based on interference from all atoms in the crystal

Used to calculate wavelength or

spacing

Used to describe diffraction in three dimensions

Commonly used in numerical

problems

Mainly used in crystallography and theoretical

analysis

Exam-Oriented Key Points

BCC (Body-Centered Cubic): Atoms are present at the 8 corners and one atom at the

center of the cube.

FCC (Face-Centered Cubic): Atoms are present at the 8 corners and at the center of

each of the 6 faces.

Reciprocal Lattice: An imaginary lattice used to analyze diffraction patterns.

Important Result: Reciprocal lattice of BCC is FCC, and reciprocal lattice of FCC is

BCC.

Bragg's Law: Used to calculate the wavelength of X-rays or the spacing between

crystal planes. In this problem, the wavelength is 0.2 nm.

Laue's Equations: Three equations that give the conditions for X-ray diffraction in

crystals and explain when constructive interference occurs.

Conclusion

Although this question combines three different concepts, they are closely connected

through X-ray diffraction. The reciprocal lattice provides a mathematical way to study

diffraction, Bragg's Law helps calculate quantities such as wavelength or interplanar

spacing, and Laue's equations describe the general conditions required for diffraction to

occur. Together, these concepts enable scientists to determine the internal arrangement of

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atoms in crystalline materials, making them fundamental tools in solid-state physics and

crystallography.

SECTION-C

5. Discuss the Einstein's theory of specific heat of solids and obtain an expression for

specific heat of solids at constant volume. What are the limitations of this theory?

Ans: Introduction

When we heat a solid object like an iron rod, a copper wire, or an aluminum spoon, it

becomes hot because it absorbs heat energy. But have you ever wondered how much heat

is required to increase the temperature of a solid by 1°C or 1 K? This property is called the

specific heat of a solid.

Scientists noticed that the specific heat of solids changes with temperature. At high

temperatures, it remains nearly constant, but at very low temperatures it decreases rapidly.

The older Dulong and Petit Law could explain the behavior only at high temperatures and

completely failed at low temperatures.

To solve this problem, the famous physicist Albert Einstein proposed the Einstein Theory of

Specific Heat of Solids in 1907. It was the first successful quantum theory that explained

why the specific heat of solids changes with temperature.

What is Specific Heat?

Specific heat is the amount of heat required to raise the temperature of 1 kg of a substance

by 1 Kelvin (or 1°C).

For solids, we generally study the specific heat at constant volume (Cv) because the volume

of a solid changes very little when heated.

Basic Idea of Einstein's Theory

Imagine a solid is made up of millions of tiny atoms arranged in a regular pattern called a

crystal lattice.

Instead of remaining fixed, every atom continuously vibrates around its mean (equilibrium)

position.

Simple Analogy

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Think of children standing in fixed places while jumping up and down.

They cannot leave their positions.

They only vibrate around the same point.

Similarly,

Atoms remain fixed in the crystal.

They only vibrate continuously.

Einstein assumed that:

Every atom behaves like a tiny harmonic oscillator.

Every atom vibrates with the same frequency.

Energy is not absorbed continuously.

Energy is absorbed or emitted only in small packets called quanta.

Diagram of Crystal Lattice

Atoms in a Solid

●──●──●──●

│ │ │ │

●──●──●──●

│ │ │ │

●──●──●──●

Each atom vibrates about

its fixed position.

↑

●

 │ 

←────┼────→

 │ 

↓

The arrows show that every atom vibrates around its equilibrium position.

Assumptions of Einstein's Theory

Einstein made the following assumptions:

1. Solid contains a large number of atoms.

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Each atom behaves like an independent oscillator.

2. Every atom vibrates with the same frequency (ν).

This is called the Einstein frequency.

3. Vibrations are independent.

One atom does not affect the vibration of another atom.

4. Energy is quantized.

The energy of one oscillator is





 

where

h = Planck's constant

ν = frequency

n = 0,1,2,3,...

Thus energy is absorbed only in fixed packets.

5. Every atom has three directions of vibration.

An atom can vibrate

along x-axis

along y-axis

along z-axis

Therefore,

Total oscillators = 3N

where

N = number of atoms.

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Average Energy of One Oscillator

According to quantum statistics,

The average energy of one oscillator is















where

h = Planck constant

k = Boltzmann constant

T = Absolute temperature

Total Internal Energy of the Solid

Since there are 3N oscillators,

The total internal energy becomes

  









This represents the total thermal energy stored in the vibrating atoms.

Expression for Specific Heat at Constant Volume

Specific heat at constant volume is obtained by differentiating the internal energy with

respect to temperature:

















Substituting the expression for and differentiating gives:





 























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This is the Einstein expression for the specific heat of solids at constant volume.

Einstein Temperature

Einstein introduced a characteristic temperature called the Einstein temperature:











Using this, the formula becomes





 

































This form is easier to use when comparing different solids.

Behavior at High Temperature

When the temperature is much higher than the Einstein temperature

󰇛

  



󰇜









 







After simplification,





 

Since

  

(for one mole),





 

This agrees with the Dulong and Petit Law, showing that at high temperatures the molar

specific heat of most solids approaches .

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Behavior at Low Temperature

When the temperature is very low

󰇛

  



󰇜









becomes extremely large.

Therefore,





 

This means that the specific heat decreases rapidly as the temperature approaches absolute

zero. Einstein's theory correctly predicted this qualitative behavior, unlike the classical

theory.

Why Does Specific Heat Decrease at Low Temperature?

At very low temperatures:

Atoms have very little thermal energy.

Most atoms remain in their lowest quantum energy state.

They cannot absorb much additional heat.

Therefore, only a few atoms contribute to heat absorption, so the specific heat becomes

very small.

Advantages (Successes) of Einstein's Theory

It was the first quantum theory of specific heat.

It explained why classical theory fails at low temperatures.

It correctly predicts that specific heat tends to zero as temperature approaches

absolute zero.

It agrees with the DulongPetit law at high temperatures.

It introduced the important concept of quantized lattice vibrations.

Limitations of Einstein's Theory

Although Einstein's theory was a major breakthrough, it has several limitations:

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1. All atoms are assumed to vibrate with the same frequency. In reality, atoms in a

crystal vibrate with a range of frequencies.

2. Oscillations are treated as independent. Real atoms interact with neighboring

atoms, affecting their vibrations.

3. The theory does not accurately explain very low-temperature behavior.

Experiments show that the specific heat varies approximately as 



at very low

temperatures, whereas Einstein's prediction does not match this dependence.

4. It works best for simple solids. More complex crystals require a better description of

lattice vibrations.

5. It ignores collective vibrations (phonons). Later, the Debye theory included these

collective vibrational modes and provided much better agreement with experiments,

especially at low temperatures.

Conclusion

Einstein's Theory of Specific Heat of Solids was a landmark in the development of quantum

physics. It proposed that atoms in a solid vibrate as quantized harmonic oscillators and

derived the expression for the specific heat at constant volume:





 

































The theory successfully explained why the specific heat of solids approaches at high

temperatures and decreases toward zero at low temperatures. Although its assumption that

all atoms vibrate with the same frequency limits its accuracy, it laid the foundation for more

advanced theories, especially the Debye theory, and remains an important milestone in

understanding the thermal properties of solids.

6. (a) Explain the inelastic scattering of photons by phonons.

(b) Determine the Debye temperature if the Debye frequency for a vibration mode in a

solid is 6 × 10

Hz.

Ans: (a) Inelastic Scattering of Photons by Phonons

To understand this topic, imagine you are standing in a quiet room filled with people.

Suppose someone throws a tennis ball across the room. If the ball hits a person who is

standing still, the ball may lose some energy and slow down. If it hits someone who is

already moving, the ball may even gain some energy.

A similar process happens inside a solid crystal.

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A crystal is made up of millions of atoms arranged in a regular pattern called a crystal

lattice. These atoms are never completely at rest. They continuously vibrate about their

fixed positions.

These vibrations are called lattice vibrations.

In quantum physics, these vibrations are represented as tiny packets of energy known as

phonons.

Similarly, light is made of tiny particles called photons.

When light enters a crystal, photons interact with these vibrating atoms (phonons). During

this interaction, energy may be exchanged between the photon and the phonon.

This phenomenon is known as inelastic scattering of photons by phonons.

What is Scattering?

Scattering simply means that light changes its direction after hitting something.

There are two types of scattering:

1. Elastic Scattering

The photon changes only its direction.

Its energy remains the same.

Its frequency remains unchanged.

Energy before scattering = Energy after scattering

2. Inelastic Scattering

The photon changes its direction.

It either gains or loses energy.

Therefore, its frequency also changes.

This is called inelastic scattering.

Role of Phonons

A phonon represents the vibrational energy of atoms inside the crystal.

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When a photon collides with these vibrations, two situations are possible.

Case 1: Photon Gives Energy to the Crystal

The photon transfers some of its energy to the vibrating atoms.

A new phonon is created.

As a result,

Photon energy decreases.

Photon frequency decreases.

Wavelength increases.

This process is called Stokes Scattering.

Simple Diagram

Before Collision

Photon (High Energy)

↓

-------------------------

| Vibrating Crystal |

-------------------------

After Collision

Photon (Lower Energy) 

+ Phonon Created

Case 2: Photon Receives Energy from the Crystal

Here, the crystal already contains vibrating atoms.

The photon absorbs energy from an existing phonon.

As a result,

Photon energy increases.

Frequency increases.

Wavelength decreases.

This process is called Anti-Stokes Scattering.

Diagram

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Existing Phonon

Crystal + Phonon

↑

Photon enters

↓

Photon leaves with Higher Energy

Phonon disappears

Why Does Frequency Change?

The energy of a photon is directly proportional to its frequency.

The relation is

  

where

E = Energy

h = Planck's Constant

f = Frequency

If energy changes, frequency must also change.

That is why inelastic scattering produces light of a different frequency than the incident

light.

Raman Effect

The inelastic scattering of photons by phonons is the basic principle of the Raman Effect.

When monochromatic light (single-color light) falls on a crystal,

Most photons undergo elastic scattering.

A small number undergo inelastic scattering.

The scattered light contains new frequencies.

Scientists study these frequencies to learn about:

Crystal structure

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Molecular vibrations

Chemical composition

Material properties

Raman spectroscopy is widely used in:

Material science

Nanotechnology

Chemistry

Medical diagnosis

Semiconductor industry

Key Points to Remember

A photon is a particle of light.

A phonon is the quantum of lattice vibration.

In inelastic scattering, energy is exchanged between photons and phonons.

If the photon loses energy → Stokes Scattering.

If the photon gains energy → Anti-Stokes Scattering.

Frequency changes because photon energy changes.

This phenomenon is the basis of Raman spectroscopy.

(b) Calculation of Debye Temperature

The Debye temperature is calculated using the relation













where





= Debye Temperature (K)

  



J\cdotps





 



  



J/K

Step 1: Substitute the given values







󰇛





󰇜󰇛



󰇜





Step 2: Multiply the numerator

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  





 



Step 3: Divide by Boltzmann constant



















  K

Final Answer

(a) Inelastic scattering of photons by phonons is the process in which a photon interacts

with the vibrating atoms (phonons) of a crystal and exchanges energy with them. If the

photon loses energy, Stokes scattering occurs. If it gains energy, Anti-Stokes scattering

occurs. This changes the frequency of the scattered light and forms the basis of the Raman

Effect, which is used to study the internal structure and vibrational properties of materials.

(b) Debye Temperature





  K

So, the Debye temperature of the solid is approximately 29 K.

SECTION-D

7. Explain the formation of energy bands in solids on the basis of Kronig-Penny model.

Ans: Formation of Energy Bands in Solids on the Basis of KronigPenney Model

The KronigPenney Model is a simple theoretical model used to explain how energy bands

and energy gaps are formed in solids. Although the mathematics behind this model can be

difficult, the basic idea is actually very easy to understand if we imagine what happens when

many atoms come together to form a solid.

1. What is an Energy Band?

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Every isolated atom has electrons that can occupy only certain fixed energy levels. These

energy levels are called discrete energy levels because electrons cannot have any energy

between them.

For example:

Single Atom

E3 ─────────

E2 ─────────

E1 ─────────

An electron can stay only on these levels.

However, in real life, solids are made of millions or billions of atoms placed very close

together. When atoms come close, their outer electrons begin to interact with each other.

Because of this interaction, each single energy level splits into many very closely spaced

energy levels.

When these tiny levels become extremely close together, they appear as one continuous

range of energies called an energy band.

2. What is the KronigPenney Model?

The KronigPenney Model is a mathematical model developed by physicists Ralph Kronig

and William Penney.

The model assumes that electrons move inside a solid through a series of repeating

potential barriers created by positively charged atomic nuclei.

Instead of studying the complicated electric field of every atom, the model simplifies the

crystal by assuming a periodic potential, meaning the same pattern repeats again and again.

It looks like this:

Potential Energy

│ ││ ││ ││ │

│______││______││______││______│______→ Distance

Barrier Well Barrier Well

Here,

Barrier = Region near the atomic nucleus where potential energy is high.

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Well = Space between atoms where electrons move more easily.

This repeating structure is called a periodic potential.

3. Basic Assumptions of the KronigPenney Model

The model is based on a few simple assumptions:

A solid contains a large number of atoms arranged in a regular pattern.

Electrons move through a repeating periodic potential.

Potential barriers are identical and equally spaced.

Electrons behave according to quantum mechanics rather than classical physics.

These assumptions make the analysis much simpler while still explaining the behavior of

electrons in solids.

4. How are Energy Bands Formed?

Imagine there is only one atom.

It has only a few fixed energy levels.

Now imagine two atoms coming close together.

Because of interaction between their electrons, each energy level splits into two levels.

1 Atom

E2 ─────

E1 ─────

2 Atoms

E2a ─────

E2b ─────

E1a ─────

E1b ─────

Now imagine 10 atoms.

Each energy level splits into 10 very close energy levels.

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If one million atoms come together, each energy level splits into one million closely spaced

levels.

These levels become so close that they appear as one continuous energy band.

One Energy Level

────────

↓

Many Atoms

═══════════════════════

Energy Band

═══════════════════════

This is exactly how energy bands are formed inside solids.

5. Formation of Forbidden Energy Gap

According to the KronigPenney Model, not every energy value is allowed for electrons.

When the Schrödinger wave equation is solved for electrons moving in a periodic potential,

two situations appear:

Some energies satisfy the equation → electrons can exist there. These form the

allowed energy bands.

Some energies do not satisfy the equation → electrons cannot exist there. These

regions are called the forbidden energy gaps (band gaps).

Diagram:

Higher Energy

↑

Allowed Band

══════════════════════

Forbidden Gap

--------------

(No electrons)

Allowed Band

══════════════════════

Forbidden Gap

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--------------

Allowed Band

══════════════════════

Lower Energy

Thus, the electron is allowed to move only inside the allowed bands and never inside the

forbidden gaps.

6. Why do Allowed and Forbidden Bands Appear?

This happens because electrons behave like waves.

Inside the crystal, electron waves reflect from the regularly spaced atoms.

At certain energies, the reflected waves reinforce each other (constructive

interference), allowing electrons to move through the crystal. These energies form

the allowed bands.

At other energies, the reflected waves cancel each other (destructive interference),

preventing electron motion. These energies become forbidden bands.

Therefore, the periodic arrangement of atoms naturally creates alternating allowed and

forbidden energy regions.

7. Importance of Energy Bands

The concept of energy bands explains why different materials behave differently.

(a) Conductors

In metals, the valence band overlaps with the conduction band or the band gap is almost

zero.

Valence Band

══════════════

██████████████

Conduction Band

══════════════

Electrons move easily, so electricity flows freely.

(b) Semiconductors

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In semiconductors, the forbidden gap is small.

Conduction Band

══════════════

Small Gap

Valence Band

══════════════

A small amount of heat or light can move electrons into the conduction band, allowing

controlled electrical conduction.

In insulators, the forbidden gap is very large.

Conduction Band

══════════════

Large Forbidden Gap

Valence Band

══════════════

Electrons cannot easily reach the conduction band, so electricity does not flow.

8. Advantages of the KronigPenney Model

The KronigPenney Model successfully explains:

The formation of allowed and forbidden energy bands.

Why electrons cannot possess every possible energy.

The difference between conductors, semiconductors, and insulators.

Electron motion inside crystalline solids.

The electronic properties of materials used in modern devices such as transistors,

diodes, LEDs, and computer chips.

Conclusion

The KronigPenney Model provides a simple and powerful explanation of how energy

bands are formed in solids. An isolated atom has discrete energy levels, but when a very

large number of atoms combine to form a crystal, these levels split into millions of closely

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spaced levels, creating allowed energy bands. Due to the periodic potential inside the

crystal, some energy values are not permitted, resulting in forbidden energy gaps (band

gaps). The size of these energy gaps determines whether a material behaves as a conductor,

semiconductor, or insulator. Thus, the KronigPenney Model forms the foundation of

modern solid-state physics and helps us understand the working of almost all electronic

materials and devices.