Prn – 2023017001468046 VIBRATIONAL ENERGY LEVELS OF A DIATOMIC MOLECULE

1] assuming SHO
Atomic and molecular physics
1] Spectra types

2] Models of atom Thus, all the successive transitions give the same
frequency of emission or absorption. ΔE= hv.

ν=
ΔE 1
=
h 2π √ k
μ
2] for real molecules
3] Hydrogen spectrum…. LBPBP

4] Quantum numbers

1) Principal quantum number (n)

Rotation vibration
2) Orbital or azimuthal quantum number (l)

3) Magnetic quantum number or magnetic orbital

quantum number (ml or lz) [proj of l on field direcn z]

4) Spin quantum number(s)

5) Magnetic spin quantum number (ms or sz)

6) Total angular momentum quantum number (j) CREDIT 01-UNIT 03:ONE AND TWO ELECTRON
7) Magnetic total angular momentum quantum number
SPECTRA
(mj or jz). It is projection of j on z axis [field direction]
ESR [g is gyromagnetic constant]
Magnetic moment of electron µ

5] Electronic configuration
Interaction energy of electron in magnetic field
6] selection rules

Δn = any Δl = ±1 Δs = 0 Δ𝑗 = 0 or 0 ±1
𝑗 0↮𝑗 0 Δml = 0 or 0 ±1 Δ𝑚s = 0 Δmj = 0 or 0 ±1 resonance conditionfor ESR observation.

Δn = any ; Δl = ±1 ; Δs = 0; Δ𝑗 = 0 or 0 ±1 but 𝑗 0↮𝑗 0

-Thus, ESR spectroscopy is based upon the absorption
ROTATIONAL ENERGY LEVELS OF A DIATOMIC MOLECULES of microwave radiation by an unpaired electron when
exposed to a strong magnetic field.

For nuclear spin QN [I]

J=0,1,2,…… is called the rotational quantum number ESR spectrometer

 ELECTRODYNAMICS

Balanced codition – power flows in arm 2 and 3

Unbalanced condn – power appears in 4th [signal det]
SYSTEMATIC EXPANSION OF POTENTIAL OF
X ray diffration ARBITRARY LOCALIZED CHARGE DISTRIBUTION

- Bragg’s diffraction condition 2𝑑𝑠𝑖𝑛 𝜃 = 𝑛𝜆

- Bragg’s diffraction condition in reciprocal
lattice 2 k ⋅G=G2 . Use law of cosines
𝐺⃗ is the reciprocal lattice vector,
𝑘⃗ ia wave vector of the incident/scattered X-ray.

by binomial expansion we get legender polynomial

- Atomic sattering factor -The ratio of the radiation [Pn]
amplitude scattered by the charge distribution in an
atom to that scattered by a point electron is called the
atomic scattering factor oratomic form factor. The
atomic scattering factor is the measure of the
efficiency of an atom in scattering X-rays. We define
the scattering factor for the jthatom as,

n 0 1 2
The quantity𝑓 is a measure of the scattering power of Pn 1 cos𝛉1.5cos2𝛉 -
the jthatom in the unit cell. Where 𝐺⃗ is the reciprocal 0.5
lattice vector and 𝑛 (𝑟⃗) is the electron concentration pole mono dipole quadrapole
associated with jthatom.
dxdydz = r sinθdrdφdθ ; da=r2sinθdφdθ
2

Allowed [hkl]
SCC All
BCC H+k+l=even
FCC Hkl all odd or all even

ELECTRIC FIELD OF DIPOLE

C-01-U-02: LINEAR QUADRUPOLE POTENTIAL AND FIELD
Proof-

Let a light pulse starts from common origin.

Use vector resolution, ;

BT
y’ = y ; z’ = z ; x 2−C 2 t 2=x '2−C2 t ' 2
qⅆ cos θ
v dip= 2 let 𝑥’ =λ(𝑥−𝑣𝑡) put in 𝑥 = λ′(𝑥’ +𝑣𝑡′) we get
4 Π ε0 r
use dipole term n=1

put x’ and t’ in x 2−C 2 t 2=x '2−C2 t ' 2 we get

Vm = Vquad = 0

Equating terms of x2, xt and t2 to 0 we get λ = λ’ =

MAXWELL’S EQUATIONS 1

G
√ 1−
v2
c
2

And other lerentz transformations

F
Velocity addn
A u = velocity of object
v = velocity of frame
u’= relative velo of obj wrt frame

relativity Doppler shift - Source moving away u +ve, λ↑,Red

shift

• The Galilean transformation

Length Time dilation
contraction
Lorentz Transformation equation:

1 I<0 Timelike same place [d=0]

2 I>0 Spacelik Same time [t=0]
 e From boundary condn
3 I=0 lightlike

To find A we normalise WF in box

QUANTUM MEHANICS

PARTIC
LE IN BOX

1] infinite petential wall

Time independent schrodinger equn

Above equn is a very important result and tells us

that:
For V = 0
1. The energy of a particle is quantized means particle
in box is quantised.

2. The lowest possible energy of a particle isNOTzero.

This is called thezero–point energyand means the
particle can never be at rest because it always has
some kinetic energy.
From boundary condition B = 0

By double diff and comparing with SWE

02:FINITE POTENTIAL WELL

From boundary conditions

For E = V0 onlt ψ c changes

TRANSMISSION AND REFLECTION

3] potential barrier

T = t2
E=V0

E<V0
solution of the Schrödinger equation is [quantum
tunnlling]
 E>V0 defined habit faces
[R=1-T] Growth from Targets crystal growth at low
Water temperatures and atmospheric
Solution pressure
Better growth control and
stability compared to high-
temperature methods
The classical result of perfect transmission without Suitable for materials sensitive to
any reflection (T=1,R=0) is reproduced when high temperatures
May have slow growth rates and
1] E ≫V0 and 2]k1a = 𝑛𝜋 solvent inclusion issues
Techniques Slow cooling or solvent
HARMONIC OSCILLATOR
for Large evaporation with precise control
Crystal of supersaturation
Low Focuses on soluble crystals grown
time–independent Schrӧdinger equation gives, Temperature via slow cooling or solvent
Solution evaporation
Growth Stable growth conditions and
suitable solvents, often water, are
Solving above equn we get [n=0,1,2…] crucial
Temp diff Two zones : a cool growth area
method and a hot nutrient saturated area.
energy levels are equidistant,
unlike particle in box. Habit of crystal- Crystals with balanced growth are
Nn is normalizn informative and minimize defects. Large faces isolate
constant imperfections, but needle-like shapes may propagate
flaws. Solvents affect crystal habits
Hn(y) is a Hermite
polynomial of degree h k l
n. eg crystallographic plane satisfies… x + y+ z=1
a b c
where [a/h, b/k, l/c]*K are the intercepts of the plane
on x, y, and z axes. a,b,c are the unit cell lengths

MI=(643:hkl)
Cell length=abc=4A,3A,8A
6 4 3
Equn x+ y+ z =1
4 8 3
FMS
Interplanar angle
Aspect Description
Crystal Growth from Solution, Growth Interplanar
Growth from Melt, Growth from Vapor distance
Methods Phase, Solid State Growth Crystallographic directions [uvw] are length of
Growth from Involves dissolving crystal
position vector projected on axes multiplied by
Solution components in a solvent to create
common factor.
a saturated solution
Solution made supersaturated by Projecn = ½ 1 0
solvent evaporation or Multiply = 1 2 0 = uvw
temperature changes
Produces crystals with well-
 Mixed Combines edge and screw components,
forms curved boundaries between
Reciprocal lattice deformed and undeformed crystal areas.
a*, b* and c* are the cell edges of the reciprocal
lattice GIBB’S PHASE RULE D = DOF; P = no of
D+P=C+2 phases;
a* = 1/acosθ
C = no of components
Clausius clapeyron Δv = specific volume
L = latent heat
𝑉=𝒂.𝒃×𝒄 Δs= specifiv entropy

Bragg’s Law
2dsinθ = nλ
Classification of Defects.

1] Point defects

CREEP

 Creep: Time-dependent deformation; significant at

temperatures about 0.4 to 0.5 of ceramics' melting
points; can occur at low stresses.

 Creep Curve: Obtained from creep test; three

regions: primary, secondary (steady-state), tertiary
Equilibrium number of creep; reflects strain rate changes.
Shottkey defects
 Creep Rupture: Occurs if test runs sufficiently long;
Equilibrium number of
fracture of specimen.
frenkel defects
Where  Observations: Not all stages always observed; curve
shape varies with material and test conditions.
- np is number of vacancy pairs and
 Importance of Steady State: Part spends most life in
- Ep is the energy of a vacancy pair which is the energy
this stage; longest and crucial for component life
required to remove a molecule (anion and cation)
prediction.
from within the crystal and carry it to its surface.
 Primary Creep: Seen at low temps and stresses;
- Ev is energy required to remove an atom from a
creep rate decreases with time; strain may increase
lattice site to the surface.
logarithmically or parabolically.
Dislcn Description
 Secondary Creep: Longest stage; majority of strain
Type
produced; constant strain rate.
Edge Dislocn moves parallel to stress
Screw Involves a ramp-like distortion  Steady State Strain: Total strain equals time
perpendicular to stress. multiplied by steady-state creep rate.
 Thermal Activation: Steady-state creep is thermally 1. Phase Changes in Liquid Glass Melt:
activated; creep rate varies with temperature.
1.1. Liquid-Liquid Phase Separation:
A] Homogeneous nucleation and growth-
morphology is droplets of one liquid dispersed in the
other liquid
B] Spinodal decomposition - the two phases are
continuously distributed throughout the whole
volume. With time the phase boundaries sharpen
and eventually the droplets in a liquid morphology
are attained.
- The following are the deformation mechanisms
which occur during the creep of the crystalline solids 1.2. Crystallization:
(metals and ceramics):
Homogeneous or heterogeneous nucleation.
(i) Dislocation glides
(ii) Dislocation climbs due to diffusion 1.3. Solidification into Rigid Glass:
(iii) Mass transport and change of shape due to
Followed by crystallization to glass-ceramic via heat
diffusion
treatment.
(iv) Grain boundary sliding
2. Liquid-Liquid Phase Separation Mechanism:
- Two types of primary creep
2.1. Based on Free Energy vs. Composition Plot:
Two minima and a maximum lead to stable or
metastable immiscibility.
n
σ =stres s Binodal curve represents minima locus.
Δ H c =¿activation Spinodal curve represents inflexion points locus.
'
ε =strain rat e energy for creep
3. Phase Separation Processes:
3.1. Nucleation and Growth:
 Activation Energy Determination: Obtained by
Occurs between binodal and spinodal curves.
comparing strain rates at two temperatures;
3.2. Fluctuations within Spinodal Curve:
 Steady State Creep Mechanisms: Involve dislocation
glide assisted by diffusion-controlled climb, diffusion Lead to spontaneous phase separation.
creep, and grain boundary sliding. Outcome:
 Diffusion Creep: Prominent at high temperatures Spontaneous fluctuations grow into two phases with
and low stresses; driven by stress-induced vacancy corresponding compositions.
concentration gradient, prompting atom diffusion. Glass is thermodynamically unstable and transforms
 Heering-Nabarro and Coble Creep: Differ by lattice to stable crystalline form below liquid forming
temperature
or grain boundary diffusion.
1. Glass Crystallization and Stability:
 Grain Boundary Sliding in Ceramics: Facilitated by
low viscosity of glassy phase at high temperatures, - Below liquid forming temp, glass transforms to
contributing to creep. stable crystalline form.
2. Glass Ceramics:
 Creep Rate Expression: Single equation integrates
diffusion coefficient, temperature, stress, and grain - produced by controlled crystallization of glass.
size for comprehensive material behavior - Glass partially crystallizes, desired amount
understanding. depends on application.
GLASS - Many contain Li2O.
3. Microstructure Design:

- Desired: fine crystals < 1 μm, evenly dispersed in

glass.

- High nucleation density needed, achieved with

nucleating agents.

- Common agents: TiO2, ZrO2, P2O5, Pt group,

noble metals, fluorides.

4. Glass Ceramic Production:

- Batch melting, shaping, cooling, controlled

heating for nucleation.

- Gradual crystal growth, then cooling to room

temp.

5. Advantages of Glass Ceramics:

- Shaping in glassy state, then heat treatment for

desired properties.

- Tailoring properties by adjusting composition and

heat treatment schedule.

- Not possible with ceramic materials, which

require powder processing.