2. What is angular momentum?
Angular momentum in classical mechanics
Angular momentum in quantum mechanics
Addition of angular momentum
Pauli ’s exclusion principle
Determinant form of wave functions (SLATER DETERMINANT)
3. Angular momentum/moment of momentum or rotational momentum is a
vector quantity that can be used to describe the overall state of a
physical system. The angular momentum L of a particle with respect to
some point of origin is
Where r is the particle's position from the origin & p = mv is its linear
momentum and × denotes the cross product.
S.I unit of angular momentum are Newton meter per sec (N·m·s−1 or
kg·m2s−1) or Joule seconds(J·s).
4. For an object with a fixed mass that is rotating about a fixed symmetry
axis, the angular momentum is expressed as the product of the moment
of inertia of the object and its angular velocity vector
Where I is the moment of inertia of the object and ω is the angular
velocity.
N.B:- Classical mechanics puts no restriction on the magnitude of the
angular momentum and its 3 components Lx, Ly &Lz can be determined
simultaneously & precisely.
5. The classical definition of angular momentum as L=r × p can be carried over to
quantum mechanics, by reinterpreting r as the quantum position operator and p as
the quantum momentum operator L is then an operator, specifically called the
orbital angular momentum operator.
However, in quantum physics, there is another type of angular momentum, called
spin angular momentum, represented by the spin operator S. Almost all
elementary particle have spin. Spin is an intrinsic property of a particle,
fundamentally different from orbital angular momentum. All elementary particles
have a characteristic spin, for example electrons always have "spin 1/2" while
photons always have "spin 1".
Finally, there is total angular momentum J, which combines both the spin and
orbital angular momentum of all particles and fields.
J=L+S
Conservation of angular momentum applies to J, but not to L or S; for example,
the spin-orbit interaction allows angular momentum to transfer back and forth
between L and S, with the total remaining constant.
6. When we have two sources of angular momentum J1 and J2 (e.g., L1 and L2
due to orbital motion of two electrons, or L and S due to orbital and spin
motions of the same electron in an atom) their resultant J and their
components are defined as follows:
J = J1 + J2
J2 = J2
x + J2
y + J2
z
And Jk =J1k + J2k, (k = x, y or z)
7. The total angular momentum is represented as a vector J of length
with j, the quantum no., having any of the values
permitted by Clebsh-Gordan series
j1 + j2 , j1 + j2 -1 , j1 + j2 -2, …
The vectors will lie anywhere along the surface of a cone around z-
axis, the component Jz being specified and Jx and Jy remain
unspecified.
The length of contributing vectors J1 and J2 will have definite
value and respectively.
The component Jz will have the magnitude mj where mj can have any
of the 2j+1 values ranging from + j to – j.
8. In a two electron atom the orbital functions and may be
same or different. In the ground state of He, e.g., the orbital for both
the electrons are the same (1s), the combined wave function being
=1s(1) 1s(2)
In the excited state, one of the electrons may have the orbital 1s and
the other 2s or vice versa
= 1s(1) 2s(2)
or = 1s(2) 1s(1)
Both of which are good eigen functions of the Hemiltonian with the
same eigen values (doubly degenerate).
However, the product of the functions differ, their squares are also
different i.e., different probability distributions for the same two
electrons in the same state. This makes no physical sense as
electrons are indistinguishable from each other or probability must
not change on merely interchanging electron positions.
= 2
=
Contd….
2
9. This means that two functions may either be same or negative of other.
Symmetric: If function remains unchanged on interchange of electron position.
Antisymmetric: If function changes sign on interchange of electron position.
=
=
Which of the two wave functions is used to describe the excited state of He as
both lead to same energy? Apparently there is no definite answer in favor of
any of these s.
As no. of electron increases, the spatial arrangement is not only important
but spin arrangement is also going to affect . Here comes the concept
of electron spin.
10. The spin is a property intrinsic to electron and it is assumed that the electron
spins around its axis. It leads to the generation of spin angular momentum
which can be given as
S =
Where s, called the spin number, could have only one value ½.
The spin angular momentum also has 3 component vectors and only one of
them can have specified value.
Spin –Orbitals:
Corresponding to an orbital which occupies an electrons. There will be two
spin orbitals and .
For a two electron system the total spin angular momentum will be a resultant of
individual spin angular momentum of electrons. Likewise the operator for the z-
component of the total spin angular momentum Sz can be written as a linear
sum of the individual electron.
Sz = S1z + S2z
Contd….
12. It is clear from this presentation that last two results shown above are neither
symmetric nor antisymmetric. Therefore we have to generate their linear
combination.
Linear Combination:
Symmetric-
Antisymmetric-
So problem is to select that which will represent the system. For this we have
a rule
PAULI EXCLUSION PRINCIPLE
“ For a system of two or more electrons the complete wave
function including spin must be antisymmetric with respect
to interchange of any two electron positions”.
13. In quantum mechanics, a Slater determinant is an expression that
describes the wave function of a multi- fermionic system that satisfies
anti-symmetry requirements and consequently the Pauli exclusion
principle by changing sign upon exchange of fermions. It is named for
its discoverer, John C. Slater, who published Slater determinants as a
means of ensuring the antisymmetry of a wave function through the use
of matrices
To explain this lets take the example of Lithium atom in ground state
Φ1 corresponds A (clockwise spin in 1s orbital)
Φ2 corresponds B (anticlockwise spin in 1s orbital)
Φ3 corresponds C (clockwise spin in 2s orbital)
There are six possibilities by which electron can be filled in these orbits
(3×2×1). Similarly for five electron system 120 possibilities exist.
So we represent the antisymmetric wave function of multi electron
system in detriminantal form as it occupies lesser space.
14. In 3 electron system we can represent Ψ as
Here
So this is a better and convenient way of representing the system and
give the same result as we get for Ψ antisymmetric by combination of 6
terms as
Ψ- = N
15. In this presentation of antisymmetric wave function, Pauli exclusion
principle are insured automatically on its own by 2 important property of
determinant:
If any 2 rows or columns of a determinant are interchanged the
resulting determinant is negative of the original.
If any 2 rows or columns in the determinant are same then the
determinant vanishes.
19. What is electron Diffraction ?
History
Use of electron Diffraction Studies
Electron interaction with matter
Intensity of diffracted beams
What is the need of using Electron diffraction ?
Principle involved
Set up for electron diffraction
Interaction of Electron Beam and Gas Molecule
Electron Diffraction pattern
Types of electron diffraction
20. Electron diffraction refers to the wave nature of
electrons. It may be regarded as a technique used to
study matter by firing electrons at a sample and
observing the resulting interference pattern. This
phenomenon is commonly known as the wave-particle
duality, which states that the behavior of a particle of
matter (in this case the incident electron) can be
described by a wave.
Both properties are used in the electron diffraction
experiment, which gives information about distances
between atoms in gas-phase molecules.
21. The de Broglie Hypothesis, formulated
in 1924, proposed by French physicist
Louis de Broglie & predicts that
particles should also behave as waves.
The diffraction of electrons was first
shown by Davisson and Germer in 1927
The first gas electron diffraction (GED)
investigation of molecular structure,
that of carbon tetrachloride, was
reported by Mark and Wierl in 1930.
22. Unlike other types of radiation used in
diffraction studies of materials, such as X-rays
and neutrons, electrons are charged particles
and interact with matter through the Coulomb
forces. This means that the incident electrons
feel the influence of both the positively
charged atomic nuclei and the surrounding
electrons.
In comparison, X-rays interact with the
spatial distribution of the valence electrons,
while neutrons are scattered by the atomic
nuclei through the strong nuclear forces. In
addition, the magnetic moment of neutrons is
non-zero, and they are therefore also
scattered by magnetic fields.
23. In the kinematical approximation for electron diffraction, the
intensity of a diffracted beam is given by:
Here is the wave function of the diffracted beam and is
the so called structure factor which is given by:
where g is the scattering vector of the diffracted beam, is
the position of an atom in the unit cell, and is the scattering
power of the atom. The sum is over all atoms in the unit cell.
The structure factor describes the way in which an incident
beam of electrons is scattered by the atoms of a crystal unit
cell, taking into account the different scattering power of the
elements through the term . Since the atoms are spatially
distributed in the unit cell, there will be a difference in phase
when considering the scattered amplitude from two atoms.
This phase shift is taken into account by the exponential
term in the equation.
24. For a single crystal x-ray diffraction, size
required is 0.05mm (diam) otherwise
intensity of diffracted beam are too weak
to be detected clearly.
Reason:-
1.Efficiency with which x-ray are diffracted
is very low.
2.Crystal as large as 0.05mm cannot be
prepared.
Electron diffraction in such cases is helpful
as it uses wave property of electron and as
its scattering efficiency is high and small
sample may be used.
25. Molecular geometry: relative position of atomic nuclei in the
molecule
Energy difference between conformers
Intramolecular motion
Information about molecular energetic
Information about electron density distribution
Electron diffraction is most frequently used in solid state physics
and chemistry to study the crystal structure of solids.
Short range order of amorphous solids
Unit cell & space group determination:
Only reliable method for crystal smaller than 0.01-0.02mm (diam).
Phase Identification:
Only when x-ray powder diffraction method are unavailable. Useful
when
1.small quantities are available
2.thin film samples
Detecting small amount of impurity present in the sample.
Evaluating bond angles and bond length in relatively small gaseous
sample.
26. Secondary diffraction:
As scattering efficiency of electron is high, the
diffracted beams are strong. Secondary
diffraction occurs when these diffracted beam
effectively become the incident beam and are
diffracted by another set of lattice planes.
Consequences:
1.Extra spots may appear in diffraction pattern and
care is needed in interpretation of diffraction data.
New experimental techniques have, however greatly
improved resolution of bands.
2.Intensity of diffracted beams are unreliable and
cannot be used for crystal structure determination.
27. Structure can be determined by analyzing the
scattering pattern produced when a beam of electrons
interacts with the sample.
Electron beam penetrate gases and produce diffraction
pattern as a result of interaction with gas molecules
Since electron are they are scattered strongly by their
interaction with nuclei and electrons of the atoms of
sample. Hence, cannot be used to study the interiors of
sample. However used for studying molecules in the
gaseous state held on surfaces and in thin films.
N.B:- Electron beam fail to penetrate beyond the
surface of solids and liquids.
Strong interaction between electron beam and
atom because charge present on the electron.
28. Diffraction:
Diffraction refers to various phenomena which occur when a
wave encounters an obstacle.
The diffraction phenomenon is described as the apparent
bending of waves around small obstacles and the spreading out
of waves,
Scattering:
Change in direction of electromagnetic waves
Scattering efficiency and behavior depends on size of
scatterers relative to wavelengths of radiation
Define size parameter as ratio of characteristic particle
diameter to wavelength
Treats particles as identical spheres
29. Scattering efficiency of a particle, customarily denoted by
Qb [non-dimensional], is defined by the following equation:
Qb = Cb / G
where Cb [length2] is the scattering cross section of the
particle, and G [length2] is the area of a geometrical cross
section of the particle in a plane perpendicular to the
direction of the incident light (i.e. the particle shadow).
The scattering efficiency may assume values greater than
unity (which conflicts with the traditionally accepted
meaning of this term, implying that its maximum value is
unity), i.e. a particle may scatter from the incident beam
more light power that falls on its geometrical cross section.
30. Diffraction of x-ray by crystal
depends upon the spacing
between the layers while the
diffraction of electrons by
gaseous molecules depends
upon the distances between the
atoms in a molecule.
31. An electron beam is produced by drawing electrons out of the cathode plate by
means of applied voltage and directing them to anode.
Beam pass across a potential difference of V volts, each electron acquires kinetic
energy as a result of the acceleration in electric field.
mv2 = eV
Thus momentum,
mv =
And for an electron wavelength λ from de Broglie relation is
λ = = = pm
Therefore accelerating voltage of 40 kV corresponds to 6 pm.
Such wavelength leads to interference effect when a accelerated beam passes
through a sample containing scattering centres seperated by the interatomic
distances between the atoms of molecule.
32. When a beam of high energy electrons passes through a
chamber containing gas molecules the charge of nuclei of
molecule will interact with the incoming beam.
Each atom of gas molecule will act as a radiation scattering
center.
Since particles of the electron beam carry a charge , the
amount of scattering resulting from interaction is relatively
large.
SCATTERING OF ELECTRON BEAM:
Coherent Scattering- No energy exchange between the beam
and scattering centre.
Incoherent Scattering- Energy exchange; change in the
wavelength and phase of the scattered electron beam
33.
34. An electron diffraction pattern consists of pattern of rings of varying
intensity. The position of darkened rings on the photographic plate are
estimated visually. For convenient comparison with calculated scattering
curves, a plot is sketched for the plate darkening as a function of parameter
s defined by
s = sin
Diffraction pattern shows that intensity at a scattering angle θ can be
expressed in terms of function s. The expression for net scattering by two
atoms i and j separated by distance rij and randomly oriented in the path of
electron beam is given by Wierl Equation.
I(s) =fifj (1+ sin srij/ srij)
Where f i and fj are the scattering factors for the atom i and j
respy.
It can be extended to polyatomic molecule as
I(s) = ∑ij fifj (sin srij/ srij)
For any assumed molecular structure the contribution from all pairs of atoms
can be calculated and plot of I (s) versus s called a theoritical scattering
curve can be made.
The theoritical curves are then compared with experimental curves.
35.
36. Gas electron diffraction:
The target of this method is the determination of the
structure of gaseous molecules i.e. the geometrical
arrangement of the atoms from which a molecule is built up.
Low-energy electron diffraction (LEED):
It is a technique for the determination of the surface
structure of crystalline materials by bombardment with a
collimated beam of low energy electrons (20-200eV) and
observation of diffracted electrons as spots on a fluorescent
screen.
Reflection high-energy electron diffraction (RHEED):
It is a technique used to characterize the surface of
crystalline materials. RHEED systems gather information only
from the surface layer of the sample, which distinguishes
RHEED from other materials characterization methods that
also rely on diffraction of high-energy electrons.
LEED is also surface sensitive, but LEED achieves surface
sensitivity through the use of low energy electrons.
37. Physical Chemistry
Gordon M. Barrow
Principles of physical Chemistry
Puri, Sharma & Pathania
An Introduction To Crystallography
F.C.Phillips
spectroscopy.chemistry.ohiostate.edu/institute
/weber1.pdf
en.wikipedia.org/wiki/Electron_diffraction
