Topic 3 - Review of Fundamentals
Atomic And Nuclear Structure
|
 |
 |
Outline
- Atomic
Structure
- Rutherford’s
Atom
- Bohr’s
Model
- Modification
of Bohr’s Model
- Periodic
Table
- Characteristic
X-rays
- Nuclear
Structure
- Isotopes
/ Nuclides
- Binding
Energy
- Nuclear
Models
- Nuclear
Stability
Rutherford’s Nuclear Atom
- 1911
- Postulated
positive charge in center of atom, called NUCLEUS
- Hypothesis
tested by Geiger and Marsden
- Gold foil
experiment
- Alpha
particles
- Experimental
result
Rutherford’s Atom

Gold Foil Experiment

Bohr’s Atomic Model
- Predecessor
model: solar system concept
- Issues:
classical theory predicted model unstable
- Resolution:
1913, Neils Bohr – denied validity of classical electromagnetic theory
- New: (Max
Planck) quantum theory of radiation
- Bohr
adopted Planck’s theory
Bohr’s Model for Hydrogen
- Explained
observed spectrum
- Assumed
for electron:
- travels
around nucleus in a circular orbit
- energy
in an orbit is proportional to its distance from the nucleus. The further
from the nucleus, the more energy it has
- if
it stays in a stationary orbit, it does not emit radiation
- Only
a limited number of orbits with certain energies are allowed (quantization
of orbits using Planck's hypothesis)
- orbits
have angular momentum as integral multiple of Planck's constant divided
by 2<p.
- can
only pass from one stationary state to another.
- Radiation
is absorbed when one jumps to a higher energy orbit
- emitted
when an one falls into a lower energy orbit.
- The
energy of the radiation emitted or absorbed is exactly equal to the difference
between the energies of the orbits
Bohr Atomic Model

Modification of the Bohr Atom
- Bohr
theory inadequate for even H atom
- Splitting
of spectral lines
- Sublevels
of energy
- Orbits
are elliptical, not spherical, restricted by quantum conditions
- Describing
atom by 4 quantum numbers
Quantum Numbers
- A>ny
of several quantities that identify the state of a physical system
such as an atom, a nucleus, or a subatomic particle
- he principal
quantum number for atomic electrons indicates the energy state
and the probability of finding the electrons at various distances
from the nucleus
- The
larger this number, the greater the energy is and the farther the electron
is likely to be from the nucleus
- The
four quantum number are sufficient to uniquely characterize uniquely each
atomic electron
Symbol
|
Quantum Number |
Value |
n
|
Principal
|
1,2,……… |
l
|
Azimuthal
|
0 to n -1 |
m
|
Magnetic
|
-l to +l |
s
|
Spin
|
-1/2, + 1/2 |
- Three
quantum numbers are needed to specify each orbital in an
atom
- the
most important is the principal quantum number, n (the same that Bohr introduced)
- The
principal quantum number specifies the energy of the electron in the orbital
- As n increases
from its lowest value 1 through its allowed values 2, 3, . . . , the energies
of the corresponding orbitals increase.
Principle Quantum Number
- n labels the shell of the atom.
- Each
shell
- has n2 individual
orbitals with the same principal quantum number
- has
orbitals that lie at approximately the same distance from the nucleus.
- esembles
the layers of an onion, with successive shells surrounding the inner shells.
Principal Quantum Number
- Hydrogen
ground state
- n = 1
- 1
electron in the orbital closest to the nucleus.
- Ionization
energy
- energy
required to elevate the electron from n = 1 to n = ¥
- energy
required to remove the electron completely from the atom.
Azimuthal Quantum Number
- orbital
angular momentum quantum
number.
- It
represents the magnitude of the orbital angular momentum of the electron
around the nucleus.
- as l increases,
the rate at which the electron circulates around the nucleus increases.
- The
values of l in a shell of principal quantum number n are
limited to the n
values 0, 1, 2, . . . , n - 1
- the
value of l in a given shell determines the subshell to which that orbital belongs.
- n subshells in a shell of principal
quantum number n.
- 2l +
1 orbitals in a given subshell.
- conventional
labels for subshells:
- l = 0 is called an s subshell,
- l = 1 is called a p subshell,
- l = 2 is called a d subshell.
- Other
subshells
- The
three subshells of the shell with n = 3 are called the 3s, 3p, and
3 d
Magnetic Quantum Number
- Subshell l consists
of 2 l + 1 individual orbitals.
- s subshell (l = 0) consists
of a single s orbital
- p subshell (l = 1) consists
of three p orbitals;
- d subshell (l = 2) consists
of five d orbitals
- The
individual orbitals are labeled with the magnetic quantum number, ml, which
can take the 2 l + 1 values l, l - 1, . . . , -l.
Quantum Recap
- n
(principal quantum number) defines electron shells. Values are integers (n = 1,2,3,4,...).
- l (azimuthal quantum number)defines atomic orbitals. Values are integers from 0
to n-1.
- m
(magnetic quantum number) defines number of orbitals of a given kind. Values
are integers ranging from -l through 0 to +l.
- s
(spin quantum number) has values of +½ and -½ (spin up, spin
down)
- From
Pauli Exclusion Principle: no two electrons can have the same set of four
quantum numbers
s orbital example

The spherical boundary surface of an s orbital. This sphere
shows the region of space in which there is the highest probability of finding
an electron that is described by the corresponding wavefunction
p orbitals

The boundary surfaces of the three p orbitals of a given shell.
They are labeled according to their orientation relative to the three axes.
An electron described by one of these wavefunctions will not be found at
the nucleus; there is a nodal plane running through the nucleus between
the two lobes.
d orbitals

The boundary surfaces of the five d orbitals of a given shell,
appropriately labeled.
Periodic Table of the Elements
- Constructed
with Bohr’s atomic model by application of Pauli exclusion principle
Periodic Table of the Elements

Periodic Table, again

Electronic Structure
- Inner
complete electron shells constitute the core or kernel
- Binding
energy of the kernel electrons is much higher than that of the valence
or conduction electrons.
- Kernel
electrons remain practically undisturbed in most of the processes in
which the atom participates.
Consequence of Periodicity

Characteristic X-rays
- Electron
ejected from the kernel (e.g, K shell)
- States
with n=1,2,3,4 constitute the K,L,M,M,.. shells
- Empty
state (or hole) is left in K-shell
- Another
electron in a higher energy level can fall into the vacant state
- Radiation
emitted by electron "falling" into vacant state is in the X-ray
spectral region
- Electron
from: L,M,N,... valence shells
- Series
of X-ray lines may be produced designated as a, b etc.
Atomic Shell Terminology
- Optical
Spectroscopists use n = 1, 2, 3, ..
- X-ray
spectroscopists use K, L, M, N, O . .
- When
an electron is removed from a particular shell,
- electrons
from all the higher-energy shells fill that vacancy
- result
in a series that appears inverted as compared with the hydrogen series.
- Different
angular momentum states for a given shell cause energy sublevels within
each shell;
- these subshells are labeled by Roman numerals according
to their energies.
Origin of Characteristic X-rays

The Nucleus
- Terminology
- Binding
Energies
- Nuclear
Models
Isotopes and Nuclides

- A is atomic mass number (protons + neutrons)
- Z is atomic number (protons)
- N is neutron number
- X is chemical symbol

-
Isotopes
(U)
- Nuclides
(U, Fe)
Binding Energy
Hypothetical
energy released if a nuclide were synthesized from Z separate H atoms
and N (equal to A - Z) separate neutrons.
Average
binding energy per nucleon shows a maximum at 56Fe falling
off gradually on both sides to about 7 MeV at 4He and to about
7.4 MeV for the most massive nuclei known.
Most
of the naturally occurring nuclei are not stable in an absolute
nuclear sense.
Mass defect
- W
= Zm(H) + (A-Z)m(N)
- weight
of sum of parts > actual weight
- mass defect
- d = W – M
- represents
mass equivalent of work done to separate nucleus into components,
- in energy
units, called Binding energy (BE)
- BE=
(W – M)amu x 931 MeV/amu
- Eb =
931(W-M)/A; MeV/nucleon
Binding Energy per Nucleon

Binding Energy
- Nuclei
heavier than iron gain energy by degrading into products closer to iron
- Only
for heavy elements does alpha decay and spontaneous fission attain observable
rates.
- Nuclear
energy is gained by fusion of most elements lighter than iron
- coulombic
repulsion keeps fusion rates low
- exception:
if nuclei are subjected to greater than 107 K in hot cores
of stars, thermonuclear bombs, controlled fusion plasmas
Nuclear Models
- Building blocks known (n, p, subatomics)
- No definite structure defined
- Analogous to atomic system
- Three major models:
- Liquid drop
- Shell model
- Combined model
Liquid Drop Model
Formulated
(1936) by Niels Bohr and used (1939) by him and John A. Wheeler to explain
nuclear fission.
Nucleons
(neutrons and protons) behave like the molecules in a drop of liquid.
If
given sufficient extra energy (as by the absorption of a neutron), the
spherical nucleus may be distorted into a dumbbell shape and then split
at the neck into two nearly equal fragments, releasing energy.
Liquid Drop Model– Bohr & Wheeler
Although
inadequate to explain all nuclear phenomena, provides excellent estimates
of average properties of nuclei
Key
points:
- Nucleus
homogenous mixture of nucleons
- Internal
energy equally distributed
- Surface
tension keeps nucleus spherical
- Explains
nuclear fission
- Allows
calculation of atomic masses
- Nuclear
binding energy can be understood with the model of a charged liquid drop.
Shell Nuclear Model
- Description
of nuclei similar to Bohr atomic model of electron energy levels.
- Developed
by the American physicist Robert Hofstadter in the 1950s.
- Constituent
nuclear particles are paired
- neutron
with neutron and proton with proton
- magic
numbers: filled nuclear-energy levels when the number of protons or neutrons
equals 2, 8, 20, 28, 50, 82, or 126indicate especially stable nuclei.
- The unpaired
neutrons and protons account for the properties of a particular species
of nucleus as valence electrons account for the chemical properties of the
various elements.
- The shell
model accurately predicts certain properties of normal nuclei, such
as their angular momentum;
Collective model
- Also
called UNIFIED MODEL, Incorporates
aspects of both the shell
model and
the liquid-drop model Explains
certain magnetic and electric properties that neither of the two
separately can explain:
- High-energy
states of the nucleus
- certain
magnetic and electric properties (magnetic and quadrupole moments)
- explained
by the motion of the nucleons outside the closed shells
combined with the motion of the paired nucleons in the core.
- Nuclear
core
- thought
of as a liquid drop on whose surface circulates a stable
tidal bulge directed toward the rotating unpaired nucleons outside
the bulge.
- Tide
of protons (positively charged particles) constitutes a current
that in turn contributes to the magnetic properties of the nucleus,
- the
greater deformation of the nucleus as the number of unpaired
nucleons increases accounts for the measured electric
quadrupole moment (an
index of nuclear shape - a measure of how much
the charge in space departs from spherical symmetry).
Nuclear Decay Schemes

Nuclear Stability
- Binding energies
- Proton vs neutron number
- Table of isotopes
Chart of the Nuclides

Nuclear binding energies, shown as a function of atomic mass number
Nuclear Stability Curve

|