Download Chapter 29: Atomic Structure What will we learn in this chapter?

Chapter 29: Atomic Structure
What will we learn in this chapter?
Contents:
Electrons in atoms
Wave functions
Electron spin
Pauli exclusion principle
Atomic structure
Periodic table
W. Pauli & N. Bohr
Note:
Both electron spin and the Pauli exclusion principle are the
missing ingredients for the understanding of the periodic table of
elements.
Electrons in atoms: wave functions
Last class:
When treating electrons in atomistic length scales we are unable
to determine both position and momentum precisely.
We need to introduce a new description in term of
probabilities.
Definition: Wave function
Similarly to classical strings, describe electrons by wave functions.
A wave function Ψ(�r1 , �r2 , . . . , �rn ) describes n electrons with
space coordinates �ri = (xi , yi , zi ). The state of the n-electron
system is fully described by Ψ .
The wave function corresponds to the probability density to find
the particles at a given point in space. Think of it as a “cloud”.
If, for example, the particles have charge, the wave function
describes also the charge distribution of the particles.
Wave functions contd.
The following quantities can be computed from a wave function:
Average position of a particle
Average velocity of a particle
Dynamic quantities (momentum, energy, angular momentum).
Example: spherically-symmetric states of the hydrogen atom
probabilities
densities
(projection)
recall Bohr’s orbits
Selected states of the hydrogen atom
How are wave functions determined?
Schrödinger equation (1926):
∂
�2 2
i� Ψ(�r, t) = −
∇ Ψ(�r, t) + V (�r, t)Ψ(�r, t)
∂t
2m
Note:
The ∇ 2 corresponds to a second-order derivative in the space
coordinate.
The physical information of the system is fully described by the
potential V between the particles. If V is known, the properties of
the system can, in principle, be computed.
The solutions of the Schrödinger equation represent various
possible states of the system.
For many systems solutions only exists for given energy levels, i.e.,
quantization is introduced (e.g., angular momentum and energy).
Quantization of the angular momentum
Bohr’s model:
The angular momentum quantization was done ad hoc.
Schrödinger equation:
The angular momentum quantization is a necessary condition for
the existence of solutions.
Permitted values of angular momentum: The possible values for L of an
electron with energy En and quantum number n in a hydrogen atom are
�
�
L = � l(l + 1) L = |L|
l = 0, 1, 2, . . . n − 1
The component of L in a given direction (e.g., z component) can only
have the set of values
Lz = �ml
The magnitude of ml cannot exceed l.
ml = 0, ±1, ±2, . . . , ±l
Hydrogen atom – angular momentum
Within the full quantum-mechanical description, the hydrogen atom
is described now by four quantum numbers: n, l, ml, s:
Principal quantum number:! ! ! ! ! n
Angular momentum quantum number:!! l
Magnetic quantum number:! ! ! ! ! ml
Electron spin (later…):! ! ! ! ! ! s
Note:
For each energy level En, there are several distinct states with the
same energy.
The “magnetic quantum number” describes small shifts in the
energy levels (and thus in the wavelengths) when the atom is
placed in a magnetic field.
The component Lz can never be quite as large as L.
Angular momentum
Note:
�
While we can determine precisely L and Lz, the component of L
perpendicular to the z axis cannot be determined and is confined
to a cone of direction (Pauli exclusion between L components).
Electron spin
In the 1920s it was noticed that energy levels would split even when
no external field was present.
This suggests that the electron has an intrinsic angular momentum, in
addition to the angular momentum associated with the orbital
motion. Picture electrons as charged spinning spheres.
The spin is also quantized. For the quantization along the z-axis
� �
√
�
1
3
1 1
�
Sz = ± � = s� S =
�=�
+1
S = |S|
2
2
2 2
Electrons are also called spin-1/2 particles (or Fermions).
Simple analogy (solar system):
Both earth and sun spin around their axes and earth spins around
the sun. The total angular momentum is the vector sum of these.
Cross section of selected states
n=3
n=2
n=1
n determines the energy, l and ml the angular momentum:
l=0
l=1
l=1
l=2
Many-electron atoms
So far:
We have only studied the simplest atom with one electron, H.
When an atom has more electrons, electron-electron interactions
need to be included.
Although the Schrödinger equation can describe these systems,
analytical solutions do not even exist for Helium (He)!
Central field approximation:
Assume that one electron feels the average field of the other
electrons and the nucleus.
In this case the quantum numbers of the angular momentum are
the same as for the hydrogen atom.
The energy levels now depend on n and l with the the usual
restrictions on (n, l, ml, s).
Pauli exclusion principle
To implement the central field approximation, we need to take into
account exclusion principles.
Naive approach:
One would expect that in the ground state of an atom all
electrons are in the same lowest state.
If this was the case then the behavior of atoms with increasing
number of electrons should show gradual changes in physical and
chemical properties.
Reality proves this naive approach wrong:
Fluorine (9 electrons) is a halogen and very reactive.
Neon (10 electrons) is an inert gas (no compounds).
Sodium (11 electrons) is an alkali metal.
Pauli exclusion principle contd.
The key to the change in the behavior of atoms when the number of
electrons is increased was discovered by Wolfgang Pauli (1925).
Pauli exclusion principle: No two electrons in an atom can occupy the
same quantum-mechanical state. Alternatively, no two electrons in an
atom can have the same values of all four of their quantum numbers.
Note:
We can now generate a list of all possible sets of quantum
numbers and thus possible states of the electrons in an atom.
Customary notation: the value of l is designated by a letter:
l = 0!! s state
l = 1!! p state
l = 2!! d state
l = 3!! f state! ! ! followed by g, …
First four shells of electron quantum states
factor 2
due to spin
X-ray
levels
States with the same n form a shell, with the same l a subshell.
Atomic structure
Structure of a neutral atom:
The atom has Z electrons, Z protons and a number of neutrons in
the core.
Note:
Z is called the atomic number.
Neutrons have no charge.
Quantum regions closes to the nucleus have lowest energies.
The outer electrons determine the behavior of the atom.
Building an atom:
Start with a bare nucleus with Z protons.
Successively add electrons paying attention to the exclusion rules.
Building atoms…
Z = 1 (hydrogen):
Ground state: 1s (n = 1, l = 0, ml = 0, s = ±1/2).
Z = 2 (helium):
Ground state: 1s2 (both electrons in a 1s state with opposed spin).
The K shell is completely filled (inert gas, forms no compounds).
Z = 3 (lithium):
Ground state: 1s2 2s (2 electrons fill 1s state, one in 2s).
The 2s electron is farther away from the core and thus loosely
bound feeling a net +e charge (5.4 eV vs 13.6 eV for the H atom).
Lithium is an alkali metal forming ionic bonds with valence +1.
Z = 4 (beryllium):
Ground state: 1s2 2s2 (2 electrons K shell, 2 in L shell).
Alkaline-earth element with forming ionic bonds with valence +2.
Building atoms contd.
…
Z = 9 (fluorine):
One vacancy in the L shell.
Affinity to gain an extra electron to fill this shell.
Forms ionic compounds with valence –1.
Z = 10 (neon):
Both K and L shell full.
Neon is a noble gas with no compounds.
Z = 11 (sodium):
K and L shells full, one electron in the M shell.
Similar to lithium. Indeed, also an alkali metal with valence +1
…
Ground-state electron configurations
Ground-state electron configurations contd.
lala
The periodic table of elements
Proceeding with the atom building, we can understand the regularities
seen in the periodic table of elements for each column.
A slight problem occurs with the M and N shells because the 3d and
4s subshells overlap in energy:
Theses elements have 1 or 2 electrons in the 4s state and
increasing numbers in the 3d states.
These elements are all transition metals for Z = 21 to 30.
Similarly, for Z = 57 to 71 two electrons are in the 6s level but the 4f
and 5d levels are only partially filled. These are the rare earth
elements.
At Z = 91 another such series starts with the actinides.
X-ray energy levels
Outer electrons of an atom:
Responsible for optical spectra.
Emit in the near-visible region with energies between 2 – 3 eV.
Vacancies in the inner shells of complex atoms:
Responsible for x-ray energy levels.
Because these electrons are close to the nucleus, binding energies
are larger and more energy is required to remove these.
Suppose an electron is knocked out of the K shell. It can be filled
with an electron of the L, M, N, … shells emitting a photon.
If the outermost electrons are in the N shell, the x-ray spectrum
has 3 lines from transitions from the L, M, N to the K shell.
K series: