Transcript Document

Ch5 Many-electron atoms
•
•
•
•
•
•
Zeeman effect
Exclusion principle
Periodical table
LS coupling
jj coupling
Two electron spectra (singlet and triplet)
Some Words
•
•
•
•
•
•
•
Zeeman effect
Normal & Anomalous
Shell & subshell
Pauli exclusion principle
Electron configuration
Fermions, Bosons
Symmetric &
antisymmetric
• Subscript, superscript
• Periodic table
• Group, period
•
•
•
•
•
•
•
•
•
•
Helium (氦), neon(氖)
Lithium (锂), sodium(钠),
Alkaline earth (碱土)
lanthanide (rare earths)
(稀土)
Fluorine (氟),
halogen(卤素)
Nitrogen (氮)
Hund’s rules
Lande’s rule
Inert (noble) gas
Ion （离子）
The Zeeman effect
• The splitting of spectral lines due to the splitting
of the energy terms of atoms in a magnetic field
is called the Zeeman effect.
• Selection rule:
m  0,  1
• It is classified as the ordinary Zeeman effect and
the anomalous Zeeman effect.
– Ordinary: The splitting of states with pure orbital
angular momentum (S=0), triplet,
E  ml  B B
– Anomalous: The splitting of states with the resultant
angular momentum and non-zero spin.
The Sodium Zeeman Effect
Many electrons in an atom
• The electrons interact not only with the
nucleus but also among themselves. It is
difficult to get the wave function.
• Electron configuration: How do the
electrons fill the shells and subshells? How
to get its ground state?
• Atoms with a close atomic number will
have a smooth or regular variation in the
atomic properties? Z=10, 11, 9?
The Pauli exclusion principle
• The Pauli exclusion principle requires that
only one electron can be in a given state,
which is labeled by quantum numbers that
indicate the energy, orbital angular
momentum, and spin angular momentum
of an electron in that state (n,l,ml,ms).
• This principle applies to fermions (next
slide).
• In the ground states of atoms, electrons
occupy the lowest energy states available
consistent with the exclusion principle.
Fermions and Bosons
• Particles of half-integral spin are often referred
to as Fermi particles or fermions, such as protons,
electrons and neutrons. They have wave
functions that are antisymmetric to an exchange
of any pair of them.
 A  12 [ a (1) b (2)  a (2) b (1)]
• Particles of 0 or an integral spin are referred to as
Bose particles or Bosons, such as photons, α
particles and helium atoms. They have wave
functions that are symmetric
to an exchange of
1
 S  2 [ a (1) b (2)  a (2) b (1)]
any pair of them.
Electron configuration (1)
• Two basic rules determine the electron structure
of many-electron atoms.
– A system of particles is stable when its total energy is a
minimum.
– Only one electron can exist in any particular quantum
state in an atom.
• Atomic shells with different principle quantum
number n are denoted by capital letters:
– n=1(K), n=2(L), n=3(M), n=4(N), n=5(O),
• Each subshell is identified by its principle
quantum number n followed the letter
corresponding to its orbital quantum number l,
denoted by s,p,d,f, g,h and so on.
Electron configuration (2)
• Each subshell can contain a maximum of 2(2l+1)
electrons, s(2), p(6), d(10)…
• A superscript after the letter indicates the number
of electrons in that subshell. For example, the
electron configuration of sodium is expressed as
1s22s22p63s1.
• The total number of electrons a shell can contain
is equal to the number of electrons in all its
closed subshells. The maximum number of
electrons in the nth shell is equal to 2n2, n=1(2),
n=2(8), n=3(18)…
• With a given electron configuration, how to
determine the atomic states?
Total angular momentum
• Each electron in an atom has a certain orbital
angular momentum L and a certain spin angular
momentum S, both of which contribute to the
total angular momentum J of the atom. How?
• In terms of vector addition, there are several
ways to obtain the total angular momentum. The
two main ways of coupling are LS coupling and
jj coupling.
– LS coupling holds for most atoms and for weak
magnetic fields.
– Jj coupling holds for the heavier atoms and for strong
magnetic fields.
LS coupling
• When the orbit-spin coupling in an individual
electron is smaller than the mutual interaction of l
or s angular momenta, LS coupling takes places.
• The orbital angular momenta Li of various
electrons are coupled electrically into a single
resultant L. So are the spin angular momenta Si
into a single resultant S. Then the L & S interact
magnetically to form the total angular momentum
by orbit-spin coupling.
L   Li S   Si J  L  S
2 S 1
LJ
jj coupling
• When the individual orbit-spin coupling is stronger
than the mutual interaction of l or s angular
momenta of different electron, j coupling takes
places.
• For heavy atoms, the nuclear charge becomes great
enough to produce spin-orbit interaction within an
electron. Each electron has a total angular
momentum Ji resulting from the vector sum of Li
and Si. Then Ji are Jcombined
together
to form
the

L

S
J

J

i
i
i
i
total angular momentum of the atom J.
Atomic state : ( J i , J j ) J
Example
• Problem: Find the possible values of the total
angular momentum number under LS coupling of
two atomic electrons whose orbital quantum
numbers are l1=1 and l2=2.
• Solution: There are three ways to combine L1 and
L2 to a single vector L, L=3,2,1. In the same way,
there are two ways to combine S1 and S2, S=1,0.
As shown in the table, the five possible
values are J=0,1,2,3, and 4.
S
L
3
2
1
1
4,3,2
3,2,1
2,1,0
0
3
2
1
Atomic state of an atom
• Atomic state: 2 S 1 LJ
• For a many-electron atom, capital letters are
used to designate the entire electronic state of
an atom according to its total orbital momentum
quantum number L.
L
0
1
2
3
4
5
6
…
State
S
P
D F
G
H
I
…
• The spin angular momentum S is represented by the
multiplicity of the state placed as a superscript. The
total angular momentum quantum number is used
as a subscript. For example, 2P3/2 S=1/2, L=1,
J=3/2.
Hund’s rules
• The rules for determining the ground-state
quantum numbers for LS coupling atoms are
known as Hund’s rules.
– For a given electron configuration, the state with
maximum multiplicity (2S+1) lies lowest in
energy.
– For a given multiplicity (S), the state with the
largest value of L lies lowest in in energy
– For equivalent electrons, the states with the
smallest J have the lowest energy in ‘normal’
multiplets, but otherwise the converse holds.
“Normal” means that the subshells are less than
half full.
(np)(n’p)
•
l1  1, l2  1, L  2,1,0
s1  12 , s2  12 , S  1,0
1
S 0 ,1P1 ,1D2 ,3S1 ,3P0,1, 2 ,3D1, 2,3
S=0
1S
0
1P
1
1D
2
When n=n’, they
are equivalent
electrons.
L+S=even number
npn’p
S=1
3S
1
3P
2,1,0
3D
3,2,1
(existing states)
States in orange
are forbidden.
Example
• Problem: Use Hund’s rules to find the groundstate quantum number of nitrogen.
• Solutions: The electron configuration of the atom
is 1s22s22p3. Three electrons in p subshell are
permitted to have ms=1/2, and the maximum
value of S is 3/2. Each electron has quantum
number (2,1,ml,1/2). ml can only be =1, 0, -1,
resulting in ML=0, L=0. Thus, L=0, S=3/2 &
J=3/2 are the ground-state quantum numbers for
4
nitrogen, with an atomic
S 3 state:
2
The periodic table (1)
• When the elements are listed in order of atomic
number, elements with similar chemical and
physical properties recur at regular interval,
known as the periodical law.
• Elements with similar properties form the groups
shown as vertical columns in the table.
• The horizontal rows in the table are called periods.
Across each period is a more or less steady
transition from an active metal through less active
metals and weakly active nonmetals to highly
active nonmetals and finally to an inert gas.
The periodic table (2)
• For inert gases, atoms contain only closed shells.
The atoms do not easily donate electrons to or
accept electrons from other elements.
• s-subshell elements form the first two column
(groups) with the alkalis (ns1) and alkaline earths
(ns2). Alkali metals have a single s electron in its
outer shell, which can be easily lost. Elements in
this group often form singly positive ions.
• Transition metals are placed in the three rows in
which the d subshell is filling.
• The lanthanide (rare earths) series involves
completing mainly the 5d and 4f subshells.
Z=9, 10,11
• Z=10, neon, an inert gas, shell & subshells are
full. It does not readily give up or to accept an
electron, generally not combining with other
elements to form compounds. Its boiling point
are low and ionization energy is high.
• Z=9, F, fluorine, halogens (卤素), p-subshell
element np5, It is one electron shy of being full.
It easily accepts an electron from other elements
to form a compound, highly reactive. F-,
Negative ions.
• Z=11, Na, sodium, alkali, s-subshell element np1,
The electron can be easily detached to other
elements in a chemical reaction, highly reactive.
Na+, Positive ions.
Two-electron spectra
• Selection rules under LS coupling are:
S  0, L  0,1, J  0,1
• There is a division into singlet and triplet
states. In singlet states, the two electrons are
antiparallel (S=0). Triplet states have the
two electrons in parallel (S=1).
• According to the selection rules, no
transition between singlet states and triplet
states is allowed.