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[
General Glossary,
Quantum Glossary,
Atomic Glossary
]
Basic Definitions & Conventions
Quantum Numbers
Consider the following quantum numbers
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Symbol
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Quantum Number
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Allowed Values
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n
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principal
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n is a positive integer
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l
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orbital angular momentum
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l
= 0, 1, ..., n-1
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L
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total orbital angular momentum
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SUMi li
(where SUM is over all i
individual electrons)
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S
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total spin angular momentum
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SUMi si
(where SUM is over all i
individual electrons)
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J
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total angular momentum
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J =
L + S
ie J =
SUMi ji
where j =
l + s
(and SUM is over all i
individual electrons)
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M
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magnetic
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projection of
J in a magnetic field
More specifically:
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ml
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where
ml =
-l,
-(l+1), ...,
l
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ms
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where
ms =
+/- 1/2
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Remember, the Pauli exclusion principle prohibits two or more
electrons having the same set of quantum numbers.
States, Levels, Shells etc
Below are the commonly used definitions for both the structual
entity and transitions between them:
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Definition
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Quantum Numbers
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Notes
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A (Zeeman) State
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n,
l,
S,
L,
J,
M
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Transitions between different
states
is known as a
Line Component
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A Level
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n,
l,
S,
L,
J
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A level contains
2J+1
states
The set of transitions between different
levels
is known as a
Line
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A Term
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n,
l,
S,
L,
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A term contains
(2S+1)x(2L+1)
levels
The set of transitions between different
terms
is known as a
Muliplet
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A Subshell
(or Configuration)
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n,
l
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The set of transitions between different
subshells
is known as a
Transition Array
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A Shell
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n
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Consisting of several
subshells
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.
For multi-electron atoms, electron levels having
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... the same n
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belong to the same
shell
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... the same n
and l
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belong to the same
subshell
all electrons in a subshell
referred to as equivalent
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By the Pauli exclusion principle,
the maximum number of electrons in a given
subshell
is
2(2l+1)
Shorthand Notation for Shells
For convenience & brevity,
the shells are often specified
by an upper-case letter
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value of n
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1
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2
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3
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4
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5
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6
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7
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letter code for shell
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K
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L
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M
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N
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O
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P
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Q
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So, for example one might see expressions like
K-shell photoelectric absorption edge...
Shorthand Notation for Subshell Configurations
The configuration of
the
N equivalent electrons
in a given subshell is
usually written as
where
- n is the
principal quantum number
(a positive integer, as above)
- l is a letter
used to specify the value of the
orbital angular momentum quantum number
(which has allowed values
l = 0, 1, ...,
n-1)
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value of l
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0
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1
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2
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3
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4
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5
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6
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7
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8
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letter code for l
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s
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p
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d
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f
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g
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h
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i
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k
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l
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The superscript value for
N is often omitted
if N=1.
More details, including examples
Number of Levels in Shells & Subshells
A level
has 2J+1 unique
states
(since there are
2J+1 possible values of
magnetic quantum number).
However the energies of all these
2J+1
states
in the absence of a magnetic field.
Thus the level
is said to be (2J+1)-fold degenerate.
The number of states
(ie max number of electrons allowed) in each
subshell is
|
n
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Values of L
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#Levels
(2L+1)x(2S+1)
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Full Subshell Config
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1 (K-shell)
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0
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2
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1s2
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2 (L-shell)
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0
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2
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2s2
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1
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6
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2p6
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3 (M-shell)
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0
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2
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3s2
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1
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6
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3p6
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2
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10
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3d10
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etc
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etc
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etc
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etc
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(since S=1/2)
However
is is important to note that the energy levels
associated with the various
higher subshells
state to "overlap"
(starting with
4s
(N-shell) level being lower than the
3d
(M-shell) level).
Thus whilst one can think of simply filling the
1s,
2s,
2p,
3s ...
subshells
when considering the ground states of
elements of increasing atomic number
(Z),
this actually breaks down beyond
Argon (Ar, Z=18).
- Ar indeed has a ground-state configuration
of 1s2
2s2
2p6
3s2
3p6
ie 18 electrons, with all the
subshells
filled.
- However
the next atom,
Potassium (K, Z=19)
has the ground-state configuration
1s2
2s2
2p6
3s2
3p6
4s1
ie the
4s subshell
has a lower energy than the
3d subshell
The energy levels and "fullness" of the various
subshells
is of course the basis of why the periodic table
is organized the way it is (& a large chunk of chemistry).
- For instance since the outer electron in the
ground-state of Potassium is in an "s"
level (ie the 4s subshell), its a
"Group IA" element.
- Since the the outer electrons in the
ground-state of the next element Calcium
(Ca, Z=20) form a
closed subshell)
(ie have 4s2), its
a
"Group IIA" element
- just like
Beryllium,
Be, Z=4,
with
1s2
2s2
and
Magnesium,
Mg, Z=12,
with
1s2
2s2
2p6
3s2
-
(Helium,
He, Z=2,
with
1s2
is different since the entire K-shell is filled)
- Once the
4s subshell
has been filled, then the
3d subshell
does in fact have the next lowest energy.
Thus for
Scandium (Sc, Z=21)
through to
Zinc (Zn, Z=30)
the 10 possible levels of the
3d subshell
start getting filled -
hence why the Periodic Table "gets wide" with Sc.
- Things get even more "messy" with
Lanthanium (La, Z=57)
onwards due to the overlap between the
5d and
5f subshells
in the O-shell (n=5)
and various
subshells
in the P-shell (n=6)
See also
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