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Principle quantum number (n)
It was proposed by Bohr.
It determines the average distance between
electron and nucleus, i.e size of the orbit.
The maximum number of electrons in an orbit
represented by this quantum number is 2n2 ie
2,8,18 etc.
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Azimuthal quantum number( ℓ )
• Azimuthal or angular quantum number.
• Proposed by Sommerfeld.
• It determines the number and shape of sub shells or
sublevels to which the electron belongs.
• The value of ℓ is (n-1) . ℓ =0,1,2,3 ...corresponds
to s, p, d, f.... subshells and contain ( 2, 6, 10,14 .......
Electrons respectively).
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Magnetic quantum number (m)
• It was proposed by Zeeman.
• It gives the number of permitted orientation of
subshells.
• The value of m varies from - ℓ to + ℓ through
zero.
• For a given value of ℓ the total value of ‘m’ is
equal to (2ℓ + 1).
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Spin quantum numbers (s)
• It was proposed by Goldshmidt & Ulen Back.
• The value of s is + 1/2 and -1/2, which signifies
the spin.
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SPACE QUANTIZATION
•The orbital quantum number ‘l’ tells us only the
magnitude of the orbital angular momentum ‘L’.
•To describe angular momentum fully the
direction of this momentum is required.
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•We know that when the electron revolves
around the nucleus gives rise to current loop and
a magnetic field is associated with it.
•Hence atomic electron possessing an angular
momentum interacts with this magnetic field.
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• The magnetic quantum number :
• ml specifies the direction of ‘L’ by finding its
component in the direction of field. This
phenomenon is called SPACE QUANTIZATION.
• Let the direction of magnetic field be parallel to
z-axis, then
• Lz = ml ħ
ml = 0, ±1, ±2, ±3 ……… ±l
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There are 2l+1 possible values of ml ranging from +l through 0
to –l.
If l = 0, Lz = ml ħ (ml =2l+1) can have only single value of 0.
If l = 1 , Lz has three values -ħ , 0 and ħ .
If l = 2 : Lz has five values -2ħ, -ħ , 0 and ħ, 2ħ
Magnitude of L is given by :
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Magnetic quantum number for spin motions:
The spin motion of an electron around its own axis also
produces a magnetic field and z-component of the spin
angular momentum is given as
S z = ms ħ
here ms is magnetic quantum number for spin motion
and = ±1\2.
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Total Angular Momentum :
When the orbital angular momentum and spin angular
momentum are coupled, the total angular momentum
is the quantized angular momentum
where the total angular momentum quantum number
is
This gives a z-component of angular momentum
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• As long as external interactions are not
extremely strong, the total angular momentum
of an electron can be considered to be
conserved and j is said to be a "good quantum
number".
• This quantum number is used to characterize
the splitting of atomic energy levels, such as
the spin-orbit splitting which leads to the
sodium doublet.
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Since Jz , Sz and Lz are scalar quantities so
Jz = Lz ± Sz
mj = ml ± ms
since ml is an integer and ms is 1\2 so mj must be half integral.
Two ways to combine L and S are
This phenomenon is called
L-S coupling.
m j  (2 j  1) values
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spin multiplicity:
•It indicates the number of possible quantum states of a
system for a given spin quantum number S.
•These states are distinguished by the spin quantum
number Sz, which can take the values -S, -S+1, ..., S-1, S.
•Therefore, the multiplicity is 2S+1, where S is the
(number of singly occupied electrons multiplied by
electron Spin Quantum Number, ms ).
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A system with S=0 has exactly one possible state; it is
therefore in a singlet state. A system with S=1/2 is a
doublet; S=1 is a triplet, and so on.
The most important application is to electrons.
A single free electron has S=1/2; it is therefore
always in a doublet state.
Two electrons can pair up in a singlet or in a triplet
state. Normally the singlet is the ground state.
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• For example, oxygen has two singly occupied
electrons which could have spin multiplicity of 3. This
means that the spins could be up up or up down or
down down, total 3 possibilities.
• Using the formula, Spin Multiplicity of oxygen = 2S+1
= 2(2*1/2)+1 = 3, where S= two singly occupied
electrons*ms (ms always equal to ½).
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Coupling of orbital angular momenta
• For two electron case
• l1  orbital angular momentum for first electron
• l2  orbital angular momentum for second electron
• After coupling resultant angular momentum:
L = (l1 - l2) to (l1 +l2)
If l1 = 1 and l2 = 1 then L = 0, 1, 2
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Coupling of spin angular momenta:
• s1  spin angular momentum for first electron
• s2  spin angular momentum for second
After coupling resultant angular momentum:
S = (s1 - s2) to (s1 +s2)
• If s1 = 1/2 and s2 = 1/2 then S = 0, 1
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SPIN ORBIT COUPLING
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J = (L-S) TO (L+S)
J is the total inner quantum number.
If L=1 and S= 0 ,1
L=1, S=0  J = 1
L=1, S=1  J = 0, 1, 2 .
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STATE DESIGNATION: n MLJ
• n  Principle quantum number
• L  TOTAL ORBITAL QUANTUM NUMBER
• J  TOTAL INNER QUANTUM NUMBER
• M  Spin Multiplicity = (2S+1) , S  TOTAL SPIN
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• IF S = 0 M = 1  SINGLET STATE
• IF S = ½ M = 2  DOUBLET STATE
• IF S = 1 M = 3  TRIPLET STATE
• Atoms with one electron in the outermost shell
will give rise doublets in atomic spectrum
• Atoms with two electron in the outermost shell
will give rise singlet and triplet in atomic
spectrum.
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H atom in ground state:
n=1
l=0 means L=0 means S state
s=1/2
j=1/2
(l+s)
M=2
(spin multiplicity = 2S+1)
Designation or Term symbol:
1 2S1/2
n MLJ
H atom in first excited state:
n=2
l=0, 1 as l=0 to (n-1) or L= 0,1 so it has S and P states
1) l=0, s=1/2, j=1/2,
M=2
so 2 2S1/2
2) l=1, s=1/2, j=1/2, 3/2 M=2
so 2 2P1/2, 2 2P3/2
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2 2P3/2
2 2P1/2
2 2S1/2
EXCITED STATE
1 2S1/2
GROUND STATE
WHAT WILL BE THE DESIGNATION FOR n = 3 level
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He ground state 1s2
• n=1
• l1=0 l2 = 0
• L= 0
S state
• s1 =1/2 s2 =1/2 S = 0, 1
• FOR GROUND STATE ONLY S = 0 STATE IS POSSIBLE
a. L=0, S=0
b. L=0, S=1
• M= 1, J = 0
M= 3, J = 1
• DESIGNATION
11S0
13S1
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HELIUM EXCITED STATE
• ONE ELECTRON IN n=2 STATE AND THE OTHER IN
n=1 STATE
• 1s1 2s1
• 1s1 2p1
1. 1s1 2s1
l1=0, l2=0
L=0
s1=1/2, s2=1/2
S=0,1
a. L=0, S=0, J=0, M=1
DESIGNATION
b. L=0, S=1, J=1, M=3
DESIGNATION
21S0
23S1
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1s1 2p1
2. 1s1 2p1
l1=0, l2=1
L=1
s1=1/2, s2=1/2
S=0,1
a. L=1, S=0, J=1, M=1
DESIGNATION 21P1
Singlet
b. L=1, S=1, J=0,1, 2 M=3 DESIGNATION
23P0,
23P1,
23P2,
Triplet
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• Singlet –singlet transitions are more probable.
• Triplet -triplet transition are also more probable.
• Singlet-triplet and triplet-singlet transition are.
rarely probable.
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