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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. 1 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). 2 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). 3 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. 4 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. 5 •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. 6 • 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 7 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 : 8 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. 9 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 10 • 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. 11 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 12 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 ). 13 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. 14 • 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). 15 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 16 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 17 SPIN ORBIT COUPLING • • • • • 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 . 18 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 19 • 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. 20 • • • • • • • • • • • • 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 21 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 22 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 23 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 24 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 25 • Singlet –singlet transitions are more probable. • Triplet -triplet transition are also more probable. • Singlet-triplet and triplet-singlet transition are. rarely probable. 26