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Using the Hund rules,find the basic terms atom whose paratially filled subshell contains (a) three p electrons, (b) four p electrons. |
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Answer» Solution :(a) when the partially filled shell contains three `p` electrons, the total spin `S` must equal `S=(1)/(2) or (3)/(2)`. The state `S=(3)/(2)` has maximum spin and is totally symmetric under exchange of spin labels. By Pauli's exclusion principle this implies that the angular part of the wave function a `p` electron is vector of `vec(r)_(1)` the total wavefunction of three `p` electrons is the totally antisymmetric combination of `vev(r )_(1),vec(r )_(2)`, and `vec(r )_(3)`. The only such combination is `vec(r )_(1).(vec(r )_(2)xxvec(r )_(3))=|(x_(1),x_(2),x_(3)),(y_(1),y_(2),y_(3)),(Ƶ_(1),Ƶ_(2),Ƶ_(3))|` This combination is a SCALAR and hence has `L = 0`. The spectral term of the ground state is then `.^(4)S_(3/2)` since `J = (3)/(2)` (b) We can think of four `P` electrons as consisting of a FULL `p` shell with two `p` holes. The state of maximum spin `S` is then `S = 1`. By Pauli's principle the orbital angular momentum part must be antisymmertic and can only have the form `vec(r )_(1)xxvec(r )_(2)` where `vec(r )_(1),vec(r )_(2)` are the coordinates of holes. Four `p` electrons can have `S = 0,1,2` but the `S = 2` state is totally symmetric. The corresponding angular wavefunction must be toatlly atisymmetric. But this is imposible: there is no quantity which is antisymetric in four vectors. Thus the maximum ALLOWED `S is S=1`. We can CONSTRUCT such a state by coupling the spins of electrons 1& 2 to S=1 and of electrons 3 & 4 to S=1 and then coupling the resultant spin states to `S=1`. Such a state is symmetric undr the exchange of spin of 1& 2nd 3 and `4` but antisymmetric under the simultaneous exchange of (1,2) & (3,4). the CONJUGATE angular wavelength must be antisymmetric under the exchange of `(1,2)` and under the exchange of `(1,2)` and `(3,4)`. (This is beacuse two exchange of electrons are involved.) The required angular wavefunction then has the from `(vec(r )_(1))xx(vec(r )_(3)xxvec(r )_(4))` and is a vector, `L=1`. Thus, using also the fact that the shell is more than half full, we find the spectral term `.^(3)P_(2)` `(J=L+S)`. |
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