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gL = 1; gS = – gE = 2

a)lowest level in 2D multiplet (input from slides):
J = 3/2, from (2s+1)L(J) atomic energy state formalism, it can be deduced that S = 1/2 & L = 2

Thus, X = (J(J+1) – S(S+1) + L(L+1))/2J(J+1) = (15/4 – 3/4 + 6)/(15/2) = 9/(15/2) = 6/5 = 1,2

&

Y = (J(J+1) + S(S+1) – L(L+1))/2J(J+1) = (15/4 + 3/4 – 6)/(15/2) = (4,5-6)/15/2 = -(3/2)/(15/2) = -3/15 = -0,2

and gJ = gL*X + gS*Y = 1*1,2 + 2*(-0.2) = 0,8

b)highest level in 2D multiplet (input from slides):
J = 5/2, S = 1/2 & L = 2

Thus, X = (J(J+1) – S(S+1) + L(L+1))/2J(J+1) = (35/4 – 3/4 + 6)/(35/2) = 14/(35/2) = 4/5 = 0,8

&

Y = (J(J+1) + S(S+1) – L(L+1))/2J(J+1) = (35/4 + 3/4 – 6)/35/2 = (7/2)/(35/2) = 1/5 = 0,2

and gJ = gL*X + gS*Y = 1*0,8 + 2*(0,4) = 1,2

c)ground state of the la atom when it is inside lanthanum metal

The electron configuration remains the same, thus L, S & J values remain the same. I don’t know if the g-factors would be influenced by adjustments of the electron cloud due to the neighboring atoms.

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