SEMESTER-4th, Paper(404), Date:23/03/2023 Expression for Zeeman Sh Let the wavelength of the original spectral line be 2., and let B be the value of the magnetic field applied. The change in wavelength or Zeeman shift 8A. in the radiations emitted by the electrons moving in the plane perpendicular to the magnetic field B may be calculated as follows. Let v be the uniform linear speed and 69 the angular velocity an electron of mass m and charge € moving in a circular orbit of radius r my? Centripetal force experienced by electron, f= 2” = mra? r —) Let a magnetic field B be applied perpendicular to the plane of rotation of the electron (plane of the diagram) and directed toward the reader. The electron will experience an additional force Bev This force will be along the radius and directed towards the centre of the circle if the rotation is in the anticlockwise direction, as in fig. (a). This is the case with the electron A in figure. The additional force will be directed away from the centre if the rotation of the electron is in the clockwise direction as in Fig. 1.1(b). This is the case with the electron B, This additional force on the electron will produce a small change 8q in its angular velocity, Bev A 8 O (a) (b) Figl.1 Thus for the electron moving in the anticlockwise direction, by eqn. (1), F+Bey = mr(o+do)? mro?+Bero = mro?+2mr@$o+ mrba?. since v= ra. = mro’?+2mro $0, neglecting 5w? being two small a)~ Be = 2mdo Be o, 60 = 2 a 2m Here electron gets accelerated Similarly for the electron B, moving in the clockwise direction, F-Bey = mr(w+$u Be bo = -— 2m ‘The negative sign indicates that the electron gets retarded ‘The above two cases may be represented by the single, equation, Be & = £— — 2m If Ais the wavelength and the frequency of the original line, ae Ri lae la c bo = a From (2) and (3), 2 Zeeman shift 64 =. + 2& 4nme Q)Theory of Normal Zeeman Effect (based on Quantum Mechanics): Consider an atom in the energy state Eo having only orbital angular _ momentum in the energy state Eg. Spin of the electron is not taken into account. Let the atom be subjected to an external magnetic field B along the Z-axis. The interaction energy AE of the atom in the magnetic field B is given by AE = -p,.B=-p, Beos® — where L is the orbital angular momentum, 4; is the orbital magnetic moment and @ is the angle between pi, and B. But mw. = -5—L cos 8 . chen =" by Bm, where "= He In the presence of the field, the total energy of the system E. = Eot+ AE = Ey+pg Bm, (2) mi can have (2/ + 1) values, viz, - /,- 1+ 1, .. 0, (= 1), Correspondingly energy E also can have (2/ + 1) values. Thus, for 1=0, m;=0, E=Eo. I=1,m=0, +1, E=Eo, E, + AE or E,+ pyB 1=2, m=0, £1, £2, E=E, E,+p,B, E,+ 2u,B GB)Transitions occur between states having different / values. Selection rules relating to these transitions are, AJ = +1, Am, =0, +1. Am, = 0 corresponds to the component line polarized parallel to the magnetic field and Am, =+1 correspond to components polarized perpendicular to the field. For transitions betwene the levels / = 0 (s - state) and / = 1. (p-state) E = Ep, and E= E,+ 1B In terms of frequency, ch hy = hyo, and hve=hv, + 5—B ch i = =hv, + ——B ie, hv =hvo,and hv re Thus the frequencies of the three lines are, cB cB Y= Vor Yo" Gam’ Y°* arm No field eet Fig.2 The splitting of energy levels in a magnetic field for the = 0 and +1 states (ie, s and p) of an atom having only orbital angular momentum is shown in Fig. 2 a)Anomalous Zeeman Effect The splitting of spectral lines into more than three components in ordinary weak magnetic fields is called anomalous Zeeman effect. This anomalous nature of the pattern is due to the spin of the electrons. This phenomenon is more complex than the normal effect and cannot be explained by classical theory but can be explained on the basis of quantum mechanics, taking into consideration the spin of the electron also. Consider an atomic system specified by its total orbital angular momentum L, total spin angular momentum S and total angular momentum J. In the absence of the field let Eq be the energy of the system. Ep is (23 + 1) fold degenerate. When placed in a weak magnetic field B acting along the Z-axis, the interaction energy AE is given by, SE = -(u,+H,).B m= “AE = e = —(L+2S).B mm e*28) ; The field being weak, L - S coupling will be very strong. Using the principles of quantum mechanics, (L + 2S) may be expressed as, L+28 = g.J~ (1) Here g is a constant known as Lande’s g factor. (also known as Lande splitting factor). Taking the dot product of eqn.(1) with J J.(L+28) = gh —0 ))Simplifying the LHS of eqn. (2) J.(L+28) = J.(+8) = PtI.s J = L+s = J-8 LL = f+¥-25,8 P+s'-v J.s —@) 2 Using this value in ¢q.(2) tag? op? J.(L+28) = py itsok Using this value in eq.(2) a ygtap py EHS a 2 Putting J = /J+(T+1) A L=/L(L+i)A and S=JS(S+i) A (J +1)+S(S+1)-L(L+1)]? rasta pL GHDeSE+) Leen) EIS *» )) =g90+De 4 2@+1)+S(S+1)-L(L+1) 2(3+1) gis a dimensionless number. Its value depends on the state of the system. When the atom is placed in a weak external magnetic field B along the z-axis, the energy due to the field is small compared to the spin-orbit coupling. The energy due to the interaction of B with the system is given by g=l —) (6)AE = ~ Hy B= gJ.B= <8 3B cos —6) 6 is the angle between J and B, Also cos ont and J, = mh v AB SE pak ming 2m 2m fe tal: Be ie, AE= gH, Bm,| 57 = He ‘The energy state E is given by E = E,+A4B=E,+gp, Bm, m, can take (2) + 1) values. fey mj he (Hj t DN pernes pee Ms Hence there can be (2j + 1) equally spaced Zeeman sublevels. The inital and final levels are denoted by | and 2 respectively. Thus the energy states are represented as, Elo = El+g'p, Bm} EB o= Ej+g'p, Bm; By selection rules for mj, Amj=0, +1 But Am;= 0 is not allowed if Aj=0 -E = (E,-E2)+H, B(e'm) -8'm}) In terms of frequency B v = v, +H (g'm; -g*m7) Thus it can be concluded that the introduction of the concept of spin leads to a satisfactory explanation of anomalous Zeeman effect. ”)oo