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Zeeman Effect B • Let us have a magnetic dipole in a magnetic field B. • Torque on a magnetic dipole in the magnetic field of density B is • = B sin • Where is the angle between and B. B • The torque is maximum when the dipole is perpendicular to the field, and torque is minimum when dipole is parallel to it. • Potential energy of dipole inside magnetic field depends on the magnitude of of magnetic moment and the orientation of the moment with respect to the field. • So potential energy at any other orientation due to applied field B is 2 2 U m d B sin d B cos •When points in the same direction as B then Um=-B Means its minimum value. •Potential energy Um is zero at = 90° means when is perpendicular to B. • The magnetic moment of the orbital electron in a hydrogen atom depends on its angular momentum L. • When electron is moving in a circular orbit, for that current loop the magnetic moment is = IA, where I is the current and A is the area it encloses. • Again an electron that makes f rev/sec in a circular orbit of radius r is equivalent to a current of –ef and so its magnetic moment is = -efr2 …………….(1) • Linear speed v of the electron is v = r = 2fr • Angular momentum L = mvr = 2fmr2 …………(2) • Comparing (1) and (2) • = -efr2 = - (e/2m)L…………(3) • - (e/2m) is referred as gyromagnetic ratio. –ve sign means that is in opposite direction to that of L. L B -e • The magnetic potential energy of an atom in a magnetic field is • Um = -B cos = (e/2m)LB cos • We know that LZ = L cos • Means cos ml l l 1 • So e U m ml B ml B B 2m Here the quantity under bracket is called Bohr magneton • So in a magnetic field the energy of a perticular atomic state depends on the value of ml as well as on n. • A state of total quantum number n breaks up into several substates when the atom is in a magnetic field and their energies are slightly more or slightly less than the energy of the state in the absence of the flield. • this phenomenon leads to s splitting of individual spectral lines into separate lines when atoms radiate ina magnetic field. • The spacing of the lines depends on the magnitude of the field. • The splitting of spectral lines by magnetic field is called the Zeeman effect. • Changes in ml are restricted to Δml =0, ±1 • Normal Zeeman effect consists of the splitting of a spectral line of frequency 0 into three components whose frequencies are B e 1 0 B 0 B h 4m 2 0 B e 3 0 B 0 B h 4m Zeeman Effect: Splits m values • Orbital magnetic moment L interacts with an external magnetic field B and separates m=1 degenerate energy levels. m=0 m = –1 l=1 NO B Field l=0 B Field m=0 With magnetic field ml = 2 ml = 1 No field l=2 ml = 0 ml = -1 ml = -2 h0 ml = 1 ml = 0 ml = -1 l=1 Δml = -1 Δml = 0 Δml = +1 With magnetic field ml = 2 ml = 1 No field l=2 ml = 0 ml = -1 h0 ml = -2 eB h 0 2m h 0 h 0 eB 2m ml = 1 ml = 0 ml = -1 l=1 Δml = -1 Δml = 0 Δml = +1 • A sample of Certain element is placed in a 0.300-T magnetic field and suitably excited. How far apart are the Zeeman components of the 450-nm spectral line of this element? • Seperation of the Zeeman component is eB 4m c , d cd 2 2 eB c 4mc 2