Derivation of Selection Rules for Magnetic Dipole Transitions A. B. Sannigrahi Indian Institute of Technology, Kharagpur-721302, India The derivation of selection rules for electric dipole transitions is outltned in almost all undergraduate texthooks on quantum chemistry and spectrostopy. Recently. Sannigmhi and 1)ns1have renorted a novel derwatiun or 3 few sw h bnsic selection rules. However, it is surprising that the selection rules for magnetic dipole transitions, which form the basis of magnetic resonance spectroscopy, are often mentioned in elementary textbooks'-4 without any derivation. The purpose of the present article is to show that the relevant derivation is very simple indeed, and it is not necessary to refer to advanced textbooks for this purpose. According to the quantum mechanical theory of interaction of matter and electromagnetic radiation,Qhe transition probability between two stationary states, I$> and I$'>, of a system is proportional to 1 1 2, where V is the time-indeoendent art of the oerturbation term representing the energy 1,f interaction. A pure magnetic dipole transition can uccur u,hen the radiation field is so aenk that it can brtng about a change only in the spin orientation d a s y s t r m without affrstine its othrr enerm .. levels. In the ahsence uf an extrrnnl mngnetic field, the spin states nf a system having H fixed value of the spin angular momentum b u dilferinl: ~ tn spin orientations are degenerate. This degeneracy is lifted when an external magnetic field is applied. In spin resonance spectroscopy (NMR, ESR etc.), the system is subjected simultaneously to an external magnetic field and the oscillating magnetic field ot'the radiation actingat right angles toearhuthrr. Anoscillatine, magnetic field applied dong the same direction as the external field merely modulates the energy levels of the spin system and causes no energy absorption. In order to derive the selection rules for magnetic dipole transitions, let us first consider a nucleus with an intrinsic spin I. When the external magnetic field, H , acts along the z-direction, the Hamiltonian representing the interaction of the spinning nucleus and the magnetic field is given by4 ~~~
~~
~
A pair of adjoint operators is now defined by
Using the well-known commutation relations for angular momentum operators, it can be shown that
which indicate that I+ and i- act as step-up and step-down operator, respectively, with respect to the eigenvalues o f f , without affecting the eigenvalues of f2. Since the eigenvalues of f, are changed by f 1 by the application of f*, we can write
where N* are the normalization constants. T o derive the selection rule, we have to see under what conditions the integral, ($1 V I P ) is nonzero. Omitting the constant term - ~ N P N H 'in (eqn. (2)) we can write
v
Since f, = (f+ where g~ is the nuclear g-factor, PN is the nuclear Bohr magneton, and f, is the operator corresponding to the z component of the nuclear spin angular momentum. A weak electromagnetic radiation of frequency u is now applied to the system with its oscillating magnetic field, H', acting along the x-direction. Since Hz' = H'cos(2?rvt), and H,' =Hz' = 0, the perturbation Hamiltonian is given by
where,
+ f-)/2,
we have from eqn. (lo),
Because of the orthogonality of the st?tes characterized by different MI'S it is easily seen that ($1 VI $') does not vanish only when MI = MI' f 1. Thus, the selectioin rule is AM1 = f1. The selection rule for the absorption and emission of energy by a spinning electron can be derived in a similar manner. In this case, I is to he replaced by s, MI by m,,- g~ by 2.0, and PN by 8 , the Bobr magneton. The selection rule is Am, = fl.
According to the quantum mechanical theory of angular momentum,2~3a particular state of the nucleus with a spin I may he represented by ] I , MI>, where MI is a quantum number which determines the z-component of the spin angular momentum. For a given value of I , MI can take on values I , I - 1, I - 2, . . . , - I. The state II, MI> is an eigenfunction of the spin operators f2 andI,, and of &. Using atomic units the corresponding eigenvalue equations can be written as
' Sannigrahi, A. B. and Das, R., J. CHEM.EDUC., 57,786 (1980).
Pilar. F. L., "Elementary Quantum Chemistry." McGraw-Hill Book Company. New York, 1968. Anderson, J. M.. "Introduction to Quantum Chemistry," W. A. Benjamin Inc.. New York. 1969. Chang. R.. "Basic Principles of Spectroscopy:' McGrawHill Book Company. New York, 1971. Volume 59
Number 10 October 1982
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