Page 1 : Elements, , 156, , of Quantum, , Mechanics, Atomic and Molecule., , ular Spectra, , in its, , atomic, , ., , state. As fora., , ple,, when the, through sodium n, passed, is, present, n, been, s, o, u, r, c, e, have, solid, incandescent, lines, D,, the, place, D,,, an, of, 18sion, exactly at, if light from, dark lines a p p e a r, when light from a sa rce, two, obtained, atomic state,, of, lines are similarly, series of dark, vapour., iodine, A, spectra., through, wavelengths is passed, continuous, has precision and measurema, ents, since line spectrum, absorbant, , is, , excited, , o, , hand a single spectrum, Out of all types of spectra,, On the other, line, it is important., hence, level diagrams f, the, energy, can be made overit;, In this way, , signifies, , a, , particular, , energy, , state., , with the help of, can be plotted, different types of atoms, the energy, line spectra to plot, In details, the study of, , absorption lino, levels is given below, low, , emission, , and, , spectra., , in the next section., , IDEA ABOUT CONSTITUTION, , OF OTHER ATOMS, , 4.11. GENERAL, K, L, M, N SHELLS, , of their model of hydrogen atom for, Bohr and Stoner proposed extension, atom consists of a central positive nucleus, heavier elements. According to this, each, mass, of the atom,, + Ze and almost entire, containing the entire positive charge, which is equal to the number of, being atomic number. The number of electrons,, protons in the nucleus, which, in turn, is equal to the atomic number 2, revolve, around the nucleus in various stationary orbits or shells-thus giving the atom a, neutral character. These shells are called the K, L, M, N, .., shells, K being the, innermost i.e., nearest to nucleus. K orbit can contain a maximum of two electrons,, , the next L can contain eight while the next M eighteen. In fact, according to Paulis, exclusion principle, the maximum number of electrons in any shell is 2n where n, gives the order of the shell. The maximum number of electrons in any shell (or orbit), is as under:, Name of the shell+, Quantum number Number of electrons>, , K, 2, , L, , M, , N, , O, , 2, , 3, 18, , 4, , 5, , 42, , 50, , 8, , P, 72, , The exception to the above scheme is that the outermost orbit cannot contain, more than eight electrons and the inner orbit next to it, eighteen. In a given atom,, all orbits may not be complete. The electrons in the outermost orbit are called the, , valence electrons while the nucleus plus completed shells of electrons are termed as, the core., , 4.12., , SHORTCOMINGS OF BOHR'S THEORY, Inspite of the success achieved by Bohr's theory in, the, explaining, spectrum and giving valuable information about atomic, structure, it hydrogen, failed on, , certain, , points, , as, , given below, , (1) Bohr's theory predicts that spectral lines are, separate single lines, while, close observations with instruments of, high resolving power indicate that eac, spectral line is accompanied by a number of faint lines, i.e., the, individual spectra, lines are a combination of several, closely, packed, fine, lines., This, is called "tine, structure' of spectral lines and could not be, explained by Bohr's theory as sucn

Page 2 : Atomic Spectra, , 157, , because Bohr's theory assumes only one orbit for each, quantum number n whereas, the observed fine structure suggests that there, might be several orbits of slightly, different energies for any given quantum number n., (2) Bohr's theory could not explain the variation in, intensity of the spectral, lines of an element., (3) Bohr's theory could only explain the spectroscopic problem of one electron, atom such, , as, , hydrogen, hydrogen isotopes,, , ionised helium, , etc., , It failed in, , explaining the spectra of complex atoms., (4) Bohr's theory could not explain the anomalous Zeeman effect, Stark effect, etc., (5) It could not explain satisfactorily the distribution of electrons in atoms., , 4.13., , SOMMERFELD'S EXTENSION OF BOHR'S MODEL, , EXPLANATION OF FINE STRUCTURE, In an attempt to explain the fine, , structure', , of, spectral lines,, Sommerfeld extended Bohr's model by, assuming that electron could move in, , (r, 0) P, U, , elliptical orbits (in addition to Bohr's, circular orbits) with one focus at the, Q, nucleus of the atom (fig. 4.9)., , Following, , Sommerfeld,, , let, , m, , Ha-:9, , us, , consider the motion of an electron (mass, m, charge -, , nucleus, , b, , Nucleus, , e) in an elliptical orbit with, , (charge, , +, , Ze), , at, , one, , focus. The, , Fig. 4.9, , instantaneous position of electron may, be described by polar coordinates r and 0 where r is the radial distance of electron, from the nucleus and 0 is the azimuthal angle (angle between r and major axis of, ellipse). Thus electron has two degrees offreedom and according to Sommerfeld each, of these degrees of freedom must be quantized separately., , If Po and P, are the angular and radial momenta of the electron, then according, to Wilson-Sommerfeld quantization rules, Po de= kh, , .. (1), , P , dr = n,h, , .. (2), , where k = 1, 2, 3,.. is called angular or azimuthal quantum number and, n, = 0, 1, 2,...is the radial quantum number., , Since n, and kare both integers, we have, n, +k=n, where n = 1, 2, 3...is the principal quantum number., , If a and b are respectively the semi-major axis and semi-minor axis of ellipse,, then from the above condition, we get, , .(1)

Page 3 : Elements of Quantunn Mechanics, Atomic and Mol., , lecular Spectra, , 158, , Thus only those elliptical orbits are allowed for which the ratio of sem, ratio of, , is equal to the, axis to the semi-major axis, number., the principal quantum, , azimuthal, , quant, , amber to, , If we calculate the total energy of the electron in a quantized ellintin., , elliptical orbit, it, , comes out to be, , E=_, mZe, E, 8en h, , (2), , Substituting for H2, 2 = 1 and the value of Rydberg constant, me", , R, E, , We get, , 8Ech, Rhc, =, , (3), , n, , These equations (2) and (3) indicate that the total energy of the atom is inversely, proportional to the square of principal quantum number n i.e., E «, , They, , further show that the total energy obtained by assuming Sommerfeld elliptical, orbits is exactly the same as for the Bohr's circular orbits., Thus, the introduction of elliptical orbits in place of circular orbits has not, introduced any new energy states for the hydrogen atom. However, the electron can, now move in a number of orbits with the same energy. For a given value of, , n =k+ n,) k can take n different values, i.e., there aren orbits of diferent, eccentricities, , according to the condition=, , which can be occupied by the, , a, , electron. For a given value of n, the major axis 'a' of the ellipse is independent of the, azimuthal quantum number k, the minor axis 6 does depend upon k. Thus there, will be different ellipses for different value of k., Obviously, the azimuthal quantum number k defines the shape of the electron, orbit. For a given value of n, the azimutual quantum number kcan have n different, , values (k =1, 2, 3,.. n), Therefoe, according to quantum condition, , =#, for any value of n there are n, a, , sub-orbits, , possible which have different, eccentricities. Thus three types of orbits are, , n=3, k=3, , possible., All these orbits have been, dipicted in, fig. 4.10. It may be seen that the orbit having, the lowest value ofkis the most, elliptical., As already mentioned, this, multiplicity of, orbits of different eccentricities (due to, different values of b) does not add new energy, levels and hence fails to explain the fine, structure of spectral lines., , n=3, k=1, , Nucleus, , n=3, k=2, , Fig. 4.10

Page 4 : Atomic Spectra, , 159, , Relativistic Correction in Sommerfeld Model. In order to, explain the, observed fine structure of spectral lines, Sommerfield applied special theory of, relativity to account the variation of mass with velocity., Since the velocity of electron in innermost Bohr orbit (ground state, n = 1) is, 2.2 x 10° m/s, which is 0.007 of the, velocity of light (c 8x 10" m/s) and in elliptic, =, , orbit the velocity of electron varies, being maximum when it is nearest to nucleus, , (at perihelion); the variation of mass with velocity can not be ignored. The theory of, relativity shows that when velocity of a particle increases, its mass also increases., This effect makes the energy of the electron in a more elliptic orbit (i.e., of greater, , eccentricity) greater than that in a less, elliptic, orbit. Thus the different k orbits, with a given value of n have slightly different energies., According to Einstein's theory of relativity, the mass of electron moving with, velocity v (comparable to c) is given by, m, , mo, , =, , 1-, , c2, , where mo is the rest mass of electron., The kinetic energy of electron, T, , = (m - mo) c, , mo, , =, , mo, , = mge, , -1, , 2, Using this expression for kinetic energy, it was shown by Sommerfeld that total, n [Equation (2)] becomes, energy of atom for a given principal quantum number, E=_, m2e1, , ze, , 8ch n 4ch*n2\, Substituting R =, , me, , (Rydberg constant), , -- Rhe Z?L,2'a/1_3, , where a, , 2e,ch, , .(28), , i s called the fine structure constant., 137, , not only depends upon the, It may now be seen that the energy of the system, azimuthal quantum number k. Thus for a, principal number n, but also upon the, values of energy., definite value of n, we have n different orbits with different

Page 5 : Elements, , 160, Since the, , of Quantum Mechanics, Atomic and Mo, , lecular Spectra, , at the perihelion, it, in mass is greatest, perihelion of, electron orbit. The, , change, , precession of the, the major axis of, the time passes on i.e.,, the ellipse rotates as, the nucleus in the plane of the ellipse, the ellipse turns about, electron is not a simple ellipse,, and the resultant path of the, known as rosette (fig. 4.11)., but a complicated curve, : Relativistic, Structure, Explanation of Fine, Sommerfeld theory shows that an orbit, , results in, , a, , correction, applied to, , with, , different energy, , given value of n, is associated, levels (depending upon the value of k =1, 2, 3,... n),, The, transitions between which a r e possible., with, , a, , one, , the fine, , rules, interpret, spectral lines in the, , k=1, , selection, , of the, , structure, , hydrogen or other spectra., Now each level (or shell) consists of a number, of sublevels (or subshells) and transitions can, take place from one subshell to another. The, second, , orbit, , k-3, k-2, , to, , energy state, , subjected to appropriate, , Fig. 4.11, , the transition of, to another, when, , spectral lines which arise due, electrons from, , n, , (n = 2, , has, , two, , k=2, , k=2, , k=1, , sublevels, , corresponding to k =1, 2 (viz., 2, and 2) while the, sublevels, third orbit (n 3), has, three, corresponding to k =1, 2, 3 (viz., 31,3, and 33)., , Ha-Line, , =, , Fig. 4.12, , Thus six different transitions are possible from n = 3 to n = 2 which constitutes the, , Ha line (first member of Balmer series) and are given by 3 2g, 3, , 2, , 3222» 3221,3 22,3,2But the number of lines observed experimentally is less than that predicted, , theoretically. Hence a selection rule was introduced according to which only those, transitions are possible for which the azimuthal quantum number changes by 1, i.e., Ak =t1 Thus three transitions, , 33213, , 22 and 3,, , 2, , are forbidden (not allowed) and hence there are only three possible transitions in, case of H, lines which are:, , However,, , in, component lines., , 4.14., , 3322 322 and 3, > 2, with experimental observations, Ha, has, , accordance, , SHORTCOMINGS, , OF BOHR, , SOMMERFELD THEORY, , Bohr Sommerfeld atom model met, , following shortcomings:, , only with partial, , success and, , nau, , he, , (1) It is based on the rival theories. The motion of, electrons in orbit obeyeu the, law of classical mechanics while the, existence of stationary orbit and the, of, emitted, radiations are governed, frequencies, by quantum mechanics., The, model, could, (2), solve the, gen, sprectroscopic problems of simplest hyar, type atom, but failed in case of complex atoms, having large number or va, alence