{"id":51,"date":"2018-11-14T05:04:21","date_gmt":"2018-11-14T05:04:21","guid":{"rendered":"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=51"},"modified":"2019-04-30T06:49:34","modified_gmt":"2019-04-30T06:49:34","slug":"coupling-schemes-i","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/chapter\/coupling-schemes-i\/","title":{"rendered":"Coupling Schemes-I"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/BemRURdxy1k\" target=\"_blank\" rel=\"noopener\"><img src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a>\r\n<\/span><\/div>\r\n<div>\r\n\r\n<strong>Contents:<\/strong>\r\n\r\n&nbsp;\r\n\r\n1.\u00a0\u00a0\u00a0\u00a0 Angular coupling momentum\r\n\r\n2.\u00a0\u00a0\u00a0\u00a0 LS coupling\r\n\r\n3.\u00a0\u00a0\u00a0\u00a0 JJ coupling\r\n\r\n4.\u00a0\u00a0\u00a0\u00a0 Spin -spin coupling\r\n\r\n&nbsp;\r\n\r\nThe students will be able to learn about the various coupling schemes.\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>I.\u00a0<\/strong><strong>Angular momentum coupling<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong>As the name suggests, constructing eigen states of total angular momentum out of eigen-states of separate angular momenta is called angular momentum coupling. The orbit and spin of a single particle can interact through <\/strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Spin%E2%80%93orbit_interaction\"><strong>spin\u2013orbit interaction, <\/strong><\/a><strong>where the complete physical picture includes spin-orbit coupling.<\/strong><\/p>\r\n&nbsp;\r\n\r\nOr\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Two charged particles, each with a well-defined angular momentum, may interact by <a href=\"https:\/\/en.wikipedia.org\/wiki\/Electrostatic#Coulomb.27s_law\">Coulomb forces, <\/a>where coupling of the two one-particle angular momenta to a total angular momentum is a useful step in the solution of the two-particle <a href=\"https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation\">Schr\u00f6dinger Equation.<\/a><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">In both the cases the separate angular momenta are no longer constants of motion, but the sum of the two angular momenta usually is still.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Angular momentum coupling in atoms is of importance in atomic spectroscopy while angular momentum coupling of electron spins is of importance in quantum chemistry. Likewise, angular momentum coupling in\u00a0<\/span><span style=\"font-size: 1em\">the nuclear shell model is ubiquitous.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">In astronomy, spin-orbit coupling reflects the general law of conservation of angular momentum that also holds for celestial systems.\u00a0<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The direction of the <\/span>angular momentum vector <span style=\"text-align: initial;font-size: 1em\">is neglected in simple cases. The spin-orbit coupling is the ratio between the frequency with which a <\/span>planet <span style=\"text-align: initial;font-size: 1em\">or other <\/span>celestial body <span style=\"text-align: initial;font-size: 1em\">spins about its own axis to that with which it orbits another body. This is commonly known as orbital resonance.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">1.\u00a0<\/strong><strong style=\"text-align: initial;font-size: 1em\">Angular momentum conservation<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The total angular momentum of a system has a constant magnitude and direction if the system is subjected to no external <\/span><strong>torque<\/strong>. <strong>Angular momentum<\/strong> <span style=\"text-align: initial;font-size: 1em\">is considered as a property of a physical system:<\/span><\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The system experiences a spherically symmetric potential field <\/span><\/li>\r\n \t<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The system moves in isotropic space<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In both the cases the angular momentum operator commutes with the Hamiltonian of the system. Thus the angular momentum and the energy (eigenvalue of the Hamiltonian) can be measured at the same time by using Heisenberg's Uncertainty principle.<\/p>\r\n&nbsp;\r\n\r\nAn example of the first situation is :\r\n<p style=\"text-align: justify\">an atom whose electrons only experience the Coulomb force of its atomic nucleus. If the electron-electron interaction (and other small interactions such as spin orbit coupling) is ignored, the <em>orbital angular momentum<\/em> <strong>l<\/strong> of each electron commutes with the total Hamiltonian. In this model the atomic Hamiltonian is a sum of kinetic energies of the electrons and the spherically symmetric electron-nucleus interactions. The individual electron angular momenta <strong>l<\/strong><em>i<\/em> commute with this Hamiltonian. That is, they are conserved properties of this approximate model of the atom.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">An example of the second situation is a rigid rotator moving in field-free space. A rigid rotor has a well-defined, time-independent, angular momentum.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">These two situations originate in classical mechanics. The third kind of conserved angular momentum, associated with spin, does not have a classical counterpart. However, all rules of angular momentum coupling apply to spin as well.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">In general the conservation of angular momentum implies full rotational symmetry and, conversely, spherical symmetry implies conservation of angular momentum. If two or more physical systems have conserved angular momenta, it can be useful to\u00a0<span style=\"text-align: initial;font-size: 1em\">combine these momenta to a total angular momentum of the combined system\u2014a conserved property of the total system. The building of eigen-states of the total conserved angular momentum from the angular momentum eigen-states of the individual subsystems is referred to as <\/span><em style=\"text-align: initial;font-size: 1em\">angular momentum coupling<\/em><span style=\"text-align: initial;font-size: 1em\">.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Application of angular momentum coupling is useful when there is an interaction between subsystems that, without interaction, would have conserved angular momentum. By the very interaction the spherical symmetry of the subsystems is broken, but the angular momentum of the total system remains a constant of motion. Use of the latter fact is helpful in the solution of the Schr\u00f6dinger equation.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Examples<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">As an example we consider two electrons, 1 and 2, in an atom (Helium atom). If there is no electron-electron interaction, but only electron-nucleus interaction, the two electrons can be rotated around the nucleus independently of each other; nothing happens to their energy. Both operators, <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">1 and <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">2, are conserved. However, if we switch on the electron-electron interaction that depends on the distance <\/span><em style=\"text-align: initial;font-size: 1em\">d<\/em><span style=\"text-align: initial;font-size: 1em\"> (1,2) between the electrons, then only a simultaneous and equal rotation of the two electrons will leave <\/span><em style=\"text-align: initial;font-size: 1em\">d<\/em><span style=\"text-align: initial;font-size: 1em\">(1,2) invariant. In such a case neither <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">1 nor <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">2 is a constant of motion in general, but the total orbital angular momentum is <\/span><strong style=\"text-align: initial;font-size: 1em\">L<\/strong><span style=\"text-align: initial;font-size: 1em\"> = <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">1 + <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">2 . Given the eigenstates of <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">1 and <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">2, the construction of eigenstates of <\/span><strong style=\"text-align: initial;font-size: 1em\">L<\/strong><span style=\"text-align: initial;font-size: 1em\"> (that still is conserved) is the <\/span><em style=\"text-align: initial;font-size: 1em\">coupling of the angular momenta of electrons 1 and 2<\/em><span style=\"text-align: initial;font-size: 1em\">.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The total orbital angular momentum quantum number <\/span><strong style=\"text-align: initial;font-size: 1em\">L<\/strong><span style=\"text-align: initial;font-size: 1em\"> is restricted to integer values. In quantum mechanics, coupling also exists between angular momenta belonging to different Hilbert spaces of a single object, e.g. its spin and its orbital angular momentum. If the spin has half-integer values, such as 1\/2 for an electron, then the total (orbital plus spin) angular momentum will also be restricted to half-integer values.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Further, the quantum states of composed systems (i.e. like two <\/span>Hydrogen <span style=\"text-align: initial;font-size: 1em\">atoms or two electrons) in basis sets which are made of tensor products of quantum states that in turn describe the subsystems individually, can be extended. We assume that the states of the subsystems can be chosen as eigenstates of their angular momentum operators (and of their component along any arbitrary <\/span><em style=\"text-align: initial;font-size: 1em\">z<\/em><span style=\"text-align: initial;font-size: 1em\"> axis).<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The subsystems are therefore correctly described by a set of <\/span><em style=\"text-align: initial;font-size: 1em\">\u2113<\/em><span style=\"text-align: initial;font-size: 1em\">, <\/span><em style=\"text-align: initial;font-size: 1em\">m<\/em><span style=\"text-align: initial;font-size: 1em\"> quantum numbers, then there is interaction among the subsystems, the total Hamiltonian contains terms that do not commute with the angular operators acting on the subsystems only. However, these terms <\/span><em style=\"text-align: initial;font-size: 1em\">do<\/em><span style=\"text-align: initial;font-size: 1em\"> commute with the <\/span><em style=\"text-align: initial;font-size: 1em\">total<\/em><span style=\"text-align: initial;font-size: 1em\"> angular momentum operator. Sometimes one refers to the non-commuting interaction terms in the Hamiltonian as <\/span><em style=\"text-align: initial;font-size: 1em\">angular momentum coupling terms<\/em><span style=\"text-align: initial;font-size: 1em\">, because they necessitate the angular momentum coupling.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Spin-orbit coupling<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The behavior of atoms and smaller particles is well described by the theory of quantum mechanics, in which each particle has an intrinsic angular momentum called spin and specific configurations e.g. electrons in an atom are described by a set of quantum numbers. Collections of particles also have angular momenta and corresponding quantum numbers, and under different circumstances the angular momenta of the parts couple in different ways to form the angular momentum of the whole. Angular momentum coupling is a category including some of the ways that subatomic particles can interact with each other.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In atomic physics, spin-orbit coupling, also known as <\/span><strong style=\"text-align: initial;font-size: 1em\">spin-pairing<\/strong><span style=\"text-align: initial;font-size: 1em\">, describes a weak magnetic interaction, or coupling, of the particle spin and the orbital motion of this particle, e.g. the electron spin and its motion around an atomic nucleus . One of its effects is to separate the energy of internal states of the atom, e.g. spin-aligned and spin-antialigned that would otherwise be identical in energy.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>L-S coupling and J-J coupling<\/strong>\r\n\r\n&nbsp;\r\n\r\nHamiltonian of the system depends on\r\n\r\n&nbsp;\r\n\r\n1.\u00a0\u00a0\u00a0\u00a0 Kinetic energy of the electrons\r\n\r\n2.\u00a0\u00a0\u00a0\u00a0 Potential energy of the electrons due to the interaction between the nucleus and the electrons\r\n\r\nFor these two factors Hamiltonian act as unperturbed Hamiltonian\r\n\r\n3.\u00a0\u00a0\u00a0\u00a0 Energy due to interaction between the electrons\r\n\r\n4.\u00a0\u00a0\u00a0\u00a0 Spin-spin correlation energy\r\n\r\n5.\u00a0\u00a0\u00a0\u00a0 Spin-orbit interaction\r\n\r\nFor these three factors Hamiltonian act as perturbed Hamiltonian\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">For <strong>lighter elements<\/strong> the spin orbit interaction energy is much less as compared to the interaction energy between electrons and spin-spin correlation energy and there will be L-S coupling for lighter element.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">For <strong>heavier elements<\/strong> the spin orbit interaction energy is much greater than interaction energy between electrons and spin- spin correlation energy and there will be j-j coupling for heavier elements.<\/p>\r\n&nbsp;\r\n\r\n<strong>L-S coupling<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In L-S coupling it is assumed that, when several electrons are present in an atom, each with a definite value of orbital angular momentum li* and spin angular momentum si* such that li*= li(li+1)h\/2\u03c0 and si*= si(si+1) h\/2\u03c0.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Due to the domination of interaction energy between electrons the orbital angular momentum of indivisible electrons will intract with each other and the total orbital angular momentum is given by L = L(L+1) h\/2\u03c0, where L varies from | l1+l2+l3\u2026\u2026..|minimum to | l1+l2+l3\u2026\u2026.|maximum the states with different values of L have fairly large energy difference, the state of largest L of being lowest energy. The different levels are designated as S, P, D, F, G\u2026\u2026..according as L=0,1,2,3,4,5\u2026\u2026. Thus:<\/p>\r\n&nbsp;\r\n\r\nFor 3p 4d electrons: l1=1, l2=2.\r\n\r\n\u2234\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 L = |l1-l2|, | l1-l2|+1, \u2026\u2026( l1+l2)\r\n\r\n=\u00a0 1, 2, 3 (P, D, F states).\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">For 3p, 4d, 5f electrons: l1=1, l2=2, l3=3. The electrons which are tightly bound to the nucleus will interact first.<\/p>\r\n\u2234\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 L = |l1-l2|, | l1-l2|+1, \u2026\u2026( l1+l2)\r\n\r\n=\u00a0 1, 2, 3 (P, D, F states).\r\n\r\n&nbsp;\r\n\r\nNow L combining with l3=3\r\n\r\n\u2234\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 L = 0, 1, 2, 3, 4, 5, 6(S, P, D, F, G, H, I states)\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">2.\u00a0\u00a0 Due to the domination of spin-spin correlation energy the spin angular momentum of individual electron will interact with each other and the total spin angular momentum\u00a0 \u00a0is given\u00a0 by\u00a0 S=\u221aS(S+1)h2\u03c0<\/p>\r\n&nbsp;\r\n\r\nThe quantum number S can take the values :\r\n<p style=\"text-align: justify\">S = |s1+s2+s3+\u2026\u2026\u2026|min to | s1+s2+s3+\u2026\u2026\u2026|max. . The states with different values of S have considerable energy difference, the state of highest S being of lowest energy. The different levels are designated by their multiplicity, (2S+1). Thus:<\/p>\r\n<p style=\"text-align: justify\">For one electron: S= s = \u00bd.<\/p>\r\n\u2234\u00a0 (2S+1) = 2 (doublet level).\r\n\r\n&nbsp;\r\n\r\nFor two electrons: s1 = \u00bd, s2 = \u00bd.\r\n\r\n\u2234\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 S = |s1 - s2|, |s1 - s2| + 1, \u2026\u2026\u2026.( s1+s2) = 0, 1.\r\n\r\n\u2234\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (2S+1) = 2 (singlet and triplet levels)\r\n\r\n&nbsp;\r\n\r\nFor three electrons: s1 = \u00bd, s2 = \u00bd, s3 = \u00bd.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">To combine three spins, we first combine two of them to obtain S\u2019 = 0, 1, and then combine the third S3= \u00bd to each of them. Thus, if we couple s3 = \u00bd to s\u2019 =0; we get s= \u00bd; and if we couple s3 = \u00bd to s\u2019 =1, we get s= 1\/2, 3\/2. Thus for three electrons, we get S =1\/2, 1\/2, 3\/2 (two sets of doublets, and one set of quartets).<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The followings branching diagram illustrates the possible total spin quantum numbers which can be obtained by combining several independent electron spins.<\/p>\r\n<img class=\"size-full wp-image-54 aligncenter\" src=\"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/93\/2018\/11\/1-18.png\" alt=\"\" width=\"555\" height=\"236\" \/>\r\n\r\n<\/div>\r\n<div>\r\n<p style=\"text-align: justify\"><img class=\"size-full wp-image-55 aligncenter\" src=\"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/93\/2018\/11\/1-19.png\" alt=\"\" width=\"692\" height=\"193\" \/><span style=\"font-size: 1em;text-align: initial\">In the general case of N electrons, the possible values of S are <\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">S=0,1,2,\u2026\u2026\u2026..N\/2forevenN.<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">S= 1\/2, 3\/2, 5\/2\u2026\u2026.N\/2 for odd N.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The dominant spin-spin correlation and the residual electrostatic interaction having been taken into accounts as a first peturbation, the smaller spin- orbit interaction is included in L-S coupling as an additional perturbation.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">As a result of smaller spin \u2013 orbit magnetic interaction, the resultant orbital angular momentum vector L and the resultant spin angular momentum vector S are less strongly coupled with each other to form a total angular momentum vector J of the atom:<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center\">J = L + S<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">the magnitude of J, L and S remaining constant. The magnitude of J is \u221aJ(J+1) h\/2\u03c0, where the quantum number J takes the value:<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center\">J=\u00a0 |L-S|, |L-S|+1, \u2026\u2026\u2026..(L-S).<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">J is integral or half \u2013 integral according as S is integral or half integral. The number of J values is (2S+1) when L &gt; S, or (2L+1) when S &gt; L. This mean that due to spin orbit magnet interaction; a level characterized by given values of Land S is further broken up into comparative closer (2S+1) or (2L+1) levels, each characterized by a different J value. The group of these J-levels forms a fine structure multiplet\u2019. The relative spacing of the fine structure levels with in a multiplet is governed by Lande interval rule.<\/p>\r\n&nbsp;\r\n\r\nLande Interval Rule\r\n\r\n&nbsp;\r\n\r\nUnder L-S coupling, the spin-orbit interaction energy is of the form\r\n\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">\u2206Es,l = a(L.S),<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Where a is an interaction constant. Let us write<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">J = L+S.<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Taking the scalar self product we have<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">J.J\u00a0 = L.L + S.S + 2L.S.<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">| J |2 = | L |2 + | S |2 +2L.S.<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">\u2234\u00a0\u00a0 \u2206Es,l = a\/2[J(J+1) \u2013 L(L+1)- S(S+1)]h2\/4\u03c02,<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Because J = J (J+1) h\/2\u03c0, and so on. We can write it as \u2206Es,l = A [J(J+1) \u2013 L(L+1)- S(S+1)] , Where A is another constant.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The various fine-structure levels of a Russell-Saunders multiplet have the same values of L and S, and differ only in the value of J. Hence, the energy difference between two fine-structure levels corresponding to J and J+1 is<\/p>\r\n<p style=\"text-align: justify\">Ej+1 - Ej = A [(J+1) (J+2)-J (J+1)]<\/p>\r\n=2A (J+1).\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Thus, the energy interval (spacing) between consecutive levels J and J+1 of a fine-structure multiplet is proportional to J+1, that is, to the larger of the two J-values involved. This is \u2018Lande interval rule\u2019.<\/p>\r\n&nbsp;\r\n\r\nNORMAL AND INVERTED MULTIPLETS\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Within a given multiplet, usually the level with smallest J value lies lowest. Such a multiplet is a \u2018Normal\u2019 multiplet. There are, however, multiplets in which the largest J level lies lowest. Such multiplets are \u2018inverted\u2019 multiplets.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">It is easy to understand how smallest J level lies lowest. The magnet field B produced by the orbital motion of the electron in the electric field of the nucleus is in the same direction as the orbital angular momentum L. In this field the most stable state, that is, the state of the lowest energy must be one in which the spin magnetic moment \u03bcs of\u00a0<span style=\"font-size: 1em;text-align: initial\">the electron lies up in the direction of B. We know that \u03bcs is directed opposite to S because the electron is negatively charged.<\/span><\/p>\r\n\r\n<\/div>\r\n<img class=\"size-full wp-image-56 aligncenter\" src=\"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/93\/2018\/11\/1-20.png\" alt=\"\" width=\"554\" height=\"290\" \/>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Thus, in the lowest energy state, L and S are in opposite directions and so the value of J is lowest. When L and S are in the same direction, which corresponds to highest J, then \u03bcs is opposite to B and the state is least stable.<\/p>\r\n&nbsp;\r\n\r\nThe inverted multiplets arise due to some perturbing influences.\r\n\r\n&nbsp;\r\n\r\n<strong>DETERMINATION OF SPECTRAL TERMS FOR L-S COUPLING<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>(1)\u00a0\u00a0 <\/strong><strong>Atoms with one optical electron: <\/strong>For hydrogen-like atom, the ground state configuration is\r\n<p style=\"text-align: justify\"><strong>\u00a0<\/strong><\/p>\r\n<strong>1s.<\/strong>\r\n\r\n&nbsp;\r\n\r\nFor this, we have\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 s = \u00bd; l = 0,\r\n\r\nSo that\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 S = s = \u00bd; multiplicity 2S+1 = 2,\r\n\r\nL = l = 0(S-state),\r\n\r\n&nbsp;\r\n\r\nAnd\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 J = |L-S|, \u2026\u2026\u2026\u2026\u2026\u2026\u2026 (L+S) = \u00bd.\r\n\r\n&nbsp;\r\n\r\nThus, the ground state term of a hydrogen-like atom is\r\n\r\n&nbsp;\r\n\r\n2<sub>S1<\/sub>\/2.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The excited state configuration and the corresponding terms for a hydrogen-like atom would be<\/p>\r\n<p style=\"text-align: justify\">2s, 3s, 4s, \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..<sup>2<\/sup>S1\/2<\/p>\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">2p,\u00a0\u00a0 3p, 4p, \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...<sup>2<\/sup>p1\/2, <sup>2<\/sup>p3\/2.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\nThe ground state configuration for all alkali atoms is <sup>2<\/sup>S1\/2.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">(2)\u00a0\u00a0 Atoms with two or more Non-equivalent optical electrons: 4p 4d For this, we have<\/p>\r\n&nbsp;\r\n\r\ns1 = \u00bd, s2 = \u00bd, l1 = 1, l2 = 2.\r\n\r\n&nbsp;\r\n\r\nThe possible values of S and L are :\r\n\r\n&nbsp;\r\n\r\n<strong>S <\/strong>= |<strong>s<\/strong><strong>1<\/strong><strong> - s<\/strong><strong>2<\/strong><strong>|, <\/strong>|<strong>s<\/strong><strong>1<\/strong><strong> - s<\/strong><strong>2<\/strong><strong>| + 1, \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026...( s<\/strong><strong>1<\/strong><strong>+s<\/strong><strong>2<\/strong><strong>)<\/strong>\r\n\r\n&nbsp;\r\n\r\n= 0, 1; multiplicity (2S+1) = 1,3\r\n\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">And<\/span>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">L = |l<\/strong><strong style=\"text-align: initial;font-size: 1em\">1<\/strong><strong style=\"text-align: initial;font-size: 1em\">-l<\/strong><strong style=\"text-align: initial;font-size: 1em\">2<\/strong><strong style=\"text-align: initial;font-size: 1em\">|, | l<\/strong><strong style=\"text-align: initial;font-size: 1em\">1<\/strong><strong style=\"text-align: initial;font-size: 1em\">-l<\/strong><strong style=\"text-align: initial;font-size: 1em\">2<\/strong><strong style=\"text-align: initial;font-size: 1em\">|+1, \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026(l<\/strong><strong style=\"text-align: initial;font-size: 1em\">1<\/strong><strong style=\"text-align: initial;font-size: 1em\">+l<\/strong><strong style=\"text-align: initial;font-size: 1em\">2<\/strong><strong style=\"text-align: initial;font-size: 1em\">).<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">= 1, 2, 3, (P, D, F, state).<\/span>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Thus, we have in all six terms, three singlet terms and three triplet terms. These terms can be written as<\/span>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\"><sup>1<\/sup>P, <sup>1<\/sup>D, <sup>1<\/sup>F, <sup>3<\/sup>P, <sup>3<\/sup>D, <sup>3<\/sup>F.<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">To take into account spin-orbit interaction, let us combine L and S to form J.<\/p>\r\n<img class=\"alignnone size-full wp-image-57\" src=\"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/93\/2018\/11\/1-21.png\" alt=\"\" width=\"553\" height=\"302\" \/>\r\n\r\n<\/div>\r\n<p style=\"text-align: justify\">Thus, a single degenerate level of configuration 4p 4d is splitted into 12 levels as shown in fig.<\/p>\r\n&nbsp;\r\n\r\n<img class=\"size-full wp-image-58 aligncenter\" src=\"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/93\/2018\/11\/1-22.png\" alt=\"\" width=\"650\" height=\"472\" \/>\r\n<p style=\"text-align: justify\">unperturbed level spin-spin correlation energy + residual electrostatic energy + spin-orbit\u00a0\u00a0\u00a0\u00a0\u00a0 magnetic energy<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Coupling Schemes-I<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/BemRURdxy1k\" target=\"_blank\" rel=\"noopener\"><img class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/BemRURdxy1k\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a><br \/>\n<\/span><\/div>\n<div>\n<p><strong>Contents:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>1.\u00a0\u00a0\u00a0\u00a0 Angular coupling momentum<\/p>\n<p>2.\u00a0\u00a0\u00a0\u00a0 LS coupling<\/p>\n<p>3.\u00a0\u00a0\u00a0\u00a0 JJ coupling<\/p>\n<p>4.\u00a0\u00a0\u00a0\u00a0 Spin -spin coupling<\/p>\n<p>&nbsp;<\/p>\n<p>The students will be able to learn about the various coupling schemes.<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>I.\u00a0<\/strong><strong>Angular momentum coupling<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong>As the name suggests, constructing eigen states of total angular momentum out of eigen-states of separate angular momenta is called angular momentum coupling. The orbit and spin of a single particle can interact through <\/strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Spin%E2%80%93orbit_interaction\"><strong>spin\u2013orbit interaction, <\/strong><\/a><strong>where the complete physical picture includes spin-orbit coupling.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Or<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Two charged particles, each with a well-defined angular momentum, may interact by <a href=\"https:\/\/en.wikipedia.org\/wiki\/Electrostatic#Coulomb.27s_law\">Coulomb forces, <\/a>where coupling of the two one-particle angular momenta to a total angular momentum is a useful step in the solution of the two-particle <a href=\"https:\/\/en.wikipedia.org\/wiki\/Schr%C3%B6dinger_equation\">Schr\u00f6dinger Equation.<\/a><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In both the cases the separate angular momenta are no longer constants of motion, but the sum of the two angular momenta usually is still.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Angular momentum coupling in atoms is of importance in atomic spectroscopy while angular momentum coupling of electron spins is of importance in quantum chemistry. Likewise, angular momentum coupling in\u00a0<\/span><span style=\"font-size: 1em\">the nuclear shell model is ubiquitous.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">In astronomy, spin-orbit coupling reflects the general law of conservation of angular momentum that also holds for celestial systems.\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The direction of the <\/span>angular momentum vector <span style=\"text-align: initial;font-size: 1em\">is neglected in simple cases. The spin-orbit coupling is the ratio between the frequency with which a <\/span>planet <span style=\"text-align: initial;font-size: 1em\">or other <\/span>celestial body <span style=\"text-align: initial;font-size: 1em\">spins about its own axis to that with which it orbits another body. This is commonly known as orbital resonance.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">1.\u00a0<\/strong><strong style=\"text-align: initial;font-size: 1em\">Angular momentum conservation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The total angular momentum of a system has a constant magnitude and direction if the system is subjected to no external <\/span><strong>torque<\/strong>. <strong>Angular momentum<\/strong> <span style=\"text-align: initial;font-size: 1em\">is considered as a property of a physical system:<\/span><\/p>\n<ul>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The system experiences a spherically symmetric potential field <\/span><\/li>\n<li style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The system moves in isotropic space<\/span><\/li>\n<\/ul>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In both the cases the angular momentum operator commutes with the Hamiltonian of the system. Thus the angular momentum and the energy (eigenvalue of the Hamiltonian) can be measured at the same time by using Heisenberg&#8217;s Uncertainty principle.<\/p>\n<p>&nbsp;<\/p>\n<p>An example of the first situation is :<\/p>\n<p style=\"text-align: justify\">an atom whose electrons only experience the Coulomb force of its atomic nucleus. If the electron-electron interaction (and other small interactions such as spin orbit coupling) is ignored, the <em>orbital angular momentum<\/em> <strong>l<\/strong> of each electron commutes with the total Hamiltonian. In this model the atomic Hamiltonian is a sum of kinetic energies of the electrons and the spherically symmetric electron-nucleus interactions. The individual electron angular momenta <strong>l<\/strong><em>i<\/em> commute with this Hamiltonian. That is, they are conserved properties of this approximate model of the atom.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">An example of the second situation is a rigid rotator moving in field-free space. A rigid rotor has a well-defined, time-independent, angular momentum.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">These two situations originate in classical mechanics. The third kind of conserved angular momentum, associated with spin, does not have a classical counterpart. However, all rules of angular momentum coupling apply to spin as well.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In general the conservation of angular momentum implies full rotational symmetry and, conversely, spherical symmetry implies conservation of angular momentum. If two or more physical systems have conserved angular momenta, it can be useful to\u00a0<span style=\"text-align: initial;font-size: 1em\">combine these momenta to a total angular momentum of the combined system\u2014a conserved property of the total system. The building of eigen-states of the total conserved angular momentum from the angular momentum eigen-states of the individual subsystems is referred to as <\/span><em style=\"text-align: initial;font-size: 1em\">angular momentum coupling<\/em><span style=\"text-align: initial;font-size: 1em\">.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Application of angular momentum coupling is useful when there is an interaction between subsystems that, without interaction, would have conserved angular momentum. By the very interaction the spherical symmetry of the subsystems is broken, but the angular momentum of the total system remains a constant of motion. Use of the latter fact is helpful in the solution of the Schr\u00f6dinger equation.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Examples<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">As an example we consider two electrons, 1 and 2, in an atom (Helium atom). If there is no electron-electron interaction, but only electron-nucleus interaction, the two electrons can be rotated around the nucleus independently of each other; nothing happens to their energy. Both operators, <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">1 and <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">2, are conserved. However, if we switch on the electron-electron interaction that depends on the distance <\/span><em style=\"text-align: initial;font-size: 1em\">d<\/em><span style=\"text-align: initial;font-size: 1em\"> (1,2) between the electrons, then only a simultaneous and equal rotation of the two electrons will leave <\/span><em style=\"text-align: initial;font-size: 1em\">d<\/em><span style=\"text-align: initial;font-size: 1em\">(1,2) invariant. In such a case neither <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">1 nor <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">2 is a constant of motion in general, but the total orbital angular momentum is <\/span><strong style=\"text-align: initial;font-size: 1em\">L<\/strong><span style=\"text-align: initial;font-size: 1em\"> = <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">1 + <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">2 . Given the eigenstates of <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">1 and <\/span><strong style=\"text-align: initial;font-size: 1em\">l<\/strong><span style=\"text-align: initial;font-size: 1em\">2, the construction of eigenstates of <\/span><strong style=\"text-align: initial;font-size: 1em\">L<\/strong><span style=\"text-align: initial;font-size: 1em\"> (that still is conserved) is the <\/span><em style=\"text-align: initial;font-size: 1em\">coupling of the angular momenta of electrons 1 and 2<\/em><span style=\"text-align: initial;font-size: 1em\">.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The total orbital angular momentum quantum number <\/span><strong style=\"text-align: initial;font-size: 1em\">L<\/strong><span style=\"text-align: initial;font-size: 1em\"> is restricted to integer values. In quantum mechanics, coupling also exists between angular momenta belonging to different Hilbert spaces of a single object, e.g. its spin and its orbital angular momentum. If the spin has half-integer values, such as 1\/2 for an electron, then the total (orbital plus spin) angular momentum will also be restricted to half-integer values.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Further, the quantum states of composed systems (i.e. like two <\/span>Hydrogen <span style=\"text-align: initial;font-size: 1em\">atoms or two electrons) in basis sets which are made of tensor products of quantum states that in turn describe the subsystems individually, can be extended. We assume that the states of the subsystems can be chosen as eigenstates of their angular momentum operators (and of their component along any arbitrary <\/span><em style=\"text-align: initial;font-size: 1em\">z<\/em><span style=\"text-align: initial;font-size: 1em\"> axis).<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The subsystems are therefore correctly described by a set of <\/span><em style=\"text-align: initial;font-size: 1em\">\u2113<\/em><span style=\"text-align: initial;font-size: 1em\">, <\/span><em style=\"text-align: initial;font-size: 1em\">m<\/em><span style=\"text-align: initial;font-size: 1em\"> quantum numbers, then there is interaction among the subsystems, the total Hamiltonian contains terms that do not commute with the angular operators acting on the subsystems only. However, these terms <\/span><em style=\"text-align: initial;font-size: 1em\">do<\/em><span style=\"text-align: initial;font-size: 1em\"> commute with the <\/span><em style=\"text-align: initial;font-size: 1em\">total<\/em><span style=\"text-align: initial;font-size: 1em\"> angular momentum operator. Sometimes one refers to the non-commuting interaction terms in the Hamiltonian as <\/span><em style=\"text-align: initial;font-size: 1em\">angular momentum coupling terms<\/em><span style=\"text-align: initial;font-size: 1em\">, because they necessitate the angular momentum coupling.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">Spin-orbit coupling<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">The behavior of atoms and smaller particles is well described by the theory of quantum mechanics, in which each particle has an intrinsic angular momentum called spin and specific configurations e.g. electrons in an atom are described by a set of quantum numbers. Collections of particles also have angular momenta and corresponding quantum numbers, and under different circumstances the angular momenta of the parts couple in different ways to form the angular momentum of the whole. Angular momentum coupling is a category including some of the ways that subatomic particles can interact with each other.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In atomic physics, spin-orbit coupling, also known as <\/span><strong style=\"text-align: initial;font-size: 1em\">spin-pairing<\/strong><span style=\"text-align: initial;font-size: 1em\">, describes a weak magnetic interaction, or coupling, of the particle spin and the orbital motion of this particle, e.g. the electron spin and its motion around an atomic nucleus . One of its effects is to separate the energy of internal states of the atom, e.g. spin-aligned and spin-antialigned that would otherwise be identical in energy.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>L-S coupling and J-J coupling<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Hamiltonian of the system depends on<\/p>\n<p>&nbsp;<\/p>\n<p>1.\u00a0\u00a0\u00a0\u00a0 Kinetic energy of the electrons<\/p>\n<p>2.\u00a0\u00a0\u00a0\u00a0 Potential energy of the electrons due to the interaction between the nucleus and the electrons<\/p>\n<p>For these two factors Hamiltonian act as unperturbed Hamiltonian<\/p>\n<p>3.\u00a0\u00a0\u00a0\u00a0 Energy due to interaction between the electrons<\/p>\n<p>4.\u00a0\u00a0\u00a0\u00a0 Spin-spin correlation energy<\/p>\n<p>5.\u00a0\u00a0\u00a0\u00a0 Spin-orbit interaction<\/p>\n<p>For these three factors Hamiltonian act as perturbed Hamiltonian<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">For <strong>lighter elements<\/strong> the spin orbit interaction energy is much less as compared to the interaction energy between electrons and spin-spin correlation energy and there will be L-S coupling for lighter element.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">For <strong>heavier elements<\/strong> the spin orbit interaction energy is much greater than interaction energy between electrons and spin- spin correlation energy and there will be j-j coupling for heavier elements.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>L-S coupling<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In L-S coupling it is assumed that, when several electrons are present in an atom, each with a definite value of orbital angular momentum li* and spin angular momentum si* such that li*= li(li+1)h\/2\u03c0 and si*= si(si+1) h\/2\u03c0.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Due to the domination of interaction energy between electrons the orbital angular momentum of indivisible electrons will intract with each other and the total orbital angular momentum is given by L = L(L+1) h\/2\u03c0, where L varies from | l1+l2+l3\u2026\u2026..|minimum to | l1+l2+l3\u2026\u2026.|maximum the states with different values of L have fairly large energy difference, the state of largest L of being lowest energy. The different levels are designated as S, P, D, F, G\u2026\u2026..according as L=0,1,2,3,4,5\u2026\u2026. Thus:<\/p>\n<p>&nbsp;<\/p>\n<p>For 3p 4d electrons: l1=1, l2=2.<\/p>\n<p>\u2234\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 L = |l1-l2|, | l1-l2|+1, \u2026\u2026( l1+l2)<\/p>\n<p>=\u00a0 1, 2, 3 (P, D, F states).<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">For 3p, 4d, 5f electrons: l1=1, l2=2, l3=3. The electrons which are tightly bound to the nucleus will interact first.<\/p>\n<p>\u2234\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 L = |l1-l2|, | l1-l2|+1, \u2026\u2026( l1+l2)<\/p>\n<p>=\u00a0 1, 2, 3 (P, D, F states).<\/p>\n<p>&nbsp;<\/p>\n<p>Now L combining with l3=3<\/p>\n<p>\u2234\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 L = 0, 1, 2, 3, 4, 5, 6(S, P, D, F, G, H, I states)<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">2.\u00a0\u00a0 Due to the domination of spin-spin correlation energy the spin angular momentum of individual electron will interact with each other and the total spin angular momentum\u00a0 \u00a0is given\u00a0 by\u00a0 S=\u221aS(S+1)h2\u03c0<\/p>\n<p>&nbsp;<\/p>\n<p>The quantum number S can take the values :<\/p>\n<p style=\"text-align: justify\">S = |s1+s2+s3+\u2026\u2026\u2026|min to | s1+s2+s3+\u2026\u2026\u2026|max. . The states with different values of S have considerable energy difference, the state of highest S being of lowest energy. The different levels are designated by their multiplicity, (2S+1). Thus:<\/p>\n<p style=\"text-align: justify\">For one electron: S= s = \u00bd.<\/p>\n<p>\u2234\u00a0 (2S+1) = 2 (doublet level).<\/p>\n<p>&nbsp;<\/p>\n<p>For two electrons: s1 = \u00bd, s2 = \u00bd.<\/p>\n<p>\u2234\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 S = |s1 &#8211; s2|, |s1 &#8211; s2| + 1, \u2026\u2026\u2026.( s1+s2) = 0, 1.<\/p>\n<p>\u2234\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (2S+1) = 2 (singlet and triplet levels)<\/p>\n<p>&nbsp;<\/p>\n<p>For three electrons: s1 = \u00bd, s2 = \u00bd, s3 = \u00bd.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">To combine three spins, we first combine two of them to obtain S\u2019 = 0, 1, and then combine the third S3= \u00bd to each of them. Thus, if we couple s3 = \u00bd to s\u2019 =0; we get s= \u00bd; and if we couple s3 = \u00bd to s\u2019 =1, we get s= 1\/2, 3\/2. Thus for three electrons, we get S =1\/2, 1\/2, 3\/2 (two sets of doublets, and one set of quartets).<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The followings branching diagram illustrates the possible total spin quantum numbers which can be obtained by combining several independent electron spins.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-54 aligncenter\" src=\"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/93\/2018\/11\/1-18.png\" alt=\"\" width=\"555\" height=\"236\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-18.png 555w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-18-300x128.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-18-65x28.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-18-225x96.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-18-350x149.png 350w\" sizes=\"auto, (max-width: 555px) 100vw, 555px\" \/><\/p>\n<\/div>\n<div>\n<p style=\"text-align: justify\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-55 aligncenter\" src=\"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/93\/2018\/11\/1-19.png\" alt=\"\" width=\"692\" height=\"193\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-19.png 692w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-19-300x84.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-19-65x18.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-19-225x63.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-19-350x98.png 350w\" sizes=\"auto, (max-width: 692px) 100vw, 692px\" \/><span style=\"font-size: 1em;text-align: initial\">In the general case of N electrons, the possible values of S are <\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">S=0,1,2,\u2026\u2026\u2026..N\/2forevenN.<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">S= 1\/2, 3\/2, 5\/2\u2026\u2026.N\/2 for odd N.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The dominant spin-spin correlation and the residual electrostatic interaction having been taken into accounts as a first peturbation, the smaller spin- orbit interaction is included in L-S coupling as an additional perturbation.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">As a result of smaller spin \u2013 orbit magnetic interaction, the resultant orbital angular momentum vector L and the resultant spin angular momentum vector S are less strongly coupled with each other to form a total angular momentum vector J of the atom:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">J = L + S<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">the magnitude of J, L and S remaining constant. The magnitude of J is \u221aJ(J+1) h\/2\u03c0, where the quantum number J takes the value:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">J=\u00a0 |L-S|, |L-S|+1, \u2026\u2026\u2026..(L-S).<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">J is integral or half \u2013 integral according as S is integral or half integral. The number of J values is (2S+1) when L &gt; S, or (2L+1) when S &gt; L. This mean that due to spin orbit magnet interaction; a level characterized by given values of Land S is further broken up into comparative closer (2S+1) or (2L+1) levels, each characterized by a different J value. The group of these J-levels forms a fine structure multiplet\u2019. The relative spacing of the fine structure levels with in a multiplet is governed by Lande interval rule.<\/p>\n<p>&nbsp;<\/p>\n<p>Lande Interval Rule<\/p>\n<p>&nbsp;<\/p>\n<p>Under L-S coupling, the spin-orbit interaction energy is of the form<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">\u2206Es,l = a(L.S),<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Where a is an interaction constant. Let us write<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">J = L+S.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Taking the scalar self product we have<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">J.J\u00a0 = L.L + S.S + 2L.S.<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">| J |2 = | L |2 + | S |2 +2L.S.<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">\u2234\u00a0\u00a0 \u2206Es,l = a\/2[J(J+1) \u2013 L(L+1)- S(S+1)]h2\/4\u03c02,<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Because J = J (J+1) h\/2\u03c0, and so on. We can write it as \u2206Es,l = A [J(J+1) \u2013 L(L+1)- S(S+1)] , Where A is another constant.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The various fine-structure levels of a Russell-Saunders multiplet have the same values of L and S, and differ only in the value of J. Hence, the energy difference between two fine-structure levels corresponding to J and J+1 is<\/p>\n<p style=\"text-align: justify\">Ej+1 &#8211; Ej = A [(J+1) (J+2)-J (J+1)]<\/p>\n<p>=2A (J+1).<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Thus, the energy interval (spacing) between consecutive levels J and J+1 of a fine-structure multiplet is proportional to J+1, that is, to the larger of the two J-values involved. This is \u2018Lande interval rule\u2019.<\/p>\n<p>&nbsp;<\/p>\n<p>NORMAL AND INVERTED MULTIPLETS<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Within a given multiplet, usually the level with smallest J value lies lowest. Such a multiplet is a \u2018Normal\u2019 multiplet. There are, however, multiplets in which the largest J level lies lowest. Such multiplets are \u2018inverted\u2019 multiplets.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">It is easy to understand how smallest J level lies lowest. The magnet field B produced by the orbital motion of the electron in the electric field of the nucleus is in the same direction as the orbital angular momentum L. In this field the most stable state, that is, the state of the lowest energy must be one in which the spin magnetic moment \u03bcs of\u00a0<span style=\"font-size: 1em;text-align: initial\">the electron lies up in the direction of B. We know that \u03bcs is directed opposite to S because the electron is negatively charged.<\/span><\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-56 aligncenter\" src=\"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/93\/2018\/11\/1-20.png\" alt=\"\" width=\"554\" height=\"290\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-20.png 554w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-20-300x157.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-20-65x34.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-20-225x118.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-20-350x183.png 350w\" sizes=\"auto, (max-width: 554px) 100vw, 554px\" \/><\/p>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Thus, in the lowest energy state, L and S are in opposite directions and so the value of J is lowest. When L and S are in the same direction, which corresponds to highest J, then \u03bcs is opposite to B and the state is least stable.<\/p>\n<p>&nbsp;<\/p>\n<p>The inverted multiplets arise due to some perturbing influences.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>DETERMINATION OF SPECTRAL TERMS FOR L-S COUPLING<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>(1)\u00a0\u00a0 <\/strong><strong>Atoms with one optical electron: <\/strong>For hydrogen-like atom, the ground state configuration is<\/p>\n<p style=\"text-align: justify\"><strong>\u00a0<\/strong><\/p>\n<p><strong>1s.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>For this, we have\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 s = \u00bd; l = 0,<\/p>\n<p>So that\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 S = s = \u00bd; multiplicity 2S+1 = 2,<\/p>\n<p>L = l = 0(S-state),<\/p>\n<p>&nbsp;<\/p>\n<p>And\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 J = |L-S|, \u2026\u2026\u2026\u2026\u2026\u2026\u2026 (L+S) = \u00bd.<\/p>\n<p>&nbsp;<\/p>\n<p>Thus, the ground state term of a hydrogen-like atom is<\/p>\n<p>&nbsp;<\/p>\n<p>2<sub>S1<\/sub>\/2.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The excited state configuration and the corresponding terms for a hydrogen-like atom would be<\/p>\n<p style=\"text-align: justify\">2s, 3s, 4s, \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..<sup>2<\/sup>S1\/2<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">2p,\u00a0\u00a0 3p, 4p, \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026&#8230;<sup>2<\/sup>p1\/2, <sup>2<\/sup>p3\/2.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>The ground state configuration for all alkali atoms is <sup>2<\/sup>S1\/2.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">(2)\u00a0\u00a0 Atoms with two or more Non-equivalent optical electrons: 4p 4d For this, we have<\/p>\n<p>&nbsp;<\/p>\n<p>s1 = \u00bd, s2 = \u00bd, l1 = 1, l2 = 2.<\/p>\n<p>&nbsp;<\/p>\n<p>The possible values of S and L are :<\/p>\n<p>&nbsp;<\/p>\n<p><strong>S <\/strong>= |<strong>s<\/strong><strong>1<\/strong><strong> &#8211; s<\/strong><strong>2<\/strong><strong>|, <\/strong>|<strong>s<\/strong><strong>1<\/strong><strong> &#8211; s<\/strong><strong>2<\/strong><strong>| + 1, \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026&#8230;( s<\/strong><strong>1<\/strong><strong>+s<\/strong><strong>2<\/strong><strong>)<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>= 0, 1; multiplicity (2S+1) = 1,3<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">And<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">L = |l<\/strong><strong style=\"text-align: initial;font-size: 1em\">1<\/strong><strong style=\"text-align: initial;font-size: 1em\">-l<\/strong><strong style=\"text-align: initial;font-size: 1em\">2<\/strong><strong style=\"text-align: initial;font-size: 1em\">|, | l<\/strong><strong style=\"text-align: initial;font-size: 1em\">1<\/strong><strong style=\"text-align: initial;font-size: 1em\">-l<\/strong><strong style=\"text-align: initial;font-size: 1em\">2<\/strong><strong style=\"text-align: initial;font-size: 1em\">|+1, \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026(l<\/strong><strong style=\"text-align: initial;font-size: 1em\">1<\/strong><strong style=\"text-align: initial;font-size: 1em\">+l<\/strong><strong style=\"text-align: initial;font-size: 1em\">2<\/strong><strong style=\"text-align: initial;font-size: 1em\">).<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">= 1, 2, 3, (P, D, F, state).<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Thus, we have in all six terms, three singlet terms and three triplet terms. These terms can be written as<\/span><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\"><sup>1<\/sup>P, <sup>1<\/sup>D, <sup>1<\/sup>F, <sup>3<\/sup>P, <sup>3<\/sup>D, <sup>3<\/sup>F.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">To take into account spin-orbit interaction, let us combine L and S to form J.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-57\" src=\"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/93\/2018\/11\/1-21.png\" alt=\"\" width=\"553\" height=\"302\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-21.png 553w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-21-300x164.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-21-65x35.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-21-225x123.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-21-350x191.png 350w\" sizes=\"auto, (max-width: 553px) 100vw, 553px\" \/><\/p>\n<\/div>\n<p style=\"text-align: justify\">Thus, a single degenerate level of configuration 4p 4d is splitted into 12 levels as shown in fig.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-58 aligncenter\" src=\"http:\/\/phyp10.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/93\/2018\/11\/1-22.png\" alt=\"\" width=\"650\" height=\"472\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-22.png 650w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-22-300x218.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-22-65x47.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-22-225x163.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-content\/uploads\/sites\/93\/2018\/11\/1-22-350x254.png 350w\" sizes=\"auto, (max-width: 650px) 100vw, 650px\" \/><\/p>\n<p style=\"text-align: justify\">unperturbed level spin-spin correlation energy + residual electrostatic energy + spin-orbit\u00a0\u00a0\u00a0\u00a0\u00a0 magnetic energy<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Coupling Schemes-I<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/BemRURdxy1k\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":3,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-51","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/chapters\/51","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":6,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/chapters\/51\/revisions"}],"predecessor-version":[{"id":671,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/chapters\/51\/revisions\/671"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/chapters\/51\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/wp\/v2\/media?parent=51"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/pressbooks\/v2\/chapter-type?post=51"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/wp\/v2\/contributor?post=51"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp10\/wp-json\/wp\/v2\/license?post=51"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}