{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Spin-Orbit - {D Ix N 0 Lu'—< I Z 9 l(D Z< I'— D...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 2
Background image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: {D Ix N 0') Lu '— < I: Z 9 l: (D Z < I: '— D Z < Z D. (I) (If p. Z LIJ E O 2 DJ .1 O E D 9 E Z (5 < E w d. 65 .c 0 s —> 00, as we did in Section 7-8 for the magnitude of the orbital angular momentum L by letting its quantum number 1—» 00. An equivalent statement is that in the classical limit the magnitude of S is completely negligible because it is so small, so spin is essentially nonclassical. This being the case, it is sometimes more harmful than helpful to think of spin in terms of a classical model like a small spinning sphere; but it must be admitted that it is difficult to avoid thinking in such terms. 8-4 THE SPIN-ORBIT INTERACTION Although spin itself is subtle, there is nothing subtle about many of the effects it produces. Perhaps the most important is that it doubles the number of electrons which the “exclusion principle” allows to populate the quantum states of multi- electron atoms. When we study this effect in Chapter 10, we shall see that the ground states of atoms would be very much altered if electrons did not have spin. This would have profound consequences on the periodic properties of atoms, and therefore on all of chemistry and solid state physics. In the present section we shall study the interaction between an electron’s spin magnetic dipole moment and the internal magnetic field of a one-electron atom. Since the internal magnetic field is related to the electron’s orbital angular momentum, this is called the spin-orbit interaction. It is a relatively weak interaction which is respon- sible, in part, for the fine structure of the excited states of one-electron atoms. The spin-orbit interaction also occurs in multielectron atoms, but in such atoms it is reasonably strong because the internal magnetic fields are very strong. Further- more, an effect completely analogous to the spin-orbit interaction occurs in nuclei. The nuclear spin-orbit interaction is so strong that it governs the periodic properties of nuclei. . The origin of the internal magnetic field experienced by an electron moving in a one-electron atom is easy to understand if we consider the motion of the nucleus from the point of view of the electron. In a frame of reference fixed on the electron, the charged nucleus moves around the electron and the electron is, in effect, located inside a current loop which produces the magnetic field. The argument is illustrated qualitatively in Figure 8-7. To make the argument quantitative, we note that the charged nucleus moving with velocity —v constitutes a current element j, where j= —Zev According to Ampere’s law, this produces a magnetic field B which, at the position of the electron, is ojxr Zeuovxr B: =_ 3 7tr3 41: r _v‘ Figure 8-7 Left: An electron moves in a circular Bohr orbit, the motion as seen by the nucleus. Right: The same motion. but as seen by the electron. From the point of view 0‘ the electron. the nucleus moves around it. The magnetic field B experienced by the electron is in the direction out of the page at the electron’s location. It is convenient to express this in terms of the electric field E acting on the electron. According to Coulomb’s law From the last two equations, we have B = *50140" x E 1 B=——c—2vxE (8-24) since c = l/‘leouo. The quantity B is the magnetic field strength experienced by the electron when it is moving with velocity v relative to the nucleus, and therefore through the electric field of strength E which the nucleus exerts on it. Equation (8-24) is actually of very general validity, and it can be derived from relativistic considera- tions. The electron and its spin magnetic dipole moment can assume difl‘erent orienta- tions in the internal magnetic field of the atom, and its potential energy is different for each of these orientations. If we evaluate the orientational potential energy of the magnetic dipole moment in this magnetic field, from an equation analogous to (8-13), we have AE= —u,-B Using (8-19), this can be written in terms of the electron’s spin angular momen- tum S as AE=9”%s-B h But this energy has been evaluated in a frame of reference in which the electron is at rest, whereas we are interested in the energy as measured in the normal frame of reference in which the nucleus is at rest. Because of an effect of the relativistic trans- formation of velocities, called the Thomas precession, the transformation back to the nuclear rest frame results in a reduction of the orientational potential energy by a factor of 2. Thus, the spin-orbit interaction energy is Mm AE = — S - B - 2 h (8 25) The transformation leading to the factor of 2 is interesting, but rather complicated, so we shall not carry it out here. (It is carried out in Appendix 0.) We shalt find it convenient to express (8-25) in terms of S - L, the scalar, or dot, product of the spin and orbital angular momentum vectors. To this end, we use, in (8-24), the relation 4m=F between the electric field E and the force F acting on the electron of charge —e. We also use the relation _ dV(r) r dr r between the force and the potential. (The term r/r is a unit vector in the radial direc- tion which gives F its proper direction.) With these relations, (8-24) becomes Imwn «ac2 r dr F: B: vxr NOllOVHEILNI ilSHO—NldS 3H1 613 o no N m ,“J < n: z 9 t w z < n; |.- o z < z" n. w w” '— z LU 2 O 2 uJ .J O E o 9 in z o < 2 Q d. (U .C o Multiplying and dividing by the electron mass m allows us to write this in terms of the orbital angular momentum, L = r x mv = —mv x r, as 1 ldV(r) emc2 r dr Note that the strength of the magnetic field B, experienced by the electron because it is moving about the nucleus with orbital angular momentum L, is proportional to the magnitude of L, and also that the magnetic field vector is in the same direction as the angular momentum vector. With this result, we can express the spin-orbit interaction energy, (8-25), as B = L (8-26) AE _ an, 1dV<r) _ 2emc2h ? dr SiL Evaluating g5 and a,” we obtain _ 1 1dV(r) AE— 2m2c2? dr S‘L (8—27) This equation was first derived in 1926 by Thomas, using as we have a combination of the Bohr model, Schroedinger quantum mechanics, and relativistic kinematics. However, it is in complete agreement with the results of the relativistic quantum mechanics of Dirac. It is important in the theory of multielectron atoms as well as of one-electron atoms. Furthermore, a similar equation is central to the understand- ing of the theory of the structure of nuclei, as we shall see later in the book. Example 8-3. Estimate the magnitude of the orientational potential energy AE'for the n = 2, I = 1 state of the hydrogen atom, to check whether it is of the same order of magnitude as the observed fine-structure splitting of the corresponding energy level. (There is no spin-orbit en- ergy in the n = 1 state, since for n = 1 the only possible value for lis l = 0, which means L = 0.) >The potential is 82 V = — ‘1 (r) 41t€0 r dV(r) _ e2 _2 dr _ 47:60 r e2 1 AE = — — S - L 47:602m2c2 r3 The magnitude of S - L is approximately hz since each of these angular momentum vectors has a magnitude of approximately in. The expectation value of l/r3 for the n = 2 state 15 approximately 1/(3a0)3. Thus . e2 1 m3e6 2 me8 47:602m2c2 3—3(41r€0)3h6 = 54 x (47teo)4c2h4 (9 x 109 nt—mz/coulz)4 x 9 x 10~31 kg x (1.6 x 10‘19 coul)a 54 x (3 x 108 m/sec)2 x (1.1 x 10‘34joule—sec)4 ~10‘23jou1e ~10'4 eV Since S - L can be either positive or negative, depending on the relative orientation of the two vectors, the energy level is split by roughly 2 x 10‘4 eV. Comparing this with the energy of the n = 2, l = 1 level of hydrogen, E2 = — 3.4 eV. we see that the ratio of the predicted energy splitting to the energy itself, |AE/E|, is about one part in 104. This is in reasonable agreement with the splitting required to explain the fine structure of the lines of the hydrogen spectrum associated with this level, as discussed in Section 4-10. and therefore it provides some confirmation of the theory we have developed. A more details? comparison of the theory with experiment will be made shortly. IAEI ; ...
View Full Document

{[ snackBarMessage ]}

Page1 / 3

Spin-Orbit - {D Ix N 0 Lu'—< I Z 9 l(D Z< I'— D...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online