Lecture 22 - PowerPoint Presentation

Slide1

Lecture 22Spin-orbit coupling

(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign.

This material has

been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies

.Slide2

Spin-orbit coupling

Spin makes an

electron

act like

a small magnet. An electron orbiting around the nucleus also makes a magnet.

These two magnetic moments can

interact

and

, depending on the relative orientations of the two

moments,

orbital energy can

be slightly altered.

We use the so-called Na D line as a paradigm.

We use the

first-order perturbation theory

to describe the shifts in orbital energies.

The spin-orbit interaction is a relativistic effect and its derivation is beyond the scope of this course. We treat it as a phenomenological effect explained in analogy to two interacting magnets.Slide3

Na D line

The orange color of the sodium

lamp is due to the Na 3

p

→3

s emission at ca. 17000 cm−1.A close examination of this transition reveals that the emission band consists of two bands separated by 17 cm−1.

Public-domain image from WikipediaSlide4

Spin-orbit coupling

Spin of an electron makes it a magnet. Orbital motion of the electron also makes it a magnet. These two magnetic moments can interact or “couple” (

spin-orbit coupling

) and cause energy level splitting.

N

S

N

S

N

S

S

NSlide5

Sum of angular momenta

Each electron has two angular momenta (a dual magnet): orbital angular

momentum

and spin angular momentum.

The total momentum is the most naturally defined as their vector

addition.

Total

Orbital

SpinSlide6

must be (space) quantized.

So its total angular momentum quantum number

j

is either a full or half integer in the range:

j

= jmin (0 or greater), jmin+1,…, jmax–1, jmaxSum of angular momentaSlide7

ExamplesIdentify the levels that may arise from the configurations (a)

(3

p

)

1

, (b) (3s)1.Slide8

Examples(a)

3

p

orbital

→ l = 1. j = l ± ½ = 3/2 or 1/2.(b) 3s orbital → l = 0. j = 0 + ½ = ½ (j = 0 –

½ is not allowed because j is non negative).Slide9

Spin-orbit coupling

Two magnets are the most stable when

they are antiparallel and

the least stable when they

are parallel.

In general, the energy due to the interaction of spin and orbital momenta should be

θSlide10

Spin-orbit coupling operator

The atomic Hamiltonian

does not have this:

This is because we do not have a counterpart in the classical energy, from which the Hamiltonian is derived. We

add

spin-orbit interaction operator:Slide11

Spin-orbit coupling operator

The

spin-orbit interaction operator

has the

spin-orbit coupling constant A. It is in units of cm−1, which is why hc

is multiplied.The value of

A is extracted from experiment (11.5 cm

−1 for Na 3

p

from the splitting of 17

cm

−1

) or relativistic quantum mechanics. Slide12

Homework challenge #6Study the special theory of relativity. One of the best textbooks is “Special Theory for Relativity for Beginners” by Jürgen Freund.

Study Dirac’s theory of relativistic quantum mechanics and explain how it introduces the concepts of spins and positrons from the first principles.

Study the work of

Pekka

Pyykkö on the effect of relativity on chemistry.Slide13

Spin-orbit coupling operator

The

spin-orbit interaction operator

makes the solution of the Schrödinger equation difficult.

Since A is very small (0.001 of 3p-3s energy difference), we use perturbation theory.Slide14

First-order perturbation theorySlide15

Na D lineSlide16

Na D line

4-fold degenerate

2-fold degenerateSlide17

Spin-orbit coupling constants

The measured values of

A

:

Li:

0.23 cm–1Na: 11.5 cm–1K: 38.5 cm–1Rb: 158 cm–1Cs: 370

cm–1 Spin

-orbit coupling arises from the special theory of relativity and greater for the heavier elements because the

1s electrons in high-Z

elements

can go nearly as fast as the speed of light.Slide18

Consequences of SO coupling

An electron in each orbital no longer has a well defined spin (magnetic quantum number,

α

or

β

).States are no longer pure spin-singlet, doublet, triplet, etc.Radiative transitions between singlet and triplet, between doublet and quartet, etc. become weakly allowed (phosphorescence).Nonradiative transitions between singlet and triplet, etc. become weakly allowed (intersystem crossing).These are more prominent in heavier elements.Slide19

Singlet and triplet statesThe singlet and triplet states have different spin and spin magnetic momenta. They are

orthogonal functions

if it were not for the SO interaction..

Separable because

z

operator does not act on spin part.

This is zero when

final

and

initial

states have different spin

eigenfunctions

, e.g., singlet and triplet.Slide20

Fluorescence & Phosphorescence

Fluorescence

is an emission of light between the same spin

states (e.g., singlet to singlet).

Since this is an

allowed transition, it is intense and fast.

Phosphorescence

is between different spin

states and mediated by

SO

. It is “

forbidden

” and it is weak and slow.

Both public-domain images from Wikipedia

F

luorescence

Phosphorescence

Intersystem crossingSlide21

Summary

Spin angular momentum as a magnet and orbital angular momentum as another magnet interact (spin-orbit coupling).

Spin-orbit coupling is a relativistic effect and is greater for heavier elements.

It causes splitting of subshell states, phosphorescence, and intersystem crossing.

The first-order perturbation theory describes the coupling.