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Home , Azimuthal quantum number, Zeeman effect

15. Crafting the : Chap 7, Conclusion. 1. Unsettled Questions of Atomic Physics. A. The and number mysteries.

• 1896: Zeeman observes a splitting of the doublet in the spectrum of a sodium in the presence of a weak .

Pieter Zeeman

double spectral lines (doublet) in absence of magnetic field...

...split into many more lines in presence of magnetic field.

The Zeeman effect: A splitting of the line spectra of an atom in a weak magnetic field. • 1897: Lorentz derives an explanation based on his theory.

- Suppose a is due to an electron with mass

me and charge e vibrating at frequency ν. - Then in the presence of a magnetic field H, this fre- quency can take one of three values: ν or ν ±Δν, where |H||e | Δν = 4πm H. A. Lorentz e

• So: In the presence of a magnetic field, a singlet in single spectral line ("singlet") of frequency absence of H... ν is split into three lines separated by an amount Δν. ...splits into triplet in presence of H.

• And: This "triplet structure" is actually observed for and helium ! Distance between triplet lines = Δν • But: This triple-splitting for singlets isn't what is observed for doublets or triplets when a magnetic field is switched on. • The Anomalous Zeeman Effect = The more complex splitting of spectra lines in the presence of a magnetic field beyond the Lorentz singlet-to-triplet splitting. • 1907: Runge discovers the following pattern:

"The hitherto observed complex splittings of spectral lines in a mag- netic field exhibit the following perculiarity: the distances of the com- ponents from the centre are integral multiples of just a fraction of the

normal separation, a = |H||e|/4πme. So far the fractions a/2, a/3, a/4, a/6, a/7, a/11, and a/12 have been definitely observed." Karl Runge

Runge's Law: s |H||e | Δν = a ⋅ , where s and r are integers and a = . r 4πm e

• For the "normal" (singlet-to-triplet) Zeeman effect, r = 1 and s = 0 or ±1. Sommerfeld (1916) On the Zeeman Effect in Hydrogen Spectrum. • For an electron in an elliptical Bohr orbit, there are two Bohr-Sommerfeld quantum conditions: y r p dφ = kh, k = 1, 2, 3, ... ∫ φ φ x p dr = rh, r = 0, 1, 2, 3, ... ∫ r

• k = "azimuthal" : angular H momentum determines shape of orbit. K vector • r = "radial" quantum number: θ determines size of orbit. • n = k + r = "principle" quantum x number: denotes energy of orbit. • m = "magnetic" quantum number: y

determines spatial orientation of orbit with cosθ = m/k respect to an external magnetic field H. • Frequencies of hydrogen lines in magnetic field are split by

Δν = Δm |H||e|/4πme. Sommerfeld's (1920) "A Number Mystery in the Theory of the Zeeman Effect".

Combination Principle: Each spectral line is made up of two terms corres- Sublevels ("series") ponding to the energies of the intial and final states. labeled by k.

• Each is characterized by the principle quantum number n = k + r. • Each energy level is divided into sublevels ("series") given by values of k = 1, 2, 3, ... and labeled by s, p, d, and b.

Selection Principle: Only transitions with Δk = ±1 or 0 Energy levels Each transition are allowed. labeled by n. (arrow) produces a spectral line.

Possible transitions for sodium atom. • Now: Apply Combination Principle to Runge's Law: ⎛ ⎞ ⎛ ⎞ ⎜ s1 s2 ⎟ ⎜s1r2 −s2r1 ⎟ Δν = Δν − Δν = ⎜a −a ⎟ = a ⎜ ⎟. 1 2 ⎜ r r ⎟ ⎜ r r ⎟ ⎝ 1 2 ⎠ ⎝ 1 2 ⎠

• So: "Magneto-Optic Splitting Rule": Δν (s r −s r ) s = 1 2 2 1 = . a r r r 1 2

The following then accounts for observed Zeeman effects: 1. For singlet spectral lines, r = 1 for any energy sublevel (s-, p-, d-, or b-). 2. For triplets, r = 1, 2, 3, 4 for s-, p-, d-, and b-sublevels. 3. For doublets, r = 1, 3, 5, 7 for s-, p-, d-, and b-sublevels.

• Example: - For a triplet spectral line belonging to the p and s series, r = 1 ⋅ 2 = 2. - For a doublet spectral line belonging to the p and s series, r = 1 ⋅ 3 = 3. Sommerfeld's (1920) "General Spectroscopic Laws...".

• Some transitions allowed by Dotted arrows are allowed but not observed. Δk = ±1 or 0 are not observed:

• Suppose each line is assigned an "inner" quantum number j, and suppose it satisfies the same selection rule as k: Δj = ±1 or 0.

j-values k-values

• What is the physical significance of j?

"Of its geometric significance we are quite as ignorant as we are of those differences in the orbits which underlie the of the series terms." "The actual cause for the doublets and triplets and therefore also the cause of the anomolous Zeeman effect is still unclear to me. Only this much is certain, that in all whole-numbered relationships, quanta are involved."

• On model-building vs. empirical rules:

"Thus it is, that at the moment we are at a loss with the modell- mässigen meaning of the line multiplicities of the non-hydrogenic elements... All the more valuable are all the lawful regularities [Gesetzmässigkeiten] that present themselves empirically..."

• On the "Magneto-Optical Splitting Rule":

"It represents for the time being, as I have noted elsewhere, a 'number mystery'... Only so much appears to be certain: that the integral harmony of our Runge numbers has its final cause in the action of hidden quantum numbers and quantum relations." B. Modellmässig vs Gesetzmässig Reasoning • "...[I]gnorance about causes was traded for a functionalist understanding of regularities within phenomena... Sommerfeld gave up the search for modellmässig foundations in order to develop a praxis -- or craft -- involving 'half-empirical' Gesetzmässigkeiten." (Seth, pg. 212.)

Example: The Rise and Fall of Sommerfeld's (1918) "Ellipsenverein" model.

• Initial problem: How can Bohr's model explain X-ray emission and absorption, which do not occur at the same frequency. • Sommerfeld (1915): Suppose, instead of feeling the nucelar charge Ze, an electron in a given orbit feels a reduced charge (Z − l)e due to screening by lower . • But: The screening constant l should be an integer and calculations derived using Bohr's model entail it isn't! • (1918): Sommerfeld's ellipsenverein model: - Describes n electrons on n identical ellipses, with each ellipse spaced at an angle of 360/n. - At any moment, each electron stands at the corner of a regular n-sided polygon. - Collective takes the form of a single circular orbit expanding and contracting around the nucleus. - Entails non-integer screening constant l!

"For my feeling, the artful interlocking of the n elec- tronic paths in our 'Ellipsenverein' is nothing un- natural; I see much more a sign therein for the high harmony of motion that must rule within the atom." • But: Essential characteristic of the ellipsenverien: the orbits don't all belong to the same atom!

"One must always keep in view the fact that the processes..., even if they occur in the very same element, must take place in different atoms."

• 1921: Bohr declares this cannot be right under the correspondence principle:

All orbitals, of whatever shape, that correspond to a given element, must be found within a single atom of that element.

• 1924: Sommerfeld concedes:

"In the present state of theory, it seems to me to be most secure to put the question of model-based meaning in the background and to first bring the empirical relations to their simplest arithmetical and geometrical form." • Former student Pauli on 4th edition of Sommerfeld's Atombau:

"I found it particularly beautiful in the presentation of the com- plex structure that you have left all modellmässig considerations to one side. The model idea now finds itself in a difficult, fundamental crisis... One now has the impression with all models, that we speak there a language that is not sufficiently adequate for the simplicity and beauty of the quantum world."

Sommerfeld's method as a craft: • "...A craft, in that, working directly with the data one extrapolates and interpolates, drawing conclusions not from model-based deductions, but from arithmetic and graphical approximations, drawing on special experience to strike a balance between different sets of the always-insufficient information from spectroscopic data." (Seth, pg. 225.) 2. From the Old World of Waves to the New World of Quanta. Contrast between Bohr and Sommerfeld: • "While Bohr sought an analogy and hence some form of connection between [electromagnetism and the quantum theory],...Sommerfeld strove...to keep them apart." (Seth, pg. 226)

Example: Sommerfeld's Spherical Wave Theory • Recall: In classical electromagnetism, a moving electron emits radiation through constant coupling to the electromagnetic field (i.e., "aether"). • In Bohr's model, an electron moving in a stationary orbit does not couple to the "aether"; only during a transition between stationary orbits does an electron emit radiation. • This requires an "abstract mode of expression":

"We must speak not of an electron but of a solution to Maxwell's equations, which is determined by conditions of coupling in the process of emission between the atom and the ether. The more abstract mode of expression, to which we are forced, is inevitable if we wish to follow out logically the view of the quantum theory." • During a transition, what produces radiation is not a classical electron, but a "solution to Maxwell's equations". • Quantum theory describes production of radiation (during transitions); classical theory describes propagation of radiation (after it has been emitted). • In particular: The emitted radiation takes the form of a classical spherical electromagnetic wave.

• Recall: Bohr's relation between frequency νn',n'' of emitted radiation and

energy difference En' − En'' of transition:

hνn',n'' = (En' − En'')

describes frequency of describes quantum classical spherical EM wave states of atom

"The aether demands its hνn',n'', the atom furnishes it by giving up

an amount of energy En' − En''." • Development of spherical wave theory: 1919-1922 in first 3 editions of Atomic Structure and Spectral Lines. • But: Abandoned in 4th edition. • Why? Compton's (1922) "A Quantum Theory of the Scattering of X-rays by Light Elements".

The "Compton Effect" • Describes scattering of light by an electron. • Suppose: Planck's quantum hypothesis applies to light (Einstein 1905). • And: A quantum of light (a photon) with frequency ν and energy E has momentum p = E/c = hν/c. • Can now model the scattering of light by an electron as an elastic collision between a photon and an electron in which momentum and energy are conserved... • Conservation of momentum requires: pi = pf + pe

• So: pe ⋅ pe = (pi − pf) ⋅ (pi − pf) • Or: p 2 = p 2 + p 2 − p p cosθ (∗) e i f i f p = hν /c • Conservation of energy requires: i i

2 2 2 2 hν + m c = hν + (m c ) +(p c) pf = hνf/c i e f e e

2 • Or: p 2 = hν /c −hν /c + m −m2 c2 (∗∗) e ( i f e ) e

• Now: Subtract (∗∗) from (∗): • Upshot: Radiation behaves like discrete particles (photons) with h λ −λ = (1− cosθ) momenta and energy! f i m c e • Additional implication: Matter (electrons) has associated Wavelength of The "Compton wave- photon λ = c/ν length" of an electron wavelength! "Whereas earlier, I had sought to maintain the wave theory for pure propagation phenomena for as long as possible, I have been pushed ever more to the ground of the extreme quantum theory of light."

• Major blow to Sommerfeld: Spherical wave theory was his alternative to Bohr's "magic wand" (correspondence principle). • But: Still critical of Bohr's modellmässig approach...

"The difficulties that emerge ever more clearly in atomic physics appear to me to arise less from an exaggerated application of the quantum theory and much more from a perhaps exagger- ated belief in the reality of concepts of models." 3. The Pauli Exclusion Principle. • 1921: Landé's improved formula for Zeeman splitting (in terms of energy differences between spectral lines): ΔE = mgωh

• m = integer, and ω = 2πν, where ν = |H||e|/4πme. • g = "g-factor", calculated by comparison with experimental data:

singlets: g = 1 2j Bravo, you are able doublets: g = , j = k or k −1 to work miracles!" 2k −1 ⎧ ⎪1 +1/k, j = k ⎪ triplets: g = ⎨⎪1−1/k +1/(k −1), j = k −1 ⎪ ⎪1−1/(k −1), j = k − 2 ⎩⎪

k = j = inner quantum number • Heisenberg's Rumpf model:

K H R • Outermost electron with k encoded in vector K. • Core ("Rumpf") of inner electrons Werner with angular momentum r encoded in Heisenberg vector R. • Zeeman effect due to coupling between K and R in presence of magnetic field H.

• Splitting factor mg corresponds to sum of projections of K and R onto H.

3 1 R2 −K 2 R = r/2, K = k − 1/2, • Landé derives g = + ⋅ J = j for even multiplets and J = j − 1/2 2 2 J 2 − 1 4 for odd multiplets (j is total ang mo).

• A modellmässig explanation of the g-factor that fits experimental data! • But: Assumes R contributes twice as much to splitting factor as K: mg = |K|cos(K, H) + 2|R|cos(R, H) • And: Classical electron theory (Larmor's theorem) entails factor of 2 should not be present! • 1924. Pauli applies himself to the problem:

"...it is much too difficult for me and I wish that I were a film come- dian or something similar and had never heard anything of physics!"

Wolfgang Pauli • Turning point: Pauli calculates relativistic correction for Rumpf model.

- Rumpf electrons move at relativistic speeds. - So: Splitting of spectral lines should depend on a relativistic factor that increases as the number of Rumpf electrons increases. - And: This correction factor should be in different elements. - But: Empirical data indicates no such correction.

• Pauli's conclusion: Rumpf can't be responsible for Zeeman effect! • Zeeman effect must be due solely to outermost electron.

"The doublet structure of the alkali spectrum, as well as the violation of Larmor's theorem comes about through a peculiar, classically non- describable kind of Zweideutigkeit [ambiguity, doubled signification] of the quantum-theoretical characteristics of the light-electron." • 1925: "On the Connection of the Closing of Electron Groups in the Atom to the Complex-Structure of Spectra." • Electron states in an atom are uniquely characterized by 4 quantum numbers:

principle n, azimuthal k, and two magnetic numbers m1, m2. • These states obey an "Exclusion Principle":

"There can never be two or more equivalent electrons in an atom for which, in strong fields, the values of all quantum numbers... coincide. If an electron is to be found in an atom for which these quantum numbers (in an external field) possess determinate values, then this state is 'occupied'."

• Seth: "The rule abandons, as Pauli would emphasize to Bohr, any talk of orbits and provides instead a formal quantum rule connecting the number of terms into which a single spectral line could split with the periodic structure of the table of elements." (pg. 256.) • A victory of quantum Gesetzmässigkeiten over model-based analysis. • Explanation of fourth quantum number in terms of : Kronig, Goudsmit and Ulenbeck.

- m2 encodes two degrees of freedom of electron in outermost orbit. - Suppose these degrees of freedom are associated with the angular momentum of a spinning electron (in either a clockwise or counter-clockwise direction about a given axis).

• Pauli is resistant. - Relativistic considerations: spinning electrons and relativity. - Sommerfeld's influence: Gesetzmässigkeiten trumps modellmässig reasoning. Contemporary description of electron states in atoms:

• Electron state characterized by four properties (n, , m, ms), energy n, orbital

angular momentum , z-component of orbital angular momentum m, and

spin ms.

• n = 1, 2, ...;  = 0, 1, 2, ... (n − 1); m = −, ... 0, ..., ; ms = −, +. • Exclusion Principle: No two electrons can be in the same state (i.e., no two electrons can have all the same values of these four properties).

n: 1 2 3 4 energy shells orbitals Z Element : 0 0 1 0 1 2 0 1 2 3 K shell (n = 1) s orbital ( = 0) 1 H hydrogen 1 L shell (n = 2) p orbital ( = 1) 2 He helium 2 M shell (n = 3) d orbital ( = 2) f orbital ( = 3) 3 Li lithium 2 1 N shell (n = 4)  etc. etc. 4 Be beryllium 2 2 5 B boron 2 2 1 6 C carbon 2 2 2 Example: The 3 electrons in a 7 N nitrogen 2 2 3 lithium atom are characterized by 8 O oxygen 2 2 4 (1, 0, 0, +), 9 F fluorine 2 2 5 (1, 0, 0, −), (2, 0, 0, +). 10 Ne neon 2 2 6 4. Reconsidering Kuhnian Revolutions. • "Sommerfeld and the members of his school did not register a sense of crisis, a sense of the existence of paradoxical and insurmountable anomalies, or a sense of the occurrence of a revolution." (Seth, pg. 266.)

"The new development does not signify a revolution, but a joyful advancement of what was already in existence, with many fundamental clarifications and sharpenings." (1929)

"My memory is that everyone was so excited about the pos- sibilities of solving problems, answering questions, the mech- anism provided by the new quantum , that there was little discussion of those details of interpretation." (1963) Elder Pauli

"No, there was no great puzzle and I think this is the greatest characteristic of the Sommerfeld group; we were not made aware of great puzzles. We were given the impression that here was a wonderful tool." (1964) Hans Bethe • "Scientific revolutions, in other words, are revolutions of conceptual foundations, not puzzle-solving techniques. Most simply: Science sees revolutions of principles, not of problems." (Seth, pg. 268.)

Critique of Kuhn: • Kuhn's analysis focuses on individuals and/or communities. • "Mesoscopic level" analysis focus on small groups within communities. • Claim: To understand scientific change, both types of analysis are important. • "The result is a scientific community made up, in the majority, of those who solve problems (and eschew the pursuit of revolutions) and a much smaller group whose focus on principles and foundations means that the only change that counts is a revolutionary and fundamental one."