The Stern-Gerlach Experiment

The Stern-Gerlach experiment was originally performed by Otto Stern and Walther Gerlach in 1922, an experiment fundamental to the development of modern quantum physics. It showed evidence for the quantisation of angular momentum and led to the discovery of the atomic property ‘spin’, comparable to an angular momentum and a constituent of total angular momentum. The spin value has only two potential values, up (+ $\frac{1}{2}$

) or down (- $\frac{1}{2}$

)¹. The total angular momentum is the summation of the spin of the electron and the orbital angular momentum⁶. The experiment was performed at a time when the Bohr-Sommerfeld model of the atom was the most prevalent, a model which followed the original Bohr model which theorised that electrons circularly orbit a nucleus with predetermined sizes and energies and that these two quantities are proportional. It also showed that the energy of the orbital predicts the energy of emitted/absorbed radiation when the electrons travel between these orbits (governed by the Rydberg formula). This theory was proposed by Niels Bohr in 1915, an alteration of the pre-existing model by Ernest Rutherford⁵.

The Bohr model describes most aspects of the current atomic theory accepted currently, although contains several discrepancies. The first problem is its violation of the Heisenberg uncertainty principle, and this stems from the consideration that electrons have a simultaneously observable radius and orbit, a concept not understood at the time. Secondly, it does not accurately predict emission spectrums of elements other than single-electron atoms (H, He⁺, …)⁶. Additionally, it cannot explain the Zeeman/Stark effect. The Zeeman effect is the splitting of spectral lines into closely spaced split lines when exposed to a magnetic field. Discovered by Peter Zeeman, the Zeeman effect can now be explained by the combination of spin and the atoms orbital angular momentum¹⁰. The Stark effect discovered by Johannes Stark is very similar to the Zeeman effect; however, the Stark effect is produced under an electric field as opposed to a magnetic field⁹.

The Bohr model could not explain the spectroscopic phenomena of fine structure, which is the splitting of the spectral lines of an element into two or more lines with a slightly different wavelength. This is now known to be caused by the combination of the spin of the electron and its angular momentum. A magnetic dipole moment is produced by the electron’s own magnetic field and this rotates the electron to emit in a varied direction, according to its spin¹².

The introduction of an improved method by Arnold Sommerfeld in 1915 produced the Bohr-Sommerfeld hypothesis, which gives solutions to the previous problems of the Zeeman effect, Stark effect and fine structure in spectroscopy within Bohr’s theory¹³. Sommerfeld suggested that electron orbits could be elliptical as opposed to purely circular, and could be given by three values, for n, k and m, whereas Bohr provided only one. n is the principal quantum number as was used by Bohr and represented the energy level (n = 1, 2, 3…). k gives the shape of the orbit, and this is circular for k=n (k = 1, 2, 3, …, n). m gives the orbital angular momentum projected onto an axis, and is given by m = -k, -k+1, …, k-1, … (m $\ne$

0). This suggests a quantised angular momentum².

Figure 1: orbitals as suggested by Sommerfeld¹⁵

This reconciled most of the Bohr shortcomings as it firstly explained the Zeeman effect. With Bohr’s model, all electrons had the same orientation relative to the magnetic field as the orbitals were perfectly circular. With ellipses suggested by Sommerfeld, electrons can be at varying orientations relative to the field, and consequentially have a varying energy which predicts a Zeeman effect and this is experimentally shown¹⁷. The same is reconciled for the Stark effect with an electric field.

The theory did not however explain the anomalous Zeeman effect. The Zeeman effect is explained by the prediction that the number of lines visible can be given by 2 $l$

+ 1, where $l$

is the integer value for the orbital angular momentum (denoted as $m$

previously), suggesting that the number of lines are odd. However, in the anomalous Zeeman effect an even amount of lines are produced, often with an odd atomic number²³. This is indicative of a total angular momentum which could take half-integer values, and this was later attributed to an electron’s spin¹⁰.

Sommerfeld also introduced what is known as the fine-structure constant ( $\propto$

) in 1916 (also known as Sommerfeld’s constant), which is dimensionless and represents the magnitude of the electromagnetic force interaction between charged particles and photons. It is equivalent to $v∕c$

, where $v$

is the velocity of an electron in the first circular orbit (k=n), and $c$

is the speed of light constant ( $3\times {10}^8\mathit{ m}/s$

). The constant was found experimentally however, and this was measured within the Michelson and Morley experiment²¹.

Sommerfeld also considered particles in a relativistic sense, with varying mass as a result of varying velocity. This allowed him to produce an expression, using the Rydberg formula and his newfound fine-structure constant, for the energy of an electron within hydrogen (for a given n and k value):

This allowed for the calculation of expected frequency variations in fine structure lines of hydrogen, a famous problem at the time in 1916 and convinced many of the fine-structure constant and Sommerfeld’s additions to the Bohr theory²¹.

Sommerfeld predicted that the angular momentum of atoms is quantised, and this was to be tested by the Stern-Gerlach experiment. The experiment was done using an oven, two slits, an electromagnet and a hot wire detector.  The oven was heated to produce silver atoms which then passed through the electromagnet and interacted with the screen.

Figure 2: Original apparatus of Stern-Gerlach Experiment²²

The experiment was undertaken in an attempt to settle which theory was correct, the quantum-based theory proposed by Bohr/Sommerfeld, or the classical Larmor theory, a theory which through ‘Larmor precession’, predicted the results to show a clear bright central point at the centre of the screen as no direction of magnetic moment would be preferred²². The Bohr-Sommerfeld theory predicted two points where the atoms would, based on the orientation of the magnetic moment relative to the magnetic field from the electromagnet, be repelled/attracted by the magnet to two potential points on the screen²².

Figure 3: Results of the Stern-Gerlach experiment in 1922²²

The depicted results indicate a minimum at the centre and two clearly separated regions, the result of two beams. However, what was missed by Stern and Gerlach during testing was that the angular momentum of the electron was zero ( $L=0$

), whereas they had assumed the atoms were in a state where the angular momentum was one ( $L=1)$

.  In the actual state, the prediction from both theories was that there should be no splitting without an angular momentum, and so clearly theory needed to be improved to make sense of the experimental result. Space quantisation was observed as a result of the experiment though, and so pointed to a new quantum theory. This problem seemed to have a solution through a postulate produced by Samuel Goudsmit and George Uhlenbeck in 1925. They postulated that particles have an intrinsic angular momentum, what we now term spin, which also can produce a magnetic moment²⁸.

The result is a total angular momentum ( $J$

), consisting of the sum of the angular momentum ( $L$

) and the spin ( $S$

):

This gave the explanation for the Stern-Gerlach experiment, as the two point could be explained by the spin, and indeed is what we know it to be today. Therefore, Stern and Gerlach performed the wrong experiment, an experiment which showed a quantisation of angular momentum by spin and supported the existence for spin, which they were not aware of at the time²⁶.

The implications of the Stern-Gerlach experiment to this day are tremendous, as its findings of quantised angular momentum and spin allowed for the development of a more robust quantum theory. It was a great steppingstone in quantum mechanics being accepted by the rest of physicists. With this change in thinking, more scientists could focus on the quantum field as the next area to be developed.  The experiment consequentially led to the Pauli exclusion principle thanks to the discovery of spin. Wolfgang Pauli presented the original idea in 1925 that no two electrons could be governed by the same quantum description, and so as electrons can only have two possible spins, orbitals could only contain a maximum of two electrons in order to be unique (cannot have the same four quantum numbers)³⁰.

Additionally, it led to further experimental work such as that done by T.E. Phipps and J. B. Taylor in 1927 which exhibited the quantisation of magnetic moment as a consequence of quantised total angular momentum. This was done by showing hydrogen (a single electron) has a magnetic moment exactly equal to one Bohr magneton, a constant named after Niels Bohr³². The Bohr magneton is given by ${\mu }_B=\frac{\mathit{e\hslash }}{2m_e}$

, where $e$

is the elementary charge, $\hslash$

is Planck’s constant divided by two pi (1.055 $\times 10^{–34}{m}^2\mathit{kg}∕s$

) and $m_e$

is the mass of an electron³¹.

The Stern-Gerlach experiment also paved the way for Erwin Schrödinger to propose his quantum mechanical model of the atom in 1926 treating electron with a wave-particle duality, an idea which may not have been possible without the work of Stern and Gerlach²⁹.

Although the Stern-Gerlach experiment did not prove what it set out to initially which was to support the Bohr-Sommerfeld theory or the classic theory, the experiment provided vital information about spin which was theorised later. It is a good example of theory needing to be changed to match reliable experimental results, as good experimental results will always be reliable regardless of theory.

References:

1. Experiment S. Stern-Gerlach Experiment [Internet]. PhET. 2019 [cited 24 May 2019]. Available from: https://phet.colorado.edu/en/simulation/stern-gerlach

2. Massachusetts Institute of Technology. The Stern-Gerlach Experiment [Internet]. Massachussetts: Massachussetts Institute of Technology Physics Department; 2019 [cited 24 May 2019]. Available from: http://web.mit.edu/8.13/www/JLExperiments/JLExp18.pdf

3. Friedrich B. Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics [Internet]. Physics Today. 2003 [cited 24 May 2019]. Available from: https://physicstoday.scitation.org/doi/10.1063/1.1650229

4. The editors of Encyclopaedia Britannica. Bohr model | Description & Development [Internet]. Encyclopedia Britannica. 2019 [cited 24 May 2019]. Available from: https://www.britannica.com/science/Bohr-atomic-model

5. Helmenstine A. What Is the Bohr Model of the Atom? [Internet]. ThoughtCo. 2019 [cited 24 May 2019]. Available from: https://www.thoughtco.com/bohr-model-of-the-atom-603815

6. Halpern J. 6.3: Line Spectra and the Bohr Model [Internet]. Chemistry LibreTexts. 2019 [cited 24 May 2019]. Available from: https://chem.libretexts.org/Courses/University_of_Missouri/MU%3A__1330H_(Keller)/06._Electronic_Structure_of_Atoms/6.3%3A_Line_Spectra_and_the_Bohr_Model

7. Khan Academy. Bohr’s model of hydrogen [Internet]. Khan Academy. 2019 [cited 24 May 2019]. Available from: https://www.khanacademy.org/science/physics/quantum-physics/atoms-and-electrons/a/bohrs-model-of-hydrogen

8. Helmenstine T. What You Need to Know About the Rydberg Formula [Internet]. ThoughtCo. 2019 [cited 24 May 2019]. Available from: https://www.thoughtco.com/what-is-the-rydberg-formula-604285

9. The Editors of Encyclopaedia Brittanica. Stark effect | physics [Internet]. Encyclopedia Britannica. 2019 [cited 24 May 2019]. Available from: https://www.britannica.com/science/Stark-effect

10. Nave R. Zeeman Effect [Internet]. Hyperphysics.phy-astr.gsu.edu. 2019 [cited 24 May 2019]. Available from: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/zeeman.html

11. Chemistry LibreTexts. 8.9: The Allowed Values of J – the Total Angular Momentum Quantum Number [Internet]. Chemistry LibreTexts. 2019 [cited 24 May 2019]. Available from: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Physical_Chemistry_(McQuarrie_and_Simon)/08%3A_Multielectron_Atoms/8.09%3A_The_Allowed_Values_of_J

12. The Editors of Encyclopaedia Britannica. Fine structure | spectroscopy [Internet]. Encyclopedia Britannica. 2019 [cited 24 May 2019]. Available from: https://www.britannica.com/science/fine-structure

13. Kragh H. The Bohr–Sommerfeld Theory [Internet]. Oxfordscholarship.com. 2019 [cited 24 May 2019]. Available from: https://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199654987.001.0001/acprof-9780199654987-chapter-4

14. Chemistry LibreTexts. 349: Analysis of the Stern-Gerlach Experiment [Internet]. Chemistry LibreTexts. 2019 [cited 24 May 2019]. Available from: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Tutorials_(Rioux)/Quantum_Teleportation/349%3A_Analysis_of_the_Stern-Gerlach_Experiment

15. bhatia l. Sommerfeld atomic model,The angular momentum of electron moving in an elliptical orbit is  kh/ 2π. [Internet]. Online Chemistry tutorial that deals with Chemistry and Chemistry Concept. 2019 [cited 24 May 2019]. Available from: https://chemistryonline.guru/sommerfeld-atomic-model/

16. Niaz M. What Can the Bohr-Sommerfeld Model Show Students of Chemistry in the 21st Century [Internet]. Academia. 2019 [cited 24 May 2019]. Available from: https://www.academia.edu/8227328/What_Can_the_Bohr-Sommerfeld_Model_Show_Students_of_Chemistry_in_the_21st_Century

17. Klus H. Sommerfeld’s Atom [Internet]. The Star Garden. 2019 [cited 24 May 2019]. Available from: http://www.thestargarden.co.uk/Sommerfelds-atom.html

18. Oldershaw R. THE MEANING OF THE FINE STRUCTURE CONSTANT [Internet]. Arxiv.org. 2019 [cited 24 May 2019]. Available from: https://arxiv.org/ftp/arxiv/papers/0708/0708.3501.pdf

19. NIST. Current advances: The fine-structure constant [Internet]. Physics.nist.gov. 2019 [cited 24 May 2019]. Available from: https://physics.nist.gov/cuu/Constants/alpha.tml

20. Lamb W. Fine structure of the hydrogen atom [Internet]. Web.mit.edu. 2019 [cited 24 May 2019]. Available from: https://web.mit.edu/8.13/8.13c/references-fall/spectroscopy/lamb-nobel-lecture-1955.pdf

21. Nath B. The Fine Structure Constant [Internet]. Ias.ac.in. 2019 [cited 24 May 2019]. Available from: https://www.ias.ac.in/article/fulltext/reso/020/05/0383-0388

22. Franklin A. Experiment in Physics > Appendix 5: Right Experiment, Wrong Theory: The Stern-Gerlach Experiment (Stanford Encyclopedia of Philosophy) [Internet]. Plato.stanford.edu. 2019 [cited 24 May 2019]. Available from: https://plato.stanford.edu/entries/physics-experiment/app5.html

23. Tuckerman M. Anomalous Zeeman effect. Presentation presented at; 2019; New York University.

24. Cresser J. Particle Spin and the Stern-Gerlach Experiment. [Internet]. 2019 [cited 24 May 2019]. Available from: http://physics.mq.edu.au/~jcresser/Phys301/Chapters/Chapter6.pdf

25. Caltech. [Internet]. Caltech; 2019 [cited 24 May 2019]. Available from: http://www.sophphx.caltech.edu/Physics_7/Experiment_33.pdf

26. Weinert F. Academia [Internet]. Wrong theory—Right experiment: The significance of the Stern-Gerlach experiments. 2019 [cited 24 May 2019]. Available from: https://www.academia.edu/26358945/Wrong_theory_Right_experiment_The_significance_of_the_Stern-Gerlach_experiments

27. The Stern-Gerlach Experiment Revisited [Internet]. 2016 [cited 24 May 2019]. Available from: https://arxiv.org/pdf/1609.09311.pdf

28. Tuckerman M. The Uhlenbeck and Goudsmit proposal. Presentation presented at; 2019; New York University.

29. Khan Academy. The quantum mechanical model of the atom [Internet]. Khan Academy. 2019 [cited 24 May 2019]. Available from: https://www.khanacademy.org/science/physics/quantum-physics/quantum-numbers-and-orbitals/a/the-quantum-mechanical-model-of-the-atom

30. Chemistry LibreTexts. Pauli Exclusion Principle [Internet]. Chemistry LibreTexts. 2019 [cited 24 May 2019]. Available from: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Electronic_Structure_of_Atoms_and_Molecules/Electronic_Configurations/Pauli_Exclusion_Principle

31. Chegg. Chegg.com [Internet]. Chegg.com. 2019 [cited 24 May 2019]. Available from: https://www.chegg.com/homework-help/definitions/bohr-magneton-204

32. Phipps T, Taylor J. The Magnetic Moment of the Hydrogen Atom. Physical Review [Internet]. 1927 [cited 24 May 2019];29(2):309-320. Available from: https://journals.aps.org/pr/abstract/10.1103/PhysRev.29.309