Quantum Mechanical Spin and its Uses

What do permanent magnets, nuclear magnetic resonance spectra for chemical analysis, and magnetic resonance images for medical diagnoses have in common? They are all based on the quantum mechanical spin. In this blog post we will take a closer look at this unique quantum effect.


Table salt consists of sodium and chlorine. If we sprinkle table salt over a hot flame the sodium lights up with an intense yellow. This property of sodium is used in sodium vapor lamps, for example, to light up historical buildings with a pleasant, warm color tone. This yellow spectral line of sodium has also gained quite some importance for analytical instruments. It is the basis of measuring methods used to determine the refractive index and optical activity of substances.

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When scientists first investigated the emission spectrum of sodium using higher resolution spectrometers they saw that the yellow spectral line actually consists of two lines next to each other at 589 nanometer wavelength.

Fig. 1: Wolfgang Pauli, Austrian-Swiss physicist and Nobel-prize winner
Source: https://upload.wikimedia.org/wikipedia/commons/4/43/Pauli.jpg

In 1925 Wolfgang Pauli (Fig. 1) introduced a two-valued quantum number for the electron based on this “fine structure” property of the spectra of sodium and other alkali metals. The two values joined the three quantum numbers already known and resulted in the complete description of the electron orbitals in the atom. Several great minds pondered the meaning of this two-value quantum number until Samuel Goudsmith and George Uhlenbeck – contrary to Pauli’s opinion – ascribed it to the angular momentum of the electron and called it “spin”. However, it also became clear that this quantum mechanical spin was not a result of the rotation of the electron because the electron is much too small to produce it. There were also doubts about the value of the spin – which is +/- ½ ħ, whereby ħ is Planck’s constant divided by 2π – because up to then only whole quantum numbers were generally accepted. Finally the quantum mechanical spin was accepted as a property which cannot be derived from classical concepts. Due to its spin, the electron behaves like a tiny rod magnet: it aligns itself according to an external magnetic field – but not in the classically expected way.
The two yellow sodium emission lines arise because the orbiting of the thermally excited electron around the nucleus generates a magnetic field. Depending on whether the spin of the excited electron is parallel or antiparallel to this magnetic field, it has a slightly higher or lower energy. When the electron returns to the ground state the photon which is emitted has therefore one of two slightly different wavelengths. The cause of this is called spin-orbit interaction.

The Stern-Gerlach experiment

In 1921/22 Otto Stern and Walther Gerlach carried out experiments which resulted in a surprise (Fig. 2). They sent a beam of vaporized silver atoms through an inhomogeneous magnetic field. The classic result would have been a wide distribution of the beam according to the magnetic field. Based on the knowledge of the time, Stern and Gerlach expected a split into three beams. In fact, when the silver atoms struck the glass plate positioned after the magnetic field they were split into two beams.

Fig. 2: The Stern-Gerlach experiment: silver atoms travel through an inhomogeneous magnetic field and are deflected up or down depending on their quantum mechanical spin. 1: furnace. 2: beam of silver atoms. 3: inhomogeneous magnetic field. 4: classically expected result. 5: actual result.
Source: https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment#/media/File:Stern-Gerlach_experiment_svg.svg

This splitting into two beams occurs as follows: All except one of the electrons of the silver atoms are in completely filled orbitals. Their magnetic properties therefore add up to zero. The one unpaired electron is in a so-called s-orbital without orbital angular momentum. Therefore solely its quantum mechanical spin affects the magnetic property of the silver atom. Depending on the electron’s direction of spin the silver atom experiences a magnetic force which moves it upwards or downwards in the inhomogeneous magnetic field. This results in the splitting of the beam into two parts. Stern and Gerlach therefore discovered the quantum mechanical spin and the quantization of its direction in the magnetic field, but could not theorize its effect.

A property of the elementary particles

In 1927 during the investigation of the thermal capacity of molecular hydrogen scientists realized that the proton – the electrically positive nucleus of the hydrogen atom – must also have a quantum mechanical spin. The neutron – first discovered in 1932 as the electrically neutral parts of larger atomic nuclei – has a spin, too. In the 1960s it became clear that protons and neutrons consist of quarks. These quarks also have spin and this adds up to the total spin of the protons and neutrons and also the total quantum mechanical spin of the nucleus.

Spin-½ particles such as electrons and quarks are affected by the Pauli Exclusion Principle which states that no two such particles can occupy the same quantum state simultaneously. This principle can be explained by the quantum mechanical wave function and is the basis for accounting for the structure of atoms and molecules and for all matter. Force carriers such as photons have an integer spin and are therefore not affected by the Pauli Exclusion Principle.

Permanent magnets

The most common effect of the electron spin can be seen in permanent magnets which we use every day in thousands of different ways. Iron and certain rare earth metals have several unpaired electrons. The quantum mechanical spin of these unpaired electrons orients itself in parallel to achieve the lowest total energy state. This orientation of the spin not only occurs in the single atom but also over micrometer-wide areas referred to as “Weiss magnetic domains”. The magnetization of the magnetic domains orients itself along an externally applied, strong magnetic field. This remains even when the external magnetic field is switched off, as long as the magnets are not heated above the Curie temperature. This is how we obtain permanent magnets.

Nuclear magnetic resonance

Many atomic nuclei have a quantum mechanical spin and therefore a magnetic moment. Measurement of the nuclear spin is based on the fact that an external magnetic field splits its energy levels into two. The nuclear spin orients itself along or against the magnetic field. To flip the nuclear spin against the magnetic field requires energy. When it flips back into the direction of the magnetic field that same amount of energy is released.

With an external magnetic field of a few Tesla, the energy needed for flipping the nuclear spin is in the energy range of radio frequency photons. The magnetic field of a few Tesla – which is around one-hundred thousand times stronger than the earth’s magnetic field – is nowadays created using superconducting solenoids. If a sample in that magnetic field is irradiated at the radio frequency which corresponds with the energy required to flip the spin then the sample will absorb energy. This measurable effect is called resonance.

Fig. 3: Nuclear magnetic resonance spectrum of the hydrogen nuclei of pure ethanol. The horizontal axis shows the relative shift in parts per million of the resonance frequencies compared to the standard molecule TMS. The vertical axis shows the signal intensity in counts. The multiple splitting of the resonance lines arises due to spin-spin coupling with neighboring hydrogen nuclei. Source: https://de.wikipedia.org/wiki/Kernspinresonanzspektroskopie#/media/File:1H_NMR_Ethanol_Coupling_shown.GIF

The energy difference required to flip the nuclear spin does not solely dependent on the externally applied magnetic field but on the local magnetic field at each nucleus. Within an organic molecule, for example, the individual hydrogen nuclei are slightly differently affected because neighboring electrons and nuclei in the molecule change the local magnetic field. Therefore the hydrogen nuclei in different areas of the molecule have slightly different resonance frequencies. In addition, resonance splitting occurs due to spin-spin couplings with neighboring hydrogen nuclei (Fig. 3). By means of the nuclear magnetic resonance spectrum the structure of the molecule can be explained, molecule types can be distinguished, and concentrations determined. Today’s nuclear magnetic resonance spectrometers do much more than what has been described here but in this post we are only concerned with the basic principles.

Magnetic resonance imaging

In magnetic resonance imaging the strong external magnetic field is superimposed with a gradient – a small change in the magnetic field in an adjustable direction in space. In this way it is possible to use the quantum mechanical spin of the nucleus not only to distinguish between different types of tissue but also determine their spatial location. This is a very complex technical procedure which delivers detailed cross-sections of the insides of our bodies. It has revolutionized medical diagnostics because it provides images of soft tissue without using ionizing radiation and even gives insight into local blood flow and metabolic intensity. Figure 4 (left) shows a magnetic resonance scanner (MRI) and (right) the image of a human knee joint which was created in such a scanner. The image is so detailed that it is possible to see even the fine blood vessels clearly.

Fig. 4: Left: Nuclear magnetic resonance scanner with the superconducting magnet within which the patient is positioned. Source: https://de.wikipedia.org/wiki/Magnetresonanztomographie#/media/File:Modern_3T_MRI.JPG
Right: Magnetic resonance image (MRI) of a human knee, Source: https://upload.wikimedia.org/wikipedia/commons/6/61/Knie_mr.jpg

Summary

It was the yellow light of sodium whose fine structure led to the discovery of the strange properties of the quantum mechanical spin. Using this quantum effect we now gain answers to questions from the fields of physics, chemistry, biochemistry, and medicine – questions which seemed unanswerable only a few decades ago. That this works so well is thanks to the implementation of a soon to be 100-year-old quantum-physical principle in modern technology.

The yellow sodium line which is actually two lines is also the basis for the optical measuring methods used to determine the refractive index and optical activity of substances. If you are interested in the refractive Index of substances, please take a look at our refractometers:

The optical activity of substances can be measured with our polarimeters:

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