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Spin quantum number

physical science Maturity 11-13 Next article →

The spin quantum number is a fundamental value in quantum mechanics. It describes the intrinsic angular momentum of a particle, such as an electron. This property is an inherent characteristic of the particle itself. Scientists use the symbol $s$ to represent this total spin quantum number. Every particle of a specific type shares the exact same value for $s$. For example, every electron in the universe has the same spin value. This value determines how particles behave in magnetic fields and how they occupy space within an atom.

To understand how spin works, we must look at the spin magnetic quantum number, written as $m_s$. This value describes the component of the spin along a specific axis, usually the z-axis. The total spin is quantized, meaning it can only exist in specific, allowed amounts. For an electron, the spin magnetic quantum number can only be +1/2 or -1/2. These two states are often called "spin-up" and "spin-down." This quantization means that an electron cannot have just any amount of rotation. It must choose one of these two specific orientations relative to the chosen axis.

Quantum projection of S onto z for spin half particles
Quantum projection of S onto z for spin half particles

Particles are categorized into two main groups based on their spin values. The first group is called fermions. Fermions, such as electrons, protons, and neutrons, have half-odd-integer spin values. The second group is called bosons. Bosons, such as photons and mesons, have integer spin values. This distinction is vital for how matter is structured. When electrons pair up in an atom, they often adopt opposite spins. A pair in a "spin singlet" state has a total spin of $S = 0$. In contrast, a "triplet" state has a total spin of $S = 1$, with $m_s$ values of -1, 0, or +1.

The discovery of spin was a major step in understanding the atom. Between 1916 and 1925, scientists were working to organize the periodic table. Arnold Sommerfeld had previously proposed three quantum numbers to describe electron orbits. However, his model could not explain the anomalous Zeeman effect, which involves how atoms behave in magnetic fields. In December 1924, Wolfgang Pauli proposed a fourth quantum number to solve this. He hypothesized that outer electrons required a two-valuedness to explain the observed data. This new value became the spin magnetic quantum number.

The term "spin" was suggested by George Uhlenbeck and Samuel Goudsmit. They originally imagined electrons physically rotating like tiny tops. They soon realized this idea was physically impossible. If an electron actually rotated like a top, it would have to move faster than the speed of light. Because of this, scientists replaced the physical spinning image with an abstract mathematical description. This description is supported by the Dirac equation. In 1928, Paul Dirac developed this relativistic wave equation. It correctly predicted the spin magnetic moment while treating the electron as a point-like particle.

We know spin is a real physical phenomenon because of the Stern–Gerlach experiment. In 1922, Otto Stern and Walter Gerlach tested the theory of spatial quantization. They evaporated silver atoms in a vacuum using an electric furnace. They then guided these atoms into a beam through an in-homogeneous magnetic field. Classical physics predicted the atoms would hit a metal plate in a single solid line. Instead, the beam split into two distinct lines. This happened because the single valence electron in silver has an unbalanced spin. This spin creates a magnetic moment, making the atom act like a tiny magnet.

Spin is not limited to electrons; atomic nuclei also possess spin. The nuclear spin is a fixed property of each nucleus. It can be an integer or a half-integer depending on the number of protons and neutrons. For instance, a Nitrogen-14 nucleus has a spin of $s = 1$. This results in three possible orientations: +1, 0, and -1. Because protons and neutrons are about 1836 times heavier than electrons, the nuclear magnetic moment is much smaller. Scientists study these properties using techniques like electron paramagnetic resonance (EPR). This method observes transitions in a magnetic field where only the spin quantum number changes.

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