Physics · Atomic and quantum physics · Year 1

Spin, the Zeeman effect and MRI

Why does a beam of silver atoms split in two in a magnet, and how does MRI see the body?

The situation

In a clinic a patient is slid into an MRI scanner. The magnetic field inside it is 1.5 T — tens of thousands of times stronger than the Earth's magnetic field. Every hydrogen nucleus in our body — a proton — behaves like a tiny magnet, but this “little magnet” cannot point in any direction it likes: in 1922 Otto Stern and Walther Gerlach sent a beam of silver atoms through a non-uniform magnetic field, and the beam did not spread out evenly but split cleanly into two parts. Why does this microscopic “little magnet” — the spin — have only two states, and how does a radio wave use it to show the inside of the body?

Topic

Spin, the Zeeman effect and MRI. MRI scanner in a hospital and the Stern–Gerlach experiment

Key ideas

Spin and the Bohr magneton
Zeeman effect
Magnetic resonance

Formulas

μ_{B} = \frac{e ℏ}{2 m_{e}} = 9.274 \cdot 10^{- 24} J/Tl, μ_{z} = - g μ_{B} m

The Bohr magneton is the “unit of measurement” of an atom's magnetic moment. For orbital motion $g_{l} = 1$ , for spin $g_{s} \approx 2.0023$ , $m_{s} = \pm \frac{1}{2}$ .

F_{z} = μ_{z} \frac{\partial B}{\partial z}, z = \frac{μ_{B} \frac{\partial B}{\partial z} L (\frac{L}{2} + D)}{m v^{2}}

Stern–Gerlach: magnet length $L$ , distance from the magnet to the plate $D$ , atomic mass $m$ and speed $v$ .

Δ E = g_{J} μ_{B} B M_{J}, g_{J} = 1 + \frac{J (J + 1) + S (S + 1) - L (L + 1)}{2 J (J + 1)}

In a weak field a level splits into $2 J + 1$ sublevels (Landé factor $g_{J}$ ). For $S = 0$ , $g_{J} = 1$ — the normal Zeeman effect.

f_{L} = \frac{γ}{2 π} B_{0} = 42.58 \frac{MHz}{Tl} \cdot B_{0}, α = γ B_{1} τ

The Larmor (resonance) frequency of protons and the tilt angle produced by an RF pulse at resonance.

\hat{𝐒} = \frac{ℏ}{2} 𝝈, σ_{x} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), σ_{y} = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix}), σ_{z} = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})

The spin operator and the Pauli matrices. If an atom in the “+” state along $z$ is measured with a magnet rotated by an angle $φ$ , then $P (+) = \cos^{2} (φ / 2)$ .

Theory

Spin and magnetic moment

An electron going round a nucleus is a tiny ring-shaped current: its orbital magnetic moment is proportional to its angular momentum, $𝝁_{l} = - \frac{e}{2 m_{e}} 𝐋$ (the gyromagnetic ratio is $e / 2 m_{e}$ ), and its unit is the Bohr magneton $μ_{B} = 9.274 \cdot 10^{- 24}$ J/T. In 1925 Uhlenbeck and Goudsmit proposed that the electron has its own intrinsic angular momentum — spin $s = \frac{1}{2}$ ; its magnetic moment is twice as “strong” ( $g_{s} \approx 2.0023$ ). The spin operator is $\hat{𝐒} = \frac{ℏ}{2} 𝝈$ (Pauli matrices), with eigenvalues $\pm \frac{ℏ}{2}$ ; the Schrödinger equation that includes it is the Pauli equation (with the extra term $μ_{B} 𝝈 \cdot 𝐁$ ). The relativistic Dirac equation “gives birth” to spin and to $g = 2$ by itself and, through its negative-energy solutions, predicted the positron; the difference of $g_{s}$ from 2 (0.1 %) is a correction from quantum electrodynamics.

The Stern–Gerlach experiment

A uniform field only turns a small magnet, but a non-uniform field ( $\partial B / \partial z \neq 0$ ) also pulls it: $F_{z} = μ_{z} \partial B / \partial z$ . In a silver atom the single outer electron is in an $l = 0$ state, so the atom's magnetic moment comes from the electron spin, $μ_{z} \approx \pm μ_{B}$ . According to classical physics the moments point in all directions and the plate should show a smooth smear (try the “Classical” mode); the experiment, however, gave two spots — the projection of angular momentum is quantised. The spots are “lip”-shaped because the gradient is greatest in the middle of the magnet. The atoms have different speeds (the oven depends on $T$ ), which is why the spots are spread out: $z \propto 1 / v^{2}$ , and the most probable speed in the beam is $v_{p} = \sqrt{3 k T / m}$ ; a velocity selector makes the spots more compact.

Measurement and the Pauli matrices

In the “Sequential” experiment only the upper (“+”) beam from the first magnet is let through, and the second magnet is rotated by $φ$ about the beam axis. At $φ = 0$ all atoms go to “+” again, at $φ = 180 °$ all go to “−”, and at $φ = 90 °$ half go one way and half the other: $P (+) = \cos^{2} (φ / 2)$ . The reason is that ${\hat{S}}_{z}$ and ${\hat{S}}_{x}$ do not commute ( $[{\hat{S}}_{x}, {\hat{S}}_{z}] = - i ℏ {\hat{S}}_{y}$ ): they cannot both be measured exactly at the same time, and a measurement along $x$ erases the “memory” of $z$ . The angle between the two-component state vector $(\binom{1}{0})$ and the eigenvectors of the operator ${\hat{σ}}_{n} = \cos φ σ_{z} + \sin φ σ_{x}$ is $φ / 2$ — this is where $\cos^{2} (φ / 2)$ comes from.

The Zeeman, Paschen–Back and Stark effects

The Hamiltonian of a charged particle in a field is $\hat{H} = {(\hat{𝐩} + e 𝐀)}^{2} / 2 m_{e} + U$ ; in a uniform $𝐁$ it yields the term $μ_{B} B ({\hat{L}}_{z} + 2 {\hat{S}}_{z}) / ℏ$ , so an energy level splits into $2 J + 1$ sublevels: $Δ E = g_{J} μ_{B} B M_{J}$ . In the vector model of the atom, $𝐉 = 𝐋 + 𝐒$ precesses about the field, while $𝐋$ and $𝐒$ precess about $𝐉$ ; when angular momenta are added, $j = l \pm \frac{1}{2}$ . In an atom with zero spin ( $g = 1$ ) a line splits into 3 — the normal Zeeman effect (cadmium, 643.8 nm). When spin is present we get the complex (anomalous) effect: sodium $D_{1}$ splits into 4 components and $D_{2}$ into 6 ( $g = 2, \frac{2}{3}, \frac{4}{3}$ ). Selection rules: $Δ M = 0$ ( $π$ , polarised parallel to the field), $Δ M = \pm 1$ ( $σ$ , circularly polarised when viewed along the field, where $π$ is not visible). If the field is stronger than the fine-structure interval, $𝐋$ and $𝐒$ precess separately and the lines gather back into a triplet — the Paschen–Back effect (a few tesla are needed in lithium, tens of tesla in sodium). In an electric field, for the $n = 2$ level of hydrogen the $2 s$ and $2 p$ states are degenerate, and perturbation theory for degenerate levels gives $\pm 3 e a_{0} E$ — the linear Stark effect; in a weak field (smaller than the fine structure) the shift is quadratic.

Fine structure, the Lamb shift and magnetic resonance

According to the Dirac formula the energy level of hydrogen also depends on $j$ : $E_{n j} = - \frac{13.6 eV}{n^{2}} [1 + \frac{α^{2}}{n^{2}} (\frac{n}{j + 1 / 2} - \frac{3}{4})]$ ( $α \approx 1 / 137$ ). For $n = 2$ the $2 P_{3 / 2}$ level lies 10.97 GHz (≈ $4.5 \cdot 10^{- 5}$ eV) above $2 P_{1 / 2}$ — this is the fine structure, the origin of the multiplet structure of spectral lines. In Dirac theory $2 S_{1 / 2}$ and $2 P_{1 / 2}$ have the same energy, but in experiment $2 S_{1 / 2}$ is ≈ 1 058 MHz higher — the Lamb shift (an effect of vacuum fluctuations). In a magnetic field a radio wave excites transitions between the sublevels: for electrons this is electron paramagnetic resonance (EPR, $f = g μ_{B} B / h \approx 28$ GHz/T; in the X-band, at ≈ 9.5 GHz, $B$ ≈ 0.34 T), for protons — NMR and MRI (42.58 MHz/T). The RF field $B_{1}$ tilts the magnetisation $𝐌$ by the angle $α = γ B_{1} τ$ , after which $𝐌$ rotates about $B_{0}$ and induces a signal in the coil. Tissues differ in their relaxation times ( $T_{1}$ , $T_{2}$ ) — this is the source of the contrast in an MRI image.

Identical particles and the Pauli principle

Two electrons cannot be told apart: if we swap them, $| ψ |^{2}$ does not change. So the wave function is either symmetric (particles with integer spin — bosons: the photon, ⁴He) or antisymmetric (particles with half-integer spin — fermions: the electron, proton, neutron). An antisymmetric function becomes zero if two particles are in the same state — this is the Pauli principle: two electrons cannot be in one state (with all quantum numbers, including $m_{s}$ , the same). The periodic table and the filling of shells follow from this (in detail — in the “Many-electron atoms” experiment). The spatial function of electrons with parallel spins is antisymmetric — they stay farther apart and the Coulomb energy falls: this is the exchange interaction. In iron it lines the spins up in parallel and gives ferromagnetism (domains, spin waves — magnons); antiferromagnetic and ferrimagnetic order come from it too. In the gas of free electrons in a metal, the Pauli principle means that only spins near the Fermi surface turn in the field — a weak Pauli paramagnetism; orbital motion gives Landau diamagnetism (1/3 of it for free electrons). In detail — in “Magnetic properties of matter”.

Did you know?

The Stern–Gerlach experiment was carried out in Frankfurt in 1922, while the hypothesis of electron spin was only put forward by Uhlenbeck and Goudsmit in 1925 — spin was “seen” in an experiment before the theory existed.
Most MRI scanners work at a field of 1.5 T, and 3 T ones are also widespread. The magnetic field at the Earth's surface is roughly 25–65 µT, so the MRI field is tens of thousands of times stronger.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Rutherford's experiment — Grade 11
The atomic nucleus and binding — Grade 11
Thermonuclear fusion — Grade 11
Detecting particles — Grade 11
Elementary particles — Grade 11
Many-electron atoms — Year 1
Bands in solids — Year 1
Schrödinger and quantum tunnelling — Year 3
Quantum dots and the potential well — Year 3
The molecule: bonding and spectra — Year 3