Magnetic Hyperfine Splitting

Introduction

In a magnetically ordered solid such as metallic iron or haematite, the Mössbauer spectrum changes dramatically: the single line or doublet opens out into six lines spread over more than ten millimetres per second. This magnetic hyperfine splitting is the nuclear Zeeman effect produced by an internal magnetic field of tens of tesla at the iron nucleus. Reading the sextet reveals the strength of that field, whether a material is magnetically ordered, and even how large its crystallites are.

Core explanation

The nuclear Zeeman effect. A nucleus with spin I has a magnetic moment. In a magnetic field B, each nuclear level splits into 2I + 1 sublevels labelled by m I, with energies E = −g N μ N B m I, where μ N is the nuclear magneton and g N the nuclear g factor.

For ⁵⁷Fe: - the ground state (I = 1/2) splits into two sublevels, m I = ±1/2; - the excited state (I = 3/2) splits into four sublevels, m I = ±3/2, ±1/2.

The two states have g factors of opposite sign, so the ordering of the sublevels is inverted between them.

Selection rules. The 14.4 keV gamma transition is a magnetic dipole (M1) transition, which obeys Δm I = 0, ±1. Of the eight possible combinations between two ground and four excited sublevels, the two with Δm I = ±2 are forbidden. That leaves six allowed lines , the characteristic sextet.

Relative intensities. The line strengths follow from angular-momentum coupling coefficients. Numbering the lines from most negative to most positive velocity: - lines 1 and 6 (Δm I = ±1, involving ±3/2): relative intensity 3; - lines 2 and 5 (Δm I = 0): relative intensity 2 in a random powder; - lines 3 and 4 (Δm I = ∓1, involving ±1/2): relative intensity 1.

So an unoriented powder gives 3:2:1:1:2:3 . Lines 2 and 5 depend on the angle θ between the hyperfine field and the gamma-ray direction, with relative intensity 4sin²θ/(1 + cos²θ). They vanish when the field is parallel to the beam and reach 4 when it is perpendicular.

Where the field comes from. The internal field is not an applied field. It arises mostly from the Fermi contact term: unpaired 3d electrons polarise the paired core s electrons, giving a net spin density at the nucleus. Smaller orbital and dipolar terms add to it. For high-spin Fe(III) the field is large, typically 45–55 T in oxides; metallic α-iron has about 33 T at room temperature.

When the sextet appears. A sextet is seen only if the electronic magnetisation stays fixed for longer than the nuclear Larmor precession time (a few nanoseconds). In paramagnets at ambient temperature the electron spins fluctuate far faster, the field averages to zero, and only a singlet or doublet remains.

Step-by-step reasoning

To build a sextet from first principles:

1. Split the ground state into 2 sublevels and the excited state into 4. 2. Draw all eight transitions and strike out the two with Δm I = ±2. 3. Place the six remaining lines symmetrically about the isomer shift. 4. Assign intensities 3, 2, 1, 1, 2, 3 for a powder. 5. Add any quadrupole shift, which displaces the outer and inner pairs in opposite directions.

Visual explanation

Picture the ground level split into two rungs and the excited level into four rungs above it. Six arrows connect them. On the spectrum below, six dips appear: the outermost deepest, the next pair medium, and the central pair shallowest, like a symmetric staircase descending towards the middle.

Real-world analogy

A choir singing one note sounds like a single tone. If the singers are split into groups told to sing slightly sharp or flat by set amounts, you hear several distinct notes. The internal magnetic field is the conductor assigning those offsets, and the selection rules decide which groups are allowed to sing.

Real-world example

A thin foil of α-iron gives a sharp sextet with outer lines near ±5.3 mm s⁻¹. Because its hyperfine field is well known, this foil is the standard used to calibrate the velocity scale of almost every ⁵⁷Fe spectrometer, and isomer shifts are quoted relative to its centroid.

Why?

Why do very small iron oxide nanoparticles give a doublet at room temperature but a sextet when cooled? In tiny particles the whole magnetisation can flip thermally. If flipping is faster than nuclear precession, the field averages away; cooling slows the flipping until a sextet reappears below the so-called blocking temperature.

Common misconception

"A sextet means six different iron sites." One magnetic site already produces six lines. Several sites give overlapping sextets, recognised by extra lines or by unexpected intensity ratios.

Worked example

Question: α-Iron gives outer lines 10.62 mm s⁻¹ apart for a field of 33.0 T. A sample shows outer lines 16.54 mm s⁻¹ apart. Estimate its hyperfine field.

Reasoning: Line separations scale linearly with field. B = 33.0 T × (16.54/10.62) = 33.0 × 1.557 ≈ 51.4 T.

Answer: About 51 T, typical of high-spin Fe(III) in a magnetically ordered oxide such as haematite.

Quick check

1. Which two of the eight possible transitions between the split ⁵⁷Fe levels are missing from a sextet, and why? Answer: The two with Δm I = ±2 (between m I = ±1/2 and ∓3/2), because the M1 selection rule allows only Δm I = 0, ±1.

Exam focus

Examiners expect you to count sublevels (2 and 4), apply Δm I = 0, ±1 to obtain six lines, quote the 3:2:1:1:2:3 powder ratio, and scale a hyperfine field from α-iron calibration. Be ready to explain why a paramagnet at room temperature shows no sextet.

Advanced insight

Applying an external field of several tesla to a sample is a powerful diagnostic. In an antiferromagnet the applied field adds to one sublattice and subtracts from the other, splitting the sextet; in a ferrimagnet such as magnetite it separates tetrahedral and octahedral sites. Because the contact field is negative for iron, the measured total field actually decreases when a paramagnetic high-spin complex is magnetised parallel to the applied field.

Summary

A magnetic field at the ⁵⁷Fe nucleus splits the ground state into two and the excited state into four sublevels. The M1 selection rule Δm I = 0, ±1 allows six transitions, giving a sextet with 3:2:1:1:2:3 powder intensities. The field, mainly from core polarisation, is about 33 T in α-iron and around 50 T in iron(III) oxides, and it appears only when electron spins are static on the nuclear timescale.

Practice questions

1. How many sublevels do the ⁵⁷Fe ground and excited states split into in a magnetic field? Answer: The ground state (I = 1/2) splits into two; the excited state (I = 3/2) splits into four. 2. State the powder intensity ratio of a ⁵⁷Fe sextet and identify which lines are angle-dependent. Answer: 3:2:1:1:2:3; lines 2 and 5 (Δm I = 0) vary with the angle between the hyperfine field and the gamma-ray beam. 3. A sample shows outer lines 8.00 mm s⁻¹ apart. Using α-iron (10.62 mm s⁻¹ for 33.0 T), estimate the hyperfine field. Answer: B = 33.0 × (8.00/10.62) ≈ 24.9 T. 4. Explain why a magnetically ordered sample may give a sextet at 4 K but a doublet at room temperature. Answer: At room temperature the magnetisation fluctuates faster than the nuclear precession time, averaging the field to zero; at 4 K the fluctuations are slow enough for the static field, and hence the sextet, to be observed.