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Alexander Pines coauthored a method enhancing nuclear magnetic resonance signals from dilute nuclear spins in solids. Repeated transfer of polarization from abundant spins, usually protons, strengthens the rarer species' signal. Decoupling the abundant spins during observation then improves spectral resolution. The technique separates the problem of building a detectable spin polarization from the problem of measuring a narrow resonance, making otherwise weak solid-state spectra accessible.^[ref]

An isotope of beryllium with 4 protons and 6 neutrons. Its nuclear spin concerns the coupled nuclear quantum state, not the electronic configuration.

An isotope of boron with 5 protons and 5 neutrons. Its nuclear spin concerns the coupled nuclear quantum state, not the electronic configuration.

Fedor Jelezko works on solid-state quantum optics and spin control in diamond. His Ulm University CV identifies him in full and lists the exact 2004 nuclear-spin paper whose first-author byline is F. Jelezko, verifying the connection between the full name and the abbreviated record.^[ref]

That experiment addressed an electron spin coupled to a single Carbon-13 nuclear spin at a nitrogen-vacancy center. A laser initialized and read out the electron-spin-dependent signal, while microwave and radio frequency pulses manipulated the coupled levels. Nuclear-state changes were detected through the electron spin. The team observed nuclear Rabi oscillations and demonstrated a conditional two-qubit gate, providing a physical test of coherent control of an individually addressed spin system.^[ref]

For an original ideal pulse example, let a selected transition have resonant Rabi frequency $f_R=100\mathrm{kHz}$ and start entirely in one level. The two-level model gives transition probabilityA $2.5\mu \mathrm{s}$ pulse gives $P=1/2$, while a $5.0\mu \mathrm{s}$ pulse gives $P=1$. Scanning the pulse duration reveals an oscillation rather than a single population change. These are chosen ideal values; detuning, relaxation and readout noise limit a real measurement.

For electron spin $s=1/2$, first add orbital and spin angular momenta to obtain $j=L\pm 1/2$. For nuclear spin $I=1$, add again to obtain $F=|j-1|,|j-1|+1,\dots ,j+1$. The projection has $2F+1$ possible values. A hydrogenic bound state still requires $L\le n-1$.

An interaction between a nucleus's magnetic or electric multipole moments and its electronic surroundings. Magnetic contact and dipolar terms can couple nuclear spin to electron spin or orbital magnetic moments. The resulting small energy shifts contribute to hyperfine structure; the contact term can also mediate conduction-electron-mediated exchange between nuclear moments in a metal.

An energy level resolved by coupling the nuclear spin $I$ to electronic total angular momentum $J$. For a rotationally invariant Hamiltonian, a fixed $F$ has $2F+1$ degenerate magnetic sublevels. Spectral component counts identify the number of upper hyperfine levels only when lower hyperfine structure, selection rules, unresolved overlaps and missing lines have been accounted for.

The set of hyperfine levels for fixed nuclear spin I and electronic J. Its total angular momenta run from F=|I-J| to F=I+J; each zero-field F level contains 2F+1 magnetic substates. The multiplet is the collection, rather than a synonym for one member level.

Hyperfine structure is the small splitting of atomic or molecular energy levels caused by interactions involving the atomic nucleus's magnetic dipole moment or electric quadrupole moment. It is finer than the electronic fine structure. A leading magnetic-dipole hyperfine interaction couples the nuclear spin $I$ to the electronic angular momentum $J$; their total angular momentum $F$ labels the resulting hyperfine levels.

With dimensionless angular momentum operators, the model $H_{\mathrm{hf}}=A\mathbf{I}\cdot \mathbf{J}$ gives shifts $A[F(F+1)-I(I+1)-J(J+1)]/2$. A hydrogen atom in its electronic ground state has $I=J=1/2$, giving $F=0,1$. Resolved microwave transitions between hyperfine levels permit precise spectroscopy and provide the working transitions in many atomic clocks. Applied magnetic fields can split each hyperfine level further into magnetic sublevels.

For dimensionless angular momentum operators and a Hamiltonian $H=a\mathbf{I}\cdot \mathbf{J}$ at fixed $I,J$, addition of angular momentum gives $E_F=a[F(F+1)-I(I+1)-J(J+1)]/2$. Subtracting neighbouring energy levels proves the displayed rule. For fine structure substitute $\mathbf{L},\mathbf{S}$ and total $J$; for magnetic-dipole hyperfine structure use electronic $J$, nuclear spin $I$ and total $F$. Electric-quadrupole interactions and mixing with nearby energy levels can violate the rule.

For identical isotropic closed-shell atoms of number density $n$, with $Z$ electrons and mean squared radius $⟨r^2⟩$ per electron, the quadratic magnetic field energy is $Z{e}^2{B}^2⟨r^2⟩/(12m_e)$. It follows from the $e^2{\mathbf{A}}^2/(2m_e)$ term with $\mathbf{A}=\mathbf{B}\times \mathbf{r}/2$. Differentiating the energy with respect to field and using $B\simeq {\mu }_{0}H$ yields the negative SI volume magnetic susceptibility. For helium, $Z=2$; for helium-3, combining it with the independent spin-one-half nuclear spin response ${\mu }_{0}n{\mu }_a^2/(k_{B}T)$ gives a density-independent sign-change scale $T_0=3m_e{\mu }_a^2/(e^2{k}_B⟨r^2⟩)$.

A nuclear physicist who authored guidelines for evaluating measured nuclear-structure data. The evaluator's task is to reconcile measurements into an internally consistent set of energy levels, nuclear spins, parities and transition properties, while distinguishing a firm assignment from a tentative one. Such work makes a nuclear-data table an assessed physical model rather than a list of unexamined measurements.^[ref]

A nitrogen-vacancy center is a point defect in diamond consisting of a nitrogen atom occupying a carbon lattice site next to a vacancy defect. The surrounding diamond remains the host crystal; the adjacent impurity and missing atom create a localized electronic system. Its electron spin and optical response make individual defects accessible to quantum control.^[ref]

A2004 experiment coupled the defect's electron spin to one nearby carbon-13 nuclear spin. Laser pulses provided optical preparation/readout, while microwave and radio frequency pulses manipulated the spin system. The researchers observed coherent nuclear-spin oscillations and implemented a conditional two-qubit operation.^[ref] Detecting fluorescence thus provided access to a spin experiment inside a solid, rather than requiring electrical contact to the individual defect.

For an ideal resonantly driven two-state model, a Rabi oscillation gives transition probability $P(t)={\sin }^2(\Omega t/2)$, where $\Omega$ is the angular Rabi frequency. A pulse lasting $\pi /\Omega$ ideally transfers the population, while $2\pi /\Omega$ returns it. This illustrates controlled coherent evolution; actual fluorescence counts and gate accuracy also depend on preparation, readout and loss of coherence.

The dimensionless coefficient in the nuclear-moment relation ${\mathbf{\mu }}_I=g_I{\mu }_N\mathbf{I}$ when I is in hbar units. The saturated projection is $g_{I}I{\mu }_N$. Distinguish a ratio of these coefficients from a ratio of moments when the nuclear spins differ.

The magnetic dipole moment associated with nuclear spin, using dimensionless angular momentum operators and the nuclear magneton ${\mu }_N$. The quoted scalar moment is its expectation in the stretched quantum state, ${\mu }_I=g_I{\mu }_{N}I$. In the same electronic energy level, the leading magnetic-dipole hyperfine structure constant is proportional to ${\mu }_I/I$, apart from isotope-dependent electronic and nuclear-size corrections.

The nuclear magneton is the unit of magnetic momentwhere $m_p$ is the proton mass. For a nuclear spin operator $\mathbf{I}$, the nuclear magnetic moment is $\mathbf{\mu }=g_I{\mu }_N\mathbf{I}/\hslash$. Consequently its gyromagnetic ratio is ${\gamma }_I=g_I{\mu }_N/\hslash$, and its Larmor frequency in a uniform magnetic field of magnitude $B$ is $|{\gamma }_I|B$. The nuclear magneton is smaller than the Bohr magneton by the electron-to-proton mass ratio.

The nuclear neutron scattering length is the effective length coefficient for low-energy nuclear scattering amplitude. For the elementary coherent calculation below, use a bound coherent coefficient $b_j$ for each site, and neglect absorption so that $b_j$ is real. This coefficient describes the atomic nucleus, including its isotope and nuclear spin dependence; it is not proportional to atomic number. Replacing nuclear coefficients by electronic magnetic moments would describe a different scattering channel, with additional direction and spatial-distribution factors, as explained in magnetic neutron diffraction.

The electric quadrupole moment of an atomic nucleus, conventionally specified in its stretched nuclear spin state. It measures a nonspherical nuclear charge distribution. Its interaction with the electronic electric-field gradient contributes to hyperfine structure and generally changes the magnetic-dipole Landé interval rule. A permanent rank-two moment vanishes for nuclear spin less than one.

Restrictions linking the exchange symmetry of nuclear spin and spatial wavefunctions. For identical nuclei, the total wavefunction must be antisymmetric for fermions and symmetric for bosons. Electronic and vibrational exchange parity must be included before assigning rotational parity.

Nuclear structure concerns the states and internal organization of an atomic nucleus, including nuclear energy levels, nuclear spin, spatial distributions and the interactions among nucleons. It connects the nucleus's constituents to observable excitation and decay patterns.

A nuclear level scheme organizes measured transition energies and assignments. Gamma decay constrains differences between levels, while reaction and scattering measurements provide complementary information about allowed transitions and spatial distributions. Several models can describe different aspects of the same nucleus; matching one transition energy alone does not establish a unique microscopic arrangement.