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A description of an atomic nucleus in terms of coordinated motion of many nucleons, especially changes in shape and orientation. The nuclear shell model organizes individual nucleon states in a nuclear mean field; the collective nuclear model organizes low-lying energy levels by nuclear deformation, nuclear rotational bands and nuclear quadrupole vibrations. These are complementary organizations of the same nuclear many-body problem. A deformed nuclear mean field can support individual-particle states while its orientation supplies a slow collective coordinate.

In his 1975 lecture, Ben Roy Mottelson described coupling individual-particle motion to nuclear shape coordinates, and distinguished observed rotational spectra from vibrations about spherical equilibrium. This connects the two descriptions without treating the collective coordinates as extra constituent particles.^[ref]

James Rainwater's 1975 account identifies unexpectedly large electric quadrupole moments as a motivation for a deformable nuclear mean field. The challenge concerned their magnitude, even where spherical-shell trends had the correct sign.^[ref]

A simple original potential energy example makes the coupling tangible. Let $q$ be a dimensionless signed shape coordinate for a chosen intrinsic configuration, $C>0$ its restoring stiffness, and $f$ a configuration-dependent coupling, both with units of energy. SupposeThe minimum is $q_{\ast }=f/C$, with energy gain $f^2/(2C)$. For chosen values $C=20u$ and $f=4u$, where $u$ is an arbitrary positive energy unit, $q_{\ast }=0.20$ and the gain is $0.40u$. This illustrates how a configuration can favor deformation when the core can respond. It is not a fitted nuclear surface or a claim that every configuration chooses the same shape; competing configurations and nuclear pairing can favor a spherical minimum.

The predictive attraction is economy: one collective scale relates several energy levels, while electric quadrupole transitions independently test their common structure. Agreement in both is much more restrictive than assigning a separate free parameter to every level. Nuclear magnetic moments test magnetic structure and angular momentum assignments; they are distinct observables from the electric quadrupole moments used to infer charge deformation.

Isotope abundance, nuclear separation energies, spin and nuclear magnetic moments supply complementary tests: abundance concerns which atomic nuclei survive and accumulate, removal energy concerns binding, and magnetic properties test the angular momentum assignments. Agreement across these observables is more informative than fitting one list of numbers.

In her 1963 lecture, Maria Goeppert Mayer recalled excess abundances at $N=50$ and $82$. She contrasted the strong binding of the last neutron in oxygen isotope ${}^{16}\mathrm{O}$ with the weaker extra neutron in oxygen isotope ${}^{17}\mathrm{O}$, and displayed beta decay energies near $N=20$ and $50$. These are compiled empirical patterns, rather than readings from an apparatus she built.^[ref]

Her 1949 letter compared observed spins and nuclear magnetic moments with orbital assignments. For example, lead isotope ${}^{207}\mathrm{Pb}$ was assigned a $p_{1/2}$ neutron hole near $N=126$, whereas bismuth's odd proton began the next shell as $h_{9/2}$. The moments tested orbital character as well as spin.^[ref]

A useful comparison holds $Z$ fixed when comparing neighboring isotopes, and examines odd-even staggering separately from a shell discontinuity. Abundance additionally depends on production and decay history. Neither the abundance pattern nor a successful spin assignment directly images orbiting particles; neither fixes a unique nuclear mean field. The experimental techniques themselves include mass spectrometry and nuclear magnetic resonance, whose preparation and readout differ. The Hofstadter nuclear electron scattering experiment separately constrains nuclear size and nuclear charge distributions, rather than measuring these occupancies.

An experimental atomic physicist who coauthored the molecular-beam method for measuring nuclear magnetic moments. Instead of determining a moment solely from a beam's static deflection, the experiment found the Larmor frequency in a known magnetic field. A resonance-induced loss of focused beam intensity made this frequency measurable with high precision.^[ref]

The interaction of a nuclear magnetic moment with the electronic magnetic field at the nucleus. Within an isolated electronic J manifold, rotational symmetry gives the displayed effective Hamiltonian for angular momenta in hbar units. Its adjacent F intervals obey the Landé interval rule.

Nuclear magnetic resonance detects the response of atomic nuclei with a magnetic moment to a radio frequency magnetic field in a steady magnetic field. Its striking possibility is to measure a nuclear property through an ordinary electrical signal from a bulk sample: an enormous number of tiny magnetic moments can produce a measurable change in a receiving circuit.

For a nucleus with spin angular momentum $\mathbf{J}$, define its gyromagnetic ratio by $\mathbf{\mu }=\gamma \mathbf{J}$. Throughout this article $\gamma$ is angular, measured in radians per second per tesla; $\gamma /(2\pi )$ is the corresponding value in cycles per second per tesla. With ${\mathbf{B}}_0=B_0\hat{\mathbf{z}}$, the Hamiltonian is $H=-\gamma B_0{J}_z$. Adjacent energy levels differ byThus a measured resonance frequency and a calibrated magnetic field determine $|\gamma |$. Converting that into a nuclear magnetic moment also requires the spin quantum number: the maximum projection has magnitude $|\gamma |\hslash I$. This simple experiment does not by itself determine the sign of $\gamma$.

The Rabi molecular beam magnetic resonance experiment detects transitions through changes in a selected molecular beam. The bulk experiments below instead detect electrical consequences of nuclear magnetization in matter at ordinary density. A small Boltzmann distribution population imbalance is multiplied by a very large number of nuclei. For noninteracting spin one-half nuclei of number density $n$ and positive $\gamma$, their equilibrium magnetization isThis follows by weighting the two projections $\pm \gamma \hslash /2$ with their Boltzmann distribution probabilities. It explains both the smallness of the bulk polarization at ordinary temperature and the value of many nuclei contributing to one receiving circuit.

NMR spectroscopy measures resonant responses of atomic nuclei to applied electromagnetic radiation in a magnetic field. In residual-group monitoring of polyurethane formation, distinguishable signal integrals supply a relative residual-group readout; overlap or very weak signals can limit that inference.

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.

Use the isolated nuclear spin Zeeman effect implied by the supplied nuclear magnetic moment, with hyperfine structure neglected. The proton gyromagnetic ratio isThe two spin levels are separated by $g_p{\mu }_N{B}_0$. Resonant nuclear magnetic resonance therefore requires the Larmor frequency

The duration is a Rabi pi pulse: in a frame rotating with the resonant transverse magnetic field, the spin must rotate through $\pi$. If $B_1$ denotes the magnitude of a circularly rotating field with the correct sense of rotation, the Rabi angular frequency is ${\Omega }_R={\gamma }_p{B}_1$. The shortest pulse is then

There is an amplitude convention to specify. If the applied transverse magnetic field is instead the linearly oscillating $B_1\cos (\omega t)\hat{\mathbf{x}}$, its resonant circular component has magnitude $B_1/2$. The rotating-wave approximation gives ${\Omega }_R={\gamma }_p{B}_1/2$ and henceThe wording does not specify this polarization convention, so the two pulse durations should be distinguished. Odd multiples of either shortest Rabi pi pulse also invert the spin within the corresponding ideal model.

For hydrogen, the leading characteristic scales areOrder-one spin and nuclear magnetic factors, and state-dependent factors, are suppressed here. Gross atomic structure is Coulomb binding. Fine structure includes relativistic kinetic and spin-orbit effects, with Thomas precession affecting the spin-orbit coefficient. Hyperfine structure couples electronic motion and spin to the nuclear magnetic moment, whose inverse nuclear mass makes it much weaker. The contact spin interaction can remain in an s state where orbital spin-orbit coupling vanishes.

For positronium, the reduced mass is $m_e/2$. Its gross levels and separation scale areThus the mean ground-state electron–Positron separation is twice the infinite-proton-mass hydrogen value: both means have the same factor $3/2$ multiplying their respective Bohr radii. Retaining proton recoil makes the ratio $2m_p/(m_p+m_e)$, very slightly below two. Separation is not the distance of either positronium constituent from its centre of mass.

Interpret the spin operators in units of $\hslash$, as required by the printed Zeeman term, so $A$ has units of energy. Two spin halves give four eigenstates. At zero field, total spin $F=0,1$ and projection $m_F$ label the singlet and triplet. At high field relative to the hyperfine scale, the individual projections $m_{s,e},m_{s,p}=\pm 1/2$ become approximately good quantum numbers. Total projection remains exact at every field, but $F^2$ does not for the two mixed states.

Choose the quantization axis along $\mathbf{B}$. The aligned-spin states $|\uparrow \uparrow ⟩$ and $|\downarrow \downarrow ⟩$ both have $E=A/4$ because their Zeeman shifts cancel. In the other product-state sector,For physical positronium $A>0$. At $B=0$ there is a triplet at $A/4$ and a singlet at $-3A/4$. At large positive field the mixed states approach $|\uparrow \downarrow ⟩$ and $|\downarrow \uparrow ⟩$, with energies $-A/4+2{\mu }_{B}B$ and $-A/4-2{\mu }_{B}B$ respectively, while the aligned pair remains degenerate at $A/4$.

The dimensionless mixing parameter is $x=4{\mu }_{B}B/A$. A crossover estimate is $x\sim 1$, rather than a sharp transition. The scale $A\sim {\alpha }^4{m}_e{c}^2$ already gives a field of a few tesla. Using the physical hyperfine interval $A/h\simeq 203.39\mathrm{GHz}$ givesThe numerical hyperfine interval is external physical input, not a number supplied by the symbolic Hamiltonian.^[ref] The high-field limit here still presumes that orbital distortion is negligible compared with Coulomb binding.

The gyromagnetic ratio is the signed proportionality coefficient $\mathbf{\mu }=\gamma \mathbf{I}$ between a nuclear magnetic moment and its spin angular momentum. With the angular-frequency convention, ${\omega }_0=|\gamma |B_0$, and $\gamma$ is expressed in rad s${}^{-1}$ T${}^{-1}$.

The deuteron is the bound proton–neutron system. Its ground state has $J^P=1^{+}$ and isospin $T=0$, with binding energy approximately $2.225\mathrm{MeV}$ and no excited bound state. The small binding compared with characteristic nuclear energy scales produces an extended tail. With reduced mass $\mu \simeq m_N/2$, the asymptotic radial wavefunction decays as $e^{-\kappa r}$, whereThe decay length is not a sharp deuteron radius or the force range. It illustrates weak binding outside the region of strong attraction.

The observed total spin is consistent with aligned nucleon spins, $S=1$, and a predominantly $L=0$ spatial state, ${}^3{S}_1$. A central attraction identical in both spin–isospin channels would not distinguish this channel from a spin-singlet proton–neutron channel. In fact the triplet channel binds while the corresponding singlet channel does not. A realistic nuclear force must therefore distinguish these spin–isospin channels, for example through a term $V_{\sigma }(r){\mathbf{\sigma }}_p\cdot {\mathbf{\sigma }}_n$, whose eigenvalues are $+1$ and $-3$ in triplet and singlet states. The absence of a bound dineutron and diproton reinforces the need to distinguish channels; proton–proton Coulomb repulsion is an additional effect, not a complete explanation of the singlet nuclear interaction.

The clearest evidence against a purely central force is the nonzero electric quadrupole moment. A pure isotropic S-wave charge distribution would give zero. The deuteron quadrupole evidence for a tensor force instead points to a nuclear tensor force,It depends on how the spins point relative to the line joining the particles. It preserves total $J$ but couples orbital angular momenta differing by two, producingBoth terms have $J^P=1^{+}$, and their interference permits a quadrupole moment. The 1939 theoretical analysis explicitly connected the measured quadrupole moment with such directional dependence.^[ref]

The nuclear magnetic moment provides a second check. A pure aligned-spin S wave would give approximately ${\mu }_p+{\mu }_n\simeq 0.880{\mu }_N$, close to but larger than the observed deuteron value approximately $0.857{\mu }_N$. The discrepancy is consistent with non-S-wave structure and interaction-current corrections; it is not by itself a model-independent measurement of a D-state probability. The deuteron D-state admixture depends on the adopted Hamiltonian and electromagnetic current. The magnetic moments and mass differences used for these scales are tabulated physical constants.^[ref]

A useful schematic interaction therefore contains central, spin-dependent and tensor terms, and possibly spin–orbit terms in other channels. Long-range one-pion exchange naturally supplies a tensor component, with its finite range set by the pion mass. The deuteron establishes an attractive triplet channel and a noncentral tensor interaction; a spin-independent central potential is insufficient.

In a pure $L=0,S=1$ stretched state both spins point along the total angular momentum, with no orbital contribution. The nuclear magnetic moment would beThe measured $0.857{\mu }_N$ differs. More decisively, a pure S wave spatial state is isotropic and has zero electric quadrupole moment. Its nonzero value requires a nonspherical component of the quantum charge distribution.

Positive parity allows even $L$; coupling $S=1$ to $J=1$ permits $L=0$ and $L=2$. The state contains a deuteron D-state admixture:with radial wavefunctions implicit. A nuclear tensor force couples the orbital components. The quadrupole moment involves their interference as well as a D-state contribution, making it sensitive to a small admixture. This is a quantum angular correlation, not two particles permanently fixed in space.

In the nonrelativistic one-body magnetic-moment approximation,The D-state coupling has $⟨S_z⟩=-1/2$ and $⟨L_z⟩=3/2$ at $J_z=1$; the proton supplies approximately half the relative orbital magnetic moment. ThusThe robust conclusion is a predominantly S wave state with a small tensor-induced D wave component. The 4% estimate is model dependent: exchange currents and other corrections also affect the magnetic moment, and a wavefunction component probability is not itself a uniquely measured observable.^[ref]

For the second isotope, consecutive gap differences are approximately $56.4$ MHz, but the smallest gap is about $1.5$ times this value. The higher-$F$ values for the five gaps are therefore $3/2,5/2,7/2,9/2,11/2$. The six levels run from $F=1/2$ to $11/2$. The electronic state is unchanged, so $J=5/2$ as in d; the endpoints giveFor example, fitting ${\nu }_F$ to $F|A|/h$ with the same equal-weight convention gives $56.408$ MHz. The nearly constant gap differences are compatible with the same leading model of the magnetic-dipole hyperfine interaction and the printed rounding.

Within an unchanged electronic state, and neglecting isotope-dependent changes of the electronic wave function and hyperfine anomaly, $A\propto g_I$. Thusor the reciprocal magnitude is about $1.284$. This is a ratio of nuclear g-factors, not directly a ratio of nuclear magnetic moments: ${\mu }_I=g_{I}I{\mu }_N$, so the moment-magnitude ratio would additionally contain $I_{56}/I_{55}=3/(5/2)$ and is about $0.935$.

The positive transition frequencies do not fix either coupling sign. If a signed ratio is required, equal signs must be supplied as an additional assumption or determined by an ordering-sensitive measurement. No absolute nuclear g-factor is inferred without a calibrated electronic field coefficient. The second ladder in b illustrates the inferred half-integer $F$ sequence without turning that sign ambiguity into a falsely measured ordering.

Edward Mills Purcell, Henry Cutler Torrey, and Robert Vivian Pound placed paraffin wax in a capacitively loaded coaxial cavity near 30 MHz between magnet poles. Coupling loops supplied and extracted power; oscillating and steady fields were perpendicular. Output balanced against a generator reference fed amplification, detection and a microammeter. A slow field sweep revealed absorption above fluctuations at room temperature. Weak drive avoided population equalization. Calibration was rough; field inhomogeneity could contribute to linewidth.^[ref]

These are driven experiments with continuous radiofrequency excitation and changing static field, not demonstrations of modern pulsed Fourier transform nuclear magnetic resonance. Bloch's theoretical paper explicitly distinguishes proposed free precession after a pulse from the forced precession actually studied.^[ref]

An experimental failure need not mean absent nuclear magnetic moments. In the model below, excessive drive reduces the population imbalance through nuclear magnetic resonance saturation; waiting too briefly can leave insufficient polarization. Separately, ordinary electrical losses can cause heating without identifying a nuclear resonance. The absorption experiment measured a balanced electrical signal, rather than using heating as its readout. Useful diagnostic checks include moving off resonance, changing the frequency, checking field uniformity and reducing drive. These are general checks; the historical operations actually documented are specified above.