Van Vleck's 1948 theory of dipolar broadening computes the second and fourth moments of magnetic resonance absorption lines in crystals exactly, via the basis invariance of quantum-mechanical traces, but leaves the detailed line profile undetermined. The paper explicitly poses three unresolved problems: the full line shape for the dipolar (and dipolar-plus-exchange) spin Hamiltonian, since finitely many moments do not determine the profile; a quantitative half-width in the exchange-narrowed regime, where the paper offers only a dimensional estimate; and the case of two spin species whose Larmor frequencies differ by an amount comparable with the dipolar broadening, so that the two resonances overlap. Subsequent literature shows: the stochastic theories of Anderson-Weiss (1953) and Kubo-Tomita (1954) resolved the exchange-narrowed half-width at the level of phenomenological correlation times; the rigid-lattice profile was proven non-Gaussian (Lowe-Norberg 1957) and has since been computed only numerically or semi-classically, with first-principles free-induction-decay calculations and their long-time behavior still under active development in 2011-2024. The surviving open core is a microscopic theory of the full line profile, wings and long-time decay included, and of the overlapping two-species case.

Background

In nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR), spins in a static magnetic field H0H_0 absorb an oscillating field near the Larmor frequency νL=gμH0/h\nu_L=g\mu H_0/h, where μ\mu is the nuclear or Bohr magneton and gg the corresponding gg-factor. In a rigid, magnetically concentrated, non-ferromagnetic crystal, the dominant mechanism spreading the absorption over a band of frequencies is the magnetic dipole-dipole interaction between spins; this broadening is 'adiabatic' (the spin system alone is involved), in contrast to the collision- and migration-dominated broadening in gases and liquids treated by Bloembergen, Purcell, and Pound in 1948.

In the 1948 paper at issue, Van Vleck showed that although, in his words, 'it is prohibitively difficult to determine the precise shape of the absorption line theoretically,' the second and fourth moments of the line can be computed exactly. The trick is that both moments can be written as traces (diagonal sums) of commutators of the Hamiltonian with the transverse magnetization, and the invariance of the trace under change of basis lets one evaluate them in the free-spin basis without knowing the astronomically numerous eigenvalues. For a simple cubic lattice of spin-1/2 nuclei (the fluorine nuclei in CaF2_2), the root-mean-square width agreed excellently with the measurements of Pake and Purcell, including its dependence on the orientation of the field relative to the crystal axes; the fourth moment showed the pure-dipolar line is close to, but not exactly, Gaussian (measured ratio of root-mean-fourth to root-mean-square deviation 1.24, theory 1.25, Gaussian 1.32).

For electronic spins one must also include the isotropic exchange interaction, the coupling A~ Sj⋅Sk\tilde A\,\mathbf{S}_j\cdot\mathbf{S}_k between neighboring spins (it arises from the exclusion principle and is negligible for nuclei). Exchange does not change the second moment but strongly enhances the fourth, forcing the profile to be more sharply peaked at center with heavier wings — hence a smaller effective half-width than a Gaussian of the same second moment. Van Vleck named this 'exchange narrowing' and used it to explain why EPR lines in concentrated salts are much narrower than raw dipolar estimates. He could only estimate the half-width heuristically, by analogy with the relaxation theory of liquids, as Δν1/2∼h⟨Δν2⟩/A~\Delta\nu_{1/2}\sim h\langle\Delta\nu^2\rangle/\tilde A — an estimate he himself characterized as conveying little beyond dimensional analysis.

The paper poses the unresolved questions examined here: (i) 'a detailed theory of the line shape which we do not have at present' for the dipolar and dipolar-plus-exchange Hamiltonians; (ii) a quantitative half-width theory in the exchange-narrowed regime; (iii) the general two-species case, where the difference between the two Larmor frequencies is comparable with the dipolar broadening so the two resonances overlap — his two-species moment calculation assumes the resonances do not overlap, and he calls the general case 'an intricate mathematical problem.' Two further announced extensions — non-resonant low-frequency absorption and ferromagnetic crystals — were planned follow-up papers by his group (Wright 1949; Van Vleck 1950) rather than open questions, and the experimental data gaps he noted were rapidly filled by the growth of NMR/EPR spectroscopy.

Two later developments define the framework in which the question is still discussed. First, the absorption profile I(ν)I(\nu) is the Fourier transform of the free induction decay (FID) s(t)s(t), the pulsed-NMR time-domain signal (Lowe and Norberg 1957), so the line-shape question is equivalently a question about the complete time decay of transverse magnetization, including its long-time behavior. Second, the stochastic theories of Anderson and Weiss (1953) and Kubo and Tomita (1954) treat the resonance frequency as randomly modulated by exchange-driven spin motion ('motional narrowing'), yielding a Lorentzian-like center with faster wing falloff and a half-width set by the modulation rate; these are the standard working theories, but their correlation times are phenomenological inputs rather than derived quantities, and they do not constitute a controlled solution of the microscopic lattice Hamiltonian.

Problem Statement

Starting from the microscopic spin Hamiltonian of a rigid crystal in the high-temperature, high-field regime — Zeeman energy plus isotropic exchange plus the full magnetic dipole-dipole interaction — determine the detailed shape of the main magnetic resonance absorption line I(ν)I(\nu), equivalently the complete free induction decay s(t)s(t), going beyond the second- and fourth-moment information that the Van Vleck trace-invariance method provides exactly. In the framework of Van Vleck's moment computations (Sections II–IV of the 1948 paper), the dipolar interaction is treated in the secular approximation, meaning that interaction terms which do not commute with the dominant Zeeman energy are dropped; this is valid when the Larmor frequency — and, for two spin species, the inter-species Larmor difference — greatly exceeds the dipolar broadening, and it underlies questions (1) and (2) below. As left by Van Vleck (1948), there are three distinct open questions: (1) a detailed theory of the line shape — in his words, 'a detailed theory of the line shape which we do not have at present' — for the dipolar and dipolar-plus-exchange Hamiltonians; for the rigid dipolar lattice (no exchange) the full profile is known not to be Gaussian, and the question covers its central shape, its wings, and the associated long-time behavior of s(t)s(t); (2) a quantitative, non-heuristic theory of the half-width in the exchange-narrowed regime, replacing the dimensional estimate Δν1/2∼h⟨Δν2⟩/A~\Delta\nu_{1/2}\sim h\langle\Delta\nu^2\rangle/\tilde A (here hh is Planck's constant, ⟨Δν2⟩\langle\Delta\nu^2\rangle is the second moment of the frequency deviation of the line, and A~\tilde A is the nearest-neighbor exchange coupling constant) with a quantity derived from the Hamiltonian rather than from a phenomenological correlation time; (3) the general case of two spin species whose Larmor frequencies differ by an amount comparable with the dipolar broadening, so that the two resonances overlap and the frequency-band separation underlying the separated-resonance calculation fails; in this regime the secular truncation of the inter-species terms is no longer justified, and the appropriate Hamiltonian treatment of the overlapping resonances — including the inter-species terms dropped under the non-overlap assumption — is itself part of the open question. Any correct theory must be consistent with the exact Van Vleck second and fourth moments. These three questions define the scope of the task. Question (1) covers the full profile — central shape, wings, and the associated long-time behavior of s(t)s(t) — for the dipolar and dipolar-plus-exchange Hamiltonians, and so includes the exchange-narrowed regime that (2) isolates; question (2) is the same result exhibited as an explicit half-width formula, which follows from (1). Question (3) is the overlapping two-species case. The profile is to be obtained from the lattice spin Hamiltonian without phenomenological or fitted inputs.

References

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