Exchange Narrowing Research Articles

Nuclear Magnetic Resonance in SolidHe3—The Exchange Bath

We have performed spin-echo experiments on $\ensuremath{\beta}$-phase ${\mathrm{He}}^{3}$ and have measured the exchange interaction $J$ between ${\mathrm{He}}^{3}$ atoms in a consistent way, both by the "exchange narrowing" of the line (exchange lengthening of ${T}_{2}$) and by the strong reduction of ${T}_{1}$ at low magnetic fields ($\ensuremath{\gamma}{H}_{0}\ensuremath{\approx}J$). $J$ is found to increase very rapidly with increasing molar volume, ranging from less than 20 kc/sec at a lattice constant of 3.4 \AA{} to 90 kc/sec at 3.5 \AA{}, and to about 300 kc/sec at 3.6 \AA{}. These exchange interactions are much too small to account for the "departure from the Curie law" found by Adams, Meyer, and Fairbank. It is shown that the susceptibility is, in fact, "Curie law" (in their temperature range) and we believe that they were reporting---as a susceptibility anomaly---the effect of a long exchange bath-lattice relaxation time (observed for the first time in the present experiment). This exchange-lattice relaxation time varies as ${T}^{\ensuremath{-}n}(7<n<11)$ and is $\ensuremath{\approx}100$ sec at 1 \ifmmode^\circ\else\textdegree\fi{}K for $\ensuremath{\beta}$-phase ${\mathrm{He}}^{3}$ of 3.5 \AA{} lattice constant. The results for the exchange-lattice relaxation time are compared with a modification of a theory by Griffiths. A major mystery remains---the exchange-bath heat capacity as measured by the spin-echo method exceeds the calculated value by >${10}^{3}$. The exchange interaction is discussed and compared with the prediction of Saunders and others.

Magnetic Field Quenching of the Suhl-Nakamura Width in Antiferromagnets

In a magnetic material, the nuclear spins are coupled via the exchange of virtual magnons (Suhl-Nakamura interaction). At low temperatures this interaction is mainly transverse, i.e., Ii+Ij−, and is a main contributor to the transverse nuclear relaxation time T2 in materials with high concentrations of nuclear spins. If an external dc field, comparable to the nuclear hyperfine field, is applied perpendicularly to the direction of sublattice magnetization in an antiferromagnet the indirect interaction tends toward a scalar form. The scalar part of the interaction has a vanishing second moment and thus does not contribute to the linewidth; in fact, this term gives rise to a small exchange narrowing of the NMR line. For the F19 resonance in materials, such as FeF2 where the hyperfine field is only of the order of 40 kG, such a narrowing may be observable in moderate fields (∼20 kG).

Elektronenspinresonanz-Untersuchungen zur Aggregation von F-Zentren in KCl

The ESR absorption line of F-centers in additively colored KCl-crystals is narrowed continuously from 46 to 35 Gauss (width between points of maximum slope) by light absorption in the F-band at room temperature. It is shown that this narrowing is due to a continuous changing of the F-centers during the illumination. There is no ESR signal observed which can be attributed to any one of the known F-aggregate centers M, R and N. The changing of the ESR signal is explained as an exchange narrowing which is caused by a loose aggregation of those F-centers which are not transformed into new centers. These aggregates are either pairs or larger clusters of F-centers. According to a calculation of the exchange frequency as a function of the distance between F-centers the observed exchange narrowing corresponds to an average distance of 4 to 5 lattice distances. ENDOR measurements are consistent with this analysis. In addition several observations on the influence of annealing and bleaching at higher temperature on the ESR-spectrum of F-centers in KCl are reported.

Antiferromagnetic and Paramagnetic Resonance in CuF2·2H2O

Antiferromagnetic and paramagnetic resonance absorption has been measured in CuF2·2H2O. At 1.4°K, the antiferromagnetic resonances in zero field occur at 95.84±0.05 kMc and 95.91±0.05 kMc. The easy direction of magnetization occurs at an angle of 3.5° with the c axis. This orientation is in agreement with neutron diffraction and single-crystal susceptibility measurements. Due to strong exchange narrowing, narrow paramagnetic resonance lines were observed. The g values are gξ=2.07, gη=2.08, and gζ=2.42. The principal axes are nearly coinciding with the ligand directions. From the g values, the energy levels and wave functions of the Cu++ ions are determined. A discussion of the anisotropy is given. The dipolar part is calculated numerically, and the anisotropic exchange of pseudodipolar type is determined, using the fact that ω1∼ω2 and the known easy magnetization direction. From the total anisotropy determined in this way, and the perpendicular susceptibility measured by Williams, the antiferromagnetic resonance frequency is predicted to occur at 130 kMc. The agreement with experiment is not unreasonable in view of the reduction of this frequency by the zero-point oscillations of the spin waves. A brief discussion of the superexchange interaction in this crystal is given.

Spin-spin relaxation and magnetic resonances in two copper tutton salts

The real and imaginary components χ′ and χ″ of the paramagnetic susceptibilities of CuCS 2(SO 4) 2. 6H 2O and Cu(NH 4) 2(SO 4) 2. 6H 2O have been studied at frequencies of 0.41, 1.21, 1.76 and 3.2 GHz. The powdered samples were placed in transmission cavities (cf. Fig. 1 and 2) and nearly all measurements have been carried out at 20°K. The accuracy of the absolute values is of the order of 5%. The exchange interaction in the Cs Tutton salt is small while in the NH 4 Tutton salt (Table IV) it is of the same order as the dipole-dipole interaction. This leads to a very different magnetic behaviour as may be seen from the figures 4–9, where χ″ is given in perpendicular magnetic fields, and χ″ as well as χ′ in parallel fields. In the figures 10–13 the data obtained in some of the fields have been plotted as a function of the frequency. In zero field (Fig. 10 and 11) reasonably good agreement with Gauss curves is found for the Cs Tutton salt. The high top value of χ″ and the negative values of χ′ at high frequencies are striking. A line shape function (Form. 6) for intermediate cases having the Gauss and Lorentz shapes as the two limits is proposed. The results for the NH 4 Tutton salt fit well with the shape function which is expected to apply in view of the ratio of exchange to dipole-dipole interaction derived from the specific heat of the spin system. The decrease of width of the χ″ curve in a perpendicular field due to truncation of the Hamiltonian of dipole-dipole interaction is observed for the Cs Tutton salt as well as a weak absorption band at half the field H L of the Larmor resonance (Fig. 4). The exchange narrowing of the Larmor resonance in the NH 4 Tutton salt is observed and, in accordance with the 10/3 factor, the width is found to decrease with increasing field (Fig. 7). In large parallel fields marked resonances at the Larmor frequency and the double Larmor frequency (at constant frequency occuring at the Larmor field and half the Larmor field) are observed in the Cs Tutton salt (Fig. 5, 6 and 13). The absolute values, as well as the ratio of the intensities agree satisfactorily with the formulae taking the hyperfine interaction of the copper nuclei into account, which have been developed by Dr. Caspers in his appendix. On the whole good agreement is found with the theoretical expectations a great part of which may be found in the appendix mentioned.

Broadening of Spin-Phonon Resonance Lines by Exchange and Magnetic Dipole Interactions

The second and fourth moments of the three principal ultrasonic free-spin absorption lines are calculated using a phenomenological form for the spin-phonon interaction. Both exchange and dipole interactions are taken into account, and it is found that exchange causes increased line width in all three cases. For the line at the Larmor frequency, the moments are compared with those for the corresponding photon absorption line, for which exchange narrowing occurs.

Exchange Interaction and Cubic Crystal Field Splitting Parameter of Fe3+ in Spinel Structure

The line widths of paramagnetic resonance absorption of Fe 3+ ions in MgAl 2 O 4 and ZnAl 2 O 4 polycrystals and of Mn 2+ ions in ZnAl 2 O 4 , having the same spinel crystal structure as ferrite, are measured as a function of concentration. The cubic crystal field splitting parameters a of Fe 3+ ion at A - and B -sites are determined from the line widths extrapolated to zero concentration: a A =1.0×10 -2 cm -1 and a B =3.1×10 -2 cm -1 . a B / a A =3.1. On the other hand, the magnitudes of exchange integral between Fe 3+ ions in A - and B -sites or in B - and B -sites are estimated from the decrease of line width due to exchange narrowing as the concentration increases, namely, J A B / g β=1.32×10 5 Oe and J B B / g β=1.90×10 3 Oe. The Curie temperatures obtained from the exchange integral are in agreement with the Curie temperatures of Mg-ferrite and Zn-ferrite measured experimentally.

Spin Resonance of Charge Carriers in Graphite

The observations reported here of the electron spin resonance in quite perfect single crystals of graphite clearly establish that the resonance arises from mobile charge carriers. The line shape is of the Dysonian form which is characteristic of conduction electron spin resonance in metals. The intensity of the spin resonance agrees, both in absolute magnitude and in temperature dependence, with values calculated from the band model of graphite by McClure. The $g$ value of the resonance shows a remarkably large anisotropy which depends strongly on temperature and on the position of the Fermi level with respect to the band edge. At room temperature in pure graphite, $g$ varies from 2.0026\ifmmode\pm\else\textpm\fi{}0.0002 to 2.0495\ifmmode\pm\else\textpm\fi{}0.0002 as the magnetic field is shifted from perpendicular to parallel to the $c$ axis. The $g$-value anisotropy increases with decreasing temperature; ${g}_{\mathrm{II}}$ becomes 2.127 at 77\ifmmode^\circ\else\textdegree\fi{}K while ${g}_{\ensuremath{\perp}}$ remains constant. The line width of the resonance is a few gauss (${T}_{2}=2.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$ sec) which is extremely narrow in comparison with the field shifts caused by changes of anisotropy with temperature. This indicates that for conduction states in graphite, the $g$ value is a strong function of the wave vector and that the line is narrowed by an averaging process in $k$ space. This averaging is similar to that which occurs in motional and exchange narrowing.

PARAMAGNETIC RESONANCE LINE SHAPES

A study of the line shape of the transitions of various concentrations of Cr+++ ion in potassium cobalticyanide and potassium aluminum alum single crystals has been made at 9300 Mc/s. The line shape is found to change gradually from Lorentzian to Gaussian with chromium concentration. Lorentzian shape of gadolinium ion in lanthanum ethyl sulphate and exchange narrowing in case of free radical and copper sulphate are reported. The line width and relative intensity of the transitions in a highly dilute cyanide salt at 4.2° K are plotted and it is found that the intensity followed the maximum of the matrix element squared for various transitions. The value of g is found to be 1.992. The actual concentration of Cr+++ in dilute salts is less than the concentration of the solution from which they are grown by more than a factor of 10 as the dilution is increased.

A contribution to the theory of the exchange narrowing of spectral lines

The intensity distribution of a composite spectral line with exchange narrowing is calculated by integrating a matrix expression derived by Anderson

Theory of the Effects of Exchange on the Nuclear Fine Structure in the Paramagnetic Resonance Spectra of Liquids

A theory of the influence of exchange forces in liquids upon the nuclear hyperfine structure in paramagnetic ``spin resonance'' spectra is presented. Formulas are derived for two limiting cases: for the exchange J much larger than the hyperfine separation A, and for A≪J. It is found that as J increases the hyperfine lines broaden (exchange broadening) and start to shift towards the ``unsplit'' Zeeman frequency. For J≈A, the hyperfine lines have coalesced into a single broad line and for J>A the line narrows (exchange narrowing) about the ``unsplit'' Zeeman frequency. These calculations are discussed in considerable detail. The effect upon the spectrum of electronic-electronic and electronic-nuclear dipolar interactions and the effect of the rapid molecular motions in liquids have also been considered, as well as their interactions with the exchange forces.

Influence of Molecular Structure on Paramagnetic Resonance Line Widths

Paramagnetic resonance absorption has been observed in a series of free radical triarylaminium salts and in a series of substituted 1-picryl-2,2-diarylhydrazyls. Within each series, the members differ from each other only in the substituents placed in certain aromatic ring positions in the molecules. Line-width measurements in fields of 3080 and 2 oersteds are reported for powder samples of these radicals, together with ``g'' values measured in the larger field. A correlation has been established between the line widths of the paramagnetic resonance absorption lines of the radicals and the Hammett sigma values of their substituent groups. In general, the line widths for both series of radicals decreased with increasingly negative values of the Hammett sigma constants of the substituents. Few compounds containing substituents with positive sigma constants have been observed. Relationships between sigma and line widths for this range of sigma values have been proposed on the basis of the expected electronic effects of such substituents, and are consistent with the available data except for halogen-substituted compounds. These results are interpreted in terms of the degree of delocalization of the unpaired electrons in the various free radicals studied. The effect of delocalization upon the exchange interaction of the unpaired electrons is also discussed in the light of the theories of exchange narrowing of paramagnetic resonance absorption lines.

Some measurements on the spectral line shape and width of a paramagnetic resonance absorption line

Paramagnetic resonance of the free radical DPPH is measured at a frequency of 20 Mc s−1 by means of a simple autodyne detector. The results of the experiments are compared with the Anderson-Weiss model of exchange narrowing. In agreement with this theory a Lorentz line shape is found. The half-value width decreases with increasing temperature and is larger than the value at microwave frequencies. The width as a function of concentration shows a sharp increase going from the crystalline state to the solution in a liquid. The imaginary peak susceptibility χ″ is measured and found to be in agreement with the theoretical value.

Relaxation Effects inPara- and Ferromagnetic Resonance

Magnetic resonance experiments have been carried out at 3-cm wavelength in para-magnetic and ferromagnetic samples at very high microwave power levels, in a temperature range between 77\ifmmode^\circ\else\textdegree\fi{}K and 700\ifmmode^\circ\else\textdegree\fi{}K. Changes in the microwave susceptibility and the dc magnetization have been observed for microwave amplitudes between 1 and 50 oersted.For a para-magnetic salt, MnS${\mathrm{O}}_{4}$\ifmmode\cdot\else\textperiodcentered\fi{}4${\mathrm{H}}_{2}$O, these changes are readily interpreted in terms of a spin-lattice relaxation mechanism. The value for the spin-lattice relaxation time is derived in three different ways and agrees well with that obtained by Gorter's nonresonant method.When a large exchange interaction occurs between the spins, the situation above the Curie point can be described in terms of a conversion of magnetic into exchange energy. The magnetic and the spin-exchange systems are not always in thermal equilibrium. The characteristic time for the transfer of energy between these systems is equal to the inverse of the line width, which is given by the Van Vleck-Anderson formula for exchange narrowing. Experimental results for an organic free radical and some ferrites confirm this point of view.Below the Curie temperature the situation is more complicated. The experimental data for several ferrites and supermalloy show qualitatively the same behavior.The absorbed magnetic energy is again converted into exchange energy with a characteristic time which is always shorter than 3\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}8}$ sec. At high temperatures this time is equal to the inverse line width and the transition to the para-magnetic region is continuous. At low temperatures the relaxation time increases roughly inversely proportional to the temperature although the width remains constant. The microwave susceptibility has an anomalous decrease at high power levels. No satisfactory explanation has been found for these effects in existing theories.

The Shape of Exchange Narrowed Paramagnetic Resonance Lines

The Shape of Exchange Narrowed Paramagnetic Resonance Lines

Exchange Narrowing in Paramagnetic Resonance

In this paper the problem of the line shape in paramagnetic resonance when large exchange interaction is present is discussed from the standpoint of a simplified mathematical model. The mathematical model can be called the model of "random frequency modulation": It is assumed that the atom absorbs a single frequency, which varies over a distribution determined by the dipolar local fields, but that this frequency varies randomly in time at a rate determined by the exchange interactions.The predicted line shape in the case in which exchange is large is of resonance type in the observable center of the line, but falls off more rapidly in the wings. This line shape has been verified experimentally in a number of cases. This conclusion seems quite independent of any assumption about the type of random frequency modulation, etc. The quantitative conclusions are reached in the following way: It is suspected, since the exchange motion is the superposition of the effects of a number of neighbors which is not particularly small, that a good approximation to the modulation function is Gaussian noise with a Gaussian spectrum. This, of course, is what would result from the superposition of a large number of rather small effects. Under this assumption both the second moment (which is independent of exchange) and the fourth moment of the line shape can be calculated. This kind of modulation is the simplest one which does give a finite fourth moment; a Markoffian, or "jump," type of modulation, which might seem more reasonable at first, does not. These moments are then compared with the moments computed by Van Vleck [Phys. Rev. 74, 1168 (1948)] to fix the two adjustable parameters, mean square frequency, and average rate of change of frequency, of the theory.The result as to line breadth, which is essentially $\ensuremath{\Delta}\ensuremath{\cong}\frac{{〈(\ensuremath{\Delta}{\ensuremath{\omega}}^{2})〉}_{\mathrm{Av}} \mathrm{d}\mathrm{i}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}\ensuremath{-}\mathrm{d}\mathrm{i}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}}{\frac{J}{\ensuremath{\hbar}}},$ if $J$ is the exchange integral, can be compared with observed line breadths by estimating $J$ from Curie-Weiss constants for a number of materials. The results are quite satisfactory if the theory is extended in two ways: (a) When the exchange frequency is larger than the resonance frequency, it can be shown that the off-diagonal elements of the dipolar interaction must be included, leading to a line-width larger by a factor of roughly 10/3; (b) in a number of cases hyperfine and Stark splitting is contributing importantly to the width.The good agreement with experiment in the cases we have investigated leads us to believe that a quantitative approach to the paramagnetic resonance line breadth problem, using only the already known concepts of dipolar interaction, exchange narrowing, and fine structure splitting, will probably explain all the observed phenomena.

Concerning the Theory of Ferromagnetic Resonance Absorption

The first part of this paper gives a simple quantum-mechanical derivation of Kittel's formula for the resonance frequency of ferromagnetic materials. The interactions between the elementary dipoles are handled directly, rather than through the ad hoc introduction of macroscopic demagnetization factors, as is usually done. In Section 3, Kittel's corrections for the effect of anisotropy on frequency are derived from the microscopic standpoint with a model having quadrupolar coupling between atoms. Section 4 discusses qualitatively the effect of exchange narrowing on the line-width. In Section 5 it is shown that Kittel's relation $g\ensuremath{-}2=2\ensuremath{-}{g}^{\ensuremath{'}}$ connecting the spectroscopic splitting factor $g$ and the gyromagnetic ratio ${g}^{\ensuremath{'}}$ is a general consequence of first-order perturbation theory. Throughout the paper it is stressed that ferromagnetic bodies may have important short-range forces of dipolar structure which arise from anisotropic exchange rather than from true magnetic coupling. It is shown that in the first approximation, inclusion of these anomalous forces does not modify Kittel's results.

Hyperfine Structure and Exchange Narrowing of Paramagnetic Resonance

Discussion of electronic paramagnetic resonance for the free radical a, a-diphenyl β-picryl hydrazyl as observed by its effect on the transmission of microwaves through a TE {sub 01} cavity with a small amount of the free radical placed approximately on the axis of the cavity; the half-width of this resonance at half maximum absorption was 1.45 oersteds.

Some Aspects of Paramagnetic Relaxation

This paper concerns the effect of interactions inside the spin system in giving a finite line width to the energy absorption lines in an oscillating magnetic field. The principal calculations are of absorption coefficients at low frequencies for copper salts. These absorption coefficients refer in most cases to the aperiodic line near zero frequency and not to the Larmor line. The first step is a discussion of the general procedure for reconstructing a shape function $f(\ensuremath{\nu})$ from its moments. The special case in which the zeroth, second, and fourth moments are known arises in the absence of a constant magnetic field, and at the Larmor frequency in a constant magnetic field perpendicular to the oscillating field. These cases are discussed and compared with experiment in Section III; extensive calculations have already been made by Van Vleck in the second case. The magnitude of the exchange coupling is the decisive factor, as pointed out by Gorter and Van Vleck; and various methods of calculating this magnitude from experimental data are given. Low frequency absorption in a perpendicular field is discussed on the basis of a Gaussian approximation in Section IV, and the agreement with the experiments of Volger, Vrijer, and Gorter is good. In Section V it is shown that quantitative calculations only emphasize the discrepancy, pointed out by Broer, between theory and experiment for low frequency absorption in a parallel constant field. An explanation of the discrepancy is given in terms of the difficulty in resolving out the different lines in this case, due to the exchange broadening of the Larmor line, in contrast to the exchange narrowing of the Larmor line in a perpendicular field. In Section VI a calculation of the isolated susceptibility of a spin system is given for strong fields; it is found to be 0.80 of the thermodynamic or adiabatic susceptibility of Casimir and du Pr\'e, even in the absence of exchange.

The Dipolar Broadening of Magnetic Resonance Lines in Crystals

In regular crystals, the width of the absorption lines arising from the magnetic moment of the electron or nucleus is caused primarily by the interaction between the magnetic dipoles. It is prohibitively difficult to determine the precise shape of the absorption line theoretically, but the invariance of the diagonal sum in quantum mechanics permits the calculation of the second moment of the frequency deviation, and hence the r.m.s. line breadth. The latter agrees excellently with the observations of Pake and Purcell on the magnetic absorption of the F nucleus in Ca${\mathrm{F}}_{2}$, both in absolute magnitude, and in the dependence on the direction between the magnetic field and the principal cubic axes. The fourth moment was also computed to examine how good an approximation is the conventional assumption of a Gaussian shape. As long as no is present (the nuclear case) the Gaussian model is moderately good. For the 100 direction in a cubic crystal, the theoretical ratio of root mean fourth to root mean square breadth is 1.25. Pake and Purcell's measurements yield 1.24. A Gaussian model would require 1.32. The theory is extended to include crystals with two kinds of spin moments (two types of nuclei, or simultaneous nuclear and electronic spin). Coupling between unlike moments is less effective (by a factor ⅔ in the r.m.s. width) than that between like in broadening the lines.In the paramagnetic absorption caused by electronic spin, it is imperative to include the effect of coupling. This interaction does not contribute to the second moment, but greatly increases the fourth. As a result, the lines are peaked much more sharply than one would compute from the second moment with the Gaussian model for line shape. This exchange narrowing explains why microwave paramagnetic absorption lines are much narrower than one first conjectures from the amount of dipolar coupling.The theoretical calculations are given in Sections II-IV. The final sections V-VI give the comparison with the experiments of Pake and Purcell, and with the model of Bloembergen, Purcell, and Pound, for r-f absorption in liquids.