Observation of Barnett fields in solids by nuclear magnetic resonance

There are two physical mechanisms in which magnetism and mechanical rotation are combined: the Einstein–de Haas (EdH) effect^1,2) and the Barnett effect.^3,4) The former refers to the phenomenon in which, when a material body is magnetized by applying a magnetic field, it starts rotating as a consequence of conservation of the angular momentum of the magnetization and the body. The EdH effect has shown promise in application to nanomechanics and spintronics.^5–8) The reciprocal of the EdH effect, the Barnett effect, posits that an electrically neutral body is magnetized when it is rotated. This magnetization was observed and discussed phenomenologically by introducing an emergent magnetic field (Fig. 1).^3,4) Then the microscopic mechanism of this field was rigorously formulated in terms of quantum mechanics and general relativity.^9–11) According to these theories, the emergent field, which originates from the coupling between an electron spin and mechanical rotation, is the essence of the magnetization related to the Barnett effect.^10,11) This emergent field is hereafter called the Barnett field.

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The Barnett field acting on a spin emerges by coordinate transformation from the laboratory frame to the rotating one:^10–12)

$$\begin{align} H & = UH_{0}U^{\dagger} - i\hbar U \frac{\partial U^{\dagger}}{\partial t} = H_{0} -\mathbf{J}\cdot \mathbf{\Omega}\\ & = \frac{\mathbf{p}^{2}}{2m} - (\mathbf{r}\times\mathbf{p})\cdot\mathbf{\Omega} + \mathbf{I}\cdot\gamma (\mathbf{B} + \mathbf{B}_{\Omega}), \end{align} \tag{ 1 }$$

where H₀ = p²/2m − I · γB is the Hamiltonian of a spin in the laboratory frame, m is the particle's mass, I is the total spin, and B is the external magnetic field. Further, γ = gq/2m is the gyromagnetic ratio, where g is the g-factor, and q is the charge. The unitary transformation is given by U = exp[iJ · Ωt/ħ], where J = r × p + I is the generator of the rotation, and Ω is the angular velocity. The last term in Eq. (1) indicates that the magnetic field B is modified by the Barnett field:

$$\begin{equation} \mathbf{B}_{\Omega} = \frac{2m}{gq}\mathbf{\Omega}. \end{equation} \tag{ 2 }$$

The Barnett field is proportional to the particle's rest mass. Because atomic nuclei are much more massive than electrons, relatively large Barnett fields are expected to act on nuclei. In addition, the Barnett field acting on the nucleus depends on the nuclear g-factor. Thus, a difference in the g-factor causes the magnitude of the Barnett field to vary even in the same material. In this report, we describe the direct observation of Barnett fields as a resonance shift in the nuclear magnetic resonance (NMR) under sample rotation.

The phenomenon of NMR proceeds by energy absorption and emission of an oscillating radio frequency (RF) field, B_x(ω), applied perpendicular to the direction of a static magnetic field, B₀. The condition for NMR is the well-known formula

$$\begin{equation} \omega_{\text{n}} = 2\pi f_{\text{n}} = |\gamma_{\text{n}}\mathbf{B}_{\text{n}}|, \end{equation} \tag{ 3 }$$

where f_n, γ_n, and B_n are the frequency of the RF field, the nuclear gyromagnetic ratio, and the local static field acting on a nucleus, respectively.¹³⁾ Neglecting small contributions from surrounding electron clouds, the local static field acting on the nucleus is the sum of an external field and the Barnett field; that is, B_n = B₀ + B_Ω. Thus, the value of B_Ω can be measured precisely as a resonance shift by measuring either the frequency for resonance under a constant field B₀ or the field for resonance under a constant frequency.¹⁴⁾ To measure B_Ω using NMR, however, the sample and the NMR detection coil should be in the same frame of reference.¹⁵⁾ Therefore, by further extending the coil-spinning technique,¹⁶⁾ both the sample and the tuning circuit for NMR in this experiment were rotated rapidly with respect to the direction of B₀ at exactly the same angular velocity.

In Fig. 2(a), we show a schematic illustration of the experimental assembly used in this study. The assembly consists of two components: the stationary coil placed along B₀ and connected to an NMR spectrometer, and a high-speed rotor consisting of a cylindrical capsule in which a specially arranged tuning circuit is installed. This circuit is composed of two small coils placed perpendicularly and connected in series, and a small capacitor. One of the two coils is arranged parallel to the stationary coil to establish coupling by mutual inductance between the tuning circuit and the stationary coil. We call this the coupling coil. The other coil, the sample coil, holds a sample inside. The RF field in the coupling coil is transmitted to the sample coil and generates the oscillating RF field B_x(ω) to induce an NMR signal. Under this configuration, the sample coil rotates at exactly the same angular velocity as the sample, thereby fulfilling the conditions for observing B_Ω. The rotor is placed inside the stationary coil, and during measurements, it is rotated up to |Ω/2π| = 10 kHz. For Ω ∥ B₀, the sign of Ω is defined as positive.

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Powdered samples of LiF, NaCl, InP, SnO₂, and Si were used for the NMR measurements. To obtain NMR signals with high S/N ratios, ¹¹⁹Sn and ²⁹Si were enriched. Although ²³Na and ⁷Li, with I = 3/2, and ¹¹⁵In, with I = 9/2, have quadrupole moments, no quadrupole splitting was observed in the NMR spectra because of the local cubic symmetry at the nuclear sites in these compounds. SnO₂ has a tetragonal rutile crystalline structure, but ¹¹⁹Sn does not have a quadrupole moment. For signal acquisition, we employed a standard single π/2 pulse technique. The NMR spectra were obtained by fast Fourier transformation (FFT) of the free induction decay signals. All the measurements were performed at room temperature.

Our first step in observing the Barnett field was to record the ¹¹⁵In NMR shifts in the nonmetallic, nonmagnetic compound InP. To check the NMR detection using the tuning circuit described above, we measured the ¹¹⁵In NMR spectra as a function of the externally applied field B₀ without sample rotation. The results are shown in Fig. 2(b), using a white scale for the spectrum profile. From a linear fit of the NMR center frequency to Eq. (3) with B_Ω = 0, we found γ_n for ¹¹⁵In in InP to be |¹¹⁵γ_n|/2π = 9.330 MHz/T, which agrees well with the reported value of 9.3858 MHz/T.¹⁷⁾ This fact indicates that the system was working properly to observe Barnett fields.

We then rotated the rotor while monitoring the NMR signal. In Fig. 2(c), we show the external field dependence of the NMR center frequency at an angular velocity of Ω/2π = +10 kHz parallel to the external field. The NMR center frequencies increase under rotation; that is, the nuclei feel an extra field in addition to the external field. If there is no change in ¹¹⁵γ_n with mechanical rotation, the data can be fitted to f_n = |¹¹⁵γ_n(B₀ + B_Ω)|/2π. The Barnett field B_Ω is estimated to be +1.1 ± 0.10 mT, which is represented by the orange arrows in Figs. 2(c) and 2(d). This is the first direct measurement of the Barnett field. We have confirmed that the Barnett field acting on the nuclei is independent of the external magnetic field [Fig. 2(e)], which implies that the Barnett field emerges because of rotation even under zero external magnetic field.

In Fig. 3(a), we plot the ¹¹⁵In NMR spectra at various values of the angular velocity Ω. The external magnetic field is fixed at B₀ = 2.75 T. Clearly, the NMR center frequency increases linearly with Ω as given by Eqs. (2) and (3). Furthermore, by reversing the rotation direction (Ω < 0), the direction of the NMR shift is also reversed [compare red and blue curves in Fig. 3(a)]; thus, the sign of B_Ω is reversed. The Barnett field acting on each nucleus is proportional to the angular velocity, including its sign.

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The sign of the g-factor of ¹¹⁵In is known to be positive;¹⁷⁾ that is, the nuclear magnetic moment is parallel to its angular momentum. Next, we measured the shifts for nuclei with negative g-factors. Equation (2) predicts that the Barnett fields reveal the sign of the g-factors. From the NMR spectra for ²⁹Si, which has a negative g-factor,¹⁷⁾ the direction of the NMR shift [Fig. 3(b)] is clearly opposite to that for ¹¹⁵In, indicating that the emergent Barnett field is opposite in direction to that for ¹¹⁵In. We collected the NMR shifts for various nuclei as a function of Ω (Fig. 4). In each sample, the NMR shift exhibits linear dependence on Ω. The g-factors for ⁷Li, ¹⁹F, ²³Na, and ¹¹⁵In are known to be positive, whereas those for ²⁹Si and ¹¹⁹Sn are negative.¹⁷⁾ All the nuclei with positive g-factors display positive slopes (red line in Fig. 4), whereas all the nuclei with negative g-factors display a negative slope (blue line), which is consistent with Eq. (2).

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We emphasize that the proposed technique realizes a new way of determining the sign of the g-factor in solids. In some nuclei, the sign is still difficult to determine.^18–20) The scope of solid-state science is currently expanding to involve heavy elements. The recent discovery of NMR signals from ²³⁹Pu was highly newsworthy,¹⁸⁾ and NMR in many nuclei remains to be discovered. For such nuclei, only minute amounts of solid are available for sampling; thus, the proposed technique may provide an exclusive method of resolving this issue because coil spinning is a unique method that enables NMR measurement of minute samples.¹⁶⁾

In the above discussion, we assumed that, for Eq. (2), the renormalization of the rotation effect on nuclear spins is absent in the zeroth-order approximation. With this assumption, Eq. (2) explains the experimental results within the resolution of these measurements, as demonstrated by Fig. 4. The next step is to introduce the renormalization for higher-order coupling.²¹⁾ This effect modifies the nuclear g-factors; namely, the slope deviates from unity in Fig. 4. When considering these modifications, it is helpful to refer to EdH studies of electronic systems.^22–24) In electronic systems, the spectroscopic g-factors are generally different from the gyroscopic g-factors. This difference arises from the fact that the gyroscopic effect combines the total angular momentum with rotation, whereas the Zeeman effect combines the total magnetic moment with a magnetic field.¹⁴⁾ This difference enables estimation of the orbital angular momentum of electrons in solids. In contrast, modification of the nuclear g-factors reveals richer physics. The nucleus contains protons, neutrons, and mesons. All of them contribute to the orbital angular momentum, but only some can contribute to the orbital magnetic moments because neutrons are electrically neutral.^25–27) Therefore, the Barnett field will provide a way to investigate such features of the nuclei, although the experimental resolution is currently not sufficient to estimate this deviation. Improving the field resolution is thus certainly worthwhile, as is applying the proposed technique to a wide range of issues.

In conclusion, we developed a new coil-spinning NMR method of detecting the Barnett field. The Barnett field is observed as an NMR shift proportional to the rotation speed. The sign of the Barnett field depends on both the sign of the nuclear magnetic moment and the rotation direction against the external magnetic field. These features can be used to determine the unknown sign of the nuclear magnetic moment. Our findings provide direct evidence of the coupling between the nuclear magnetic moment and mechanical rotation, and will produce new technology in which mechanical motion manipulates the nuclear spin angular momentum as well as the electron spin angular momentum.

Acknowledgments