Nuclear Spin Relaxation and Electric Quadrupole Frequency Shift of ¹²⁹ Xe- ¹³¹ Xe in Cubic Stemless MEMS Atomic Vapor Cells

Abstract

Research on the electric quadrupole frequency shift mechanism and nuclear spin relaxation of ¹²⁹Xe–¹³¹Xe in alkali metal atomic vapor cells fabricated using microelectromechanical systems (MEMS) technology remains in its early stages. Therefore, an in-depth investigation in this field is crucial for enhancing the performance of miniaturized atomic gyroscopes and other quantum devices. This study provides a comprehensive analysis of the electric quadrupole frequency shift and relaxation of ¹²⁹Xe–¹³¹Xe nuclear spins in a stemless cubic MEMS vapor cell. To elucidate the microscopic characteristics of wall interactions, nuclear spin quadrupole interactions with mixed silicon and glass surfaces in MEMS vapor cells were examined. The influence of the angle between the vapor cell’s geometric axis and the applied magnetic field on the electric quadrupole frequency shift was investigated, leading to the determination of the vapor cell’s asymmetry parameter η and the average twist angle ⟨θ⟩ of nuclear spin-wall collisions. Furthermore, the temperature dependence of the electric quadrupole frequency shift in ¹³¹Xe atoms was characterized, enabling the extraction of the desorption activation energy (E _A ) of ¹³¹Xe on the MEMS vapor cell surface. By analyzing the free induction decay (FID) signals of ¹²⁹Xe and ¹³¹Xe at varying temperatures, the relationship between their relaxation rates and temperature was established. These findings offer valuable insights for optimizing nuclear spin relaxation times in microatomic gyroscopes.

1. Introduction

Hyperpolarized rare gas nuclear spins have important applications in many fields, such as magnetic resonance imaging, − rare gas comagnetometry, nuclear magnetic resonance gyroscope (NMRG), spin-exchange-relaxation-free (SERF) gyroscope, fundamental symmetry investigations, , and searches for long-range nuclear-spin-dependent forces. Among these, for polarized antimagnetic atoms with nuclear spins I ≥ 1 (e.g., ²⁰¹Hg, ¹⁰⁹Cd, ¹³¹Xe, ⁸³Kr, and ²¹Ne), which have a quadrupolar coupling between the nuclear electric quadrupole moment and the electric field gradient (EFG) at the vapor cell wall, this coupling serves as the primary mechanism governing relaxation and quadrupolar frequency shifts. , Studying nuclear spin relaxation and frequency shifts is essential for enhancing the performance of atomic devices.

Cohen–Tannoudji first introduced the theory of nuclear spin relaxation in antimagnetic atoms due to electric quadrupole interactions in his study on the spin relaxation of ²⁰¹Hg. The detection of an additional frequency progression in the FID signal of ²⁰¹Hg indicates unequal splitting of the nuclear Zeeman energy levels. The observed progression frequency varies with the angle between the cell symmetry axis and the applied magnetic field. These characteristics are consistent with quadrupole interactions, suggesting that the primary wall interaction of antimagnetic atoms arises from the quadrupole coupling between the nuclear quadrupole moment and the EFG at the vapor cell wall. Volk et al. observed similar phenomena in spin-polarized ⁸³Kr and ¹³¹Xe nuclei and developed a semiquantitative theory describing coherent wall quadrupole interactions in noble gas nuclear spins, , which, however, did not account for the dependence of the progression frequency on the asymmetry of the vapor cell.

Wu et al. observed stronger quadrupolar interactions in a series of studies conducted by the Happer group at Princeton. − By using highly asymmetric cells, they were able to distinguish quadrupolar resonance lines with clarity. They also developed a more comprehensive micro-perturbation theory of wall interactions that accounts for pressure-dependent diffusion and cell shape, with particular emphasis on coherent quadrupolar interactions. Butscher et al. described the electric quadrupole shift and relaxation rates as temperature-dependent phenomena using a surface interaction model for Xe atoms adsorbed on glass surfaces. Feng, Sheng et al. systematically investigated the influence of the relative orientation between the cubic vapor cell’s geometrical axis and the external magnetic field on the electric quadrupole shift. They also developed the methodology for quantifying and controlling this variation.

Because of the large surface-to-volume ratios of micro atomic vapor cells, the development of these cells has increased the importance of studying the electric quadrupole wall interactions of polarized antimagnetic atoms. To optimize and amplify nuclear quadrupole resonance (NQR) in the vapor cell and achieve a large nuclear quadrupole shift, Donley et al. used a 1 mm³ volume MEMS atomic vapor cell made of a different material, ensuring that the cubic cell no longer had cubic symmetry. This approach was employed to investigate the crossover between the NMR and NQR interaction regimes. Chen et al. investigated the electric quadrupole interaction between ¹³¹Xe and silicon using a micro vapor cell fabricated through the MEMS process , and obtained the desorption activation energy required to escape from the silicon surface as E _si = 0.009 eV. In summary, research on the spin relaxation and electric quadrupole interactions of Xe nuclei inside a still-developing micro vapor cell made using MEMS-based techniques is extremely limited. Note that current micro vapor cells typically use silicon and glass materials. There are no micro vapor cells fabricated purely from glass. Although traditional glass-blown vapor cells are fabricated without other materials, these vapor cells include a stem, which could contribute to quadrupolar frequency shifts. Moreover, the stem cannot be properly controlled as it is handmade using a torch.

Therefore, in this paper, the nuclear spin relaxation and electric quadrupole frequency shift of ¹²⁹Xe–¹³¹Xe atoms in the fabricated MEMS cubic atomic vapor cell, which is nearly entirely fabricated by glass are systematically investigated. The effect of the angle ψ between the geometrical and magnetic field axes of the cubic vapor cell on the frequency shift of the NQR was investigated by varying the direction of the magnetic field. The cosine relationship between the frequency shift and the azimuthal angle ψ is elucidated to obtain the asymmetric parameter of the atomic vapor cell and the average twisting angle ⟨θ⟩ = 35.1 μrad of the ¹³¹Xe spin for each wall collision. The temperature behavior of the quadrupole interaction of ¹³¹Xe atoms with the wall surface was described by varying the temperature and detecting the FID signal of ¹³¹Xe. This facilitated the determination of the desorption activation energy of ¹³¹Xe atoms on the wall surface, which was found to be E _A = −0.102 ± 0.01 eV. By measuring the relaxation times of ¹²⁹Xe and ¹³¹Xe at different temperatures, the relationship between the linear variation of the relaxation rate of ¹²⁹Xe with the number density of alkali-metal atoms and the change rule, where the relaxation rate of ¹³¹Xe decreases exponentially and then increases with temperature have been obtained, and the physical mechanism leading to this rule has been revealed.

2. Experimental Section

To investigate the electric quadrupole interaction between the ¹³¹Xe nuclear spin and the walls of the MEMS vapor cell, it is essential to first develop a theoretical interaction model. Figure presents a schematic depiction of the electric quadrupole interaction occurring when ¹³¹Xe collides with the vapor cell wall.

1.

Schematic diagram of ¹³¹Xe colliding with the vapor cell wall.

The ¹³¹Xe collision rate τ _v with the vapor cell wall is given by kinetic theory. If the energy loss during the inelastic collision exceeds the kinetic energy of the incident atoms, then ¹³¹Xe atoms become adsorbed on the surface. In such wall collisions, the average dwell time on the wall, τ _s, is limited by the rate of thermally activated desorption. Furthermore, the relative movement of the adsorbed ¹³¹Xe atoms, which results from thermal vibrations and surface diffusion, causes the interaction potential experienced by the nuclei to fluctuate over time. Additionally, thermal activation causes ¹³¹Xe to jump with a jump time τ′_s between nearby energy vantage points. The interaction times for each phase in Figure are defined as follows:

$$\{\begin{array}{l}\frac{1}{{\tau }_v}=\frac{{\mathit{v̅}}_{131}S}{4V}\\ \frac{1}{{\tau }_s}=\frac{1}{{\tau }_{s0}}\exp (-E_{\mathit{A}}/k_{\mathrm{B}}T)\\ \frac{1}{{{\tau }^{\prime }}_s}=\frac{1}{{{\tau }^{\prime }}_{s0}}\exp (-E_D/k_{\mathrm{B}}T)\end{array}$$ 1

where ${\mathit{v̅}}_{131}=\sqrt{\frac{8k_{\mathrm{B}}T}{\pi m_{131}}}$ is the average molecular thermal movement velocity of ¹³¹Xe; k _B = 1.380649 × 10^–23 J/K is the Boltzmann constant; S and V are the surface area and volume of the cell chamber; E _A is the desorption activation energy of ¹³¹Xe at the wall surface; and 1/τ_{s 0} is the escape rate of the atoms away from the surface, which is related to the vibrational frequency of the adsorbed atoms perpendicular to the surface in the surface potential well. E _D is the diffusion activation energy of ¹³¹Xe, and 1/τ′₀ is the ¹³¹Xe atom migration rate at the surface, which is related to the vibrational frequency of the adsorbed atoms parallel to the surface in the surface potential well. During the dwell time τ _s of ¹³¹Xe on the vapor cell wall, the spin-polarized nuclei undergo a process of electric quadrupole interaction with the EFG. This interaction affects the magnetic resonance behavior of the polarized nuclear spin ensemble, resulting in shifts of the progression frequency, as well as wall-induced contributions to nuclear polarization relaxation.

The Hamiltonian for the electric quadrupole interaction of ¹³¹Xe with the vapor cell wall is ,

$$H_Q=\frac{1}{6}\sum _{i,j}{Q}_{ij}\frac{{\partial }^2{V}_{\mathrm{wall}}}{\partial x_i\partial x_j}$$ 2

where Q_ij is the electric quadrupole moment tensor of the atom, $\frac{{\partial }^2{V}_{wall}}{\partial x_i\partial x_j}$ is the microscopic EFG at the surface of the vapor cell wall, V_wall is the coupling of the surface potential to the electric quadrupole moment tensor Q_ij of the adsorbed atom. The microscopic EFG is generated by polar groups such as −OH and −ONa on the surface and electrons in the conductive band on the metal surface. These field gradients fluctuate with time due to the motion of adsorbed atoms on the walls.

For ¹³¹Xe, the second-order transition frequency is

$$\{\begin{array}{l}{\mathrm{\Omega }}_{-3/2,-1/2}={\mathrm{\Omega }}_0-\mathrm{\Delta }\mathrm{\Omega }/2\\ {\mathrm{\Omega }}_{-1/2,1/2}={\mathrm{\Omega }}_0-\frac{3{(\mathrm{\Delta }\mathrm{\Omega })}^2}{16\mathrm{\Delta }\mathrm{\Omega }}\\ {\mathrm{\Omega }}_{1/2,3/2}={\mathrm{\Omega }}_0+\mathrm{\Delta }\mathrm{\Omega }/2\end{array}$$ 3

where ${\mathrm{\Omega }}_0={[{({\omega }_0-\omega )}^2+{\omega }_1^2]}^{1/2}$ is the nuclear progression frequency of ¹³¹Xe and ΔΩ is the electric quadrupole frequency shift of ¹³¹Xe.

Perturbation theory addresses in detail the NQR frequency shifts in NMR spectra, taking into account pressure-dependent diffusion and cell shape, and expresses the results as a microscopic description of the interactions. Neglecting the complexity of diffusion, the electric quadrupole frequency shift of |−3/2⟩⟨−1/2| and |1/2⟩⟨3/2| coherence is expressed as

$$\mathrm{\Delta }\mathrm{\Omega }=\pm \frac{{\mathit{v̅}}_{131}S}{2V}\frac{1}{2I-1}{\int }_S\frac{d{S}^{\prime }}{S}⟨\theta ⟩(\frac{3}{2}{\cos }^2\psi -\frac{1}{2})$$ 4

where ⟨θ⟩ is the average twist angle during each collision, S represents the internal surface area of the cell chamber, V is the volume of the vapor cell chamber, and ψ is the angle between the local surface normal of the cell wall and the direction of the magnetic field. S’ is the area of the different material regions of the wall. ΔΩ demonstrates that the frequency shift arises from the integral of electric quadrupole interactions over the entire cell wall, ensuring the stability of the average quadrupole interaction-induced frequency shift.

Here, ⟨θ⟩ is set in the integral to allow for the possibility that the cell wall is composed of different materials. Accordingly, based on the materials composing the walls of the fabricated cubic MEMS atomic vapor cells, the vapor cell was divided into two regions: the glass wall section and the silicon wall section, with the desorption activation energies of the two parts defined as E _g and E _Si, respectively, and the average twisting angles of each collision defined as ΔΩ and ⟨θ _Si⟩, respectively. At this point eq can be rewritten as

$$\begin{array}{l}\mathrm{\Delta }\mathrm{\Omega }=\mathrm{\Delta }{\mathrm{\Omega }}_g+\mathrm{\Delta }{\mathrm{\Omega }}_{\mathrm{Si}}\\ =\pm \frac{{\mathit{v̅}}_{131}S}{2V}\frac{1}{2I-1}{\int }_{S_g}\frac{d{S}_g}{S}⟨{\theta }_{\mathrm{g}}⟩(\frac{3}{2}{\cos }^2\psi -\frac{1}{2})\\ \pm \frac{{\mathit{v̅}}_{131}S}{2V}\frac{1}{2I-1}{\int }_{S_{\mathrm{Si}}}\frac{d{S}_{\mathrm{Si}}}{S}⟨{\theta }_{\mathrm{Si}}⟩(\frac{3}{2}{\cos }^2\psi -\frac{1}{2})\\ =\pm \frac{{\mathit{v̅}}_{131}S}{8V}(3{\cos }^2\psi -1)({\eta }_1⟨{\theta }_{\mathrm{g}}⟩+{\eta }_2⟨{\theta }_{\mathrm{Si}}⟩)\end{array}$$ 5

where η ₁ and η ₂ are the asymmetry parameters of the vapor cell, ⟨θ _g⟩ = 38 μrad for Pyrex glass.

Equations and indicate unequal splitting between the ¹³¹Xe Seeman energy, and the ΔΩ of ¹³¹Xe is found to be related to the following factors: the shape of the cell chamber, the material of the walls, the shapes and proportions occupied by the different materials, and the angle of the cell relative to the external magnetic field.

3. Results and Discussion

3.1. Experiment Establishment

The micro alkali-metal atomic vapor cell, shown in Figure a, is fabricated using MEMS technology and consists of multiple layers: top glass, silicon, intermediate glass, and bottom glass, with respective thicknesses of 0.5 mm, 0.5 mm, 2 mm, 0.5 mm, and 0.5 mm. These layers are bonded through anodic bonding, , ensuring excellent hermeticity. The vapor cell chip has an overall size of 5 × 5 × 4 mm, with an internal chamber measuring 3 × 3 × 3 mm, meeting the miniaturization and mass production requirements for quantum devices such as atomic magnetometers and multilaser channel atomic gyroscopes. The chamber is filled with an appropriate amount of Rb, 5 Torr ¹²⁹Xe, 15 Torr ¹³¹Xe, 600 Torr N₂, and 5 Torr H₂. The sidewalls of the chamber are specially molded to reduce the side roughness to a S_a of 22 nm.

2.

(a) MEMS-fabricated atomic vapor cells featuring cubic chambers; (b) schematic diagram of the vapor cell coordinate system and definition of the magnetic field variation. Test system schematic (c) and platform (d). DFB: distributed feedback laser; λ/2: half-wave plate; PBS: polarized beam splitter; CL: convex lens; λ/4: quarter-wave plate; LIA: lock-in amplifier; DAQ: data acquisition system.

As depicted in Figure c,d, a test system based on the absorption method was constructed to detect signals associated with the fabricated MEMS atomic vapor cells. A 400 μm-diameter multimode fiber, connected to the heated laser, aligns with the vapor cell, which is subsequently heated to the desired temperature using a 1550 nm wavelength laser. To enhance temperature uniformity during heating, poly­(ether ether ketone) (PEEK) is employed as thermal insulation around the vapor cell.

The vapor cell is heated using a divergent 1550 nm laser beam to ensure uniform illumination of the entire cell volume rather than localized spot heating. To further enhance the heating uniformity, dual laser heating probes are symmetrically arranged around the cell. A PID controller is implemented to regulate both the laser power and cell temperature through negative feedback control. The temperature stability of the vapor depends not only on the heating power but also on the material’s thermal properties, including heat capacity, thermal conductivity, and insulation performance. Through systematic optimization, we achieved precise temperature control with fluctuations within ±10 mK, demonstrating stability comparable to conventional nonmagnetic heating systems.

A uniform magnetic field is applied in the z-direction, while a modulated magnetic field, Bycos­(ωt), is introduced along the y-direction, along with compensating magnetic fields in all directions, generated by three sets of magnetic field coils. Magnetic shielding is achieved using a four-layer Permalloy shield combined with Mn–Zn ferrite shielding to isolate the atomic vapor cell from external magnetic interference.

Additionally, a pump laser is introduced into the vapor cell via a polarization-preserving fiber, and the alkali-metal atoms are optically pumped along the z-direction by using a circularly polarized 795 nm laser, which facilitates spin exchange to polarize the Xe nuclear spins. The combination of rubidium atomic spins, the pump laser, and a modulated magnetic field forms a magnetometer that probes the Xe nuclear spin magnetic field. The modulated magnetic field operates at 1000 Hz, significantly exceeding the nuclear spin progression frequency. The nuclear spin progression signals and relaxation times of ¹²⁹Xe and ¹³¹Xe were measured using the free induction decay (FID) method.

As illustrated in Figure b, this study explores the influence of the static magnetic field B on the electric quadrupole interaction by precisely varying the angle between the vapor cell and the magnetic field. The magnetic field direction B is controlled by adjusting the magnetic field components ΔB _x , ΔB _y , and ΔB _z along the x, y, and z axes, respectively. Here, the z-axis corresponds to both the pump laser direction and the symmetry axis of the vapor cell. The orientation of the magnetic field B is characterized by the zenith angle ψ and the azimuth angle ϕ.

Each experimental cycle comprises three distinct stages. In the first stage, polarized Rb atoms are utilized to hyperpolarize Xe through spin-exchange interactions under a bias magnetic field of B₀ = 240 nT along the z-axis for 60 s. In the second stage, a π/2 pulse is applied to tilt the polarization of ¹²⁹Xe and ¹³¹Xe into the xy-plane simultaneously. Concurrently, the bias field is adjusted to B₁ such that |B₁| = |B₀|, while its direction is modified to examine the influence of varying zenith angles ψ on Xe spin progression. During this phase, Xe atoms precess around B₁, and their spins are monitored by using an Rb magnetometer. In the final stage, the bias field is restored to B₀, and a magnetic field gradient is applied to ensure the complete depolarization of Xe atoms. Throughout the entire cycle, the pump laser remains active, while the vapor cell temperature and Rb atomic density are continuously monitored via the optical depth (OD) of the Rb absorption spectra.

3.2. FID Signal

Figure a presents typical FID signals recorded from the fabricated MEMS vapor cells at 393 K. To accurately characterize these signals, a sinusoidal fitting function incorporating four distinct frequency components was constructed. By integrating the e-exponential decay associated with spin precession for each frequency component, eq provides a precise match to the observed FID signals, ,

3.

Measured typical FID signals (a) and FFT signals (b) of the fabricated MEMS vapor cell; (c) experimental results for quadrupole shift as a function of ψ for ¹³¹Xe atoms when ϕ = 0° and 45°.

$$\begin{array}{l}M(t)=a{e}^{-t/T_{129}}\cos (2\pi f_{129}t+{\varphi }_{129})\\ +e^{-t/T_{131}}{b}_1\cos [2\pi (f_{131}+\mathrm{\Delta }\mathrm{\Omega })t+{\varphi }_{131}^1]\\ +e^{-t/T_{131}}{b}_2\cos (2\pi f_{131}t+{\varphi }_{131}^2)\\ +e^{-t/T_{131}}{b}_3\cos [2\pi (f_{131}-\mathrm{\Delta }\mathrm{\Omega })t+{\varphi }_{131}^3]\\ +D\end{array}$$ 6

where a, b ₁, b ₂, and b ₃ represent the amplitudes of the transverse magnetization of xenon for each frequency component. The relaxation rates 1/T ₁₂₉ and 1/T ₁₃₁ correspond to the transverse relaxation times of ¹²⁹Xe and ¹³¹Xe, respectively. The parameters f ₁₂₉ and f ₁₃₁ denote the precession frequencies of ¹²⁹Xe and ¹³¹Xe, while ΔΩ represents the electric quadrupole frequency shift of ¹³¹Xe. Additionally, ${\varphi }_{131}^1$ , ${\varphi }_{131}^2$ , ${\varphi }_{131}^3$ are the initial phases of each component, and D serves as a fitting constant.

Figure b presents the frequency spectrum derived from the fast Fourier transform (FFT) of the FID signal shown in Figure a. To further analyze this spectral structure, the FFT data were subsequently fitted using a set of four Lorentzian functions:

$$f(v)=\sum _{i=1}^4{A}_i\frac{{\mathrm{\Gamma }}_i/2\pi }{{(v-v_i)}^2+{({\mathrm{\Gamma }}_i/2)}^2}+b$$ 7

where A _i represents the amplitude of each vibrational component, v _i denotes the corresponding resonance frequency, and Γ _i characterizes the peak width of each resonance. Fitting the obtained FID signal using eqs and , to ensure the accuracy of the obtained relaxation time and electric quadrupole shift values, all fittings guaranteed R ² ≥ 0.999, root-mean-square error (RMSE) ≤ 0.03, and each value was the average of three sets of FID signals under the same conditions. Typically, the transverse relaxation times (T ₂) for ¹²⁹Xe and ¹³¹Xe were determined to be 6.39 and 5.83 s in Figure a, respectively. Additionally, their Larmor precession frequencies were measured at 2.77 Hz for ¹²⁹Xe and 0.83 Hz for ¹³¹Xe in Figure b. The presence of three distinct splitting peaks near 0.83 Hz reveals the characteristic quadrupole shift spectrum of ¹³¹Xe, while ¹²⁹Xe, lacking this frequency shift, exhibits only one precession peak.

3.3. Geometry Dependence of Frequency Shifts

To explore the correlation between the quadrupole interaction of ¹³¹Xe and the orientation of the magnetic field within the vapor cell, the electric quadrupole frequency shifts ΔΩ were measured at 393 K with different magnetic field directions. The magnetic field B was maintained within the yz-coordinate plane (ϕ = 0°) or the ϕ = 45° plane, and the magnetic field strength was kept constant at 240 nT. By adjusting the voltages of the triaxial magnetic field coils and scanning the azimuthal angle ψ of B in both planes, the FID signals and the corresponding ΔΩ of ¹³¹Xe were obtained, as illustrated in Figure c.

At this point, for the cubic vapor cell, the ΔΩ in eq can be expressed in the coordinate system of this figure as

$$\mathrm{\Delta }\mathrm{\Omega }=\frac{p{e}^{2}qQ}{2\hslash }[3{\cos }^2(\psi )-1+\eta {\sin }^2(\psi )\cos (2\varphi )]$$ 8

where Q represents the electric quadrupole moment, p is the coefficient, q = V _zz /e, and η denotes the asymmetry parameter of the vapor cell. ℏ, the reduced Planck’s constant, is the fundamental unit of angular momentum. Using eq to fit the electric quadrupole frequency shift variation curve at ϕ = 0° in Figure c, we obtain |η| = 0.11 ± 0.01. Two potential sources contribute to the asymmetry parameter in the vapor cell: a geometric asymmetry due to the material asymmetry caused by the silicon material on the sidewalls of the vapor cell, and a nonuniform accumulation of Rb droplets observed on the inner surface of the vapor cell walls. In the ϕ = 45° plane, the frequency shift of the electric quadrupole changes for different zenith angles ψ, and ΔΩ = 0 occurs at ψ = ±54.7° (cos ψ = 0.578). This result is in accordance with eq , which is called the “magic angle”.

In order to obtain the average twist angles ⟨θ _Si⟩ for each wall collision of ¹³¹Xe, eq was used to fit the data for ϕ = 45° in Figure c, to obtain ⟨θ_Si⟩ = 27.45 μrad, which is close to the results in ref. .

3.4. Temperature Dependence of Frequency Shifts

Experiments were carried out to examine the temperature dependence of the quadrupole interaction between ¹³¹Xe atoms and the walls of the MEMS vapor cell. By varying the heating laser intensity, the vapor cell’s OD at various temperatures and the electric quadrupole frequency shift ΔΩ of ¹³¹Xe were observed while maintaining a constant B of 240 nT along the z-axis.

As described in ref. , the quadrupole frequency shift ΔΩ is related to the dwell time τ _s of ¹³¹Xe, and its dependence on the temperature of the vapor cell wall is characterized by eq .:

Figure a,b illustrates the dependence of the electric quadrupole frequency shift ΔΩ/2πon temperature. In Figure a, ΔΩ/2π decreases exponentially with temperature, a logarithmic transformation of its reciprocal gives a linear decrease of In­(2π/ΔΩ) with temperature 1/T in Figure b. The data in Figure b were fitted separately to obtain the desorption activation energy E _A = −0.102 ± 0.01 eV. This value is lower than the desorption activation energy E _g = 0.12 eV reported in previous measurements of the Pyrex 7740 glass cell in the reference, which may be due to the smaller desorption activation energy E _si = 0.009 eV of the silicon wall leading to a reduction in the overall desorption activation energy of the fabricated MEMS vapor cells.

4.

Quadrupole frequency shift as an exponential dependence on the temperature T (a) and the temperature 1/T (b), with the magnetic field B oriented along the z-axis. (c) The transverse relaxation rate of ¹²⁹Xe varies as a function of temperature; (d) linear dependence of the ¹²⁹Xe transverse relaxation rate on the [n _Rb].

3.5. Temperature Dependence of Relaxation Rate

By measuring the FID signals of ¹²⁹Xe and ¹³¹Xe at different temperatures, the relaxation times of the two isotopes at various temperatures can be extracted separately, and then the temperature dependence of the electric quadrupole interactions between the two isotopes and the vapor cell wall can be investigated.

$${\gamma }_{129}=C_1{\sigma }_{\mathit{se}}{\mathit{v̅}}_{\mathrm{rel}}^{129}\left[n_{\mathrm{Rb}}\right]+C_2{\mathit{v̅}}_{129}{e}^{\frac{2E_{129}}{k_{\mathrm{B}}T}}+C_3$$ 9

Figure c illustrates the dependence of the relaxation rate γ ₁₂₉ of ¹²⁹Xe on temperature T, showing an exponential increase as the temperature rises. Eq was used to fit the data in Figure c, where σ _se = 2.0 × 10^–19cm^–2 is the spin-exchange collision cross-section of Xe and Rb, ${\mathit{v̅}}_{r_{\mathrm{el}}}^{129}=\sqrt{\frac{8(m_{\mathrm{Rb}}+m_{129})k_{\mathrm{B}}T}{\pi m_{\mathrm{Rb}}{m}_{129}}}$ represents the average relative velocity between ¹³¹Xe and Rb, C ₁, C ₂, and C ₃ are constant coefficients, $[n_{\mathrm{Rb}}]=\frac{1}{T}{10}^{26.178-4040/T}$ is the number density of rubidium atoms in the gas phase, E ₁₂₉ is the desorption activation energy of ¹²⁹Xe at the wall. The first term on the right-hand side of the equation corresponds to the collision-induced relaxation of ¹²⁹Xe with the alkali metal, while the second term accounts for relaxation caused by interactions with the vapor cell wall. Through fitting, the desorption activation energy E ₁₂₉ is determined to be 0.039 eV.

$$\mathrm{\Delta }\mathrm{\Omega }\propto e^{-E_{\mathit{A}}/k_{\mathrm{B}}T}$$ 10

Figure (d) shows the variation law of γ₁₂₉ with the number density of [n _Rb]. Since the vaporization thermal energy of Rb (about 72.216 kJ/mol = 0.75 eV) is much larger than the desorption activation energy E ₁₂₉ of ¹²⁹Xe, it is usually assumed that ${\gamma }_{\mathrm{wall}}^{129}=C_2{\mathit{v̅}}_{129}{e}^{2E_{\mathit{A}}/k_{\mathrm{B}}T}$ is independent of temperature within the experimental temperature range. Thus, it can be approximated that the γ₁₂₉ of ¹²⁹Xe is linearly related to [n _Rb], and eq can be converted to

$${\gamma }_{129}=C_1[n_{\mathrm{Rb}}]+{\gamma }_{\mathrm{wall}}^{129}$$ 11

Fitting the data in Figure d using eq yields the slope C ₁ = 2.51 × 10^–15 cm³·s^–1, and the magnitude of C ₁ is influenced by the pressure of the N₂ gas filling within the vapor cell, at which point the wall relaxation, ${\gamma }_{\mathrm{wall}}^{129}$ = 0.078 s^–1 is the intercept of the linear fit for [n _Rb] = 0. The main reason for the linear relationship between γ₁₂₉ and [n _Rb] is the rapid increase in the [n _Rb] with increasing temperature, the relaxation rate of ¹²⁹Xe due to collisions with alkali metals is much higher than the ${\gamma }_{\mathrm{wall}}^{129}$ , and the alkali metal forms a van der Waals molecule with ¹²⁹Xe, which in turn destroys the nuclear spin state of ¹²⁹Xe.

The nuclear spin relaxation of ¹³¹Xe results from its interaction with the EFG on the vapor cell wall surface. This relaxation is driven by a combination of random wall collisions, variations in wall dwell times τ_s , and the lateral motion of adsorbed ¹³¹Xe atoms along the wall. Figure shows the change in the relaxation rate γ₁₃₁ of ¹³¹Xe in the fabricated MEMS vapor cell as a function of temperature T, and the γ₁₃₁ shows a trend of initially decreasing and then increasing with the rise in T.

5.

Variation of the transverse relaxation rate of ¹³¹Xe with the temperature.

There are two main factors contributing to the relaxation rate γ₁₃₁. The first relaxation mechanism arises from the quadrupole interaction between the ¹³¹Xe nuclear spin and the vapor cell walls, and the second is the relaxation due to the spin-exchange collision between Rb–¹³¹Xe. Combining these two relaxation mechanisms, the following equation was used to fit the data in Figure : ,

$${\gamma }_{131}=C_4{\sigma }_{\mathit{se}}{\mathit{v̅}}_{\mathrm{rel}}^{131}\left[n_{\mathrm{Rb}}\right]+C_5{\mathit{v̅}}_{131}{e}^{\frac{\mathrm{E̅}}{k_{\mathrm{B}}T}}+C_6$$ 12

where ${\mathit{v̅}}_{\mathrm{rel}}^{131}=\sqrt{\frac{8(m_{Rb}+m_{131})k_{\mathrm{B}}T}{\pi m_{Rb}{m}_{131}}}$ represents the average relative velocity between ¹³¹Xe and Rb; C ₄, C ₅, C ₆ are the fitting constants, E̅ is the activation energy, which is defined as

$$\mathit{E̅}=\{\begin{array}{ll}{E}_D+E_{\mathit{A}}& {\tau }_s^{\prime }\ll {\tau }_s\\ {2E}_A& {\tau }_s^{\prime }\approx {\tau }_s\end{array}$$ 13

where E _D is the diffusion activation energy. Regarding the contribution of the vapor cell wall surface to the ¹³¹Xe relaxation rate γ ₁₃₁, the fit obtained an activation energy E̅ that is approximately twice the ¹³¹Xe desorption activation energy E _A , i.e., E̅ = 0.21 ± 0.02 eV ≈ 2E _A . It is concluded that E _D is approximately equal to E _A , i.e., the adsorbed atoms have a lower surface migration probability and have the same probability of jumping to a different surface position of the cell wall or escaping back to the gas phase.

Analysis reveals that at lower temperatures, the primary relaxation mechanism of ¹³¹Xe is dominated by electric quadrupole interactions at the vapor cell walls, resulting in an exponential decrease in the relaxation rate with increasing temperature. However, at higher temperatures, the relaxation rate rises due to a significant increase in alkali metal density, where collisional depolarization from Rb–¹³¹Xe spin exchange becomes the dominant relaxation mechanism. Overall, in the MEMS atomic vapor cells with cubic chambers fabricated in this study, optimal ¹³¹Xe transverse relaxation times (T ₂) are achieved within an operational temperature range of 370–390 K.

4. Discussion

The transverse relaxation time of Xe atoms is directly related to the performance of a nuclear magnetic resonance gyroscope (NMRG). The angle random walk (ARW) and bias stability (BS) of the NMRG predictions depend on the T ₂ time as follows:

$$\{\begin{array}{l}ARW=\frac{3600}{T_2\times \mathrm{SNR}\sqrt{\mathrm{\Delta }f}}\\ BS\propto k\frac{1}{\sqrt{T_1{T}_2}}\end{array}$$ 14

where SNR is the signal-to-noise ratio of the EPR magnetometer, and Δf is the phase noise bandwidth, with units in Hz. As can be seen from eq , the research work in this paper can provide guidance for suppressing quadrupole interactions, increasing the spin relaxation time of Xe nuclei, and thereby improving NMRG indicators, such as ARW and BS.

Despite significant advancements in MEMS vapor cell technology, two primary challenges persist in the miniaturization of alkali-metal vapor cell-based atomic gyroscopes: (a) System integration challenges: The cointegration of active optoelectronic components (e.g., diode lasers, photodetectors) with MEMS vapor cells and associated signal processing electronics remains technically demanding, particularly in achieving compact packaging while maintaining optical alignment stability and signal integrity. (b) Multiphysics coupling issues: Effective decoupling and suppression of complex multiphysical field interactions (including thermal, mechanical, optical, and magnetic effects) within the miniaturized gyroscope architecture present substantial engineering hurdles that limit further size reduction without performance degradation.

Table compares the main results obtained in this study with those of previous studies. The asymmetry parameters η measured in this study are significantly smaller than those of the stem-sealed rectangular cell, indicating that the cubic chamber has a suppressing effect on the electric quadrupole shift of ¹³¹Xe.

1. Comparison of the Main Results of This Study with Those of Other Relevant Studies.

Parameters	E A (eV)	⟨θ Si⟩ (μrad)	η
This work	0.102	27.45	0.11
Stem-sealed cylindrical cell	0.12	/	/
MEMS cell with different sidewall materials	/	29	/
Stem-sealed rectangular cell	0.14	/	0.22

The rigorous theoretical framework for electric quadrupole interactions on silicon-glass hybrid surfaces remains in the preliminary research stage. We plan to conduct further investigations in this direction, particularly by employing more advanced atomic-scale simulations to validate the hypothesized values of the local EFGs. This approach will provide crucial theoretical support for understanding the fundamental interactions at these complex material interfaces.

5. Conclusions

This study explores the quadrupole frequency shift and relaxation of ¹³¹Xe nuclear spins during collisions with the chamber walls in MEMS-fabricated atomic vapor cells. A theoretical model for the electric quadrupole frequency shift is developed to describe the interaction mechanism between nuclear spins and the vapor cell walls. By precisely controlling the angle between the symmetry axis of the vapor cell wall surface and the magnetic field, the cosine dependence of the quadrupole moment on the azimuthal angle ψ is revealed, allowing for the determination of the vapor cell’s asymmetry parameter and the average rotation angle of ¹³¹Xe per wall collision. Temperature-dependent FID measurements of ¹³¹Xe characterize the quadrupole interaction with the cell walls, leading to an estimation of the desorption activation energy of ¹³¹Xe at E _A = −0.102 ± 0.01 eV. Additionally, relaxation time measurements for ¹²⁹Xe and ¹³¹Xe reveal a linear dependence of ¹²⁹Xe relaxation rates on alkali metal atom density, while ¹³¹Xe relaxation rates exhibit an exponential decrease followed by an increase with temperature. The underlying physical mechanisms governing these behaviors are analyzed. This research enhances the understanding of silicon surface interactions and contributes to improving the stability and performance of the NMRG.

The authors declare no competing financial interest.