PHYSICAL REVIEW A 104, 042819 (2021) Magnetic-field-inhomogeneity-induced transverse-spin relaxation of gaseous 129 Xe in a cubic cell with a stem Deok-Young Lee , * Sangkyung Lee , † M. M. Kim , and Sin Hyuk Yim Agency of Defense Development, Daejeon 34186, Republic of Korea (Received 20 January 2021; revised 1 October 2021; accepted 4 October 2021; published 27 October 2021) We investigate the transverse-spin relaxations of 129 Xe due to diffusion in the presence of magnetic-field gradients in relation to the dimensions of a sub-cm sized cubiclike atomic gas cell with a stem. The transverse- spin-relaxation rate (Ŵ B ) of 129 Xe is measured as a function of various magnetic-field gradients, ∂ B z /∂ x, ∂ B z /∂ y, ∂ B z /∂ z, and ∂ B y /∂ y. From the measured transverse-spin-relaxation rates in five atomic gas cells with different stem sizes, the quadratic coefficients of Ŵ B with respect to the magnetic-field gradients are extracted. To investigate the effect of the dimensions of the stem, we calculate the ratio between the quadratic coefficients in each atomic gas cell, which is invariant under scaling in the cell size and change in the diffusion coefficient. We compare these ratios with those obtained analytically from a rectangular parallelepiped model for the atomic gas cells and with those obtained numerically from a more precise model taking the stems directly into account. Using a scaling argument, we provide a scheme for estimating the quadratic coefficient of a cubic atomic cell with a stem. Finally, we determine the diffusion coefficient from the measured quadratic coefficient. Compared to the analytical method for a rectangular parallelepiped, numerical analysis considering the stem provides the diffusion coefficient as a value close to the value given by Fuller’s equation. We estimate the diffusion coefficients of 129 Xe in the gas mixture of nitrogen, 129 Xe, and 131 Xe as 0.13 cm 2 /s at the standard temperature and pressure condition. DOI: 10.1103/PhysRevA.104.042819 I. INTRODUCTION Noble gas atoms have been widely applied to MRI [1–3], gyroscopes [4], magnetometers [5], and even fundamental symmetry tests [6,7], which rely on their long spin-relaxation times. These usually range from minutes to hours, depend- ing on the atom. Among noble gases, xenon is suitable for gyroscopes because of its “Goldilocks” spin-exchange cross section [8]. It has a moderate spin-relaxation time, on the order of 100 s, which means that its spin can be initialized after a relatively short time compared to other noble gas atoms. Spin-relaxation times can be classified into two categories, longitudinal-spin-relaxation time T 1 and transverse-spin-relaxation time T 2 . In this paper, T 2 represents the effective macroscopic transverse relaxation time including the dephase effect due to the field inhomogeneity, referred to as T ∗ 2 in NMR literature [9]. The longitudinal-spin-relaxation time T 1 is related to the spin-exchange rate, determining the initialization time. The transverse-spin-relaxation time T 2 is related to the duration of the macroscopic spin precession, which determines the angular random walk of a gyroscope which is proportional to 1/T 2 [4]. The transverse relaxation time T 2 is usually shortened by magnetic-field gradients as well as interatomic collisions and wall collisions. It can be written as the inverse of the trans- * Present address: Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea. † Corresponding author: sklee82@add.re.kr verse relaxation rate Ŵ tot , where Ŵ tot = Ŵ col + Ŵ wall + Ŵ B . Ŵ col is the relaxation rate due to collisions between atoms, Ŵ wall is the relaxation rate due to collisions with walls, and Ŵ B is the relaxation rate due to magnetic-field gradients originating from coils and the optical pumping beam. Usually, Ŵ B mainly contributes to the transverse-spin-relaxation rate; the diffusive motion of noble gas atoms in magnetic-field gradients causes spin phase decoherence and transitions be- tween spin states [10]. Magnetic-field-gradient-induced spin relaxation in highly symmetric cells, such as spherical cells and cubic cells, previously has been studied [10–15]. In a spherical cell of radius R, the transverse relaxation rate is given by Ŵ B,spherical ≈ 8γ 2 R 4 175D |∇B z | 2 , (1) where γ is the gyromagnetic ratio of the noble gas atoms, D is the diffusion coefficient, and ∂ B z /∂ z is the magnetic-field gradient of B z along the longitudinal direction [10]. In a cubic cell of side length L, the transverse relaxation rate is given by [11] Ŵ B,cubic ≈ γ 2 L 4 120D |∇B z | 2 . (2) The diffusion coefficients of noble gas atoms have been mea- sured using Eq. (1) in cm-sized atomic gas cells where the effect of the stem is negligible [12,15]. In terms of miniaturization and low power consumption, atomic gas cells in mm size have been increasingly used in various devices and instruments [16]. Atomic gas cells 2469-9926/2021/104(4)/042819(10) 042819-1 ©2021 American Physical Society