key: cord-0530957-6fwpo4k8
authors: Chen, Yao; Yu, Mingzhi; Ma, Yintao; Zhao, Libo; Wang, Yanbin; Guo, Ju; Lin, Qijing; Jiang, Zhuangde
title: Quadrupolar interaction induced frequency shift of 131Xe nuclear spins on the surface of silicon
date: 2021-11-14
journal: nan
DOI: nan
sha: 2520e1f460f891eb6ee48b2738cabcc2ce658088
doc_id: 530957
cord_uid: 6fwpo4k8

The combination of micro-machined technology with the Atomic Spin Gyroscope(ASG) devices could fabricated Chip Scale Atomic Spin Gyroscope(CASG). The core of the gyroscope is a micro-machined vapor cell which contains alkali metal and isotope enriched noble gases such as 129Xe and 131Xe. The quadrupolar frequency shift of 131Xe is key parameters which could affect the drift of the ASG and is related to the material of the cell in which they are contained. In micro machined technology, the typical utilized material is silicon. In this article, we studied the electric quadrupolar frequency shift of 131Xe atoms with the silicon wall of the micro-machined vapor cell. A cylinder micro-machined vapor cell is utilized in the experiment and a large part of the inner cell surface is composed of silicon material. We studied the temperature dependence of the 129Xe spin relaxation and 131Xe frequency shifts to evaluate the interaction of the nuclear spin with container wall and the alkali metal atoms. The results show that the average twisted angle of the 131Xe nuclear spins as they collide with the silicon wall is measured to be 29 *10^-6 rad. The desorption energy for the 131Xe nuclear spin to escape from the silicon surface is Esi = 0.009eV . This study could help to improve the bias stability of the CASG which is a key parameter for the gyroscope as well as may developes a method to study the surface property of various material.

Hyper polarization of isotope enriched nuclear spinsWalker and Happer [1997] could find wide range application, including atomic spin gyroscopeKornack et al. [2005] , , Larsen and Bulatowicz [2012] ,neutron spin filtersQin et al. [2021] ,magnetic resonance imaging of the lungs for COVID-19 studyLi et al. [2016a] , testing physcis beyond the standard modelJi et al. [2018] , Lee et al. [2018] , Bulatowicz et al. [2013] , etc. Isotope enriched noble gas nucleus such as 131 Xe, 21 N e and 83 Kr whose nuclear spin are larger than 3/2 own nuclear quadrupole moments. Thus, the study of the nuclear quadrupolar interaction between the nuclear spin and the surrounding enviroment toke attention in several area. For example, in a nuclear magnetic resonance gyroscope or an atomic co-magnetometerSorensen et al. [2020] , Xu et al. [2021] , the nuclear quadrupolar frequency shift and relaxation of 131 Xe through colliding with the container wall could affect the bias instability and detection sensitivity of the gyroscope respectively.

The combination of atomic devices with chip-scale fabrication technology could greatly reduce the size and cost of the atomic sensorsKitching [2018] . For atomic sensors based on an alkali vapor cell such as atomic clockKnappe et al. [2005] , atomic magnetometerKrzyzewski et al. [2019] and atomic spin gyroscopeLimes et al. [2019] , the key is an alkali vapor cell in which several kinds of gases are filled. Especially in a spin exchange relaxation free(SERF) gyroscopeLi et al. [2016b] , Duan et al. [2018] and nuclear magnetic resonance gyroscopeSorensen et al. [2020] , typically 131 Xe and 21 N e are filled. Traditionally, the glass material is utilized for alkali vapor cell fabrication, while in a micro-machined alkali vapor cell, the typically utilized material is silicon and glass. The glass-silicon anode bonding technology is usually utilized in the cell fabrication processChen et al. [2021] , Han et al. [2018] . The main body of the cell is composed of a silicon block and two covered glass ends. Ion beam deep etching is usually utilized to drill a hole for the container space of the vapor cell.

Due to the quadrupolar interaction between the 131 Xe nuclear spins and the vapor cell surface wall, the perturbation from the electric field gradient of the wall surface will cause both the frequency shift and relaxation of the nuclear spins. There are several studies about the nuclear spin-glass collidingButscher et al. [1994, 1996] while there is little study about the nuclear spin-silicon colliding. The frequency shift and relaxation of the nuclear spins relate to the temperature of the vapor cell since the desorption time of the nuclear spins on the surface would affect the nuclear quadrupolar interaction strengthButscher et al. [1994] . The desorption is thermally activated and there is a parameter named activation energy E A which is 0.12eV for the nuclear-glass surface interactionButscher et al. [1994] . Moreover, the shape of the vapor cell is also related to the quadrupolar interaction. We can control the temperature, material and shapes of the vapor cell to change the quadrupolar interaction strength. For example, in order to study the crossover between the NMR and the nuclear quadrupole resonance interaction regimes, alkali vapor cells with a different kind of material are utilized to make the nuclear quadrupolar frequency shifts more clearlyDonley et al. [2009] . A rectangular alkali vapor cell is also utilized to change the quadrupolar interaction strengthFeng et al. [2020] .

The nuclear quadrupole resonance(NQR) spectroscopy could also be utilized to identify chemicals and it is sometimes called the fingerprint of the chemicals. Thus, the NQR spectroscopy could be utilized to detect explosivesCooper et al. [2016] . In this paper, we mainly focus on the NQR spectroscopy study in ASG application. Especially in micromachined ASGs, the alkali vapor cell' material is composed of both glass and silicon. The quadrupolar interaction between the nuclear spins and the silicon surface is studied. We measured the temperature dependence of the 131 Xe frequency shift. The NMRG bias instability is close related to the NQR spectroscopy. Thus, this study could help to give solutions to improve the bias instability of the chip scale ASG.

The quadrupolar interaction happens as the nuclear spin collides with the container wall. The atoms will be absorbed on the surface for a while. This process will cause both the relaxation and frequency shift of the 131 Xe nuclear spin. The nuclear spin of 131 Xe atoms is 3/2 and there is a nuclear quadrupole moment in the nucleus. The energy level of the nuclear spin should shift if the nuclear quadrupole moment feels Electric Field Gradient(EFG) as well as the relaxation of the nuclear spin could occur if EFG fluctuation exists.

The vapor cell utilized in our paper is made of both silicon and glass. As shown in Fig.1 , The main body of the vapor cell is made of silicon and the geometry is cylindrical. The inner diameter is 3mm and the length is 2mm in our experiment. The two ends of the vapor cell is covered by glass and it is connected to the silicon through anode bonding. The details about how to fabricate the vapor cell could be found in our paperChen et al. [2021] . As the polarized nuclear spin of 131 Xe atoms colliding with the cell walls, they will be adsorbed by the cell wall for a short period as well as diffuse from site to site during the adsorbed periodButscher et al. [1994] . The mean adsorption time τ s is defined to be the average time that the atoms are adsorbed on the surface of the cell wall. The adsorption time is related to the temperature of the cell and the time should be decreases if the temperature of cell rise. There is a parameter named E A which is defined to be the activation energy of desorption to characterize τ s . The relation is 1/τ s ∝ exp(−E A /k B T ) Butscher et al. [1994] . Since both of the frequency shift and relaxation due to quadrupole interaction with cell wall are related to the desorption energy, we can do a measurement of the dependence of frequency shift and relaxation with the cell temperature.

According to this referenceWu et al. [1988] , there is a good experiment evidence that the magnitudes of the fluctuating field gradients at the wall are quite large compared to the mean value of the field gradients, and then we assume that:

where V w is the electric field potential at the wall surface, x i and x j are unit vectors in the i and j directions. i and j could be x,y or z directions. When the atoms are adsorpted on the glass surface wall of the cell, it is plausible to make an assumption that the fluctuations are nearly isotropic, i.e, we assume that the microscopic structure of the wall is sufficiently rough that any tensor components of the electric field gradient have approximately the same mean-squared amplitude as any other. We believe that the silicon surface is also rough and the EFG and its fluctuation are isotropic too. While every material has a characterize deactivation energy. It is reported that in a vapor cell with RbH surface coating, 131 Xe atoms would stay for a shorter time than that of the glass surface. Thus, the relaxation time of 131 Xe atoms could be longerWu et al. [1990] . In our experiment, we divided the vapor cell into two parts. One is the glass part and the other one is the silicon part. Both of the two parts can cause the nuclear quadrupole splitting of the energy level and relaxation. According to the referenceWu et al. [1988] , the NQR shifts for the | − 3/2 >< −1/2| and |3/2 >< 1/2| coherence as:

where ∆Ω g and ∆Ω si are the frequency shifts from collision with glass and silicon surfaces respectively. v is the velocity of the atoms, S is the inner surface area of the vapor cell and V is the inner volume of the vapor cell. I is the nuclear spin of the atoms. θ g and θ si are the average angles as the nuclear spin colliding with the glass surface and silicon surface respectively. ψ is the angle between the holding magnetic field and the normal direction to the inner surface of the vapor cell. For example, if the holding magnetic field is directed to the z direction and there is a small area dS si on the silicon surface, the angle between the holding magnetic field and the normal direction which is perpendicular to the small area is 90 degree.

The configuration of the experimental setup is shown in Fig.1 . The micro-machined alkali vapor cell is at the center of the experiment and the geometry is cylinder. The inner diameter is 3mm and the height is 2mm. A small amount of cesium metal, 5 Torr natural abundance Xe gas and 650 Torr Nitrogen gas are filled in the vapor cell. The natural abundance Xe gas contains 26.4% 129 Xe and 21.2% 131 Xe. The pump laser is circular polarized and tuned to the Cs D1 line absorption center. The beam is expanded to cover the cell as large as possible. A second 3W 1550nm heating laser is utilized to heat the vapor cell to the desired temperature. After passing through the vapor cell, a PD(photo diode) is used to accept the transmitted pump laser light. 3 sets of magnetic field coil are utilized for the holding magnetic field in the z direction, the modulation magnetic field B y Cos(ωt) in the y direction and the compensation magnetic field in all the three directions. Several layers of magnetic field shields together with a MnZn ferrite shield are utilized for shielding the vapor cell from the earth 's magnetic field. The Xe nuclear spins are hyper-polarized through spin exchange optical pumping with Cs atomic spins. The hyper-polarized Xe nuclear spin will produce magnetic field which could be experienced by the Cs electron spins. The effective magnetic field will be approximately Romalis and Cates [1998] . In the equation, B K is the magnetic field produced by the nuclear spin such as 129 Xe or 131 Xe. k 0 is an enhancement factorWalker [1989] which could enhance the magnetic field experienced by the Cs electron spins during spin exchange collision. µ K is the nuclear magnetic moment for spin species K.

[N ] is the number density of the nuclear spins and P K is the polarization of the nuclear spin species K. 129 Xe nuclear magnetic moment µ( 129 Xe) is -0.78µ N in which µ N is the nuclear magnetic moment of the neutron and 131 Xe nuclear magnetic moment µ( 131 Xe) is 0.69µ N . The magnetic field produced by 129 Xe and 131 Xe will be in the opposite directions. The Cs atom spins, the pump laser and the modulation magnetic field B y Cos(ωt) compose of a single beam absorption magnetometerShah et al. [2007] , Shah and Romalis [2009] to detect the hyper-polarized Xe nuclear spin magnetic field. The modulation frequency of the magnetic field is 1000Hz which is much larger than the nuclear spin precession frequency.

Since the vapor cell is quite small, it is hard to use free space laser beam for the experiment. A polarization maintaining fiber is utilized for guiding the pump laser to the vapor cell. A multi-mode fiber with 400µm core is utilized for the heating laser. The temperature of the vapor cell is measured by the Cs D1 absorption line. We first do a measurement of the line-width of the D1 absorption line. then the Cs atom density could be calculated by the absorption line. Thus we can get the temperature of the vapor cell through the measured number density. We also measured the temperature of the vapor cell through a temperature sensor. The temperature measured by the absorption method is around 5 degree larger than that of the temperature sensor. Here we will use the absorption method, since it directly measures the number density of the Cs atoms.

The free induction decay(FID) signal is a typical way to measure the precession frequency and relaxation of the nuclear spins. Fig.2 shows one of the FID signal. A holding magnetic field in the z direction B z is added to the system. After several minutes' spin exchange optical pumping, The polarization tends to be stable. We suddenly add a step magnetic field which is around 1/10 of the holding magnetic field in the y direction to let the nuclear spin precess around the total magnetic field. After around 1 second, we removed the y step magnetic field and the nuclear spins will precess around the holding magnetic field B z . Note that the residual magnetic field in the shield is below 5nT and we compensated this residual field by the coils with the single beam Cs atomic magnetometer. We mention that the modulation amplitude of the magnetometer only affect the relaxation of Cs atoms. We set the modulation amplitude as small as possible to do the measurement. The FID signal composes several frequencies. We did a fft analyse of the signal shown in Fig.2 . We can see that there are 4 frequencies in the signal. The precession frequency of 129 Xe is single and it is around 3 times of the 131 Xe's precession frequencies. There are 3 components in the 131 Xe's precession. Due to the quadrupolar interaction, the frequency difference of the 3 peaks are the same and equal to ∆Ω. From Fig.2 we can see that the high frequency precession is from 129 Xe. There is also the 131 Xe precession frequency with smaller frequency. The beating signal of 131 Xe is caused by the nuclear quadrupolar interaction. In order to acquire the frequencies and the relaxation times of the 4 precession components, we do a fitting to the FID experiment data with the following equation:

where A, B 1 ,B 2 and B 3 are the amplitudes of the 4 precession components. Γ 129 and Γ 131 are the decay rates of the nuclear spin. f 129 and f 131 are the precession frequencies of 129 Xe and 131 Xe without the nuclear quadrupole shift. φ 129 , φ 1 131 , φ 2 131 and φ 3 131 are the phase of the 4 components respectively. From the fitting result in Fig.2 we can see that the equation could fit well to the experiment data. In order to see the pure 131 Xe precession signal, we substract the signal of 129 Xe from the FID signal. Fig.4 shows the pure signal. We can clearly see the beating signal of 131 Xe precession. 

The nuclear quadrupolar shift is related to the temperature of the vapor cell since the temperature could affect τ s . Finally, the average angle θ g could be affected by the temperature. We changed the vapor cell temperature and then measure the 131 Xe quadrupolar shifts. The relationship between the temperature and the frequency shifts are shown in Fig. 5 . According to Equation. 2, we can simplify the equation into:

where d is the diameter of the vapor cell and h is the height of the vapor cell. Since Ln(1/ θ ) ∝ −E A /(k B T ), we set the horizontal axis to be 1/T and the vertical axis to be Ln(2π/∆Ω). As shown in this referenceButscher et al. [1994] , the average angle θ g for Pyrex glass is 45µrad as the temperature of the vapor cell is 373K. From the fitting result of Fig.5 , as the temperature of the vapor cell is 373K, we can calculate that the average angle for the silicon is 29µrad based on equation.4. This result is similar to the result in this referenceDonley et al. [2009] in which the angle for silicon is 29µrad.

We can also get the desorption energy of 131 Xe atoms on the surface of silicon E si . With the increasing of the temperature, the time that nuclear spins stay on the material will be shorter. Thus, the average angle will be reduced and finally the quadrupolar frequency shift will decrease. In equation.4, it is reasonable to suppose that the average angle θ is propotional to Exp(E/k b T ) in which E is the desorption energy of the materialButscher et al. [1994] . Suppose that there are coefficients k 1 and k 2 which connect the average angle and the desorption energy for the glass and silicon. It is reasonable to set the average thermal velocity of the nuclear spin to be 245m/s which is the velocity under 373K since the thermal velocity is weakly temperature dependent. At 373K, the average angle for the glass is 45µrad and the desorption energy is E g = 0.12eV Butscher et al. [1994] . We can calculate that k 1 is equal to 1.1 × 10 −6 . Together with the experimental condition, we can get:

where c is Ln(606061k 2 ). x is 1/T and e is E si /k b . We fit the experimental data shown in Fig.5 with equation5 . The fitting results show that e is 109 and c is 0.89. We can get that E si is 0.009 and k 1 is equal to 4.0 × 10 −6 . From the results we see that as the nuclear spin absorbed on the silicon surface, they seems to stay much shorter time than that of the glass. It seems that the EFG is larger on the surface of silicon since the factor k 2 is around 3 times larger than that of the glass.

Except the frequency shifts, we also studied the relaxation of the 129 Xe nuclear spins. We changed the number density of the Cs atoms and then measure the relaxation rate of the nuclear spins. The relaxation of 129 Xe is also measured through the FID method. The FID signal is fitted to Equation.3 and then the relaxation rate could be measured. The relationship between the Cs number density and 129 Xe relaxation rate is shown in Fig.6 . The fitting shows that the slope is 3.7 × 10 −15 cm 3 /s and the relaxation rate of 129 Xe nuclear spin tends to be 0.038 s −1 as the Cs number density is 0. Thus the wall relaxation rate of 129 Xe is 0.038. During the measurement of the relaxation rate, we find that the relaxation is strongly related to Cs polarization. This is because the polarized Cs atoms could produce magnetic field which could be experienced by the nuclear spins. As the precession frequencies of the nuclear spins and the Cs electron spins close to each other, Cs atom spins would damp the Xe precessionFang et al. [2016] . Thus, we lowered the Figure 6 : The relationship between the number density of Cs atoms and the relaxation rate of 129 Xe under low pumping power.

pump laser power as low as possible to do the experiment. We also let the holding magnetic field direct to the nuclear spin magnetic field direction, thus the electron spin of Cs atom could experience both of the nuclear spin magnetic field and the holding magnetic field. This arrangement could set the electron spin precession frequency far away from the nuclear spin precession frequency. During the experiment, we found that the electron spin magnetic field could reach around 200nT at high power.

The collision between Cs and 129 Xe could also cause the relaxation. Especially for the heave atoms, binary collision and van der Waals molecule formation could happen at the same time. The interaction time could be long as the molecule forms. Thus nitrogen gas could be filled in the vapor cell as the high density nitrogen molecule could break the Cs-129 Xe van der Waals moleculesShao et al. [2005] . According to the reference, the binary collision coefficient is 9.4 × 10 −16 cm 3 /s. For the vapor cell utilized in our experiment, there is 0.85 Amagat nitrogen gas filled. From the measured total relaxation(both of the binary collision and three body collision) result we measured, we can get that the van der Waals molecule formation induced relaxation coefficient is 2.8 × 10 −15 cm 3 /s. This result shows that the relaxation induced by the three body collision is about 3 times larger than that of the binary collision relaxation rate.

The relaxation of 131 Xe is mainly composed of two parts. The first one is the collision relaxation with Cs atoms. This relaxation process should be similar to that of the 129 Xe atoms. We define this relaxation rate to be Γ 131 se and it includes the binary spin exchange relaxation and tree body collision relaxation. We define a spin exchange rate coefficient k se and Γ 131 se is equal to k se [Cs] where [Cs] is the number density of Cs atoms. The other one is the relaxation from the quadrupolar interaction. According to the referenceWu et al. [1988] , the quadrupolar relaxation of 131 Xe for the |3/2 >< 1/2| and | − 3/2 >< −1/2| energy levels are determined to be:

In the equation, θ 2 is the average squared angle as the nuclear spin collide with the surface wall. We believe that this angle is different for the silicon and glass material. It is also reasonable again to suppose that the EFG fluctuation which could cause the relaxation is isotropic. The coefficients beside the square twisted angle is the inner surface area percentage of the two material. According to the referenceButscher et al. [1994] , the average of the squared twisted angle θ 2 is propotional to Exp(2E/k B T ). Thus, equation.6 could be written:

According to the measured results for E g = 0.12eV and θ 2 g = 3.4 × 10 −6 rad 2 , under the temperature of 373K, we can calculate k 2 to be 1.95 × 10 −9 with the relation θ 2 g = k 2 Exp( 2Eg k B T ). We substitute these results into equation.7, the quadrupolar relaxation of 131 Xe nuclear spin could be:

Now we turn to do a calculation of θ 2 g . As the cell temperature is 393.5K, the number density of Cs is 6.6 × 10 13 cm 3 . The total relaxation of 131 Xe which includes the quadrupolar relaxation with surface and the spin collision with Cs atoms is 0.21sec −1 . First, we need to substract the relaxation from collision with Cs atoms. There is rare study about the spin exchange collision relaxation between Cs and 131 Xe atoms. We can only do an empirical calculation of Cs − 131 Xe spin exchange relaxation from the result of Cs − 129 Xe. Since the angular momentum of 131 Xe is 3/2 which is 3 times larger than that of the 131 Xe. While the nuclear magnetic moments of the two isotopes are nearly the same. Thus it is reasonable to assume that the spin exchange rate coefficient k se for Cs − 129 Xe is 3 times that of the Cs − 131 Xe pair. The measured result for Cs − 129 Xe pair is 3.7 × 10 −15 cm 3 /s and thus we can calculate the spin exchange rate coefficient k 131 se to be 1.2 × 10 −15 cm 3 /s. Under the temperature of 393.5K, the spin exchange relaxation of 131 Xe is 0.08s −1 . We substract the spin exchange relaxation from the total relaxation and then we can get the quadrupolar relaxation of 131 Xe under 393.5K to be 0.13s −1 . Since the squared average angle for collision with the glass material is temperature dependent, we can calculate the angle under 393.5K to be 2.3 × 10 −6 rad 2 . According to equation6, we can calculate that the glass material induced quadrupolar relaxation to be 0.06s −1 under 393.5K. The silicon material induced quadrupolar relaxation is 0.07s −1 . Finally, we can calculate that θ 2 si under 393.5K is equal to 2.2 × 10 −6 rad 2 which is similar to that of the glass colliding.

The glass material utilized in our vapor cell is boro-silicate glass with a thermal expansion coefficient of 3.3 × 10 −6 /K. The limitation of this study is that we could not independently measure the quadrupolar interaction between 131 Xe and the glass surface. We took the parameters from other references for the desorption energy, the mean twisted angle and mean squared twisted angle. Since there is no data about the spin exchange rate between Cs and 131 Xe nuclear spins, we theoretically estimate this parameter based on the measured rate of 129 Xe. Further studies could be done to directly measure the spin exchange optical pumping rate between Cs and 131 Xe. For comparison, the spin exchange parameters between Cs and 129 Xe are well known. The other limitation of this study is that we could not measure the relaxation of 131 Xe in a wide range of temperature. At low temperature and high temperature, the 131 Xe FID signal will be very weak compared to the 129 Xe FID signal. Moreover, the FID signal is not so strong since the natural abundance Xe is utilized in our experiment. Other isotopes will contribute to a large part of the Cs spin relaxation and thus the sensitivity of the single beam atomic magnetometer will be low. Further studies could be done to just fill isotope enriched 131 Xe atoms in the vapor cell. Note that when we fabricating a micro-machined alkali vapor cell, the anode bonding chamber is quite large compared to the glass blown vapor cell fabricating system. It will waste a lot of expensive isotope enriched 131 Xe for making the micro-machined vapor cell. We are now developing a gas recycling system to improve the situation.

In conclusion, we have studied the quadrupolar frequency shift and relaxation of 131 Xe nuclear spins as they collide with the silicon container wall. The silicon material is a kind of widely utilized material in the MEMS technology. This study helps to know more about the surface property of the silicon. CSAG would be developed and this study will finally helps to improve the bias instability of the CSAGs. We divided the alkali vapor cell into the glass parts and the silicon part. Models for the frequency shift and relaxation were developed to measure the quadrupolar interaction between the nuclear spins and the surfaces. The desorption energy of 131 Xe on the silicon surface is measured to be 0.009eV. At 373K, the average twisted angle θ g as 131 Xe collide with the silicon surface wall is 29 µrad. The relaxation of 131 Xe was also studied and we acquired the average square twisted angle θ 2 si which could decide the relaxation of 131 Xe nuclear spin to be 2.2 × 10 −6 rad 2 under 393.5K.

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