CN107192633A - Under a kind of SERF states in on-line measurement atom magnetometer air chamber alkali metal density method - Google Patents
Under a kind of SERF states in on-line measurement atom magnetometer air chamber alkali metal density method Download PDFInfo
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- 150000001340 alkali metals Chemical class 0.000 title claims abstract description 73
- 229910052783 alkali metal Inorganic materials 0.000 title claims abstract description 46
- 238000000034 method Methods 0.000 title claims abstract description 24
- 238000005259 measurement Methods 0.000 title claims abstract description 23
- 238000005086 pumping Methods 0.000 claims description 25
- 238000001514 detection method Methods 0.000 claims description 22
- 230000010287 polarization Effects 0.000 claims description 18
- 238000005485 electric heating Methods 0.000 claims description 10
- 230000003287 optical effect Effects 0.000 claims description 10
- 230000005540 biological transmission Effects 0.000 claims description 3
- 238000010438 heat treatment Methods 0.000 claims description 3
- 230000005389 magnetism Effects 0.000 claims description 3
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- 230000001105 regulatory effect Effects 0.000 claims description 2
- 239000003513 alkali Substances 0.000 claims 1
- 238000001739 density measurement Methods 0.000 claims 1
- 230000003313 weakening effect Effects 0.000 claims 1
- ZLMJMSJWJFRBEC-UHFFFAOYSA-N Potassium Chemical group [K] ZLMJMSJWJFRBEC-UHFFFAOYSA-N 0.000 description 3
- 230000000694 effects Effects 0.000 description 3
- 238000001285 laser absorption spectroscopy Methods 0.000 description 3
- 238000007405 data analysis Methods 0.000 description 2
- 230000035945 sensitivity Effects 0.000 description 2
- 238000004458 analytical method Methods 0.000 description 1
- 230000007547 defect Effects 0.000 description 1
- 230000005284 excitation Effects 0.000 description 1
- 230000010355 oscillation Effects 0.000 description 1
- 230000003534 oscillatory effect Effects 0.000 description 1
- 238000010408 sweeping Methods 0.000 description 1
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- G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
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Abstract
The invention discloses the method for alkali metal density in on-line measurement atom magnetometer air chamber under a kind of SERF states (no spin-exchange relaxation state).Apply weak background magnetic field and the horizontal linear FM signal of low-frequency range on atom magnetometer, the response signal of magnetometer in the frequency sweep time is gathered using data collecting card.The magnetic resonance curve in frequency domain is obtained by Fast Fourier Transform (FFT), and analyzes the resonance line width and resonant frequency for obtaining the curve.Change background magnetic field, obtain multigroup resonance line width and resonant frequency, conic fitting processing is carried out to it and can obtain the spin-exchange time, so as to calculate alkali metal atom density in air chamber at this temperature.This method can be implemented in the case where maintaining SERF states, and realize alkali metal atom density on-line measurement in air chamber in itself merely with SERF magnetometers.In addition, linear FM signal can obtain preferable frequency resolution in low-frequency range (within 1kHz), it is adaptable to the low-frequency range swept frequency range of SERF magnetometers low-intensity magnetic field requirement.
Description
Technical Field
The invention belongs to the field of measurement and analysis of key performance parameters of an atomic magnetometer, and particularly relates to a method for measuring the density of alkali metal in an atomic magnetometer gas chamber on line in an SERF state.
Background
The SERF (spin exchange relaxation free) atomic magnetometer is an instrument which works in the SERF state and realizes magnetic field measurement by utilizing the zeeman effect of atomic spin. The alkali metal gas cell is the core sensitive element of the SERF atomic magnetometer, and the density of alkali metal atoms directly restricts the intrinsic sensitivity of the magnetometer. Therefore, under the condition of maintaining the SERF state, only the atomic magnetometer is utilized, and the method has important significance for realizing accurate measurement of the density of the alkali metal atoms in the closed air chamber.
Currently, laser absorption spectroscopy and faraday optical rotation methods are two main methods for detecting the density of alkali metal atoms in a gas chamber. The experimental results show that the difference between the experimental measurement values of the laser absorption spectrometry at 453K and the theoretical calculation results is up to 10 times. The accuracy of this method is limited by the stretching and twisting of the lorentz line shape. The Faraday optical rotation method is to measure the density of alkali metal atoms by utilizing the Faraday optical rotation effect of linearly polarized light. The result shows that the method has approximate measurement precision to the laser absorption spectroscopy. However, this method requires the application of a strong magnetic field (1.2T), which may magnetize the magnetic shielding barrel of the atomic magnetometer, and affect the magnetic shielding effect. Furthermore, both methods require the use of other devices and cannot work in the atomic magnetometer SERF regime. Therefore, the demand of online measurement of the alkali metal atom density by a SERF atomic magnetometer cannot be met.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method and the device for online measuring the density of the alkali metal in the atomic magnetometer gas chamber in the SERF state are provided to overcome the defects of the prior art, so that the SERF state needs optical pumping and polarized atomic spin; the density of alkali metal atoms is high, and the atomic spin collision rate is increased; a weak magnetic environment. Under the condition of maintaining the SERF state, the density of alkali metal atoms in the closed gas chamber is accurately measured by using only the atomic magnetometer.
The technical scheme adopted by the invention is as follows: a method for on-line measuring alkali metal density in an atomic magnetometer gas chamber in a SERF state is characterized in that a weak background magnetic field and a transverse linear frequency modulation signal are applied to the atomic magnetometer. And acquiring a response signal of the magnetometer within sweep frequency time by using a data acquisition card, obtaining a magnetic resonance curve in a frequency domain through fast Fourier transform, and analyzing to obtain the resonance line width and the resonance frequency of the curve. And changing a background magnetic field to obtain multiple groups of resonance line widths and resonance frequencies, and fitting the resonance line widths and the resonance frequencies to obtain spin exchange time so as to calculate the density of the alkali metal atoms in the air chamber at the temperature.
Wherein a weak background magnetic field within 50nT and a transverse chirp signal are applied to the atomic magnetometer through the active magnetic compensation coil.
The bandwidth of the transverse linear frequency modulation signal covers the resonance frequency and is more than 5 times of the line width of a corresponding magnetic resonance amplitude-frequency curve of the applied background magnetic field.
The multiple groups of resonance line widths and resonance frequencies are fitted according to a quadratic curve, and the quadratic coefficient is in direct proportion to the spin exchange time, so that the density of alkali metal atoms is calculated.
Wherein the invention can be realized only by using the atomic magnetometer. The middle part of the atomic magnetometer comprises an alkali metal air chamber, a non-magnetic electric heating device, a small vacuum chamber, a magnetic compensation coil and a magnetic shielding barrel, wherein the non-magnetic electric heating device is arranged on the outer layer of the alkali metal air chamber and used for heating the alkali metal air chamber; the pumping light path of the atomic magnetometer is in the z direction, and comprises a pumping laser, a beam expander, a polarizer and a quarter wave plate, and the pumping light path is used for generating circularly polarized light to polarize alkali metal atoms; the detection light path of the atomic magnetometer is in the x direction, and comprises a detection laser, a polarizer, a half wave plate, a polarization beam splitter prism, a reflector and a balance detector, and the detection light path is used for detecting the optical rotation angle of a linear polarization detection light polarization plane generated by atomic spin precession under a background magnetic field. The output signal of the balance detector is collected by a data acquisition card. The pumping light and the detection light are orthogonal to the central position of the alkali metal gas chamber.
The invention comprises two parts of testing and data analysis, and the specific steps are as follows:
step 1, according to the type of alkali metal, an alkali metal air chamber is heated to 140-200 ℃ through a non-magnetic electric heating device, a pumping laser is started, a beam expander enables pumping light to sufficiently cover the alkali metal air chamber, linearly polarized light is obtained through a polarizer, circularly polarized pumping light transmitted along the z direction is obtained through the linearly polarized light through a quarter wave plate, and the wavelength is adjusted to the D1 line of alkali metal atoms; in order to ensure the low polarizability of atomic spin, the optical power density of pumping light is regulated to be not more than 0.2mW/cm2(ii) a Starting a detection laser, obtaining linear polarization detection light propagating along the x direction through a polarizer, tuning the wavelength to a D2 line of alkali metal atoms, then detuning until the peak-to-peak value of a magnetometer response signal is maximum, and adjusting a half wave plate to enable the power density of transmission light and reflection light of a polarization beam splitter prism to be equal when the polarization beam splitter prism is not pumped, namely the output signal of a balanced detector is zero;
step 2, compensating residual magnetism in the magnetic shielding barrel through the magnetic compensation coil, and then repeating the following steps: applying a background magnetic field in the z direction, wherein the background magnetic field is not more than 50nT, applying a linear frequency modulation signal with the sweep frequency time of not less than 100s in the y direction, the frequency range of the linear frequency modulation signal comprises the resonance frequency corresponding to the background magnetic field in the z direction, and simultaneously acquiring a magnetometer response signal in the time period by a data acquisition card;
and 3, performing fast Fourier transform on the signals acquired by the acquisition card, fitting the obtained frequency domain signals to an amplitude-frequency theoretical curve responded by the SERF magnetometer to obtain a plurality of groups of resonance frequencies and line widths, wherein the fitting formula is as follows:
where a and b are coefficients used to facilitate the fit.
Step 4, fitting a plurality of groups of resonance line widths and resonance frequencies measured under different weak background magnetic fields according to a quadratic curve to obtain spin exchange time TSEThe fitting formula is as follows:
where c and d are coefficients used to facilitate the fit. The alkali metal atom density was further calculated according to the following formula:
wherein n represents the density of alkali metal atoms, and the other parameters are constant or known: sigmaSEIs the spin exchange cross-sectional area of collision, KBIs the boltzmann constant, T is the temperature in kelvin units, and M is the average mass of the alkali metal atoms.
The principle of the technical scheme of the invention is as follows:
applying an oscillating magnetic field in the y direction of an atomic magnetometerWhere B' is the amplitude of the oscillating magnetic field and ω is the oscillation frequency. This magnetic field can be decomposed into two counter-propagating rotating magnetic fields:
the z direction of the magnetometer is designed to have a background magnetic field B0Corresponding to a resonance frequency of ω0。
Wherein,is a unit vector in the z direction, γeIs the electron gyromagnetic ratio, and q (P) is a nuclear slowing factor, a constant related to the alkali metal species. The response signal of the SERF magnetometer is that two center frequencies are respectively within +/-omega0The sum of the lorentz curves of (a):
wherein,is the steady state atomic spin polarizability without any oscillatory excitation, and Δ ω is the magnetic resonance linewidth.
Because the magnetic resonance frequency of the SERF magnetometer is very low, -omega0The influence of the curve has to be taken into account when fitting. Based on this, the amplitude-frequency theoretical curve fitting formula of the SERF magnetometer response is as follows:
where a and b are coefficients used to facilitate the fit. Under low polarizability and weak background magnetic field, magnetic resonance line width and magnetic resonance center frequency omega0The quadratic of (d) is linear:
the fitting equation is thus obtained as:
where c and d are coefficients used for convenient fitting, TSEIs the spin exchange time, the inverse of which is proportional to the alkali metal atom density:
wherein n represents the density of alkali metal atoms, and the other parameters are constant or known: sigmaSEIs the spin exchange cross-sectional area of collision, KBIs the boltzmann constant, T is the temperature in kelvin units, and M is the average mass of the alkali metal atoms.
Therefore, a plurality of groups of resonance line widths and resonance frequencies measured under different weak background magnetic fields are fitted according to a quadratic curve, the spin exchange time can be obtained through quadratic coefficients, and the alkali metal atom density is further calculated.
Compared with the prior art, the invention has the advantages that:
(1) the invention realizes the online measurement of the density of alkali metal atoms in the gas chamber only by utilizing the SERF magnetometer.
(2) The invention performs the measurement while maintaining the SERF state.
(3) By fitting multiple groups of resonance line widths and resonance frequencies measured under different weak background magnetic fields according to a quadratic curve, not only can the spin exchange time be obtained, but also the limit line width under a zero magnetic field can be obtained, which is an important parameter for evaluating the theoretical sensitivity of the SERF magnetometer.
(4) The linear frequency modulation signal can obtain better frequency resolution in a low frequency band (within 1 kHz), and is suitable for a low frequency band frequency sweeping range required by a weak magnetic field of an SERF magnetometer.
Drawings
FIG. 1 is a schematic view of the system of the apparatus of the present invention;
FIG. 2 is a graph of the resonance frequency and the resonance line width measured by a potassium atom magnetometer at 170 ℃ with a background magnetic field varied from 2nT to 46nT, and a quadratic fit thereof.
The reference numbers are listed below: the device comprises a 1-pumping laser, a 2-beam expander, a 3-polarizer, a 4-quarter wave plate, a 5-magnetic compensation coil, a 6-alkali metal air chamber, a 7-non-magnetic electric heating device, a 8-small vacuum chamber, a 9-magnetic shielding barrel, a 10-detection laser, a 11-polarizer, a 12-half wave plate, a 13-polarization beam splitter prism, a 14-balance detector, a 15-data acquisition card, a 16-reflector and a 17-function signal generator.
Detailed Description
The invention is further illustrated by the accompanying drawings and the detailed description.
As shown in fig. 1, the present invention can be implemented using only the atomic magnetometer itself. The atomic magnetometer comprises an alkali metal air chamber 6, a non-magnetic electric heating device 7, a small vacuum cavity 8, a magnetic compensation coil 5 and a magnetic shielding barrel 9, wherein the non-magnetic electric heating device 7 is arranged on the outer layer of the alkali metal air chamber 6 and used for heating the alkali metal air chamber 6, the small vacuum cavity 8 is used for reducing heat convection and heat conduction, the magnetic shielding barrel 9 is used for isolating interference of an environmental magnetic field, and the magnetic compensation coil 5 is connected to a function signal generator 17 and used for generating magnetic fields in the xyz three vertical directions; the pumping light path of the atomic magnetometer is in the z direction, and comprises a pumping laser 1, a beam expander 2, a polarizer 3 and a quarter wave plate 4, and the pumping light path is used for generating circularly polarized light to polarize alkali metal atoms; the detection light path of the atomic magnetometer is in the x direction, and comprises a detection laser 10, a polarizer 11, a half wave plate 12, a polarization beam splitter prism 13, a reflecting mirror 16 and a balance detector 14, and the detection light path is used for detecting the optical rotation angle of a linear polarization detection light polarization plane generated by atomic spin precession under a background magnetic field. The output signal of the balanced detector 14 is collected by a data acquisition card 15. The pumping light and the detection light are orthogonal to the central position of the alkali metal gas cell 6.
The invention comprises two parts of testing and data analysis, and the specific steps are as follows:
step 1, according to the type of alkali metal, an alkali metal air chamber 6 is heated to 140-200 ℃ through a non-magnetic electric heating device 7, a pumping laser 1 is started, a beam expander 2 enables pumping light to sufficiently cover the alkali metal air chamber 6, linearly polarized light is obtained through a polarizer 3, circularly polarized pumping light which is transmitted along the z direction is obtained through a quarter-wave plate 4 through the linearly polarized light, and the wavelength is adjusted to the D1 line of alkali metal atoms; the optical power density of pumping light is adjusted to be not more than 0.2mW/cm for ensuring the low polarizability of atomic spin2(ii) a Starting a detection laser 10, obtaining linear polarization detection light propagating along the x direction through a polarizer 11, tuning the wavelength to a D2 line of alkali metal atoms, then detuning until the peak-to-peak value of a magnetometer response signal is maximum, and adjusting a half wave plate 12 to enable the power density of transmission light and reflection light of a polarization beam splitter prism 13 to be equal when the polarization beam splitter prism is not pumped, namely the output signal of a balance detector 14 is zero;
step 2, compensating residual magnetism in the magnetic shielding barrel 9 through the magnetic compensation coil 5, and then repeating the following steps: applying a background magnetic field in the z direction, wherein the background magnetic field is not more than 50nT, applying a linear frequency modulation signal with the sweep frequency time of not less than 100s in the y direction, the frequency range of the linear frequency modulation signal comprises the resonance frequency corresponding to the background magnetic field in the z direction, and simultaneously acquiring a magnetometer response signal in the time period by a data acquisition card (15);
step 3, performing fast Fourier transform on the signals acquired by the data acquisition card 15, fitting the obtained frequency domain signals to an amplitude-frequency theoretical curve responded by the SERF magnetometer to obtain a plurality of groups of resonance frequencies and line widths, wherein the fitting formula is as follows:
where a and b are coefficients used to facilitate the fit.
Step 4, fitting a plurality of groups of resonance line widths and resonance frequencies measured under different weak background magnetic fields according to a quadratic curve to obtain spin exchange time TSEThe fitting formula is as follows:
where c and d are coefficients used to facilitate the fit. The alkali metal atom density was further calculated according to the following formula:
wherein n represents the density of alkali metal atoms, and the other parameters are constant or known: sigmaSEIs the spin exchange cross-sectional area of collision, KBIs the boltzmann constant, T is the temperature in kelvin units, and M is the average mass of the alkali metal atoms.
As shown in FIG. 2, the resonance frequency and the resonance line width measured when the background magnetic field of the potassium atom magnetometer was changed from 2nT to 46nT at 170 ℃ were plotted and the quadratic fit thereof revealed that the spin-exchange time was about 2.1 × 10- 5s, corresponding to a potassium atom density of 3.7 × 1013cm-3。
Those skilled in the art will appreciate that the details of the invention not described in detail in this specification are well within the skill of those in the art.
Claims (8)
1. A method for online measurement of alkali metal density in an atomic magnetometer gas chamber in a SERF state is characterized in that: applying a weak background magnetic field and a transverse linear frequency modulation signal on an atomic magnetometer, collecting a response signal of the magnetometer in sweep frequency time by using a data acquisition card, and analyzing to obtain a resonance line width and a resonance frequency of a magnetic resonance curve in a frequency domain; and changing a background magnetic field to obtain multiple groups of resonance line widths and resonance frequencies, and fitting the multiple groups of resonance line widths and resonance frequencies to obtain spin exchange time so as to obtain the density of alkali metal atoms in the air chamber of the magnetometer at the working temperature.
2. The method for on-line measurement of alkali metal atom density in atomic magnetometer gas chamber under SERF state according to claim 1, wherein: a weak background magnetic field within 50nT and a transverse chirp signal are applied to the atomic magnetometer through an active magnetic compensation coil.
3. The method for on-line measurement of alkali metal atom density in atomic magnetometer gas chamber under SERF state according to claim 1, wherein: the bandwidth of the transverse linear frequency modulation signal covers the resonance frequency and is more than 5 times of the line width of the corresponding magnetic resonance amplitude-frequency curve of the weak background magnetic field.
4. The method for on-line measurement of alkali metal atom density in atomic magnetometer gas chamber under SERF state according to claim 1, wherein: the data acquisition card is used for acquiring response signals of the magnetometer within sweep frequency time, time domain signals acquired by the data acquisition card are converted into magnetic resonance curves within a frequency domain through fast Fourier transform, and then a plurality of groups of resonance frequencies and line widths are obtained through an amplitude-frequency theoretical curve responded by the SERF magnetometer.
5. The method for on-line measurement of alkali metal atom density in atomic magnetometer gas chamber under SERF state according to claim 1, wherein: and fitting a plurality of groups of resonance line widths and resonance frequencies according to a quadratic curve, wherein the quadratic coefficient is in direct proportion to the spin exchange time, and further obtaining the density of the alkali metal atoms.
6. The method for on-line measurement of alkali metal atom density in atomic magnetometer gas chamber under SERF state according to claim 1, wherein: the SERF state of the magnetometer is maintained during density measurement by utilizing the linear frequency modulation signal in a low frequency band, namely within 1 kHz.
7. The method for on-line measurement of alkali metal atom density in atomic magnetometer gas chamber under SERF state according to claim 1, wherein: the method is characterized in that the online measurement of the density of alkali metal atoms in the gas chamber is realized, namely, the measurement is realized only by using an atomic magnetometer, wherein the atomic magnetometer comprises an alkali metal gas chamber (6), a non-magnetic electric heating device (7) on the outer layer of the alkali metal gas chamber, a small vacuum cavity (8), a magnetic compensation coil (5) and a magnetic shielding barrel (9); the non-magnetic electric heating device (7) is used for heating the alkali metal gas chamber (6), the small vacuum chamber (8) is used for weakening heat convection and heat conduction, the magnetic shielding barrel (9) is used for isolating the interference of an environmental magnetic field, and the magnetic compensation coil (5) is connected to the function signal generator (17) and used for generating magnetic fields in the xyz three vertical directions; the pumping light path of the atomic magnetometer is in the z direction, and comprises a pumping laser (1), a beam expander (2), a polarizer (3) and a quarter-wave plate (4) and is used for generating circularly polarized light to polarize alkali metal atoms; the detection light path of the atomic magnetometer is in the x direction, and the detection light path comprises a detection laser (10), a polarizer (11), a half wave plate (12), a polarization beam splitter prism (13), a reflector (16) and a balance detector (14), and is used for detecting the optical rotation angle of a linear polarization detection light polarization plane generated by atomic spin precession under a background magnetic field; the output signal of the balance detector (14) is collected by a data acquisition card, and the pumping light path and the detection light path are orthogonal to the central position of the alkali metal gas chamber (6).
8. The method for on-line measurement of alkali metal atom density in atomic magnetometer gas chamber under SERF state according to claim 7, wherein: the specific steps of measurement by an atomic magnetometer are as follows:
step 1, according to the type of alkali metal, an alkali metal air chamber is heated to 140-200 ℃ through a non-magnetic electric heating device, a pumping laser is started, a beam expander enables pumping light to sufficiently cover the alkali metal air chamber, linearly polarized light is obtained through a polarizer, circularly polarized pumping light transmitted along the z direction is obtained through the linearly polarized light through a quarter wave plate, and the wavelength is adjusted to the D1 line of alkali metal atoms; in order to ensure the low polarizability of atomic spin, the optical power density of pumping light is regulated to be not more than 0.2mW/cm2(ii) a Starting a detection laser, obtaining linear polarization detection light propagating along the x direction through a polarizer, and tuning the wavelength to alkali metalThe atomic D2 line is detuned until the peak-to-peak value of the magnetometer response signal is maximum, and a half wave plate is adjusted to ensure that the power densities of the transmission light and the reflection light of the polarization beam splitter prism are equal when the polarization beam splitter prism is not pumped, namely the output signal of the balanced detector is zero;
step 2, compensating residual magnetism in the magnetic shielding barrel through the magnetic compensation coil, and then repeating the following steps: applying a background magnetic field in the z direction, wherein the background magnetic field is not more than 50nT, applying a linear frequency modulation signal with the sweep frequency time of not less than 100s in the y direction, the frequency range of the linear frequency modulation signal comprises the resonance frequency corresponding to the background magnetic field in the z direction, and simultaneously acquiring a magnetometer response signal in the time period by a data acquisition card;
and 3, performing fast Fourier transform on the signals acquired by the acquisition card, fitting the obtained frequency domain signals to an amplitude-frequency theoretical curve responded by the SERF magnetometer to obtain a plurality of groups of resonance frequencies and line widths, wherein the fitting formula is as follows:
<mrow> <msub> <mi>S</mi> <mi>x</mi> </msub> <mo>=</mo> <mi>a</mi> <mo>&times;</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>&Delta;</mi> <mi>&omega;</mi> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mi>&Delta;</mi> <mi>&omega;</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>-</mo> <msub> <mi>&omega;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>&Delta;</mi> <mi>&omega;</mi> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mi>&Delta;</mi> <mi>&omega;</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>+</mo> <msub> <mi>&omega;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>-</mo> <msub> <mi>&omega;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mi>&Delta;</mi> <mi>&omega;</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>-</mo> <msub> <mi>&omega;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>+</mo> <msub> <mi>&omega;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mi>&Delta;</mi> <mi>&omega;</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>&omega;</mi> <mo>+</mo> <msub> <mi>&omega;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>b</mi> </mrow>
where a and b are coefficients used to facilitate fitting;
step 4, fitting a plurality of groups of resonance line widths and resonance frequencies measured under different weak background magnetic fields according to a quadratic curve to obtain spin exchange time TSEThe fitting formula is as follows:
<mrow> <mi>&Delta;</mi> <mi>&omega;</mi> <mo>=</mo> <mi>c</mi> <mo>&times;</mo> <msub> <mi>T</mi> <mrow> <mi>S</mi> <mi>E</mi> </mrow> </msub> <msubsup> <mi>&omega;</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mi>d</mi> </mrow>
wherein c and d are coefficients used for convenient fitting, and further the alkali metal atom density is calculated according to the following formula:
<mrow> <mi>n</mi> <mo>=</mo> <msqrt> <mfrac> <mrow> <mi>&pi;</mi> <mi>M</mi> </mrow> <mrow> <mn>8</mn> <msub> <mi>K</mi> <mi>B</mi> </msub> <mi>T</mi> </mrow> </mfrac> </msqrt> <mo>/</mo> <msub> <mi>&sigma;</mi> <mrow> <mi>S</mi> <mi>E</mi> </mrow> </msub> <msub> <mi>T</mi> <mrow> <mi>S</mi> <mi>E</mi> </mrow> </msub> </mrow>
wherein n represents the density of alkali metal atoms, and the other parameters are constant or known: sigmaSEIs the spin exchange cross-sectional area of collision, KBIs the boltzmann constant, T is the temperature in kelvin units, and M is the average mass of the alkali metal atoms.
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