CN101458319B - Method for optimizing optical parameter of cesium-beam frequency scale median bundle - Google Patents

Abstract

The invention discloses a method for optimizing a beam optics paramenter in a caesium beam frequency standard, and belongs to the navigation technical field. The method comprises the following steps: solving a differential equation of motion of caesium atoms in a magnetic field by actual values of magnetic field gradient and effective magnetic moment, and performing an optimal solution by a Monte Carlo algorithm. The method can avoid algorithm errors caused by a gradient problem, improper integrals and the like in the existing optimization algorithm of the beam optics, and causes the computing result more accurate; and avoids the problem of lower execution efficiency of multiple integrals, and saves computing cost. In addition, physical quantities adopted in the method are all actual values, thus avoiding the disadvantages of the physical quantities such as oversimplification and idealization.

Description

A kind of method that beam optics parameter in the caesium bundle frequency marking is optimized

Technical field

The invention belongs to field of navigation technology, relate to a kind of when realizing magnetic separation attitude cesium-beam atomic clock, the optimization method of parameter in its beam optics system.

Technical background

Magnetic separation attitude cesium-beam atomic clock occupies critical role in the temporal frequency field, and it belongs to the one-level frequency marking, is applied to the various fields of modern science and technology, comprises punctual, temporal frequency metering etc., especially has a wide range of applications in navigation field.The atomic clock that is used to navigate in the worldwide navigation Positioning System (GPS) of the U.S. and Muscovite GLONASS (Global Navigation Satellite System) (GLONASS) and locatees just comprises small-sized magnetic separation attitude cesium-beam atomic clock.

At present, the country that really can accomplish small-sized magnetic separation attitude caesium clock engineering practicability has only the U.S. and Russia.China has just carried out development to small-sized magnetic separation attitude caesium clock in the sixties in last century, but does not up to the present really reach the through engineering approaches application level as yet, and wherein, it is major reason that small-sized magnetic separation attitude caesium clock index is difficult to reach the engineering request for utilization with the life-span.

For magnetic separation attitude cesium-beam atomic clock, the design of beam optics system has significant impact to index.The beam optics system comprises caesium stove, collimating apparatus, state selection magnet (comprise A magnet and B magnet, also cry A magnetic field and B magnetic field, each magnetic field is a secondary magnetic field) and detector etc., as shown in Figure 1.The beam optics system is determining the characteristic parameters such as rate distribution, signal to noise ratio (S/N ratio), chamber phase differential frequency displacement, secondary Doppler shift, adjacent line traction frequency displacement and short-term stability of beam tube gating atom.Therefore, the design of beam optics system is extremely important.

In fact the design of beam optics system is exactly the beam optics Parameter selection.These parameters comprise the position of position of detector, collimating apparatus drift angle, B magnet etc.Because between some parameter is conflicting, as reducing the signal live width, tends to lose signal intensity, reduce short-term stability etc.Therefore, when design beam optics system, need take all factors into consideration each parameter of beam optics.

In the existing beam optics method for designing, the most direct method is that the means by mathematics are optimized each parameter.Its process is: define certain objective function earlier, as the function of beam optics parameter, adopt methods such as maximum gradient method or cost function method to find out the parameter that can make objective function reach extreme value then.Because the parameter that beam optics relates to is numerous, and some parameter is discontinuous, so there is great difficulty in the optimization of beam optics.

At present, have the maximum gradient of employing method to carry out simulation, also have the cost function of employing method to carry out simulation, though obtained certain effect, its result is quite coarse, falls far short from practical application.Its reason is or simplifies in simulation process too much, is taken as constant as gradient terms and effective magnetic moment with two pole fields, and this will cause optimizing the result and produce than mistake; Or algorithm existing problems itself, this is owing to the designer and does not know the continuity of objective function and the connectedness of parameter space.As a rule, the designer would rather rely on original physics to consider, makes some preliminary parameters, studies by experiment then, determines other parameter.This method has increased design cost on the one hand, and these parameters usually can't reach the requirement of optimal design on the other hand.

Summary of the invention

The objective of the invention is in order to overcome the defective of prior art,, be difficult to problem that wherein parameter is optimized, propose a kind of method that beam optics parameter in the caesium bundle frequency marking is optimized for solving when the beam optics system design.Its core is: adopt the actual value of magnetic field gradient and effective magnetic moment to find the solution the differential equation of motion of caesium atom in magnetic field, and adopt Monte Carlo algorithm to be optimized and find the solution.

The objective of the invention is to be achieved through the following technical solutions, may further comprise the steps:

(1) position angle and the emission angle to certain caesium atom in the caesium atomic beam carries out the MonteCarlo sampling respectively.By sampling, obtain the position angle and the emission angle of this caesium atom.

(2) rate distribution of above-mentioned caesium atom is carried out Monte Carlo sampling.By sampling, obtain the speed of this caesium atom.

(3) with the result of step (1) and (2) as starting condition, adopt the Bulirsch-Stoer method to find the solution the differential equation of motion of caesium atom in A magnetic field.According to solving result, can draw the movement locus of this caesium atom in A magnetic field.

(4) repeating step (1)～(3) draw all movement locus by A magnetic field caesium atom.According to the movement locus of these caesium atoms, count number and all position distribution of the caesium atom that can pass through A magnetic field by the caesium atom in A magnetic field.Calculate the percent of pass of caesium atom according to the number of the caesium atom by A magnetic field, determine the drift angle of collimating apparatus, the parameters such as coordinate of collimator-alignment A magnetic field inlet according to percent of pass; Determine the position in B magnetic field by the position distribution of the caesium atom in A magnetic field according to all.

So far, just finished the optimization of these 3 parameters of position in coordinate, the B magnetic field of drift angle, the collimator-alignment A magnetic field inlet of collimation device when the beam optics system design, laid a good foundation thereby use for the through engineering approaches that realizes magnetic separation attitude cesium-beam atomic clock.

Beneficial effect

The inventive method, the Algorithm Error of having avoided being run in the existing beam optics optimized Algorithm of bringing as gradient problem, improper integral etc. makes result of calculation more accurate; Avoid multiple integral to carry out the lower problem of efficient, saved the computing cost.In addition, the physical quantity that adopts among the present invention is actual value, has avoided physical quantity too to simplify and Utopian disadvantage.

Description of drawings

Fig. 1 is a magnetic separation attitude caesium bundle frequency marking beam optics system schematic;

Fig. 2 is at x=± a, two magnetic fields that parallel wire produced at z=0 place;

Fig. 3 is the cartridge shape of magnet, and wherein convex surface and concave surface are on the equipotential surface of two fields of line.

Two field of line right-handed coordinate systems of Fig. 4 for setting up;

Fig. 5 is the position relation of magnet and collimating apparatus.Wherein magnet is in the XOY coordinate system, and collimating apparatus is in X ' O ' Y ' coordinate system.The nose portion and the recessed head part of magnet have also been indicated among the figure;

Fig. 6 is the movement locus of caesium atom.Wherein the magnetic field inlet is on the AB face, and outlet is on the CD face, and the unit of coordinate is m among the figure, and for the purpose of observing obviously, transverse axis has been got different ratios with the longitudinal axis;

Fig. 7 is the relation between atom percent of pass and the angle.Wherein transverse axis is represented collimating apparatus drift angle (unit for °), and the longitudinal axis is represented percent of pass;

Fig. 8 is that percent of pass η is with D point Y coordinate figure y under the situation of α=1 ° _DRelation.Wherein the unit of transverse axis is m;

Fig. 9 is that percent of pass η is with D point Y coordinate figure y under the situation of α=1.5 ° _DRelation.Wherein the unit of transverse axis is m;

Figure 10 is that percent of pass η is with D point Y coordinate figure y under the situation of α=2.0 ° _DRelation.Wherein the unit of transverse axis is m;

Figure 11 is that percent of pass η is with D point Y coordinate figure y under the situation of α=2.5 ° _DRelation.Wherein the unit of transverse axis is m;

Figure 12 is the position distribution density f of atom ₁(y) and f ₂(y).The curve representative on the left side | F=3〉attitude, the peak representative on the right | F=4〉attitude;

Embodiment

Below in conjunction with drawings and Examples preferred implementation of the present invention is described further.

After earlier the metal caesium being heated to uniform temperature (temperature is controlled at 90 ℃～120 ℃) in the caesium stove, the caesium atom is sprayed by collimating apparatus.Collimating apparatus adopts slender pipeline, is used to improve the angle distribution of caesium atomic beam, makes atomic beam possess better directivity.However, the caesium atom still can exist during by collimating apparatus and tube wall between, the interatomic collision of caesium, therefore when the caesium atom sprayed from collimating apparatus, can there be an angle in the transmit direction of caesium atom and collimating apparatus pipeline axis, this angle is called emission angle.Setting this emission angle is θ, and θ is in 0～pi/2 scope.

Definition θ ₀Be the angle of divergence of collimating apparatus, its angle value is by the geometric configuration decision of collimating apparatus pipeline, that is:

$\tan {\mathrm{\θ}}_0=\frac{2r}{l}$

Wherein r, l are respectively the inside radius and the length of collimating apparatus pipeline.Work as θ ₀In the time of≤3 °, the caesium atomicity that emits in unit solid angle can be expressed as approx with the relation that emission angle theta changes:

$I\left(\mathrm{\θ}\right)~\left\{\begin{array}{cc}{\cos }^{-1}\frac{\mathrm{\θ}}{{\mathrm{\θ}}_0}-\frac{\mathrm{\θ}}{{\mathrm{\θ}}_0}{(1-\frac{{\mathrm{\θ}}^2}{{\mathrm{\θ}}_0^2})}^{\frac{1}{2}}+\frac{2\mathrm{\θ}}{{3\mathrm{\θ}}_0}[1-{(1-\frac{{\mathrm{\θ}}^2}{{\mathrm{\θ}}_0^2})}^{\frac{3}{2}}],& [0\≤\mathrm{\θ}\≤{\mathrm{\θ}}_0]\\ \frac{{\cos }^2\mathrm{\θ}}{\sin \mathrm{\θ}},& [{\mathrm{\θ}}_0\≤\mathrm{\θ}\≤\mathrm{\π}/2]\end{array}\right.---\left(1\right)$

To emission angle theta, sample according to (1) formula.

The position angle of supposing collimating apparatus pipeline center axis is

Because

Defer to even distribution, therefore to the position angle

Sample according to even distribution,

Sample range be 0～2 π.

By sampling, obtain the position angle and the emission angle of caesium atom.

Then, the rate distribution of caesium atom is carried out Monte Karlo sampling.The speed of caesium atomic beam is deferred to following distribution:

$f\left(v\right)~v^3{e}^{-v^2/{\mathrm{\α}}^2}---\left(2\right)$

Wherein, v is a caesium atom speed size, and α is a most propable speed.The value of α is provided by following formula:

$\mathrm{\α}=\sqrt{\frac{2\mathrm{kT}}{m}}$

In the formula, T is the caesium furnace temperature, and unit is Kelvin; K is a Boltzmann constant; M is the caesium atomic mass.Speed to the caesium atom is sampled according to (2) formula.By sampling, obtain the speed of caesium atom.

The caesium atom is stressed being shown below in A magnetic field:

$\stackrel{\→}{G}=\mathrm{\μ}\▿H---\left(3\right)$

μ represents atomic magnetic moment, and H represents magnitude of field intensity.Small-sized magnetic separation attitude caesium bundle frequency marking adopts plush copper magnet and recessed the magnetic field that magnet produces usually, and Fig. 2 has shown plush copper magnet and recessed magnet.This magnetic field can be by at a distance of being that simulate in the magnetic field that two parallel wires of I are produced for 2a, by inverse current, and as shown in Figure 3, its size is:

$H(y,z)=\frac{4\mathrm{Ia}}{r_1{r}_2}---\left(4\right)$

Wherein, r ₁, r ₂For on the plane a bit with the distance of two leads.The magnetic field gradient of y direction is

$\frac{\∂H}{\∂y}=-2I\frac{2a}{r_1^3{r}_2^3}(r_1^2+r_2^2)y---\left(5\right)$

In order to draw the equation of motion of caesium atom in two fields of line, set up coordinate system as shown in Figure 4 earlier, wherein the x axle passes the axis on the face of cylinder, magnet plush copper place, and true origin is taken at plane, place, magnet porch.(3) the represented equation of motion of formula can turn in this coordinate system:

$\left\{\begin{array}{c}\stackrel{\·\·}{x}\≈0\\ \stackrel{\·\·}{y}\≈\frac{{\mathrm{\μ}}_{\mathrm{eff}}}{m}\frac{\∂H(y,z)}{\∂y}\\ \stackrel{\·\·}{z}\≈0\end{array}\right.---\left(6\right)$

Here m is the quality of caesium atom, μ e _FfBe effective moment of magnetic couple of caesium atom, it is the component of the moment of magnetic couple of caesium atom along magnetic direction, is provided by the Breit-Rabi formula:

μ wherein _BBe the Bohr magneton, the desirable ξ ≈ H/0.3284 that does of the value of parameter ξ, when the caesium atom be in (3, m _F) (6) formula is got "+" number when attitude and (4 ,-4) attitude, when be in (4, m _F) attitude and satisfy m _F≠-4 o'clock, get "-" number.

First equation and the 3rd equation draw v from (6) formula _x=constant, v _z=constant, constant are any real number.Second equation equal sign the right is two a product, and wherein one is effective magnetic moment μ _Eff, from (7) formula as seen, it is relevant with parameter ξ, and ξ and magnetic field H are proportional, H be again y, function, so μ _EffIt is the function of y; Another is a gradient terms, and from (5) formula as can be seen, this also is the function of y.It is a complex expression that contains y that this product of two causes equation equal sign the right, finds the solution to such an extent as to second equation can not be resolved.

Because the magnet inlet is limited, most of caesium atoms in the caesium atomic beam can not enter magnetic field, in addition, though some caesium atom enters magnetic field, but, really can only account for the sub-fraction of total atom number by the atom of magnetic field of magnets through beating behind the segment distance on plush copper or recessed head.η is as follows for the definition percent of pass:

$\mathrm{\η}=\frac{T_0}{T}---\left(8\right)$

Wherein T represents the total atom number of collimating apparatus ejection, T ₀Expression is by the atomicity of magnetic field of magnets.By the definition percent of pass, some beam optics parameters just can calculate.

When calculating the optimal value of collimating apparatus angle α, supposition α gets different values earlier, under every kind of α value, calculate the percent of pass η of caesium atom then respectively, can obtain η like this, just can obtain the optimal value of α according to this curved line relation with a curved line relation between the α.

The coordinate of supposing entry position, collimating apparatus axis alignment A magnetic field is y _D, calculate y now _DOptimal value.Allow y _DGet different values, calculate the value of percent of pass η under different values then, so just obtain η with y _DBetween a curved line relation, make y according to curved line relation _DOptimal value.

The position of B magnetic field inlet can provide according to location map, describes now.The caesium atom can deflect in the magnetic field of gradient is arranged, different state deflection differences.The definition atom is as follows along the normalized position distribution density function of y direction f (y):

$\frac{1}{N}\mathrm{dN}=f\left(y\right)\mathrm{dy}$

Wherein the unit of f (y) is (individual)/m, N be a certain state of caesium atom that passes through the magnetic field slit (| F=3〉attitude or | F=4〉attitude) the total atom number.Here further will | F=3〉the position distribution density of attitude is defined as f ₁(y), | F=4〉attitude is defined as f ₂(y), known f ₁(y) and f ₂(y), also just known the deflection situation of atom, known the details of caesium atom, also just known the entry position of B magnet along certain direction deflection.

Embodiment

As shown in Figure 4, magnet is cut open along plane z=0, obtained Fig. 5, wherein the hacures representative profile.Fig. 6 has also expressed coordinate system X ' O ' Y ' at collimating apparatus and place thereof, and the outlet of collimating apparatus is at O ', and its axis is an X ' axle, and it intersects at D point (the D point is exactly the entry position of collimating apparatus) with Y-axis.The caesium atom is along the ejection of the direction of X ' and enter the magnet inlet.

The radius (being the distance between the OA among Fig. 5) of getting plush copper in the calculating is 1.85mm, distance between the AB is 1.85mm, distance between the magnet entrance and exit (being the distance between the OC) is 25.4mm, collimator port (locating at O ') is 24.5mm apart from the distance of y axle, these two parameters of angle between the ordinate of O ' and collimating apparatus and the x axle are undetermined according to analog case, evenly locate owing to require collimating apparatus to point to magnetic field, thus these two parameters if determine one just much of that.Suppose collimating apparatus tan θ in addition ₀=0.1727.

(1) movement locus of caesium atom

At first calculated angle between collimating apparatus and x axle and be that 1.64 °, plush copper summit (both A points among Fig. 5) are located magnetic field intensity 15800Gs, the movement locus of atom when the caesium furnace temperature is 95 ℃.

Result of calculation as shown in Figure 6, the track among Fig. 6 is the projection of actual path on the z=0 plane, form by three parts, first section at O ' this a part of track is a straight line with between the vertical line AB, atom is not subjected to the effect of any external force herein.The second portion of track is between vertical line AB and vertical line CD, and the caesium atom is subjected to the effect in magnetic field in this zone, and track deflects.In this zone, under the trajectory bias that has, on the trajectory bias that has, this is in corresponding to the caesium atom as can be seen | F=3〉and attitude or be in | F=4〉attitude; The trajectory deflection that has is obvious, and the deflection that has is not obvious, and this is corresponding to the speed difference of atom.The third part of track is in the zone on vertical line CD the right, and atom is not subjected to the effect of any external force herein, and track is a straight line.

(2) relation between collimating apparatus angle and the atom percent of pass

Suppose that the caesium furnace temperature is that 95 ℃, the axis of the magnetic field intensity 15800G of magnet plush copper summit place, collimating apparatus are all the time by homogeneity range (apart from 1.2 * 1.85 places, the plush copper center of circle), when therefore the angle α between collimating apparatus and the x axle changes, corresponding adjustment is answered in the position of collimating apparatus, the value of α is respectively 0 °, 0.5 °, 1 °, 1.2 °, 1.45 °, 1.55 °, 1.64 °, 2.0 ° and 2.5 °, gets 3000000 atoms when calculating at every turn.Result of calculation is presented among Fig. 7.

From angle qualitatively, angle is big more, and the atomicity that can pass A magnet slit is few more, and percent of pass is low more.Actual computation show when α when 0 ° increases gradually, percent of pass μ has the trend of slow rising, begin in the time of greatly near 1.2 ° to descend, and downtrending is comparatively obvious.This phenomenon can be made description below, as α during near 0 °, although there is more atom can enter magnetic field, but because the deflection in magnetic field, more atom is beaten on salient pole head and recessed cartridge, thus this moment can be smoothly atom by magnetic field be not maximum, when α increases slightly (as near 1.2 °), reduce to some extent although enter the atomicity in magnetic field, play the also corresponding minimizing of atomicity on cartridge, the atomicity by magnetic field is slightly risen.When α further increased, the atomicity of passing slit further descended, and beat the atomicity on cartridge simultaneously because inertia effect begins rising, and percent of pass has just descended significantly.

(3) percent of pass is with the relation of entry position

As shown in Figure 6, when the axis X at collimating apparatus place ' constant with the angle of X-axis, but the Y coordinate y of ordering as D _DWhen changing, how percent of pass changes.Studied under α=1 °, 1.5 °, 2.0 ° and 2.5 ° of four kinds of situations percent of pass η below with y _DSituation of change.

Supposition caesium furnace temperature is 95 ℃ during calculating, and the magnetic field intensity 15800G of place, magnet plush copper summit gets 3000000 atoms at every turn when calculating.

First kind of situation: α=1 °, the scope y of Y coordinate figure _DBe 0.00188～0.00258.Getting 8 points calculates.Fig. 8 has shown result of calculation.Put when nearer when the entry position of collimating apparatus distance A, percent of pass η is less, along with the D point away from the A point, η rises gradually, y _DIn the time of greatly near 0.0024, it is maximum that the value of η reaches, and just begins then to descend.It should be noted that place, maximal value place has surpassed homogeneity range and far away from homogeneity range.

Second kind of situation: α=1.5 °, the scope y of Y coordinate figure _DBe 0.00185～0.00255.Getting 8 points equally calculates.Fig. 9 has shown result of calculation.Entire curve approaches among Fig. 9 substantially.Though place, maximal value place surpasses homogeneity range, compares with Fig. 9, the distance that surpasses reduces.

The third situation: α=2.0 °, the scope y of Y coordinate figure _DBe 0.00196～0.00266.Continuing to get 8 points calculates.Figure 10 has shown result of calculation.Within same scope, the shape of curve is compared with Fig. 9 with Fig. 8, and is comparatively different.Further reduce though the maximal value of curve surpasses the distance of homogeneity range, in case cross peak value, curve is just very fast to descend.

The 4th kind of situation: α=2.5th °, the scope y of Y coordinate figure _DBe 0.00187～0.00257.Continuing to get 8 points calculates.Figure 11 has shown result of calculation.Approaching with Figure 10, but peak value is near homogeneity range.

Find out that from Fig. 8～Figure 11 the peak value of percent of pass η is shifted to the left side of curve along with the increase of angle α value.Contrast 4 curves, also can find out increase, the also corresponding decline of the amplitude of entire curve along with angle α value.

According to Fig. 8～Figure 11 and in conjunction with the result of Fig. 7, collimating apparatus drift angle and y _DValue just can make.

(4) position distribution of caesium atom two states

Before pointed out according to the details of caesium atom along certain direction deflection, the entry position that can make B magnet.Research caesium atom is along the deflection situation of y direction among Fig. 5 now.

100 ℃ of temperature of supposition during calculating, angle α=1.6 °, the ordinate of D is 1.2 * 1.68mm, gets 3000000 atoms, the detecting location that atom distributes is at distance slit outlet 0.254m place.Result of calculation as shown in figure 12.

The peak on the left side is corresponding to | F=3 among Figure 12〉attitude, the right is corresponding to | F=4〉attitude.Because the caesium atom velocity magnitude, direction, the reference position difference that enter magnetic field, the caesium atom site distribution density of two states all has certain broadening.Here more noticeable is that two peaks do not separate significantly, and their sizable parts are overlapped.Suppose selection | F=3〉the caesium atom of attitude, if the magnet B inlet is placed the maximal value place (horizontal ordinate is approximately 0.075) at peak, the left side, will accept maximum this moment | F=3〉atom of attitude, but also accepted more simultaneously | and F=4〉the caesium atom of attitude.Therefore, in some cases, directly B magnet inlet is placed peak-peak place and infeasible.If require | F=3〉the caesium atom of attitude is many as far as possible, and | F=4〉the caesium atom of attitude is the least possible, and so the magnet B inlet being placed between the horizontal ordinate 0.003～0.005 may be to select preferably.Figure 13 has been arranged in a word, just can determine the position of B magnet according to the purpose of research.

Under the condition of aforementioned supposition and according to the coordinate system of Fig. 5, the result of calculation of front is summarized as follows:

Collimating apparatus drift angle: 1.6 °;

y _D：0.0022m～0.0024m；

The position of B magnet: 0.003m～0.005m.

Claims (2)

1. method that beam optics parameter in the caesium bundle frequency marking is optimized is characterized in that may further comprise the steps:

Step 1, Monte Carlo sampling is carried out at the position angle of certain caesium atom in the caesium atomic beam and emission angle respectively, obtained the position angle and the emission angle of this caesium atom;

Step 2, the rate distribution of above-mentioned caesium atom is carried out Monte Carlo sampling, obtain the speed of this caesium atom;

Step 3, set up two field of line right-handed coordinate systems, and with step 1 and two result as starting condition, adopt the Bulirsch-Stoer method to find the solution the differential equation of motion of caesium atom in A magnetic field,, draw the movement locus of this caesium atom in A magnetic field according to solving result;

Step 4, repeating step one draw all movement locus by A magnetic field caesium atom to step 3; According to the movement locus of these caesium atoms, count number and all position distribution of the caesium atom that can pass through A magnetic field by the caesium atom in A magnetic field; Calculate the percent of pass of caesium atom according to the number of the caesium atom by A magnetic field, determine the drift angle of collimating apparatus, the parameters such as coordinate of collimator-alignment A magnetic field inlet according to percent of pass; Determine the position in B magnetic field by the position distribution of the caesium atom in A magnetic field according to all.

2. a kind of method that beam optics parameter in the caesium bundle frequency marking is optimized as claimed in claim 1 is characterized in that:

For drawing the equation of motion of caesium atom in two fields of line, set up two field of line right-handed coordinate systems earlier, wherein the x axle passes the axis on the face of cylinder, magnet plush copper place, and true origin is taken at plane, place, magnet porch; H is a magnetic field intensity, and then the equation of motion can be expressed as:

$\left\{\begin{array}{c}\stackrel{\·\·}{x}\≈0\\ \stackrel{\·\·}{y}\≈\frac{{\mathrm{\μ}}_{\mathrm{eff}}}{m}\frac{\∂H(y,z)}{\∂y}\\ \stackrel{\·\·}{z}\≈0\end{array}\right.---\left(6\right)$

Wherein, m is the quality of caesium atom; μ _EffBe effective moment of magnetic couple of caesium atom, provide by the Breit-Rabi formula:

${\mathrm{\μ}}_{\mathrm{eff}}=\stackrel{-}{+}\frac{\mathrm{\ξ}+m_F/4}{{[1+m_F\mathrm{\ξ}/2+{\mathrm{\ξ}}^2]}^{1/2}}{\mathrm{\μ}}_B---\left(7\right)$

μ wherein _BBe the Bohr magneton, the value of parameter ξ is got and is made ξ ≈ H/0.3284, when the caesium atom be in (3, m _F) (6) formula is got "+" number when attitude and (4 ,-4) attitude, when be in (4, m _F) attitude and satisfy m _F≠-4 o'clock, get "-" number;

η is as follows for definition caesium atom percent of pass:

$\mathrm{\η}=\frac{T_0}{T}---\left(8\right)$

Wherein T represents the total atom number of collimating apparatus ejection, T ₀Expression is by the atomicity of magnetic field of magnets;

When calculating the optimal value of collimating apparatus angle α, supposition α gets different values earlier, calculates the percent of pass η of caesium atom then under every kind of α value respectively;

The coordinate of supposing entry position, collimating apparatus axis alignment A magnetic field is y _D, allow y _DGet different values, calculate the value of percent of pass η under different values then, obtain η with y _DBetween a curved line relation, make y according to curved line relation _DOptimal value;

Definition caesium atom is as follows along the normalized position distribution density function of y direction f (y):

$\frac{1}{N}\mathrm{dN}=f\left(y\right)\mathrm{dy}$

Wherein the unit of f (y) is (individual)/m, and N is the caesium atom that passes through the magnetic field slit | F=3 attitude or | F=4〉the total atom number of attitude; Further will | F=3〉the position distribution density of attitude is defined as f ₁(y), | F=4〉attitude is defined as f ₂(y).

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