Ultra-dilute gas transmittance revisited

Let us study a system with several particles: odd number of 2D particles, aligned parallel to the detector, spaced every 2r. The detector is placed symmetrically in the middle. We will publish these conditions later. An identical probability distribution gives the probability of locating each particle. Although we use Gaussian distributions, any probability distribution works, because (int _{-infty}^{infty}P(x)dx=1). In this way, we do not become attached to any particular form of wave packet. Again, to simplify the calculations, we project 2D particles onto the plane of the detector to work with 1D distributions. Figure 3 shows such a configuration for (N=9) particles. The solid red line marks the detector and the dotted red lines are the limits of the visibility tunnel.

We define the transmittance of the dilute gas cloud TR as proposed in Ref.11. The transmittance is the probability that a photon that the detector would have detected in the absence of a cloud passes unabsorbed through the whole NOT-cloud of elements and is detected by the detector. Collisions with individual particles are independent, so we can think of this process as a Markov chain:

$$begin{aligned} TR = prod _{n=1}^N big ( 1-G,P(o_n) big ), end{aligned}$$

(seven)

where (G,P(o_n)) is the probability that the photon is absorbed by not-th gas molecule, which is compensated by (on) of the detector, see Eq. (5) in Ref.11.

picture 3

A sample configuration of 9 identical particles evenly distributed every 2r. The solid red line marks the detector and the dashed red lines are the edges of the visibility tunnel.

Now we take advantage of periodicity. Identical pieces (of the same shape and number) of the probability distribution escape and flow into the tunnel of visibility. Thus, we can “unfold” a single distribution periodically instead of considering all distributions in one place. Then we virtually “shift” the detector NOT times (per a period of 2r) and take the product of all its positions. This way we can replace (o_n=r(2n-N-1)/2), and eq. (7) becomes:

$$begin{aligned} TR = prod _{n=1}^N left( 1-G,Pleft( rfrac{2n-N-1}{2}right) right) . end{aligned}$$

(8)

Figure 4

The idea of ​​splitting a probability distribution into several pieces, periodically every 2r as required by eq. (8). Values ​​of not, oh, (P_v(o)), (G,P_v(o)) and (1-G,P_v(o)) are overlaid in red for each part for convenience.

Figure 4 illustrates this idea. A distribution is divided into several pieces, periodically every 2r. Values ​​of not, oh, (P_v(o)), (G,P_v(o)) and (1-G,P_v(o)) are superimposed for each part for convenience.

As expected, all probabilities (P_v(o)) sum up to 1, which means that the analyzed interval contains the entire particle. Probability does not “leak” laterally. We interpret this as a conserved mass in the system.

The following applies to (G=const):

$$begin{aligned} sum _{n=1}^{N} P_v(o_n)=1 Rightarrow sum _{n=1}^{N} G,P_v(o_n)=G=const . end{aligned}$$

(9)

Transmission is the product of (1-G,P_v(o_n)), see eq. (seven). The sum of its components is always constant. (sum (1-G,P_v(o_n))=const) As shown above. However, the constant sum does not guarantee that the product is constant:

$$begin{aligned} sum _{n=1}^{N} a_n=sum _{n=1}^{N} b_n not Rightarrow prod _{n=1}^{N} a_n=prod _{n=1}^{N} b_n. end{aligned}$$

(ten)

This shows that even for closed conserved-mass systems, the transmission may change because the conservation of mass depends on a sum (of certain elements), but the transmission depends on a product (of the same elements). In general, the transmission depends on how the distributions are divided. This division depends on (i) the shapes of the probability distributions and (ii) the width of the detector.

The shape of the normal distribution depends only on its standard deviation. detector size r significantly influences the values ​​of the individual components of the product for one or the other (r,sim,stdev) Where (rinfluencing the product and, finally, the measured transmission.

In the classical case, for well localized particles (ideal gas) and a macroscopic detector, we have (stdev,ll,r). This way, any nonzero probability of a particle always arrives at a piece making all the elements of the product of Eq. (7) equal to 1, except one element. The single element less than 1 determines the value of the entire product. The product does not change when changing r because there will always be only one such element. Thus, in this case, the detector size cannot affect the transmission measurement. This explains why we do not observe any dependence of the transmission on the detector size in classical systems.

This analysis applies to any number of particles (NOT). Even for NOT the (on) substitution leading to Eq. (8) should be slightly different.

Figure 5 shows the dependence of transmittance on particle standard deviation for measurement with a fixed-size detector. The following section describes the details of the graph.

Figure 5
number 5

The graphs show the sample dependence of transmittance on standard deviation for measurement with a fixed-size detector. The solid line is the on-axis measurement of the cloud and the dotted line indicates an off-axis measurement. The unit of length is equal to the radius of the detector r. The width of the detector is 2 ((r=1)). A 1D cloud is (N=61) particles in total, and they are evenly spaced every 2r. The g coefficient is fixed at 0.7 (after (TR_{cl}=30%)). Particles have a 1D normal distribution where the standard deviation is expressed in units of detector radius (r). Detector offset for off-axis detection is 20r of the cloud axis. The top diagram shows the magnification of the left side of the bottom diagram.

Dense or inhomogeneous clouds

If there is much more than one particle per detector area (as in any real-world configuration), then we repeat the above reasoning several times in the following way. We divide the gas cloud into enough parts that each of them statistically contains only one particle per “detection area”. We calculate the (partial) transmission for each of these parts independently. From the property of independence of the probability of absorption by individual gas particles, we calculate the product of the partial transmissions, obtaining the total transmission of the whole cloud.

The same approach works for analyzing inhomogeneous gas clouds. One has to divide such a cloud into homogeneous parts, calculate the (partial) transmissions separately and take their product to obtain the total transmission.

Alternatively, we can do the trick of adjusting the constant g. We can put it equal to (1-TR_{cl})where (TR_{cl}) is the classical transmittance of the cloud. We then take a set of “artificial” particles distributed exactly every 2r as requested above. Such an artificial particle represents all the real particles present in the visibility tunnel of a given piece. Remember, however, that the spread (eg standard deviation) of this artificial particle is the same as that of any cloud particle. That is, we do not sum the masses of individual particles to calculate the propagation speed. This last method is very efficient for numerical calculations.

3d cloud

For a three-dimensional gas cloud, it must first be projected onto the plane of the detector. For such a 2D model, we require to distribute the particles evenly: one particle per detection zone. The simple way to analyze 2D is to consider the normal distribution and a square detector with side equal to 2r. For such a system: (i) an analytical solution is available, see (11) and (18) in Ref.11 and (ii) the square shape of the detector makes it possible to cover the whole plane with adjacent detectors. Then we can perform the same periodic reasoning given above for the 1D model.

An arbitrary shape of the detector makes reasoning more difficult and changes the quantitative equations. However, this is still possible because it requires only a finite area of ​​the detector. However, qualitatively, the presented principle of the dependence of the transmission on the detector area is valid.

. Transmittance of gases ultradilutes revisited scientific reports

. Ultradilute gas transmittance revisited