# Semi-Analytical Monte Carlo Method to Simulate the Signal of the VIP-2 Experiment

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## Abstract

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## 1. Introduction

## 2. Anomalous Electron Pairs

## 3. Structure of the Semi-Analytical Monte Carlo Simulation of the X-ray Signal

Algorithm 1: Structure of the semi-analytical Monte Carlo for VIP2. |

#### 3.1. Modulation Schemes

## 4. Discussion

`Nrepeats`times, but in addition to this, there is another parameter—

`nrepeat`—that acts as if the same initial configuration were run over and over again. The reason for introducing this additional option is that in exploratory runs there may not be enough X-ray events to allow an in-depth analysis of the spectral features of the X-ray signal. Taking a large value of

`nrepeat`corresponds to assuming an unrealistically large data-acquisition time, but it provides a clear view of the signal shape and of the resulting frequency spectra.

`Nrepeats = 10`(10 different initial configurations) and

`nrepeat = ${10}^{4}$`(each configuration is used 10,000 times as the initial configuration), for different initial conditions and modulation types. In these simulations, we take

`T0 = 86,400`(a full current modulation period lasts 86,400 s, i.e., one day), so that each plot displays a total of 10 full current modulation periods (i.e., 10 days). The resulting total data-taking time of ${10}^{6}$ days is clearly impossible to achieve with a single target like the one simulated here; however, this points to a possible, very ambitious variant of the experiment, where the single target—with its associated detector—is replicated many times to produce summed signals such as those in Figure 3—with 10,000 replicas one could obtain the same total data-taking time in just 100 days.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Parameter List

`const`types. Apart from

`modtype`and

`sdtype`, which define the modulation type, all parameters are fixed in the runs reported in this paper.

// 1. physical constants const double kBT = 1.38e-23 * 300.; // J const double eC = 1.602e-19; // C const double mc2 = 511e3; // electron mass, keV const double me = 9.109e-31; // electron mass, kg const double c = 3e8; // speed of light, m/s const double h = 6.626e-34; // Planck’s constant const double NA = 6.022e23; // Avogadro’s constant // 2. material constants (Cu) const double density = 8.96e3; // density, kg/m**3 const double A = 63.546; // atomic weight const double rho = 16.78e-9; // resistivity, Ohm*m const double sigma = 1/rho; // conductivity *) const double eF = 7.00; // Fermi energy, eV const double n = 6.846e27 * (2 *pow(eF,1.5)/3); // electron density at 0K,m**-3 const double Dc = sigma*kBT/(n* eC*eC); // Diffusion coefficient const double vF = 0.75*c*sqrt(2*eF/mc2); // Fermi velocity // other derived constants const double mfp = me*vF*sigma/(n*eC*eC); const double tau = mfp/vF; const double lambdae = h/(me*vF); // electron wavelength const double mfpce = 1/(n*M_PI*(lambdae/2)*(lambdae/2) ); const double taue = mfpce/vF; // mean time between close encounters // 3. experimental constants (VIP2, new target) const double l = 0.071; // m const double w = 0.03; // m const double z = 5e-5; // m const double V = l*w*z; // target volume const double M = density*V; // target mass const double Natoms = 1000*NA*M/A; // atoms in the target const double taue = mfpce/vF; // mean time between close encounters // 4. current modulation const int modtype = 2; // modulation type (0 = constant; 1 = sine wave; 2 = square wave) const double ItotMax = 180.; // max current (A) const double ItotMin = 0.; // min current (A) const double vdMax = ItotMax/( n*eC*z*w); // max vd (m/s) const double T0 = 86400.; // modulation period (s) // 5. MC setup const int Nrepeats = 10; // number of iterations of main loop const double tmax = 10.*T0; // max observation time (s) const int nrepeat = 10000; // number of repetitions with exactly the same initial conditions const int sdtype = 1; // initial spatial distribution (1=equidistributed; 2=random uniform) const double len = vdMax*tmax; // max length associated with drift (m) const double TraversalTime = l/vdMax; // shortest traversal time const double dtT = TraversalTime; // "synonim" for TraversalTime const double dt = 1.; // timestep const double DecayFactor = nrepeat*(1 - exp(-dt/Ntau)); const int ntsteps = floor(tmax/dt); // number of timesteps const double xstart = -len; const double xstop = -0.010; const double Ltot = xstop - xstart; // total depth of electron reservoir const double dxe = 1; // anomalous electrons are evenly spaced 1 m apart const int Nelettr = floor(Ltot/dxe); const double Dcc = Dc/3.; // diffusion coefficient in the longitudinal direction only

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**Figure 1.**Young tableau for an electron system that violates the Pauli Exclusion Principle. The number n of wrong-symmetry electrons is very small with respect to the total number N. (adapted from Figure 2 in [17]).

**Figure 2.**Circuit scheme for the VIP2 experiment. Upper panel: the motion of the conduction electrons is driven by a current source that produces a drift speed ${\mathbf{v}}_{d}$ in the VIP2 target. The electrons flow from a large copper reservoir into the target. The reservoir is designed to provide a sufficient flow of “new” electrons so that the fraction of anomalous electrons (those with the wrong symmetry pairing with electrons in the target) is not readily exhausted by running the same electrons again and again through the target. Lower panel: for simulation purposes, the large bulk reservoir is replaced by a very long wire, and the drift speed is uniform throughout the circuit.

**Figure 3.**Signals obtained with

`Nrepeats = 10`,

`nrepeat = 10,000`, and

`T0 = 86,400`, for different initial conditions and modulation types. All panels show the X-ray counts in 100 s time bins vs. time. Top panel: randomly distributed electrons in the initial configuration; current is modulated with a square wave between a maximum value of 180 A and a minimum of 0 A. Center panel: electrons are initially equidistributed (fixed 1 m spacing), although the position of the first electron is random; current is modulated with a square wave between a maximum value of 180 A and a minimum of 0 A. Bottom panel: randomly distributed electrons in the initial configuration; the current is modulated with a sine wave between a maximum value of 180 A and a minimum of 0 A.

**Figure 4.**Zoom on the initial part of the signals shown in Figure 3. All panels show the X-ray counts in 100 s time bins vs. time. Top panel: randomly distributed electrons in the initial configuration; current is modulated with a square wave between a maximum value of 180 A and a minimum of 0 A. Center panel: electrons are initially equidistributed (fixed 1 m spacing), although the position of the first electron is random; the current is modulated with a square wave between a maximum value of 180 A and a minimum of 0 A. Bottom panel: randomly distributed electrons in the initial configuration; the current is modulated with a sine wave between a maximum value of 180 A and a minimum of 0 A.

**Figure 5.**Power spectral densities (PSD, or periodograms) obtained from the Fourier analysis of the signals (X-ray counts) shown in Figure 3. Top panel: randomly distributed electrons in the initial configuration; current is modulated with a square wave between a maximum value of 180 A and a minimum of 0 A. Center panel: electrons are initially equidistributed (fixed 1 m spacing), although the position of the first electron is random; the current is modulated with a square wave between a maximum value of 180 A and a minimum of 0 A. Bottom panel: randomly distributed electrons in the initial configuration; the current is modulated with a sine wave between a maximum value of 180 A and a minimum of 0 A. The red arrows show the expected positions of the most relevant harmonics of the fundamental frequency ($1.1574\times {10}^{-5}$ Hz).

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**MDPI and ACS Style**

Milotti, E.; Bartalucci, S.; Bertolucci, S.; Bazzi, M.; Bragadireanu, M.; Cargnelli, M.; Clozza, A.; Curceanu, C.; De Paolis, L.; Del Grande, R.;
et al. Semi-Analytical Monte Carlo Method to Simulate the Signal of the VIP-2 Experiment. *Symmetry* **2021**, *13*, 6.
https://doi.org/10.3390/sym13010006

**AMA Style**

Milotti E, Bartalucci S, Bertolucci S, Bazzi M, Bragadireanu M, Cargnelli M, Clozza A, Curceanu C, De Paolis L, Del Grande R,
et al. Semi-Analytical Monte Carlo Method to Simulate the Signal of the VIP-2 Experiment. *Symmetry*. 2021; 13(1):6.
https://doi.org/10.3390/sym13010006

**Chicago/Turabian Style**

Milotti, Edoardo, Sergio Bartalucci, Sergio Bertolucci, Massimiliano Bazzi, Mario Bragadireanu, Michael Cargnelli, Alberto Clozza, Catalina Curceanu, Luca De Paolis, Raffaele Del Grande,
and et al. 2021. "Semi-Analytical Monte Carlo Method to Simulate the Signal of the VIP-2 Experiment" *Symmetry* 13, no. 1: 6.
https://doi.org/10.3390/sym13010006