Monte Carlo Simulations
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- Roberta Harmon
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1 Monte Carlo Simulations 1
2 Homework 3: Optical Sim A little late, so I am making it easier than my original goal (still due this coming MONDAY, 9-26) Run my Ray Tracer program and determine what fraction of photons survive under the following cases o Increase to as many photons as your computer can handle (1e8, 1e9?) o How does your answer change if reflection is diffuse? EC: Study partial diffuse and partial specular. What happens at 50/50? o Randomize the initial positions of the photons (currently only direction) EC: map out the light collection efficiency in 3 dimensions o Add absorption length in the quartz of 1 cm (EC: wavelength-dependent!) o Add scattering at random angle in Xe (MFP = 20 cm) (EC: correct angles!) o Absorb photons at edges of detector (10 cm tall) EC: hemispherical bottom NOTE: exercise has multiple right answers (but close) New rule: beginning programmers can do EC on own For Friday, need volunteer with 10/10 +EC from hw2 A new tradition 2
3 Sources summer/docs/2006--degreve--reflection_refraction.pdf 3
4 Our Next Physics Topic Neutrinos! o Builds on optical photon simulations, which built on random numbers in turn Neutrinoless double-beta decay, and finding the absolute neutrino mass: 2 nuclear physics problems o Majorana versus Dirac neutrinos: are these particles their own anti-particles? o Neutrinos oscillate in flavor, which means they must have mass. What is it? Both issues can be tackled by studying ß decay o We will focus on tritium (H-3) as a classic example. Used for latter (the mass) LXe used again This is not just modern it is is CONTEMPORARY physics o Not usually covered in any course 4
5 What is Energy Resolution? In general Recombination microphysics 5
6 Plot Digitization (all systems) o Java-based Plot Digitizer for Windows, Apple Mac, and Linux / Unix (at least some flavors) If for any reason this fails to work for your machine, Googling digitizing plots as an exact phrase should provide alternative programs (For large numbers of data points, there are automated versions as well in existence later) 6
7 For Homework # 4 7
8 HW #4: Energy Resolution Your mission is to find the best model to represent the data from the last slide (you may increase do_over ) o But first, determine how low you can go in approximating a binomial as a Gaussian (current setting is 10 particles, on line 65. Traditional is 5 or 10) o We know binomial sucks as match to energy resolution but add to the final plot Test the following models (combine all into 1 final plot with different colors and shapes, with the blue and red points you digitized on there as well, labeled as data) o 0: Poisson and modified Poisson (Fr = 1 is normal Poisson). Float the pre-factor in front of Ni (currently set to as default) in steps of to find the best match to data (std dev of quanta per Ni as a function of y), creating a new loop to do so. A flat value may not work. You may need a y-dependent factor o 1: Gaussian: Set the standard deviation to be some number times Ni and then try times Ne (current default is 0.06 and Ni not Ne). Try both ways, and like step 0 above you may need to have this factor depend on y (= Ne / [Ne + Nph] ) o 2: Repeat 1 but do it with sqrt(ne) and sqrt(ni). Again find best fit(s) o 3: Dobi-Sorensen method (2 ways): binominal convolved with Gaussian OR boxcar. Ultimately, as 0-3 each have 2 options there are EIGHT different ways Extra credit: re-do steps 0 to 3 using Np and rp as your variables for stat distributions in place of Ne and 1-rP 8
9 Tips and Tricks Code already outputs y s and sigma(quanta)/ni, that go on the x- and y-axes, respectively (y = Ne / Nq) It also tries to find a good second-order poly fit for you, so you can use it to plug in y-values that actually match the x-axis coordinates of points you digitized o You may need a higher order such as quartic, or finer precision. Modify code, though both of these ideas will slow It down immensely o You may also find some models work only on left half of plot (y < ~0.5) and in that case, visualize with plots, and restrict to below 0.5 and do it anyway Determine the best fit by calculating lowest meansquared error. Example is already provided for you by having a quadratic fitter included at code s bottom The bonus: first replace Np as Nx plus rp * Ni and then do Ne=Nq Np, the opposite of how it s done right now o Partial EC: you do not have to do all of 0-3 x2 ways to get full the 5% in points 9
10 Introducing Optimization Goodness of fit determiners: SSE (sum squared-err) and MSE (mean squared-err) SSE = a sum over ( theory experiment )^2 MSE = SSE divided by number of data points In the future, we will learn GoF parameters which included uncertainties (like chi^2) 10
11 What Physics is Happening? N q N x N i P r N p N e You are either an exciton or an ion/electron, so N q is just for book-keeping. Not a physical quantity A new electron is born from within the nucleus (n decay) o The neutrino flies away without depositing any energy (in the liquid xenon) o The H-3 (hydrogen-3 or tritium ) nucleus has become He-3 (2 protons, 1 neutron) The beta electron travels through the Xe in detector o It knocks out more electrons, generating electron-ion pairs (N i ) o But, it also excites atoms, leading to excitons too (N x ). The ratio of excitons to ions in xenon is ~0.2 (or to total of 0.17), energy and field ~independent Excited atoms lead to photons when de-exciting, and ionized atoms lead to electrons being collected (in an electric field). Higher field means more e- s ripped out o Some liberated electrons get re-captured, by parent ion or other. This is covered by an energy + field dependent recombination probability (called rp in app) 11
12 Continued è Conservation of the total (integer) number of quanta, of either type: Nq = Nx + Ni = Np + Ne- (simply due to energy conservation of course) Anti-Correlation: This means that for a fixed number of quanta Nq, if Nx goes up Ni must go down and vice versa, which leads to Np and Ne- being anticorrelated with each other: 1-1 Recombination Fluctuations: THIS is what you are modeling/simulating here. It controls the final slosh between the Np and Ne- populations (after the Nx/ Ni ratio of 0.2 determines the initial slosh, between the Nx and Ni, set to binomial and not touched) o Naively should be binomial: you EITHER excite/ionize, but data says NO Work Function: Averaging over the ionization and excitation processes, W LXe =13.7 ev (-> 73 N q /kev) 12
13 Let s Work Out an Example We will comment out one line, 144, and comment in three lines: LOTS of commenting in/out here This de-activates binominal recombination (a.k.a. electron re-capture) fluctuations and activates the modified Poisson function I invented instead We will leave the default value of the variance boost factor as Fr = * Ni and see how well it matches the data AS IS We will use not the discrete values but the values of the quadratic fit spit out as last line of my program s output to screen, to line up red s and blue s of p. 7 We will find it only works for low values of the ionization fraction (only the left side of the graph) You will find better value than.0075, including fixing the right hand side, by having value drop smoothly 13
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