# Presentation: 004109

## Particle ID with Si dE/dx

## Intro

- Our goal is to write a program that takes two track measurements, a particle’s momentum and the charge deposition it produces in our silicon detector, and outputs the certainty that the track was made by a proton,pion, muon, kaon, or electron.

## Description of Si

- Basic design of silicon detector: silicon wafers (300 microns thick) arranged dodecagonally around the beam pipe.
- 6 layers.
- Particle comes in, ionizes silicon atoms along its path.
- Amount of ionization/deposition depends on particle speed.
- Charge collected on strips.
- Up to six hits per track. Each gives:
- Charge Deposition
-  Number of Strips (1-5).

## Using dE/dx for Particle ID

- Ionization as a function of particle velocity: Given by Bethe-Bloch Equation.(see graph on upper right)
- This can also be drawn in momentum space; get one curve per particle.
- Particle identity determined by the curve our momentum/ ionization measurements land on.

## The Landau Distribution 

- 6 hits along single track fall into the Landau Distribution
- High tail resulting from secondary ionization, which occurs when the electrons deflected by a passing particle cause additional, “secondary”, electrons to be knocked free.
- Cannot be expressed in closed form
- No mean

## Hit Clustering; Multiple Strips 

- Can use spatial information about the charge deposition (number of strips) to detect and compensate for presence of secondary ionization.
- More strips mean more secondary ionization, so we can ignore hits with too many strips.

## Fitted Single Hit Distributions for ns(number of strips)=1,2,3,4,5

## Combining the Track’s Hits  

- We would like to associate with each track a single value for the ionization-i.e. we need a way of combining the six hits in a way that accurately represents the track as a whole.
- The most obvious way of doing this is to average the six hits.
- As it turns out, this is an extremely poor measure of the track ionization, since the Landau itself has no mean.

## Combining the Track’s Hits (Alternatives to the Average)

- Truncated Mean (averages a subset of the track’s hits)
- Trunc: discards hit with largest deposition
- Usens: discards hits with more than 3 strips
- Trunc3: like usens, but also discards 3-strip cluster w/ largest charge
- Slowing power: =1/<1/Ii>
- Maximum Likelihood
- Best option; haven’t fully implemented it yet, but from what we seen, holds most promise as far as distinguishing particles. Will be explained in detail later.

## Single Hit and Track Distributions

- If we send many of the same particle into our detector at fixed momentum, we expect their ionization to be the value predicted by the Bethe-Bloch equation for that curve.
- Measurements actually fall into a distribution around expected value.
- Two types of distributions which will be discussed:
- Single Hit: (e.g. The Landau) over hit readings taken directly from the detector, no  combination of 6 hits.
- Track Distributions:  over combined ionization for individual tracks. Particular method (trunc, trunc3, etc.) used here must be taken into account.
- We will be interested in the quantity
-        (I_measured(particle) – I_expected(particle))/ 
-       which tells us, for a specific particle, the number of standard deviations separating the measured track Ionization (using some combination of the six hits) from the expected track ionization given by the Bethe-Bloch Equation.
- We can see from the track distributions on the next page (given for protons at low momenta) that the mean ionization follows the shape of the Bethe-Bloch curve, decreasing with increasing momentum.
- Notice also that the width of the distribution decreases.

## Examples of Track Distributions

## The Bethe-Bloch Equation and the Universal Curve

- Using B-B eqn, devised our own parametrization to UC.
- Fit to data from detector (usens shown here: red=protons, blue=pions,green=electrons).
- Use param values from fit in our program.
- This is our basis for determining I_expected for a particular particle with a known momentum.

## Uncertainties

- Reminder: we would like to know the quantity
-               (I_measured – I_expected)/ 
- for a given particle track.
- In order to determine this, we need to know , the track uncertainty.
- Need to know upper and lower uncertainties as functions of mean ionization.
- Collected track distributions for fixed momenta (for trunc, trunc3, trunc3_ISL).
- Fit a bifurcated Gaussian to distribution for each momentum. Found mean (), upper and lower widths (2 , 1, respectively).
- Plotted upper, lower widths with respect to mean, to find variation of uncertainties with mean ionization.
- We then used the parametrizations:
- 1 = (5.2+(k122))
- 2 = (5.2+(k222))
- to fit the plot.

## Uncertainty Plots (revisited)

## Separations

- When the Bethe-Bloch curves for two particles get too close (relative to the width of the track distribution), it is impossible to tell one type of particle from the other.
- The appropriate measure of distance between two curves is given by the separation, which is defined as the difference in ADC counts divided by the width of the track distribution.
- Since the distances between the curves is fixed by Bethe-Bloch, we require **small uncertainties** to be able to classify a measurement as belonging on one particle’s curve or another’s.

## Maximum Likelihood Technique

- Is a way of finding the underlying probability distribution for a set of data drawn randomly from the distribution.
- Requires:
- data points x1,...,xn taken from an unknown probability distribution.
- an initial guess P(x) at the distribution’s general functional form, with at least one free parameter u.
- We define the likelihood function L() as the product of the probability densities at each xi:
-        L() = P(x1)...P(xn)
- In order to find the u that makes the set x1,..., xn most probable, we need  to find the value of u at which L has its absolute maximum.
- Instead of maximizing L, however, it is completely equivalent, and often much easier, to minimize -ln(L), which is given by:
- -ln(L) = -ln(P(x1))- ...-ln(P(xn)).

## Maximum Likelihood as an Alternative to Trunc, Usens, etc.

- Both common sense and experience suggest that the  Maximum Likelihood Technique is the best way to combine the six hits along a particular track. Its only drawback is that it is much more difficult to apply.
- As an approximation to the Landau, and as a parametrization for Maximum Likelihood fitting, we use a piecewise defined function that is a bifurcated Gaussian with an exponential tail.
- This function has 5 parameters, and ultimately we would like to express them all in terms of the MPV, so that we can perform a Likelihood fit on each track; i,.e. find the value of the MPV that maximizes the Likelihood function for a given set of hits

## Finding u1(u1ag), u2(u1ag), s1(u1ag), and s2(u1ag)

- Use fits to Single Hit Distributions shown above (for aggregate and 5 ns distributions) to express all parameters as linear functions of u1ag.
- Now able to perform Likelihood fit.
-             STDEV/AVG ->

|  | Av(s1/mu1ag) | Av(s2/mu1ag) | Av(mu1/mu1ag) | Av(mu2/mu1ag) |
| --- | --- | --- | --- | --- |
| ag | 0.247916 | 0.328999 | 1 | 1.346366 |
| ns=1 | 0.230004 | 0.178224 | 0.829174 | 1.047041 |
| ns=2 | 0.188371 | 0.232363 | 0.962608 | 1.220573 |
| ns=3 | 0.190532 | 0.285314 | 1.143752 | 1.390308 |
| ns=4 | 0.213832 | 0.631839 | 1.298738 | 1.935205 |
| ns=5 | 0.355758 | 1.558563 | 1.726436 | 3.282429 |

|  |  |  |  |  |
| --- | --- | --- | --- | --- |
| ag | 0.04483 | 0.122604 | 0 | 0.071549 |
| ns=1 | 0.027966 | 0.205839 | 0.040664 | 0.12298 |
| ns=2 | 0.016562 | 0.020457 | 0.017109 | 0.015171 |
| ns=3 | 0.067405 | 0.201227 | 0.012474 | 0.082183 |
| ns=4 | 0.101398 | 0.224923 | 0.048367 | 0.150548 |
| ns=5 | 0.269718 | 0.385868 | 0.256614 | 0.342063 |

## Simulation

- Used random number generator in computer to simulate tracks in the detector.
- Generated 1000 hits, found width of distribution to be substantially better than anything we’ve tried before.
- Will fit bifurcated Gaussians to track distributions, as before. Expecting smaller uncertainty with Likelihood.

## Conclusion

- State of Project: Actual code already written. Attempting to refine uncertainty measurements with Likelihood.
- For Trunc and Trunc3, we’ve found resolutions of 14.8 and 15.6 percent, respectively.
- Compare to 11% for COT and 12% for Likelihood.