Difference between revisions of "Simulations of Particle Interactions with Matter"
Line 1,038: | Line 1,038: | ||
: Here a thin absorber is one whose thickness is less than the charge equilibrium distance defined as the distance traveled until the projectile's velocity is <math> v \ll zc\alpha</math> | : Here a thin absorber is one whose thickness is less than the charge equilibrium distance defined as the distance traveled until the projectile's velocity is <math> v \ll zc\alpha</math> | ||
− | A rule of thumb is that for low energy ions thickness <math>\sim \frac{5 \frac{\mu g}{cm^2}{\rho}</math> | + | A rule of thumb is that for low energy ions thickness <math>\sim \frac{5 \frac{\mu g}{cm^2}}{\rho}</math> |
=== Multiple Scattering === | === Multiple Scattering === |
Revision as of 23:15, 26 September 2007
Overview
Particle Detection
A device detects a particle only after the particle transfers energy to the device.
Energy intrinsic to a device depends on the material used in a device
Some device of material with an average atomic number (
) is at some temperature ( ). The materials atoms are in constant thermal motion (unless T = zero degrees Klevin).Statistical Thermodynamics tells us that the canonical energy distribution of the atoms is given by the Maxwell-Boltzmann statistics such that
represents the probability of any atom in the system having an energy where
Note: You may be more familiar with the Maxwell-Boltzmann distribution in the form
where
would represent the molesules in the gas sample with speeds between andExample 1: P(E=5 eV)
- What is the probability that an atom in a 12.011 gram block of carbon would have and energy of 5 eV?
First lets check that the probability distribution is Normailized; ie: does
?
is calculated by integrating P(E) over some energy interval ( ie: ). I will arbitrarily choose 4.9 eV to 5.1 eV as a starting point.
assuming a room empterature of
then
and
or in other words the precise mathematical calculation of the probability may be approximated by just using the distribution function alone
This approximation breaks down as
Since we have 12.011 grams of carbon and 1 mole of carbon = 12.011 g =
carbon atomsWe do not expect to see a 5 eV carbon atom in a sample size of
carbon atoms when the probability of observing such an atom isThe energy we expect to see would be calculated by
If you used this block of carbon as a detector you would easily notice an event in which a carbon atom absorbed 5 eV of energy as compared to the energy of a typical atom in the carbon block.
- Silicon detectors and Ionization chambers are two commonly used devices for detecting radiation.
approximately 1 eV of energy is all that you need to create an electron-ion pair in Silicon
approximately 10 eV of energy is needed to ionize an atom in a gas chamber
The low probability of having an atom with 10 eV of energy means that an ionization chamber would have a better Signal to Noise ratio (SNR) for detecting 10 eV radiation than a silicon detector
But if you cool the silicon detector to 200 degrees Kelvin (200 K) then
So cooling your detector will slow the atoms down making it more noticable when one of the atoms absorbs energy.
also, if the radiation flux is large, more electron-hole pairs are created and you get a more noticeable signal.
Unfortunately, with some detectore, like silicon, you can cause radiation damage that diminishes it's quantum efficiency for absorbing energy.
The Monte Carlo method
- Stochastic
- from the greek word "stachos"
- a means of, relating to, or characterized by conjecture and randomness.
A stochastic process is one whose behavior is non-deterministic in that the next state of the process is partially determined.
Quantun Mechanics is perhaps the best example such a non-deterministic systems. The canonical systems in Thermodynamics is another example.
Basically the monte-carlo method uses a random number generator (RNG) to generate a distribution (gaussian, uniform, Poission,...) which is used to solve a stochastic process based on an astochastic description.
Example 2 Calculation of
- Astochastic description
- may be measured as the ratio of the area of a circle of radius divided by the area of a square of length
You can measure the value of
if you physically measure the above ratios.- Stochastic description
- Construct a dart board representing the above geometry, throw several darts at it, and look at a ratio of the number of darts in the circle to the total number of darts thrown (assuming you always hit the dart board).
- Monte-Carlo Method
- Here is an outline of a program to calulate using the Monte-Carlo method with the above Stochastic description
begin loop x=rnd y=rnd dist=sqrt(x*x+y*y) if dist <= 1.0 then numbCircHits+=1.0 numbSquareHist += 1.0 end loop print PI = 4*numbCircHits/numbSquareHits
A Unix Primer
To get our feet wet using the UNIX operating system, we will try to solve example 2 above using a RNG under UNIX
List of important Commands
- ls
- pwd
- cd
- df
- ssh
- scp
- mkdir
- printenv
- emacs, vi, vim
- make, gcc
- man
- less
- rm
Most of the commands executed within a shell under UNIX have command line arguments (switches) which tell the command to print information about using the command to the screen. The common forms of these switches are "-h", "--h", or "--help"
ls --help ssh -h
the switch deponds on your flavor of UNIX
if using the switch doesn;t help you can try the "man" (sort for manual) pages (if they were installed). Try
man -k pwd
the above command will search the manual for the key word "pwd"
Example 3: using UNIX
Step
- login to inca.
click here for a description of logging in if using windows - mkdir src
- cd src
- cp -R ~tforest/NucSim/Day1 ./
- ls
- cd Day1
- make
- ./rndtest
Here is a web link to the source files you can copy in case the above doesn't work
A Root Primer
Example 1: Create Ntuple and Draw Histogram
Cross Sections
Definitions
- Total cross section
- =
- Differential cross section
- =
- Solid Angle
- = surface area of a sphere covered by the detector
- ie;the detectors area projected onto the surface of a sphere
- A= surface area of detector
- r=distance from interaction point to detector
- sterradians
- if your detector was a hollow ball
- sterradians
- Units
- Cross-sections have the units of Area
- 1 barn =
- [units of ] =
- Fixed target scattering
- is the area of the ring of incident particles
= # of particles in =
- = # particles in a ring of radius and thickness
You can measure
if you measure the # of particles detected in a known detector solid angle from a known incident particle Flux ( ) as
Alternatively if you have a theory which tells you
which you want to test experimentally with a beam of flux then you would measure counts (particles)
- Units
- = # of particles
- or for a count rate divide both sides by time and you get beam current on the RHS
- integrate and you have the total number of counts
- Classical Scattering
- In classical scattering you get the same number of particle out that you put in (no capture, conversion,..)
- tells you how the impact parameter changes with scattering angle
Example 4: Elastic Scattering
This example is an example of classical scattering.
Our goal is to find
for an elastic collision of 2 impenetrable spheres of diameter . To solve this elastic scattering problem we will describe the collision using the Center of Mas (C.M.) coordinate system in terms of the reduced mass. As we shall see, by using C.M. coordinate system the 2-body collision becomes a 1-body problem. Then we will describe the motion of the reduced mass in the C.M. Frame.
Media:SPIM_ElasCollis_Lab_CM_Frame.xfig.txt
- Variable definitions
- = impact parameter ; distance of closest approach
- = mass of incoming ball
- = mass of target ball
- = iniital velocity of incoming ball in Lab Frame
- = final velocity of in Lab Frame
- = scattering angle of in Lab frame after collision
- = iniital velocity of in C.M. Frame
- = final velocity of in C.M. Frame
- = iniital velocity of in C.M. Frame
- = final velocity of in C.M. Frame
- = scattering angle of in C.M. frame after collision
- Determining the reduced mass
- vector definitions
- = a position vector pointing to the location of
- = a position vector pointing to the location of
- = a position vector pointing to the center of mass of the two ball system
- = the magnitude of this vector is the distance between the two masses
In the C.M. reference frame the above vectors have the following relationships
solving the above equations for
and and defining the reduced mass as- reduced mass
leads to
We can use the above reduced mass relationships to construct the Lagrangian in terms of
instead of and thereby reducing the problem from a 2-body problem to a 1-body problem.- Construct the Lagrangian
The Lagrangian is defined as:
where
kinetic energy of the system
Potential energy of the system which describes the interaction
- =
after substituting derivative of the expressions for
and- = The 2-body problem is now described by a 1-body Lagrangian
Lagranges equations of motion are given by
where
represents on of the coordinate (cannonical variables).To get the classical scattering cross section we are interested in finding an expression for the dependence of the impact parameter on the scattering angle,
.Now lets redraw the collision in terms of a reference frame fixed on
(before collision its the Lab Frame but not after collision). Media:SPIM_ElasColls_CMFrame_xfig.txt
The C.M. Frame rides along the center of mass, the above coordinate system though has its origin on
and only overlaps in space with the CM frame at the collision point sufficiently to illustrate . If then there is no collision ( ), otherwise a collision happens when r=a (the distance between the balls is equal to their diameter). A head on collision is defined as ( ).- Observation
- as gets smaller, gets bigger
Using plane polar coordinates (
) we can describe the problem in the lab frame as:
Lagranges Equation of Motion:
there is a constant of motion ( Constant angular momentum)
substitute
into
The two equations above are in terms of
and whereas our goal is to find an expression for . Since is related to and is related to ( ; see figure above) we should try and find expressions for in terms of- Trick
- or
We now need an expression for
in order to integrate the above equation to determine the functional dependence of and hence .Since Energy is conserved (Elastic Scattering), we may define the Hamiltonian as
solving for
substituting the above into the equation for
and integrating:
For
:
substituting this expression for
into the last expression for above :
- Integral Table
let
then
or
- Now substitue the above into the expression for
drop the negative sign, sqrt in denominator allows this, and use the trig identity
- compare with result from definition
- = scattering cross-section
- number of particles scattered = number of incident particles
- Area =
Lab Frame Cross Sections
The C.M. frame is often chosen to theoretically calculate cross-sections even though experiments are conducted in the Lab frame. In such cases you will need to transform cross-sections between two frames.
The total cross-section should be frame independent
or
where
is in the CM frame and is in the Lab frame.
- A non-relativistic transformation
The transformation is governed by the dependence of
onLets return back to our picture of the scattering Process
if we superimpose the vectors
and we haveTrig identities (non-relativistic Gallilean transformation) tell us
solving for
For an elastic collision only the directions change in the CM Frame:
&- From the definition of the C.M.
- conservation of momentum in CM Frame
- Gallilean Coordinate transformation
- another experission for
using the above gallilean transformation we can do the following
or
after a little trig substitution
constant
now use the chain rule to find
- constant
after substitution:
For the above equation to be more useful one would prefer to recast it in terms of only
and masses.Stopping Power
Ann. Phys. vol. 5, 325, (1930)
Bethe Equation
Classical Energy Loss
Consider the energy lost when a particle of charge (
) traveling at speed is scattered by a target of charge ( ). Assume only the coulomb force causes the particle to scatter from the target as shown below.- Notice
- as is scattered the horizontal component of the coulomb force ( ) flips direction; ie no horizontal force
where
- k =
- r = distance between incident projectile and target atom
- b= impact parameter of collision
Using the definition of Impulse one can determine the momentum change of as
Let's assume that the energy lost by the incident particle
is absorbed by an electron in the target atom. This energy may be cast in terms of the incident particles momentum change asBy calculating the change in momentum (
) of the incident particle we can infer the amount of energy lost by the incident particle and absorbed one of the target materials atomic electrons.using
we havecasting this in terms of the classical atomic electron radius
- just equate
Then
and
- : = 1 here because I shall assume the energy is lost to just the electron and the Atom is a spectator
Now let's calculate an expression representing the average energy lost for an incident particle traversing a material of some thickness.
Let
- = Probability of an interaction taking place which results in an energy loss
If we let
Z = Atomic Number = # electrons in target Atom = number of protons in an Atom
N = Avagadros number =
A = Atomic mass =
= probability of hitting an atomic electron in the area of an annulus of radius ( ) with an energy transfer between and
Then
- = energy lost by the incident particle per distance traversed through the material
I am just adding up all the energy losses weighted by the probability of the energy loss to find the total energy loss.
- : classically
- =
- =
- =
where
if A=1The limits of the above integral should be more physical in order to reflect the limits of the physics interaction. Let b_{min} and b_{max} represent the minimum and maximum possible impact parameter where the physics is discribed, as shown above, by the coulomb force.
- What is ?
if
then diverges and the energy transfer . Physically there is a maximum energy that may be transferred before the physics of the problem changes (ie: nuclear excitation, jet production, ...). The de Borglie wavelength of the atom is used to estimate a value for such that- What is ?
As
gets bigger the interaction is "softer" and longer. If the interaction time ( ) is so long that it is equivalent to an electron orbit ( ) then the atom looks more like it is neutrally charged. You move from an interaction in which the electron orbit is perturbed adiabatically such that there is no orbit change and the minimum amount of energy is transferred to no interaction taking place because the atom is neutral.Let
- : fields at high velocities get Lorentz contracted
- : I mean excitation energy of target material ( )
Condition for
:
Example 5: Find for a 10 MeV proton hitting a liquid hydrogen ( ) target
A = Z=z=1
= 0.511 MeV
I = 21.6 eV : see solid data point From Figure 27.5 on pg 6 of PDG below.
Just need to know
and"a 10 MeV proton"
Kinetic Energy (K.E.) = 10 MeV =Proton is not relativistic
Plugging in the numbers:
- How much energy is lost after 0.3 cm?
Notice that the units for energy loss are normalized by the density of the material
To get the actual energy lost I need to multiply by the density. So for any given atom the energy loss will depend on the state (solid, gas, liqid) of the atom as this effects the density of the material.
- = 2.2 MeV
File:SPIM HydrogenStoppingPower.pdf Compare with Triumf Kinematics Handbook, 2nd edition, September 1987, L.G. Greeniaus
Bethe-Bloch Equation
While the classical equation above works in a limited kinematic regime, the Bethe-Bloch equation includes the corrections needed to cover most kinematic regimes for heavy particle energy loss.
where
- = Max K.E. transferable to the Target of mass in a single collision.
- = correction for electron spin and very distant collisions which deform the electron atomic orbits each process reducing dE/dx by
- = density correction term: in the classical derivation the material is treated as just a system of atoms uniformly distributed in space. These Atoms, however, give the material polarizability which can reduce the electric field (dielectric).
GEANT 4 implementation
The GEANT4 file (version 4.8.p01)
source/processes/electromagnetic/standard/src/G4BetheBlockModel.cc
is used to calculate hadron energy loss.
line 132
where
line 143
- = density corection =
line 148
- = shell correction, corrects for the classical asumption that the atomic electron velicity is initially zero; or the the incident particles velocity is far greater than the atomic electron's velocity.
line 154
Energy Dependence
The above curve shows the energy loss per disntace traveled (
) as a function of the incident particles energy. There are three basic regions. At low incident energies ( < 10^5 eV) the incident particle tends to excite or even ionize the atoms in the material it is penetrating. The maximum amount of energy loss per distance traveled is defined at as the Bragg peak. The region after the Bragg peak in which the energy loss per distance traveled reaches its smalest value is reffered to as the point of minimum ionizing. Minimimum ionizing particles will have incident energies corresponding to this value or larger. The characteristic of the minimum ionizing particles is that their energy loss per distance traveled is essentially constant making simulations easier until the drop below the minimum ionizing energy level as they are passing through the material.In general the Bethe-Bloch equation breaks down at low energies (below the Bragg peak) and is a good description (to within 10%) for
- and < 26 (Iron)
the
term in the Bethe-Bloch equation dominates between the Bragg peak and the minimum ionization region.the
term and its corrections influence the dependence of as you move up in energy beyond the minimum ionization point.Energy Straggling
While the Bethe-Bloch formula gives you a way to quantify the amount of energy a heavy charged particle looses as a function of the distance traveled, you should realize that when you calculate the total energy lost via
you are only determining the AVERAGE energy loss. In other words, Bethe-Bloch is the Astochastic process describing energy loss.
In reality the energy loss process is a stochastic process because of the statistical fluctuations which occur in the actual number of collisions which take place.
Thick Absorber
A thick absorber is one in which a large number of collisions takes place. In this situation the central limit theorem from statistics tells you that the larger number of random variables
involved will result in observables which are distributed in a Gaussian manner.The gaussian probability function is defined as
where the Full Width at Half Max (FWHM) of the distribution =
In the case of energy loss, the variance using the Bethe-Bloch equation should be
the realitivistic variance is
for very thick absorbers see
C. Tschaler, NIM 64, (1968) 237 ; ibid, 61, (1968) 141
When simulating energy loss of heavy charged particles the Bethe-Bloch equation may be used to calculate a \frac{dE}{dx} which can determine the average energy loss at the given kinetic energy of the particle. This average is then smeared according to a gaussian distribution of variance
Thin Absorbers
In thin absorbers the number of collisions is small preventing the use of the central limit theorem to describe the stochatic process of energy loss in terms of a Gaussian distribution. The Large energy transfers that are possible cause the energy loss distribution to look like a Gaussian one with a high energy tail (or foot).
The skewness of the resulting energy loss distribution is quantified as
- = lead term in Bethe Bloch equation
= density of absorbing material.
- = max energy transfered in 1 collision (headon / knock out collision)
This comes from the relativistic kinematics of an Elastic Collisions.
Conservation of Momentum
:Conservation of Energy
:using conservation of E & P as well as substituting for
you can showthis will lead to the equation
- : cons of E
- : cons of P
solving for
(Landau THeory)
Landau assumed
- is max energy transfer
- electrons are free (energy fransfer is so large you can neglect binding)
- incident particle maintains velocity (large momentum transfer from big mass to small mass) (bowling ball hits ping pong ball)
L. Landau, "On the Energy Loss of Fast Particles by Ionization", J. Phys., vol 8 (1944), pg 201
instead of a gaussian distribution Landau used
where
File:SPIM Landau ThinkAbsorberDist.jpg
(Vavilou's Theory)
Vavilous paper
P.V. Vavilou, "Ionization losses of High Energy Heavy Particles", Soviet Physics JETP, vol 5 (1950? )pg 749
describe the physics for the case
The distribution function derived is shown below as well as a conceptual overlay of Vavilou's and Landau's distributions. (The
in the picture should be a )
where
GEANT4's implementation
GEANT 4 uses the skewness parameter
to determine if it will use a "fluctuations model" to calculate energy straggling or the gaussian model described in section 2.2.1.kappa > 10
If
- > 10
and we have a thick absorber ( large step size) then the Gausian function in 2.2.1 is used to calculate energy straggling.
What happens is
is calculated via then the actual energy loss predicted by the simulation is chosen from a Gaussian distribution to account for energy straggling such that the of this Gaussian distribution is given by:where
- = electron density of the medium
- = charge of the incident particle
- = step size
- = cutoff kinetic energy for -electrons
tells GEANT where to put the cutoff for using the Gaussian distribution for energy straggling. This tells the simulation the low energy cutoff where Bethe-Bloch starts to fail due to ionization.
Delta-electrons
What is a
- electron?- electrons are also known as "knock -on" electrons and delta rays.
As heavy particles traverse a medium they can ionize electrons from atoms.
In a cloud chamber such and event would look like:
The physics of ionization is different from the physics used to calculate Bethe-Bloch energy loss. Remember Bethe-Bloch starts to break down at low energies below the Bragg peak.
Because of this GEANT 4 sets the cutoff for this proces to be
- > 1 keV
Note: The BE energies of an electron in Hydrogen is 13.6 ev and the electrons in Argon have binding energies between 15.7 eV and 3.2 keV.
Fluctuations Model: kappa < 10
If
Then GEANT 4 uses a "Fluctuations Model" to determine energy loss instead of Bethe-Bloch.
- Fluctuations Model
-
- the atom is assumed to have on 2 energy levels and
- you can excite the atom and lose either or or you can ionize the atom and lose energy according to a function.
The total energy loss in a step will be
where
- , , and are the number of collisions which are sampled froma poison distribution
- : rand = random number between 0 and 1
- = mean ionization energy
The fluctuation model was comparted with data in
K. Lassila-Perini and L. Urban, NIM, A362 (1995) pg 416
The cross sections used for excitation and ionization may be found in
H. Bichel, Rev. Mod. Phys., vol 60 (1988) pg 663
Range Straggling
- Def of Range (R)
- The distance traveled before all the particles energy is lost.
- = theoretical calucation of the path length traveled by a particle of incident energy
- Note units:
the Energy Straggling introduced in the previous section results is particles penetrating material to different depths. The energy straggling results in Range straggling.
If we do a shielding experiment where we have a source of incident particles of energy E and we count how many "punch" through a material of thickness (x) we would see a trnsmission coeffient
which would look likeFractional Range Straggling
fractional range stragling
Assuming the energy loss of a non-relativistic heavy ion through matter follows a Gaussian (thick absorber)
Then it can be shown that
where
- = mass of the target electrons
- = atomic mass of the Projectile
since
- kg
and
- 1 a.m.u. = kg
then
- = 1.17 % if using a proton (A=1)
The above is a "back of the envelope" estimate. The experimentaly measured values for Cu, Al, and Be target using a proton projectile are
If the incident projectile is an electron then
making electron range straggling a vague concept.Therre are several definitions of electron range
- 1.) Maximum Range ( )
- This range is defined using the continuous slowing down approximation (CSDA) in the the electrons are assumed to have many collisions over very small dimensions making it appear to be continuous energy loss instead of discrete. The range is then calculated by integrating over these average energy losses .
- 2.) Practical Range ( )
- This stopping distance is defined by extrapolating the electron transmission cure to zero (see below).
Electron Capture and Loss
Bohr Criterion
- "A rapidly moving nucleus is fully ionized if its velocity exceeds that of its most tightly bound electron"
The Bohr Model:
for the inner most electron (
)
- the fine structure constant
If
the the nucleus is fully ionizedor
if
alternatively if the ion is moving through a material with a speed such that
Then electrons may be captured by the projectile and lost by the target.
Z-effective
Describing the charge state of your heavy ion traveling through matter at a velocity below the Bohr criterion is very complicated. There is a competition between electron capture and loss. Accurate cross sections are needed to simulate the process reliably.
Some insight into this process can be found using the Thomas-Fermi model to describe a nucleaus (the charged heavy projectile) moving slow enough so it has captured many electrons but fast enough so its not neutral.
let
- effective charge f the projectile =
- = number of protons
- = average number of captured electrons
for the purpose of simulations you would like a relationship for
in terms of andwhen
MeV the data indicates thatwhere
When calculating stopping power for E < 10 MeV you use Zeff in the Bethe=Bloch equation.
Note: As the ions charge state fluctuates while is slows down (or if accelerated through materials) you will need to recalcualte the energy loss, and as a result you will get larger energy loss fluctuations in this energy range.
For thin aboserber you will look for stripping and loss cross sections.
- Here a thin absorber is one whose thickness is less than the charge equilibrium distance defined as the distance traveled until the projectile's velocity is
A rule of thumb is that for low energy ions thickness