Skip to content

GitLab

  • Projects
  • Groups
  • Snippets
  • Help
    • Loading...
  • Help
    • Help
    • Support
    • Community forum
    • Submit feedback
  • Sign in
O OPALManualWiki
  • Project overview
    • Project overview
    • Details
    • Activity
    • Releases
  • Repository
    • Repository
    • Files
    • Commits
    • Branches
    • Tags
    • Contributors
    • Graph
    • Compare
    • Locked Files
  • Issues 0
    • Issues 0
    • List
    • Boards
    • Labels
    • Service Desk
    • Milestones
    • Iterations
  • Merge requests 0
    • Merge requests 0
  • CI/CD
    • CI/CD
    • Pipelines
    • Jobs
    • Schedules
  • Operations
    • Operations
    • Incidents
    • Environments
  • Analytics
    • Analytics
    • CI/CD
    • Code Review
    • Issue
    • Repository
    • Value Stream
  • Wiki
    • Wiki
  • Snippets
    • Snippets
  • Members
    • Members
  • Activity
  • Graph
  • Create a new issue
  • Jobs
  • Commits
  • Issue Boards
Collapse sidebar
  • snuverink_j
  • OPALManualWiki
  • Wiki
  • partmatter

Last edited by Jochem Snuverink Sep 13, 2017
Page history

partmatter

Table of Contents
  • 1. Physics Models Used in the Particle Matter Interaction Model
    • 1.1. The Energy Loss
    • 1.2. The Coulomb Scattering
    • 1.3. The Flow Diagram of CollimatorPhysics Class in OPAL
    • 1.4. Available Materials in OPAL
    • 1.5. Example of an Input File
    • 1.6. A Simple Test

1. Physics Models Used in the Particle Matter Interaction Model

The command to define the particle matter interacton is PARTICLEMATTERINTERACTION.

MATERIAL

The material of the surface.

ENABLERUTHERFORD

Switch to disable Rutherford scattering, default true.

The so defined instance has then to be added to an element using the attribute

1.1. The Energy Loss

The energy loss is simulated using the Bethe-Bloch equation.

-\frac{\mathrm{d} E}{\mathrm{d} x}=\frac{K z^2 Z}{A \beta^2}\left[\frac{1}{2} \ln{\frac{2 m_e c^2\beta^2 \gamma^2 Tmax}{I^2}}-\beta^2 \right],

where Z is the atomic number of absorber, A is the atomic mass of absorber, m_e is the electron mass, z is the charge number of the incident particle, K=4\pi N_Ar_e^2m_ec^2, r_e is the classical electron radius, N_A is the Avogadro’s number, I is the mean excitation energy. \beta and \gamma are kinematic variables. T_{max} is the maximum kinetic energy which can be imparted to a free electron in a single collision.

T_{max}=\frac{2m_ec^2\beta^2\gamma^2}{1+2\gamma m_e/M+(m_e/M)^2},

where M is the incident particle mass.

The stopping power is compared with PSTAR program of NIST in Figure 1.

Figure 1 : The comparison of stopping power with PSTAR.

dEdx

Energy straggling: For relatively thick absorbers such that the number of collisions is large, the energy loss distribution is shown to be Gaussian in form. For non-relativistic heavy particles the spread \sigma_0 of the Gaussian distribution is calculated by:

\sigma_0^2=4\pi N_Ar_e^2(m_ec^2)^2\rho\frac{Z}{A}\Delta s,

where \rho is the density, \Delta s is the thickness.

1.2. The Coulomb Scattering

The Coulomb scattering is treated as two independent events: the multiple Coulomb scattering and the large angle Rutherford scattering.
Using the distribution given in Classical Electrodynamics, by J. D. Jackson, the multiple- and single-scattering distributions can be written:

P_M(\alpha) \;\mathrm{d} \alpha=\frac{1}{\sqrt{\pi}}e^{-\alpha^2}\;\mathrm{d}\alpha,
P_S(\alpha) \;\mathrm{d} \alpha=\frac{1}{8 \ln(204 Z^{-1/3})} \frac{1}{\alpha^3}\;\mathrm{d}\alpha,

where \alpha=\frac{\theta}{<\Theta^2>^{1/2}}=\frac{\theta}{\sqrt 2 \theta_0}.

The transition point is \theta=2.5 \sqrt 2 \theta_0\approx3.5 \theta_0,

\theta_0=\frac{{13.6}{MeV}}{\beta c p} z \sqrt{\Delta s/X_0} [1+0.038 \ln(\Delta s/X_0)],

where p is the momentum, \Delta s is the step size, and X_0 is the radiation length.

1.2.1. Multiple Coulomb Scattering

Generate two independent Gaussian random variables with mean zero and variance one: z_1 and z_2. If z_2 \theta_0>3.5 \theta_0, start over. Otherwise,

x=x+\Delta s p_x+z_1 \Delta s \theta_0/\sqrt{12}+z_2 \Delta s \theta_0/2,
p_x=p_x+z_2 \theta_0.

Generate two independent Gaussian random variables with mean zero and variance one: z_3 and z_4. If z_4 \theta_0>3.5 \theta_0, start over. Otherwise,

y=y+\Delta s p_y+z_3 \Delta s \theta_0/\sqrt{12}+z_4 \Delta s \theta_0/2,
p_y=p_y+z_4 \theta_0.

1.2.2. Large Angle Rutherford Scattering

Generate a random number \xi_1, if \xi_1 < \frac{\int_{2.5}^\infty P_S(\alpha)d\alpha}{\int_{0}^{2.5} P_M(\alpha)\;\mathrm{d}\alpha+\int_{2.5}^\infty P_S(\alpha)\;\mathrm{d}\alpha}=0.0047, sampling the large angle Rutherford scattering.

The cumulative distribution function of the large angle Rutherford scattering is

F(\alpha)=\frac{\int_{2.5}^\alpha P_S(\alpha) \;\mathrm{d} \alpha}{\int_{2.5}^\infty P_S(\alpha) \;\mathrm{d} \alpha}=\xi,

where \xi is a random variable. So

\alpha=\pm 2.5 \sqrt{\frac{1}{1-\xi}}=\pm 2.5 \sqrt{\frac{1}{\xi}}.

Generate a random variable P_3,
if P_3>0.5

\theta_{Ru}=2.5 \sqrt{\frac{1}{\xi}} \sqrt{2}\theta_0,

else

\theta_{Ru}=-2.5 \sqrt{\frac{1}{\xi}} \sqrt{2}\theta_0.

The angle distribution after Coulomb scattering is shown in Figure 2. The line is from Jackson’s formula, and the points are simulations with Matlab. For a thickness of \Delta s=1e-4 m, \theta=0.5349 \alpha (in degree).

Figure 2: The comparison of Coulomb scattering with Jackson’s book.

10steps

1.3. The Flow Diagram of CollimatorPhysics Class in OPAL

Figure 3: The diagram of CollimatorPhysics in OPAL.

diagram

Figure 4: The diagram of CollimatorPhysics in OPAL (continued).

Diagram2

1.3.1. The Substeps

Small step is needed in the routine of CollimatorPhysics.

If a large step is given in the main input file, in the file CollimatorPhysics.cpp, it is divided by a integer number n to make the step size using for the calculation of collimator physics less than 1.01e-12 s. As shown by Figures 3 and 4 in the previous section, first we track one step for the particles already in the collimator and the newcomers, then another (n-1) steps to make sure the particles in the collimator experience the same time as the ones in the main bunch.

Now, if the particle leave the collimator during the (n-1) steps, we track it as in a drift and put it back to the main bunch when finishing (n-1) steps.

1.4. Available Materials in OPAL

Table 1. List of materials with their parameters implemented in OPAL.
Material Z A \rho [g/cm^3] X0 [g/cm^2] A2 A3 A4 A5 OPAL Name

Aluminum

13

26.98

2.7

24.01

4.739

2766

164.5

2.023E-02

Aluminum

AluminaAl2O3

50

101.96

3.97

27.94

7.227

11210

386.4

4.474e-3

AluminaAl2O3

Copper

29

63.54

8.96

12.86

4.194

4649

81.13

2.242E-02

Copper

Graphite

6

12.0172

2.210

42.7

2.601

1701

1279

1.638E-02

Graphite

GraphiteR6710

6

12.0172

1.88

42.7

2.601

1701

1279

1.638E-02

GraphiteR6710

Titan

22

47.8

4.54

16.16

5.489

5260

651.1

8.930E-03

Titan

Air

7

14

0.0012

37.99

3.350

1683

1900

2.513E-02

Air

Kapton

6

12

1.4

39.95

2.601

1701

1279

1.638E-02

Kapton

Gold

79

197

19.3

6.46

5.458

7852

975.8

2.077E-02

Gold

Water

10

18

1

36.08

2.199

2393

2699

1.568E-02

Water

Mylar

6.702

12.88

1.4

39.95

3.35

1683

1900

2.513E-02

Mylar

Berilium

4

9.012

1.848

65.19

2.590

966.0

153.8

3.475E-02

Berilium

Molybdenum

42

95.94

10.22

9.8

7.248

9545

480.2

5.376E-03

Molybdenum

1.5. Example of an Input File

KX1IPHYS: ParticleMatterInteraction, TYPE="Collimator",MATERIAL="Copper";

KX2IPHYS: ParticleMatterInteraction, TYPE="Collimator",MATERIAL="Graphite";

KX0I: ECollimator, L=0.09, ELEMEDGE=0.01, APERTURE={0.003,0.003},OUTFN="KX0I.h5", 
      PARTICLEMATTERINTERACTION='KX1IPHYS';

FX5:  Slit, L=0.09, ELEMEDGE=0.01, APERTURE={0.005,0.003}, 
      PARTICLEMATTERINTERACTION='KX2IPHYS';

FX16: Slit, L=0.09, ELEMEDGE=0.01, APERTURE={-0.005,-0.003}, 
      PARTICLEMATTERINTERACTION='KX2IPHYS';

FX5 is a slit in x direction, the APERTURE is POSITIVE, the first value in APERTURE is the left part, the second value is the right part. FX16 is a slit in y direction, the APERTURE is NEGATIVE, the first value in APERTURE is the down part, the second value is the up part.

1.6. A Simple Test

A cold Gaussian beam with \sigma_x=\sigma_y=5 mm. The position of the collimator is from 0.01 m to 0.1 m, the half aperture in y direction is 3 mm. Figure 5 shows the trajectory of particles which are either absorbed or deflected by a copper slit. As a benchmark of the collimator model in OPAL, Figure 6 shows the energy spectrum and angle deviation at z=0.1 m after an elliptic collimator.

Figure 5: The passage of protons through the collimator.

longcoll6

Figure 6: The energy spectrum and scattering angle at z=0.1 m

spectandscatter

Clone repository
  • autophase
  • beam command
  • benchmarks
  • control
  • conventions
  • distribution
  • elements
  • fieldmaps
  • fieldsolvers
  • format
  • geometry
  • Home
  • introduction
  • lines
  • opal madx
View All Pages