1
Master thesis
Monte Carlo simulation of low energy electrons and
positrons in liquid water
Klas Marcks von Würtemberg
Stockholm 2003
2
Abstract
An advanced simulation code, LEEPS (Low Energy Electron Positron Simulation), has been
adapted to simulation of electrons and positrons in liquid water for energies down to 50 eV.
Different scattering parameters and results from simulations are compared with existing data
in the literature.
Several programs including a subroutine package for simulation of secondary electrons
created in binary like collisions have been developed in purpose of charting different
characteristics of the energy deposition.
A toy model for DNA damage is presented as an example of how LEEPS possibly can be
used for future investigation of cellular damage due to radiation.
Sammanfattning
En avancerad simuleringskod, LEEPS (Low Energy Electron Positron Simulation), har
anpassats för simulering av elektroner och positroner i vatten för energier ner till 50 eV.
Spridningsparametrar och resultat från simuleringar jämförs med litteraturvärden.
Flera program, bl. a. ett subbrutinpaket för simulering av sekundärelektroner skapade i
binära kollisioner, har utvecklats för att kartlägga olika aspekter av energidepositionen.
En enkel model för DNA skador presenteras som ett exempel på hur LEEPS möjligen kan
användas för framtida studier av strålningsinducerade cellskador.
3
Contents
1………Cellular effects of radiation
2………Leeps
2.1………….Monte Carlo simulation of radiation transport
2.2………….Input data of LEEPS
2.3………….Output data of LEEPS
2.4………….Program organisation
2.5………….The main program
3………Modefications of LEEPS
3.1………….Change in how data is tabulated
3.2………….Corrected use of Fermi energy in MCLEES
4………Scattering data for liquid water
4.1………….Calculation of PWA-file for liquid water
4.2………….OOS-density for water
4.3………….Choice of switch energy
4.4………….Mean ionization potential
5………Comparison of data
5.1………….Scattering parameters
5.2………….Data generated by simulation
6………Simple model for secondary electrons
6.1………….Secondary electron yield
6.2………….Program organisation for secondary electrons
7………Energy deposition
7.1………….Toy model for DNA damage
8………Programs
8.1………….RSIM
8.2………….PARSIM
8.3………….RADAXSIM
8.4………….RADAXSIM2
8.5………….PATHSIM
8.6………….TRAJSIM
8.7………….TRAJSIM2
Appendix
References
4
1 Cellular effects of radiation
It is a well-known fact that animal cells show a loss of reproductive ability after being
exposed to radiation. Such “reproductive death” is the goal of radiotherapy. An effective
radiotherapy therefore demands deep knowledge of the processes causing cell death and of
how to control these processes. Two questions are hereby raised.
What parts of the cell are most sensitive to irradiation if we want to inhibit its reproductive
ability?
How does a certain kind of radiation interact with the cell on microscopic level?
There seems to be no definite answer to the first question although the DNA molecule is
suggested as the most likely target for cell killing [11]. Research has shown that when
different plants are exposed to radiation, the lethal dose varies with species. If the dose
however is expressed in terms of energy absorption per chromosome an almost constant value
is observed. Clearly this indicates that the DNA molecule is a crucial target for cell killing.
Deeper investigations [11,12] show that, although they are rare, double strand breaks in DNA
are the most likely to have lethal consequence.
Deoxyribonucleic acid, DNA, consists of two spiral strands twisted around one another
forming a double helix. This double helix is approximately 23 Å in diameter. An estimated
energy threshold for breaking a strand in the double helix is 17.5 eV [11]. Knowing this it is
desirable to know what the probability for incident radiation, i.e. an electron or positron, with
initial energy E to deposit energy in a way that will cause a double strand break in a DNA
molecule.
23 Å
Strand
Figure 1: Schematic illustration of the DNA helix.
Suppose the physics behind a scattering process (the probabilities for all elastic and inelastic
transitions) is completely known. There is still one problem to be solved: what should a good
model of a cell look like? Is it possible to build a reasonable model with all cell membranes,
macroscopic molecules, proteins etc.? Since a cell to large extent consists of water the more
or less obvious answer is that pure water should represent a good model of a cell.
Equipped with vital information of the constitution of the DNA molecules together with a
good theory for the interaction between a particle and a water molecule, it should be possible
to get a first glimpse of how a cell is affected when it is irradiated.
The aim of this work has been to adapt an advanced simulation code (LEEPS) to simulation
of electrons and positrons in liquid water and to develop simple and user-friendly tools
(programs) suited for investigation of radiation transport characteristics
5
Schematic illustration of the program structure when LEEPS is used
Main program
Link to
Link to
Read
Support
MCLEES or MCLEPS
PHY.file
2 LEEPS
LEEPS (Low Energy Electron and Positron Simulation) is a package of subroutines written
for a Fortran 77 compiler. These subroutines are constructed for high precision Monte Carlo
simulation of electron and positron transport in different elements and compounds. LEEPS
uses two different subroutine packages: MCLEES and MCLEPS for electron and positron
simulation respectively. An additional subroutine package, SUPPORT, contains vital
integration algorithms. The subroutines in these packages are called from a main program
designed by the user. At present LEEPS is capable of simulating electrons and positrons with
kinetic energy down to 50 eV.
LEEPS uses Dirac partial wave analysis to compute the elastic DCS [13].
For the inelastic DCS two different models based on optical data are used to extend the
optical oscillator strength (OOS) to the generalised oscillator strength (GOS): The Liljequist
δ - oscillator model [7] suitable for inner shell excitations, and the two modes model [4]
designed for outer shell (valence electron) excitations.
The details of the advanced theories used in LEEPS are not described here. Detailed
descriptions of the methods used [5-8,10] are given by the constructors of LEEPS: J.M
Fernández-Varea, D. Liljequist and F. Salvat.
2.1 Monte Carlo simulation of radiation transport
A Monte Carlo simulation uses random numbers to sample events from probability
distributions describing the physics of a setting. In a scattering process these probability
distribution functions (PDFs) are generated by the elastic and inelastic differential cross
sections (DCSs).
pin (E , W , Θ ) =
d 2σ in (E , W , Θ )
σ in (E )
dWdΩ
p el (Θ ) =
1
1 dσ (Θ )
σ el dΩ
(1)
(2)
6
Here E is the kinetic energy of a particle and W its energy loss in a scattering event. Θ is
the polar scattering angle. Considering that the molecules in the medium are oriented at
random the azimuthal scattering angle is uniformly distributed in the interval [0,2π ] .
The elastic and inelastic DCSs contains all information needed for performing a random
track particle simulation. The total cross section of a molecule is
σ T = σ el + σ in
(3)
If a particle travels through a medium with particle density N , the probability for a scattering
event to occur in an interval ds is Nσ T ds . The interaction probability per unit path length is
then Nσ T . The mean free path between two interactions can thus be written.
1
λT ≡
Nσ T
(4)
It is now possible to calculate the PDF of the path length between two scattering events.
The probability that the particle travels a path length s without interacting is
∞
s
1 − ∫ p (s ′)ds ′ = ∫ p (s ′)ds ′
0
s
The probability of having next collision in the interval [s, s + ds ] is p (s )ds . This must equal
the product of the probability of arriving at s without interaction and the probability of
interacting within ds . That is
p (s )ds =
ds
λT
∞
∫ p(s ′)ds ′
(6)
s
The solution to this equation is
p (s ) =
1
λT
−
e
s
λT
(7)
Since most random sampling algorithms deliver random numbers uniformly distributed in
the interval [0,1] , there is need for a method which allows such random numbers to sample
from non-uniform PDFs. There are numerous methods one can use for this purpose. It lies
however outside the frames of this thesis to give any details of these. To sample a path length
s from (7) one can use a method some times referred to as the “inverse transform method”. If
R is a random number uniformly distributed in the interval [0,1] , this method gives
s = −λT ln R
(8)
Given a particle characterised by its position r ( x, y, z ) , its energy E and its direction of
motion d = (i, j , k ) , the algorithm for simulating its motion in a material may look as follows:
7
•
•
•
•
Calculate DCSs for current particle energy.
Sample path length to next scattering event according to (8).
Calculate next scattering position r + s ∗ d .
Sample collision type from the respective total probabilities:
p eltot =
•
•
σ el
σT
p intot =
σ in
σT
(9)
Sample scattering angles and energy loss from the PDF:s generated by the DCS:s.
Calculate new energy and new direction of motion.
Indeed this is the correct procedure to follow in a main program using LEEPS.
2.2 Input data of LEEPS
Whatever the purpose of a LEEPS main program is, some basic data (in-data) is required for
LEEPS to function properly. These data contain the physics of a simulation and is made
available to LEEPS through a file of type (name).PHY. An example is shown in table(1).
Table 1: PHY-file for liquid water
Water Electrons
10
18.0154
0.994
0.0
540.0
h2oe-pwa.dat
81
ooswater.dat
634
TITLE
NUMBER OF ELECTRONS PER MOLECULE
MOLECULAR WEIGHT (G/MOL)
DENSITY AT 37 C (G/CM3)
FERMI ENERGY (EV)
SWITCH ENERGY (EV)
ELASTIC SCATTERING DATA FILE
NUMBER OF ENERGY GRID POINTS(.LE.81)
OOS DATA FILE
NUMBER OF OOS GRID POINTS(.LE.1500)
Fermi energy
The two-modes model is based on the theory of interaction between an incident charged
particle and a free electron gas. The model may also be used to approximately describe the
interaction with the loosely bound electrons in semiconductors and insulators. Trying to
perform simulations in pure water, which is an insulator, however gives rise to problems since
the concept “Fermi energy” is then physically not well defined. However a value of this
parameter must be chosen. A more detailed discussion about this can be found in section 5.2.
The Fermi energy only affects the calculations in MCLEES (i.e. when electrons are
simulated). Positrons are distinguishable from electrons so the Pauli principle has no role in
the scattering processes simulated by MCLEPS. Mark however that the fermi energy still is an
input-parameter in the PHY-files used by MCLEPS. Perhaps the reason for this is that some
of the output-parameters of LEEPS are tabulated for kinetic energies measured with reference
to the fermi energy, a confusing convention that has been changed in MCKLAS and
MCKLASP (section 3.1).
8
Switch energy
The switch energy determines when to switch from the δ - oscillator model to the twomodes model. If the resonance energy of a sampled oscillator in a collision is greater than the
switch energy, the δ - oscillator model will be used. The two-modes model is used for
oscillators with lower energy. A reasonable choice is to set the switch energy equal to the
lowest threshold for inner shell excitations.
PWA.DAT file
This is the name of the file containing the PWA (Partial Wave Analysis) data for elastic
scattering. As convention an “e” before the “-“-sign (see table 1) indicates electron data and a
“p” indicates positron data. The file is created with the program PWADIR [13] and contains
2000 phase shifts for 81 energy grid points between 50 eV and 100 keV. LEEPS reads the file
and performs linear log-log interpolation in the table of data during simulation.
OOS.DAT file
This is the name of the file that contains the OOS-data. It may contain up to 1500 grid
points. LEEPS reads the file and performs linear log-log interpolation in the table of data
during simulation. The data in the OOS-file for water used in this work is due to M.
Dingfelder et al [4].
2.3 Output data of LEEPS
When LEEPS is run, an output-data file (name.PAR) is created. This file contains vital
simulation parameters including a table of important quantities calculated via interpolation in
the PWA-and OOS-data. No simulation has to be performed to create the table. The table
consists of elastic mean free path (empf), 1:st transport mean free path, 2:nd transport mean
free path, inelastic mean free path (imfp), stopping power (stp) and range tabulated for
different energies. An example (PARSIM.PAR) of the out-data file is shown in appendix.
2.4 Program organisation
In a main program only those subroutines of LEEPS listed below need to be called. In their
turn they will call other subroutines in LEEPS and SUPPORT
•
SUBROUTINE START
Reads the scattering database and creates the PAR-file.
•
SUBROUTINE CROSS(E,CSEL,CSIN)
Delivers the inverse mean free paths (in cm-1) for elastic and inelastic scattering for
electrons or positrons with energy E.
9
•
SUBROUTINE RSELAS(COSTH)
Delivers a value of the polar scattering angle sampled from the elastic scattering DCS for
electrons or positrons with energy E entered in the last call to subroutine CROSS.
•
SUBROUTINE RSINEL(DE,COSTH)
Delivers values of the energy loss and polar scattering angle sampled from the inelastic
DCS for electrons or positrons with energy E entered in the last call to CROSS.
•
SUBROUTINE DIRECT(COSTH,PHI,U,V,W)
Performs the standard trigonometric calculation of the new direction of motion (U,V,W)
after collision using the polar and azimuthal scattering angles COSTH and PHI.
•
FUNCTION RAND
The random number generator used by LEEPS. The statement “RAND (0.0D0)” generates
a random number uniformly distributed in the interval [0,1] .
As an alternative to subroutine DIRECT, the subroutine NEWDIR(DIR,COSTH,R) may be
used. Here DIR is an array that contains the three components of the direction of motion
(U,V,W) and R is a random number used to sample the azimuthal scattering angle. The new
direction of motion is returned in DIR. It is a matter of taste which subroutine to use.
NEWDIR is however used in all programs developed during this work since:
1) It is based on simpler mathematics than DIRECT
2) It makes programs easier to overview. The programmer does not have to be concerned
about the azimuthal scattering angle. All necessary calculation is performed inside the
subroutine, which is not the case in DIRECT.
NEWDIR is not a part of LEEPS and must be included separately in the main program if
used. The subroutine is found in some of the programs in appendix. See for example
RADAXSIM.FOR.
2.5 The main program
In table 3 an example of a basic main program is shown. It illustrates some of the principles
of how to use LEEPS. Below is a list of important details to remember when constructing a
main program.
•
•
•
After declaring the variables to be used, it is very important to add the variable “RND” to
the common block “RSEED”. This will link the random seed, chosen by the user, to all
necessary subroutines in LEEPS.
The random seed, preferably a large integer, is read from a name.DAT-file (an example
of a DAT-file is shown in table (2)) created by the user. This file also contains parameters
like the initial energy of a particle (E0), the name of the PHY-file (SCFILE), the number
of particles to be simulated (ITRACK) and the absorption energy (EABS) which marks
the lower limit of the simulation.
When the energy of a particle is lower than “EABS” the simulation is stopped. “EABS”
can at present not be lower than 50 eV since PWA-data are not implemented below this
energy.
10
•
•
•
The variable “RND” must be associated with the random seed. This will convert the
integer seed to a floating point, which is necessary.
The PHY-file (CSFILE) must be opened with unit “7” if LEEPS shall work. Likewise the
output-file “SIM.PAR” is to be opened with unit “6” if the scattering parameters shall be
successfully stored.
The main program must always be linked to the subroutine SUPPORT which contains
vital integration algorithms.
Table 2: Example of a dat-file.
20000.
100.
1000
1354489
H2Oe.phy
INITIAL ENERGY
ABSORPTION ENERGY
NUMBER OF TRACKS
RANDOM SEED
IN-DATA FILE
With the code in table 3 as basis, all possible calculations based on single particle
trajectories can be performed. Any information about the particle may be extracted and stored
during simulation. The reader is advised to carefully study the code before he/she writes own
programs. In appendix some programs used during this work are presented. These can
hopefully give some inspiration.
Table 3: Example of main program.
* PROGRAM SIM
* MONTECARLO SIMULATION OF A PARTICLE IN A MEDIUM USING LEEPS
*
* EO
INITIAL ENERGY (EV)
* EABS
ABSORBTION ENERGY (EV)
* ITRACK
NUMBER OF PARTICLES SIMULATED
* ISEED
RANDOM NUMBER GENERATOR SEED
* CSFILE
CROSS SECTION FILE (PHY FILE)
* POS(3)
POSITION COORDINATES FOR PARTICLE BEING SIMULATED (X,Y,Z)
* DIR(3)
DIRECTION OF MOTION FOR SIMULATED PARTICLE (NORMALIZED
VECTOR)
PROGRAM SIM
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
IMPLICIT INTEGER*4 (I-N)
CHARACTER*40 CSFILE
DOUBLE PRECISION POS(3),DIR(3)
PARAMETER (PI=3.1415926535897932D0)
COMMON/RSEED/RND
********** READ INPUT PARAMTERS
OPEN(1,FILE='SIM.DAT',TYPE='OLD')
READ(1,*) E0
READ(1,*) EABS
READ(1,*) ISEED
READ(1,*) ITRACK
READ(1,'(A40)') CSFILE
11
CLOSE(1)
RND=ISEED
OPEN(7,FILE=CSFILE,TYPE='OLD')
OPEN(6,FILE='SIM.PAR',TYPE='NEW')
********* READ SCATTERING FILES
CALL START
CLOSE(7)
CLOSE(6)
********** BEGIN PARTICLE SIMULATION
DO 100 I=1,ITRACK
ENERGY=E0
********** SET INITIAL POSITION AND DIRECTION OF A PARTICLE
POS(1)=0.D0
POS(2)=0.D0
POS(3)=0.D0
DIR(1)=0.D0
DIR(2)=0.D0
DIR(3)=1.D0
50 CONTINUE
CALL CROSS(ENERGY,CSEL,CSIN)
********* CALCULATE MACROSCOPIC CROSS SECTION
CSTOT=CSEL+CSIN
********* PATHLENGTH TO NEXT SCATTERING EVENT
DS=-DLOG(1.0D0-RAND(0.0D0))/CSTOT
********* NEW POSITION
POS(1)=POS(1)+DIR(1)*DS
POS(2)=POS(2)+DIR(2)*DS
POS(3)=POS(3)+DIR(3)*DS
********* COLLISION TYPE
R=RAND(0.0D0)
!RANDOM NUMBER FOR SAMPLING AZIMUTHAL ANGLE
IF (RAND(0.0D0)*CSTOT.LT.CSEL) THEN
********* ELASTIC SCATTERING
CALL RSELAS(CT)
CALL NEWDIR(DIR,CT,R)
GOTO 50
ELSE
12
********* INELASTIC SCATTERING
CALL RSINEL(DE,CT)
CALL NEWDIR(DIR,CT,R)
ENERGY=ENERGY-DE
IF (ENERGY.LT.EABS) THEN
********* STOP SIMULATION IF ENERGY LOWER THAN EABS
GOTO 100
ELSE
********* CONTINUE SIMULATION IF ENERGY GRATER THAN EABS
GOTO 50
END IF
END IF
100
CONTINUE
END
3 Modifications of LEEPS
During this work some modifications of LEEPS have been done in order to make
comparison with data easier. Moreover an inconsistent use of the fermi energy in MCLEES
has been corrected. With these changes MCLEES and MCLEPS are transformed to the
subroutine packages MCKLAS and MCKLASP respectively. (The users of LEEPS are
encouraged to change these names)
3.1 Change in how out-data is tabulated
In the original version of LEEPS the out-data are tabulated for predetermined energy grid
points in the interval 50 eV – 100 keV. The modifications made allow the data to be tabulated
in an optional energy interval chosen during runtime. The space between two grid points is
also optional.
When START is called a message will appear on the screen:
“ENTER ENERGY INTERVAL FOR CALCULATING PARAMETERS”
“START WITH THE LOWER LIMIT”
“ENTER START ENERGY IN EV”
The user must enter a value not lower than 50 eV
Example: 50
“ENTER STOP ENERGY IN EV”
The user must enter a value not lower than 50 eV and higher than the value previously entered
or the program will crash.
Example: 190
13
“ENTER SAMPLING INTERVAL”
The desired distance between two energy grid points is entered.
Example: 20
Table 4 shows how the data are tabulated if the examples above are entered.
Table 4: Scattering parameters for liquid water tabulated by MCKLAS
ENERGY
EL. MFP
1st TMFP
2nd TMFP
IN. MFP
ST. POWER
RANGE
(eV)
(cm)
(cm)
(cm)
(cm)
(eV/cm)
(cm)
--------------------------------------------------------------------------5.000E+01 1.289E-07 3.241E-07 1.960E-07 1.615E-07 1.212E+08 1.000E-35
7.000E+01 1.484E-07 4.261E-07 2.495E-07 1.201E-07 2.073E+08 1.224E-07
9.000E+01 1.661E-07 5.290E-07 3.028E-07 1.111E-07 2.526E+08 2.087E-07
1.100E+02 1.824E-07 6.334E-07 3.562E-07 1.113E-07 2.715E+08 2.844E-07
1.300E+02 1.978E-07 7.397E-07 4.099E-07 1.149E-07 2.772E+08 3.571E-07
1.500E+02 2.124E-07 8.480E-07 4.641E-07 1.200E-07 2.762E+08 4.292E-07
1.700E+02 2.263E-07 9.582E-07 5.188E-07 1.258E-07 2.719E+08 5.020E-07
1.900E+02 2.398E-07 1.071E-06 5.741E-07 1.320E-07 2.660E+08 5.763E-07
The necessary linear log-log interpolations are performed by a new subroutine, CALCPAR,
which is a modified version of CROSS [10].
3.2 Corrected use of the fermi energy in MCLEES
This section is preferably read by those very familiar with the theories and program structures
of LEEPS.
When electrons are simulated, MCLEES reads the Fermi energy, EF, from a PHY-file.
During this work it was found that the subroutines in MCLEES use EF in an inconsistent way,
which results in miscalculations. The inconsistencies were found in the subroutines DOEL,
TMEL, RSDOEL and RSTMEL. All these subroutines have the in-parameters ”EF” and “EK”
labeled “fermi energy” and “Electron kinetic energy above Fermi level” respectively. The
calculations done in these subroutines are in agreement with this notation and the theoretical
calculations [USIP Report 92-06]. Deeper inspections show however that “EK” is NOT the
kinetic energy above the fermi level but indeed the kinetic energy assumed for the supposedly
free, incident electron.
DOEL and TMEL are both called from the subroutine READIN where EK for some reason
is called EV defined by the expression: “ EV = DEXP(EEL(IE )) ”. The array EEL is created in
the subroutine READEL. The elements of EEL are the logarithm of the energy grid points
read from the PWA file. The conclusion is that if everything shall be consistent, PWA.DAT
must be tabulated for energies above some Fermi level and not for absolute energy, which is
the case.
Since DOEL and TMEL are not called during simulation it may be argued that this
inconsistency merely creates a shift in the data produced by DOEL and TMEL, which indeed
it does.
A more disturbing fact is that the same inconsistencies occur in RSDOEL and RSTMEL,
both involved during simulation. RSDOEL and RSTMEL are called from RSINEL. The
parameter “EV” is defined in CROSS as “electron kinetic energy” and is passed to RSINEL
via a common block COMMON/CALC. Form RSINEL it is passed on to RSDOEL and
14
RSTMEL where it is identified as “electron kinetic energy above fermi level”. Again there is
an inconsistency this time causing error during simulation. The errors are corrected in
MCKLAS simply by changing the expression “ E = EV + EK ” to “ E = EK ” in the affected
subroutines.
In table 5 data from simulation of 200 eV electrons in Si obtained with MCLEES and
MCKLAS are compared. The differences are small but noticeable and support the arguments
above. The use of MCLEES gives somewhat higher values when the simulated particle
energy is Esim = Ekin + EF and not Esim = Ekin as in MCKLAS
Different path lengths (Å)
Path length
Penetration depth
Axial penetration depth
Radial Penetration depth
MCLEES
42.0
24.2
13.1
14.1
MCKLAS
40.8
23.7
13.0
13.7
Table 5: Different path lengths obtained with the program PATHSIM (section) using MCLEES and MCKLAS.
The data is from simulation of 5*104 electrons with initial energy 200 eV in Si. Si is used since the fermi energy
12.5 eV [5] is relatively high and therefore should have noticeable effect.
4 Scattering data for water
4.1 Calculation of the PWA-file for water
The PWA-files used by LEEPS for calculation of elastic DCSs are created with the program
PWADIR (DIRac Partial Wave Analysis) [13]. In the beginning of a PWA file are five lines
with basic information of the file. These lines are not read by LEEPS and therefore do not
affect the calculations. After these lines follows a large number of data blocks with data for
specific energies in the interval 50 eV – 500 keV. The first row in each block holds the
quantities: Energy, elastic cross section, 1:st transport cross section and 2:nd transport cross
section.
The remaining data consists of 150 unequally spaced grid points of the single scattering polar
angle PDF for a particle with the energy given in the first row.
Since PWADIR only generates data for elements (which are assumed to have spherical
symmetric potentials) the PWA-file for water cannot be directly created. PWA-files for
oxygen and hydrogen are however available to us. Thus an approximate solution to the
problem is to consider water as a mixture of free hydrogen and oxygen atoms. The total
elastic cross section for a water molecule is then:
σ T = 2σ H + σ O
(10)
The probability of interaction with the different atoms in a scattering event is:
p Htot =
2σ H
σT
and
pOtot =
σO
σT
(11)
The single scattering polar angle PDF for water is thus given by:
p H 2O (E , Θ ) = p Htot ∗ p H (E , Θ ) + pOtot ∗ p O (E , Θ )
(12)
15
The program COMBINE2 (se appendix) performs the necessary calculations.
The PWA-data files for water are called H2OE-PWA.DAT and H2OP-PWA.DAT for
electrons and positrons respectively.
4.2 OOS density for water
The inelastic scattering model in LEEPS is based on the OOS (optical oscillator strength)
df (W )
density,
, of the scattering target. The OOS density could in words be described as the
dW
effective number of electrons per unit energy that are excited in the optical limit at excitation
energy W. The OOS density for water [4] is shown in figure 2.
0
10
Figure 2: OOS density for water
-1
10
-2
10
df(w)/dw (1/eV)
The OOS density data file for water
“ooswater” contains data in the interval
6.3 eV - 104 eV. The edge at 539 eV is
caused by excitation of the k-shell in
the oxygen atom.
-3
10
-4
10
-5
10
-6
10
0
1
10
10
4.3 Choice of switch energy
2
10
W (eV)
3
4
10
10
In a previous work by Emfietzoglou et al [1] a model for inelastic scattering was based on a
division of the electron subsystem into a valence band representing the smearing of the four
outer shells of H2O and a core shell representing the oxygen K-shell. The large difference in
binding energy between valence and core shells (see table 8) provides justification for this
approximation. With this argument as ground the switch energy, Ws, should be assigned a
value of about 540 eV. Other choices have also been tested (se below)
4.4 Mean ionization potential
The mean ionization potential (I) used in the famous Bethe-Bloch formula is defined by
∞
∫ ln(W )
ln I =
0
df (W )
dW
dW
df (W )
∫0 dW dW
∞
(13)
The value of I calculated for H2O by LEEPS is 78.3 eV (it is one of the parameters calculated
when START is called). It can be compared to [2] 80.9 eV, 81.8 V (Dingfelder et al [4]) and
the experimental value 79.75 ± 0.5 eV measured by Bichsel and Hiraoka from which it differs
16
only by a few percent. This indicates that the OOS-data by Dingfelder makes a good ground
for the inelastic scattering model in LEEPS
5 Comparison of data
One of the main objectives of this work has been to compare data generated by LEEPS for
scattering in water with results of other works. When simulating electron and positrons in
solids like Al, Cu, Ag and Au the results of LEEPS are in good agreement with experimental
data [8]. This does not necessarily mean that the data obtained from simulations in water will
be of the same quality.
5.1 Scattering parameters
The scattering parameters (stp, mfp etc.) make a good set of benchmarking data when
different scattering models are evaluated. Data calculated by LEEPS are compared with
results of previous [9], [4] works in figure 3-10
Since transitions with final target electron energy below EF are forbidden in the models used
by LEEPS [10], higher values of EF yield lower stopping power and longer inelastic mfp. It is
found (figure 3 and 4) that variation of EF between 0 eV and 15 eV has a noticeable but small
effect on electrons with energy below 100 eV. Above 100 eV no effect is visible. Hence we
will for consistency with positron scattering assign the value 0 eV to EF.
As expected [2] (figure 5) consequent use of the δ - oscillator model (Ws = 0) exaggerates
the stopping power in the lower energy region while the inelastic mfp is reduced for all
energies (figure 6). Using the two modes model throughout has little effect compared with
data for Ws = 540 eV.
8
3
x 10
Stopping power (ev/cm)
2.5
2
1.5
1
Leeps
Dingfelder
Ashley
Akkerman
0.5
0
0
50
100
150
Energy (ev)
200
Figure 3: Stopping power for EF = 0 eV and EF = 15 eV. Ws = 540 eV
Higher value of EF yields a slightly lower stopping power.
250
17
-7
x 10
3
Leeps
Dingfelder
Ashley
Akkerman
Inelastic mean free path (cm)
2.5
2
1.5
1
0.5
0
0
20
40
60
80
Energy (ev)
100
120
140
160
Figure 4: Inelastic mean free path for EF=0 eV and EF= 15 eV. Ws = 540 eV
Higher values of EF yields longer inelastic mfp.
8
4
x 10
Leeps
Dingfelder
Ashley
Akkerman
3.5
Ws=0 eV
Stopping power (ev/cm)
3
2.5
2
1.5
1
Ws=540 eV
Ws=10000 eV
0.5
0
0
500
1000
1500
Energy (ev)
2000
Figure 5: Stopping power for Ws = 0 eV, 540 eV and 10000 eV. EF = 0 in all
Three cases. Using δ - oscillator model throughout exaggerates stopping power
in the lower energy region.
18
-7
x 10
8
Leeps
Dingfelder
Ashley
Akkerman
Inelastic mean free path (cm)
7
Ws=540 eV and 10000 eV
6
5
4
3
Ws=0 eV
2
1
0
0
200
400
600
800
1000 1200
Energy (ev)
1400
1600
1800
2000
Figure 6: Inelastic mfp for Ws = 0 eV, 540 e V and 10000 eV. EF = 0 in all
three cases. Using the δ - oscillator model throughout reduces the inelastic mfp.
9
10
Stopping power (eV/cm)
Leeps
Dingfelder
Ashley
Akkerman
8
10
7
10
1
10
2
3
10
10
Energy (eV)
Figure 7: Stopping power for electrons. EF = 0 eV, Ws = 540 eV
4
10
19
-5
10
Inelastic mean free path (cm)
Leeps
Dingfelder
Ashley
Akkerman
-6
10
-7
10
1
2
10
3
10
4
10
10
Energy (eV)
Inelastic mean free path micro g/cm2
Figure 8: Inelastic mfp for electrons. EF = 0 eV, Ws = 540 eV.
LEEPS electrons
LEEPS positrons
Pimblott electrons
Pimblott positrons
0
10
-1
10
0
500
1000
1500
2000
Energy ev
2500
Figure 9: Inelastic mfp for positrons and electrons.
3000
3500
20
300
LEEPS electrons
LEEPS positrons
Pimblott electrons
Pimblott positrons
range micro g/cm2
250
200
150
100
50
0
0
1000
2000
3000
4000 5000 6000
Energy ev
7000
8000
9000
10000
Figure 10: Range for positrons and electrons. LEEPS gives systematically a slightly
shorter range which may be trivially due to a higher cut off energy.
In figure 7 and 8 stp and inelastic mfp calculated with EF = 0 eV and Ws = 540 eV are
plotted and compared with data for energies up to 10 keV. A subtle effect of the models used
in LEEPS is that for energies below 300 eV the maximum in stp and minimum in inelastic
mfp are higher and lower respectively than the data to which they are compared. For high
energies the agreement is good (fig 5-8)
Unfortunately comparison of positron data for the scattering parameters for water is rather
limited since little material has been found on this subject. Figure 9 and 10 show comparison
of inelastic mfp and range with data by Pimblott et al [3]. LEEPS gives slightly lower values
mainly due to a higher cut off energy. In the same work there exist some data from
simulations of positrons and LEEPS is compared with these in next section.
5.2 Data generated by simulation
There are of course much data that can be obtained from actual simulations of particles. For
example the purpose of this work is to map by simulation the energy deposition of electrons
and positrons in liquid water. An important detail of a scattering process that is best evaluated
by track simulation is the angular PDF:s. There are four different measures of the distance
travelled by a particle that indirectly gives information of these (see figure 11 for illustration).
1.
2.
3.
4.
The path length.
The penetration distance between the initial position (Xi) and final position (Xf).
The axial component of the vector Xf - Xi (i.e. parallel to the initial electron trajectory)
The absolute value of the radial component of the vector Xf – Xi (i.e perpendicular to the
initial electron trajectory).
21
Axial distance
Xf
Radial
distance
Xi
Initial direction
Figure 11: Schematic illustration of different distances.
With the program PATHSIM (section 8.5) these distances are calculated from track
simulations. In table 6 and 7 the results are compared with the data by Pimblott et al. The
differences are significant. In the lower region (E<300 eV) this is explained by the lower
energy cut-off used by Pimblott (25 eV) with greater path length and penetration depth as
consequence. For intermediate energies (1 keV – 10 keV) it can be seen that although the path
length for positrons simulated by LEEPS remains somewhat lower, the penetration depth is
greater compared to Pimblott. In the high-energy region (E>30 keV) both path and
penetration are again lower. This can of course be the result of many different factors. For
example the statistical errors in the Pimblott data are not known.
A physical explanation would be that the models used in LEEPS tend to favour forward
scattering (compared to Pimblots models) for some parts of the energy scale. This would
increase the axial penetration alt. reduce the radial penetration in the intermediate energy
region as observed in table 7.
Particle energy Path length
(keV)
Pimblott
Electron
0.07
0.98
0.10
1.08
0.30
1.97
1.00
6.88
3.00
35.4
10.0
268
30.0
1817
100.0
1.47*10^4
(µg/cm²)
Leeps
Positron
Electron
0.47
0.17
0.53
0.29
1.18
1.08
5.36
5.81
30.8
33.4
245
260
1693
1797
1.37*10^4 1.46*10^4
Penetration
Pimblott
Positron
Electron
0.12
0.31
0.20
0.35
0.83
0.73
4.95
3.64
29.7
21.3
239
173
1572
1190
1.31*10^4 9.96*10^3
(µg/cm²)
Positron
0.29
0.33
0.75
3.71
21.2
168
1158
9.54*10^3
Table 6: Path length and penetration for electrons and positrons in liquid water.
Leeps
Electron
0.11
0.18
0.65
3.62
21.5
173
1215
9.97*10^3
positron
0.11
0.18
0.68
3.78
21.7
169
1112
9.27*10^3
22
Particle energy (keV)
Axial penetration (µg/cm²)
Pimblott
Leeps
Electron Positron Electron
0.18
0.19
0.07
0.20
0.22
0.11
0.48
0.55
0.41
2.53
2.79
2.24
15.0
15.6
14.8
123
123
121
852
841
860
7264
7072
7105
0.07
0.10
0.30
1.00
3.00
10.0
30.0
100.0
Radial penetration
Pimblott
Electron Positron
0.23
0.18
0.26
0.21
0.47
0.43
2.23
2.08
13.1
12.4
105
98.9
723
672
5896
5532
Positron
0.09
0.15
0.54
2.87
15.9
122
811
6776
(µg/cm²)
Leeps
Electron
0.05
0.10
0.37
2.11
12.7
102
712
5852
positron
0.03
0.06
0.32
1.96
11.9
96.2
631
5313
Table 7: Axial and radial penetration for electrons and positrons in liquid water.
In figure 12 the differential inverse mean free path (DIMFP) is shown for 1 keV electron
impact in liquid water. It is useful when different OOS extension models are evaluated. In a
work by Emfietzoglou [2] the DIMFP:s of several dispersion algorithms are compared
including the δ -oscillator model. The two-modes model is however not included. The peak
value at ~20 eV is significantly smaller when the two-modes model is used, compared to
Ashley’s δ -oscillator model, even when the same optical data are used [2]. The effect is
possibly visible in figure 7 and 8 where it could explain the lower stp and longer imfp
compared to data by Dingfelder et al for energies over 0.5 keV.
Comparison of data shows that the results by LEEPS are in good agreement with previous
works. An important improvement is to add PWA data for energies down to 20 eV, which
will improve simulations in the lower energy region. With the present cut of at 50 eV and an
energy resolution around 2 eV, the resolution in range (CSDA) is approximately 2 Å.
8
7
DIMFP 1/(micro m)/eV
6
5
4
3
2
1
0
0
5
10
15
20
25
30
Energy (eV)
35
40
45
50
Figure 12: DIMFP for 1 keV electron impact in liquid water calculated by LEEPS.
The same curve calculated with Ashley’s δ -oscillator [2] gives a peak value of
9 (micro m*eV)-1.
23
6 Simple model for secondary electrons
In an inelastic collision the transferred energy may not simply “stay” at the location of the
collision. To gain a more realistic picture of the energy distribution in liquid water we must
include secondary electrons in the scattering model. The transferred energy can if large
enough ionize the target molecule and create a secondary electron with energy T = W − E I
where W is the transferred energy and E I is the ionization energy. This raises a number of
difficult problems. The OOS-density, which the inelastic model in LEEPS is based on, gives
in it self practically no information about the different energy levels below 539 eV in water
(539 eV represents the ionization energy of the K-shell in the oxygen atom) although there
exist a number of different possible transitions in this energy region, which all contribute to
the oos-density. Hence it is impossible to tell from the OOS-density alone what energy a
secondary electron will have when it is created. Moreover, since its initial quantum state is not
known the direction of motion after excitation can not be calculated. All these problems can
in principal be solved but there are more important issues that should be taken care of
(extension of elastic data to lower energy regions is one improvement) before this is done.
A very simple model based on mean energy excitations and binary collisions can however be
invoked in a simulation without changing LEEPS. At present LEEPS can only simulate
electrons and positrons down to 50 eV. If a secondary electron is to be simulated it must have
at least this energy. Secondary electrons with kinetic energies significantly higher than their
original binding energies are expected to appear through approximately binary collisions. This
also means that the momentum of such electrons (before excitation) has limited influence on
the direction of motion after collision. A model based on binary collisions is thus justified in
the present version of LEEPS.
In table 8 (Dingfelder et al. [4]) the different ionization levels in liquid water and their OOS
is shown. If the oos-density is used as weight the average ionization energy for the four levels
below the oxygen k-shell is approximately 16 eV.
Shells
1b1
3a1
1b2
2a1
k-shell
Ionization
Energy (eV)
10.79
13.39
16,05
32.30
539.00
OOS
2.25
2.06
1.61
1.21
1.79
Table 8: Different ionization energies in liquid water. Data
from Dingfelder et al. [4]
Assuming that the transitions of the outermost shells (below 539 eV) contribute to the oosdensity in a linear decreasing manner (in a log-log plot) above 539 eV (figure 13), ionization
of the oxygen k-shell will clearly dominate the oos-density abow 539 eV. This gives no
guarantee for the gos-density to show the same principle. Studying the frequency of different
energy deposition of a 20 keV electron (figure 14) it is observed that the number of inelastic
events with energy transfer in the region just above the k-edge are approximately four times
as many compared to the events just below the K-edge. Thus in a simplified model for
secondary electrons we may assume two ionization energies: E1=16 eV and E2=539 eV where
all energy transfers above 539 keV are assumed to result in excitation of the k-shell.
24
0
10
-1
10
Higher curve: original oos
Lower curve: contribution from
outermost shells subtracted
-2
df(w)/dw (1/eV)
10
-3
10
-4
10
-5
10
Assumed contribution
from outermost shells
over 539 keV
-6
10
-7
10
0
1
10
2
10
10
W (eV)
3
4
10
10
Figure 13: The assumed contribution to the oos-density from the outermost shells is
subtracted for energies over the oxygen K-edge.
0
10
-1
Events
10
-2
10
-3
10
-4
10
0
100
200
300
400
Energy transfer (eV)
500
Figure 14: Number of events with different energy transfers for electron with
initial energy 20keV. Cut off energy 600 eV.
600
25
The theory for the direction of motion of a secondary electron is based on classical
relativistic kinematics. The calculation is performed by the subroutine SECDIR. There are
two unfortunate twists to the theory due to the fact that it is based on conservation of kinetic
energy and momentum, which is not the truth since:
a) Some of the energy is “lost” to binding energy.
b) The momentum of the bound electron is by no means zero as assumed in the present
binary collision model.
Assuming that the kinetic energy of a bound electron is of the same magnitude as its binding
energy the model presented should be valid for energy transfers 160 eV (10 times the average
binding energy) – 539 eV. In the region 50 eV – 160 eV the direction of a daughter electron
will probably have a significant error if a binary collision theory is used. These electrons have
however short ranges. More questionable is the use of this model for excitations of the K-shell
in the region 600 eV – 5 keV. The secondary electrons created in these collisions have a
significant range and with the momentum argument above the directional error will for sure
be large. Perhaps a more adequate model would be to choose the direction of these electrons
randomly. The yield of such electrons is however low and can according to Emfietzoglou [2]
be safely neglected.
6.1 Secondary electron yield
What amount of secondary electrons can be expected from a primary particle with a certain
initial energy if the model above is used? If the yield is low the effect of secondary electrons
is perhaps barely noticeable. With the two ionization levels 16 eV and 539 eV the minimum
energy transfer which can create a “high energy” secondary electron, i.e. with energy over the
cut off limit (50 eV), is 66 eV. In figure 15 the number of energy depositions above 70 eV is
shown for different initial energies of positrons and electrons. Interestingly the number of
events increases linearly with initial energy. For a 100 keV particle a yield of approximately
200 high energy secondary electrons can be expected when the yield for10 keV particles is 20
electrons. Such amounts of “extra” particles are impossible to ignore and the conclusion is
that, although simple, the described model for secondary electrons is indeed justified.
Figure 16 shows the secondary electron spectra calculated with the simplified two-level
scheme compared to data from a model [2] based on Drude function representation of the oosdensity of the different ionization levels. The two-level model shows good resemblance with
comparison data down to secondary electron energies around 20 eV. Therefore there is no
need for a more advanced ionization scheme before LEEPS is capable of simulating particles
below 20 eV. The binary collision model can however still be questioned.
6.2 Program organization for secondary electrons
Trying to include simulation of secondary electrons in the basic main program in table 3 it
stands clear that some simplifications of the structure in the main program is needed. When a
secondary electron is created its initial position, energy and direction of motion must be stored
in a convenient way if it shall be shall simulated later. Moreover all daughter electrons can
create new “secondary” electrons, which initial data must be stored. This process could in
theory go on for a large number of “generations” and the subroutine package SIMPACK is
developed for easy user-friendly programs simulating all daughter electrons. SIMPACK
consists of the following five subroutines:
26
200
Electrons
Positrons
180
160
140
Events
120
100
80
60
40
20
0
0
1
2
3
4
5
6
Initial energy (eV)
7
8
9
10
4
x 10
Figure 15: Number of inelastic events with energy loss above 70 eV for electrons
and positrons with different initial energies.
1
DIMFP 1/micro m/eV
10
0
10
-1
10
-2
10
0
10
1
2
10
10
Secondary electron energy (eV)
3
10
Figure 16: Secondary electron spectra, dΣ / dW , as function of secondary electron
Kinetic energy calculated with the simplified two-level ionization scheme. Stars represent
(visually read off) data by D. Emfietzoglou [2]. Good resemblance with comparison data
justifies use of the simplified scheme down to 20 eV.
27
•
SUBROUTINE SECWR(POS,OLDPOS,DIR OLDDIR,OLDEN,DE,ISEC)
Determines if a secondary electron is created in an inelastic collision and writes its initial
data to a scratch file. If the energy transfer is in the interval (E1+50 eV) – (E2 eV) an
electron in the lower shell is excited. Energy transfers larger than E2+50 eV result in
excitation of the oxygen K-shell. POS holds the position of the event. OLDPOS holds the
position of the last event (may be necessary). DIR is the direction of the primary particle
after the scattering event. OLDDIR is the direction of the primary particle before the
scattering event. OLDEN is the energy of the primary particle before the event. DE is the
energy transfer. ISEC counts the number of secondary electrons
•
SUBROUTINE SECRE(ISEC)
Simulate secondary electrons with initial data calculated and stored by SECWR. If an
extra secondary electron is created the data for it is stored in the scratch file for
simulation.
•
SUBROUTINE PARTSIM(POS,DIR ENERGY,DE)
Simulates a particle to its next scattering event and calculates its new position, direction
and energy.
•
SUBROUTINE SECDIR(DIR,OLDDIR,OLDEN,DE)
Calculates the direction of motion of a secondary electron as if it was created in a binary
collision.
•
SUBROUTINE NEWDIR(DIR,CT,R)
Calculates the new direction of a primary electron in a scattering event.
Of these subroutines only SECWR, SECRE and PARTSIM are called from a main program.
An example of a main program called “SIM2” based on the use of SIMPACK is shown in
table 9. When a primary electron is simulated all initial secondary electron data is written to a
scratch file. First when the primary simulation is stopped the scratch file is read and the extra
electrons are simulated.
There are some important issues that must be remembered when SIMPACK is used:
•
•
•
•
The two common blocks COMMON/SECPAR and COMMON/INPAR must be declared
as in SIM2. This will transfer necessary data to SIMPACK.
Before simulating any particles a scratch file with direct access has to be opened with unit
“3”. The secondary electron data will be written to this file, whose data blocks must be
large enough for seven double precision variables: three position coordinates, three
direction vector components and the kinetic energy. Unfortunately different systems use
different standards for declaring the size (RECL) of these data blocks. The programs in
this work are developed for a system using “storage units” for specifying record lengths.
One double precision variable demands two storage units and therefore the value “14” is
assigned to RECL when the scratch file is opened.
Immediately when the simulation of a primary electron starts the value “0” must be
assigned to ISEC. This variable counts the number of created secondary electrons and is
of vital importance when storing and reading secondary electron data.
Before a scattering event the old direction and energy of a primary electron must be
assigned to OLDDIR and OLDEN. Otherwise secondary electron data cannot be correctly
calculated.
28
•
It is assumed that the user of SIMPACK is in some way interested in the data for (by) the
secondary electrons. Therefor SECRE calls the subroutine
SECDAT(POS,ENERGY,DE,NUMSEC) to which it passes information of position,
energy and energy transfer for secondary electrons. SECDAT does however not exist and
must therefor be created by the user and included in the main program. This gives a
necessary freedom to the user whose exact intentions are not known. The variable
NUMSEC holds the number of the secondary electron being simulated.
Table 9: Example of main program including simulation of secondary electrons.
* PROGRAM SIM2
* MONTECARLO SIMULATION OF PRIMARY AND SECONDARY ELECTRONS.
*
*
*
*
*
*
*
*
*
EO
EABS
ITRACK
ISEED
CSFILE
POS(3)
DIR(3)
OLDDIR(3)
OLDEN
INITIAL ENERGY (EV)
ABSORBTION ENERGY (EV)
NUMBER OF PARTICLES SIMULATED
RANDOM NUMBER GENERATOR SEED
CROSS SECTION FILE
POSITION COORDINATES FOR PARTICLE BEING SIMULATED (X,Y,Z)
DIRECTION OF MOTION FOR SIMULATED PARTICLE
DIRECTION OF PARTICLE BEFORE COLLISION
PARTICLE ENERGY BEFORE COLISSION
PROGRAM SIM2
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
IMPLICIT INTEGER*4 (I-N)
CHARACTER*40 CSFILE
DOUBLE PRECISION POS(3),OLDDIR(3),DIR(3)
PARAMETER (PI=3.1415926535897932D0)
COMMON/RSEED/RND
COMMON/SECPAR/TWOABS,ONEABS,TWOLEV,ONELEV
COMMON/INPAR/EABS
********** READ INPUT PARAMTERS
OPEN(1,FILE='SIM2.DAT',TYPE='OLD')
READ(1,*) E0
READ(1,*) EABS
READ(1,*) ITRACK
READ(1,*) ISEED
READ(1,'(A40)') CSFILE
CLOSE(1)
******** SECONDARY ELECTRON DATA USED IN SUBROUTINE SIMPACK
ONELEV=16.0D0
TWOLEV=539.0D0
! PREFERABLY READ FROM DAT FILE
ONEABS=ONELEV+EABS
TWOABS=TWOLEV+EABS
! LOWEST POSSIBEL ENERGY TRANSFER RESULTING IN
! SECONDARY ELECTRONS
RND=ISEED
OPEN(7,FILE=CSFILE,TYPE='OLD')
OPEN(6,FILE='SIM2.PAR',TYPE='NEW')
CALL START
29
CLOSE(7)
******* OPEN FILE FOR STORING SECONDARY ELECTRON DATA
OPEN(3,STATUS='SCRATCH',ACCESS='DIRECT',RECL=14)
********** BEGIN TRACK SIMULATION
DO 100 I=1,ITRACK
********** NUMBER OF SECONDARY ELECTRONS CREATED
ISEC=0
ENERGY=E0
POS(1)=0.D0
POS(2)=0.D0
POS(3)=0.D0
DIR(1)=0.D0
DIR(2)=0.D0
DIR(3)=1.D0
50 IF (ENERGY.GT.EABS) THEN
OLDDIR(1)=DIR(1)
OLDDIR(2)=DIR(2)
OLDDIR(3)=DIR(3)
OLDEN=ENERGY
******** SIMULATE PARTICLE TO NEXT SCATTERING EVENT
CALL PARTSIM(POS,DIR,ENERGY,DE)
******** WRITE SECONDARY ELECTRON DATA TO THE SCRATCH FILE OPENED WITH UNIT
7
CALL SECWR(POS,DIR,OLDDIR,OLDEN,DE,ISEC)
GOTO 50
END IF
******** SIMULATE SECONDARY ELECTRONS
CALL SECRE(ISEC)
100
CONTINUE
CLOSE(3)
END
30
7 Energy deposition
The effect of the described secondary electron model is illustrated in figure 17. It tends to
broaden the energy deposition spectrum outwards from the primary particles initial position as
can be expected. The effect is not very large for a 1 keV electron. Thus the use of the model
can be questioned. Note however that the number of secondary electrons is very small
(average number around three per primary particle) for a 1 keV electron. It is possible that the
effect is larger for high primary particle energies, which yield more secondary particles.
However as counterbalance their penetration depth is greater. The importance of secondary
electrons will become evident in the next section.
60
50
10-3
Axial distance (nm)
40
30
10-2
20
10-1
10
0
-10
-20
5
10
15
20
25
Radial distance (nm)
30
35
40
Figure 17: Energy deposition in eV/nm3 for a 1 keV primary electron as
function of radial and axial distance from the initial position. The outer curve
in each curve pair is obtained when secondary electrons are simulated.
The outermost curve pair is heavily disturbed by statistical fluctuations.
7.1 Toy model for DNA damage
Trying to estimate the amount of strand breaks in a DNA molecule caused by radiation,
different models for the spatial structure of DNA have been investigated [11,12]. One of the
simplest models is to consider the DNA molecule to be a cylinder with diameter 23 Å. The
energy threshold for a strand break in a DNA molecule is estimated to be 17.5 eV [11]. It is
therefore desirable to know what amount of energy depositions above the threshold that can
be expected if an electron with initial energy E is emitted in random direction from a point at
some distance D from the central axis of the cylinder. Any such energy deposition will in this
model result in a single strand break. The reasons for calling this a “toy model” are many: In
reality a DNA molecule has a much more complex shape than a cylinder. Nor are the effects
of free radicals included in the model.
31
Figure 18: Projection on a plane of the track of a 1keV electron in
liquid water. Two secondary electrons are visible. The circle illustrates
the cross section of a cylinder with diameter 2.3 nm.
In figure 19 the number of events with energy transfer above the threshold (17.5 eV) inside
the cylinder is shown for a 1 keV electron as function of its initial distance from the central
axis of the cylinder. The initial direction is chosen randomly. It is evident that strand breaks
caused by direct hits of primary electrons are rare. Not even an electron that starts at the
surface of the cylinder will with certainty cause a strand break. Due to the simplicity of the
model it is not advisable to draw any deeper conclusions. The model serves however the
purpose of adding some intuition to a subject, which previously was submitted to more or less
qualified guesses (for those not already familiar with the topic). Moreover it can be said that
the model very clearly shows the importance of simulating secondary electrons. The
probability for a strand break increases almost with a factor two when these are added to the
simulation.
32
Number of energy depositions E>17.5 eV inside cylinder
1
No secondary electrons
Secondary electrons included
0.9
0.8
0.7
0.6
0.5
0.5 keV e- → H2O
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Initial electron distance from cylinder (nm)
6
7
Number of energy depositions E>17.5 eV inside cylinder
1
No secondary electrons
Secondary electrons included
0.9
0.8
0.7
0.6
0.5
0.4
1 keV e- → H2O
0.3
0.2
0.1
0
0
1
2
3
4
5
Initial electron distance from cylinder (nm)
6
7
Number of energy depositions E>17.5 eV inside cylinder
1
No secondary electrons
Secondary electrons included
0.9
0.8
0.7
0.6
0.5
0.4
0.3
5 keV e- → H2O
0.2
0.1
0
0
1
2
3
4
5
Initial electron distance from cylinder (nm)
6
7
Figure 19: Probability for inelastic events with energy transfer above 17.5 eV
for primary electron as function of its initial position distance from the centre
of a cylinder with diameter 2.3 nm. Initial direction of the electrons is chosen
randomly.
33
8 Programs
Several programs for investigation of different properties of particle scattering in liquid
water, such as energy deposition and track structures, have been developed during this work.
The programs are all based on the codes in table 3 and table 9. The present main objective has
been partly to benchmark the LEEPS application liquid water, partly to develop some
hopefully useful and user-friendly tools easy to adapt to different problems.
Below the more important programs are described. The reader is advised to not just relay on
these descriptions but also to get somewhat familiar with the source codes in appendix.
All programs has an out-file name.PAR which contains the tabulated scattering parameters.
8.1 Program RSIM
The program simulates the frequency of different energy transfers on different radial
distances ( r = x 2 + y 2 + z 2 ) from the initial position of a particle. The events are sorted in
radial and energy channels whose numbers and widths are set by the user in the DAT file.
Link to: SUPPORT, MCKLAS or MCKLASP
In data file: RSIM.DAT
100000.
50.
1000
37466543
100
100
0.075D-6
2.0D0
H2OE.phy
INITIAL ENERGY
ABSORBTION ENERGY
NUMBER OF TRACKS
RANDOM SEED
NUMBER OF ENERGY CHANNELS MAX 200
NUMBER OF RADIAL CHANNELS MAX 200
RADIAL CHANNEL WIDTH CM
ENERGY CHANNEL WIDTH EV
IN-DATA FILE
Out data files: RSIM.PAR, RSIM.RES
RSIM.RES contains the particle-normalised number of events in each channel. For each
energy channel the radial data is written to the file from small to large radius.
8.2 Program PARSIM
The program calculates the scattering functions for a specific PHY-file entered by the user
during runtime. The user will be asked if the data shall be stored to PARSIM.RES for
possibility to plot it later.
Link to: SUPPORT, MCKLAS or MCKLASP
In data: Given interactively.
Out data-files: PARSIM.PAR, PARSIM.RES
PARSIM.RES contains data for the scattering functions. The data are mixed but are easily
separated when read from the plot program (matlab for example). The data is stored in the
following order.
Energy, el. mfp, 1:st tmfp, 2:nd tmfp, in. mfp, st. power, range. Data number 1,8,15,22…
holds energies, data number 2,9,16,23 hold 1:st tmfp etc.
34
8.3 Program RADAXSIM
Calculates the energy deposition in eV/nm3 as function of radial and axial distance from the
initial position of the simulated particle (see section 7). In the DAT file the user specifies the
number and width of the axial and radial channels. Since the axial distance (see figure 7) can
be negative the fraction of the axial channels representing positive and negative distances
must be given in the DAT file. The two fractions must be unity when added.
Link to: SUPPORT, MCKLAS or MCKLASP
In-data file: RADAXSIM.DAT
1000.
50.
200
37466543
100
150
0.075D-6
0.075D-6
0.6666667
0.3333333
H2OP.phy
INITIAL ENERGY
ABSORBTION ENERGY
NUMBER OF TRACKS
RANDOM SEED
NUMBER OF RADIAL CHANNELS MAX 200
NUMBER OF AXIAL CHANNELS MAX 200
RADIAL CHANNEL WIDTH CM
AXIAL CHANNEL WIDTH CM
POSITIVE PART OF AXIAL SCALE
NEGATIVE PART OF AXIAL SCALE
IN-DATA FILE
Out-data files: RADAXSIM.PAR, RADAXSIM,RES
RADAXSIM.RES contains the particle-normalized energy deposition in each channel. For
each axial channel the data for all radial channels are written to the file form small to large
radius.
8.4 Program RADAXSIM2
Calculates the energy deposition in eV/nm3 as function of radial and axial distance from the
initial position of the simulated particle. Secondary particles created with the described model
(section 7) are included. In the DAT file the user specifies the number and width of the axial
and radial channels. Since the axial distance can be negative the fraction of the axial channels
representing positive and negative distances must be given in the DAT file. The two fractions
must be unity when added. The DAT file also contains the two ionization levels used in the
secondary electron model.
Link to: SUPPORT, MCKLAS, SIMPACK
In-data file: RADAXSIM2.DAT
1000.
50.
200
37466543
100
150
0.075D-6
0.075D-6
0.6666667
0.3333333
16.0D0
539.0D0
H2OP.phy
INITIAL ENERGY
ABSORBTION ENERGY
NUMBER OF TRACKS
RANDOM SEED
NUMBER OF RADIAL CHANNELS MAX 200
NUMBER OF AXIAL CHANNELS MAX 200
RADIAL CHANNEL WIDTH CM
AXIAL CHANNEL WIDTH CM
POSITIVE PART OF AXIAL SCALE
NEGATIVE PART OF AXIAL SCALE
FIRST IONIZATION LEVEL
SECOND IONIZATION LEVEL
IN-DATA FILE
35
Out-data files: RADAXSIM2.PAR, RADAXSIM2.RES
RADAXSIM2.RES contains the particle-normalized energy deposition in each channel. For
each axial channel the data for all radial channels are written to the file form small to large
radius.
8.5 Program PATHSIM
The program calculates the average penetration distance, average path length, and average
axial penetration distance and average radial penetration distance for the simulated particles.
Moreover the distribution of all these quantities (except path length) are given as out-data.
The user must specify the number of channels and the channel widths in the DAT file. Since
the axial distance can be negative the fraction of the axial channels representing positive and
negative distances must be given in the DAT file. The two fractions must be unity when
added.
Link to: SUPPORT, MCKLAS or MCKLASP
In-data: PATHSIM.DAT
1000.
50.
20000
37466543
100
150
100
0.075D-6
0.075D-6
0.075D-6
0.6666667
0.3333333
H2OP.phy
INITIAL ENERGY
ABSORBTION ENERGY
NUMBER OF TRACKS
RANDOM SEED
NUMBER OF RADIAL CHANNELS
NUMBER OF AXIAL CHANNELS
NUMBER OF PEN DEPTH CHAN
RADIAL CHANNEL WIDTH CM
AXIAL CHANNEL WIDTH CM
PEN DEPTH CHAN WIDTH CM
POSITIVE PART OF AXIAL SCALE
NEGATIVE PART OF AXIAL SCALE
IN-DATA FILE
Out-data: The average of the four quantities mentioned above is presented on the screen when
the simulation is finished.
Out-data files: PATHSIM.PAR, PATHSIM.RES
PATHSIM.RES contains the particle-normalized number of track ends in the different
channels for the quantities mentioned. The data is written to the file in three blocks in the
following order: Penetration distance, radial penetration depth and axial penetration distance
from low to high channel number. In figure 20 examples are shown.
0.025
0.02
0.02
0.015
0.015
P(d)
P(z)
0.025
0.01
0.01
0.005
0.005
0
0
10
20
30
40
50
60
Penetraion depth, d (nm)
70
80
0
-40
-20
0
20
40
Axial penetration, z (nm)
60
Figure 20: Different distributions for a 1keV Positron scattered in liquid water. Data generated by PATHSIM.
80
36
8.6 Program TRAJSIM
The program writes the positions of all scattering events for the simulated particles to a file.
Trajectory patterns are easily obtained from this data.
Link to: SUPPORT, MCKLAS or MCKLASP
In-data file: TRAJSIM.DAT
1000.
50.
30
37466543
H2OP.phy
INITIAL ENERGY
ABSORBTION ENERGY
NUMBER OF TRACKS
RANDOM SEED
IN-DATA FILE
Out-data files: TRAJSIM.PAR, TRAJSIM.RES
TRAJSIM.RES contains the positions of all scattering events. If the particles initial position is
(0,0,0) the different particle trajectories can easily be separated. The data are sorted in the
following sequence: X1, Y1, Z1, X2, Y2, Z2…
In figure 21 is an example shown.
-6
x 10
cm
5
0
-5
5
5
-6
x 10
0
cm
0
-5
-5
cm
Figure 21: Trajectory patterns for 30 1 keV electrons in liquid
water. The particles start at (0,0,0) with initial direction (0,0,1).
-6
x 10
37
8.7 Program TRAJSIM2
The program writes the positions of all scattering events for the simulated particles to a file.
The positions of secondary particle scattering events are written to an additional file.
Trajectory patterns are easily obtained from this data.
Link to: SUPPORT, MCKLAS, SIMPACK
In-data file: TRAJSIM2.DAT
1000.
50.
30
37466543
16.0D0
539.0D0
H2OP.phy
INITIAL ENERGY
ABSORBTION ENERGY
NUMBER OF TRACKS
RANDOM SEED
FIRST IONIZATION LEVEL
SECOND IONIZATION LEVEL
IN-DATA FILE
Out-data files: TRAJSIM2.PAR, TRAJSIM2P.RES, TRAJSIM2S.RES
TRAJSIM2P.RES contains the positions of all scattering events for the primary particles.
If the particles initial position is (0,0,0) the different particle trajectories can easily be
separated. The data are sorted in the following sequence: X1, Y1, Z1, X2, Y2, Z2…
TRAJSIM2S.RES contains the positions of all scattering events for the secondary particles.
The different tracks are preceded by three zero coordinates (0,0,0) which makes the different
particle trajectories easy to separated. The data are sorted in the following sequence: X1, Y1,
Z1, X2, Y2, Z2…
An example is shown in figure 22.
-6
x 10
8
cm
6
4
2
0
6
-6
x 10
6
4
2
cm
0
2
0
4
-6
x 10
cm
Figure 22: Trajectory pattern for a 2 keV electron scattered in liquid water.
Secondary electrons are included.
38
References
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