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SRIM_2008manual.pdf
SRIM_2008manual.pdf
SRIM_2008manual.pdf
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Simulation of primary damage production by Monte Carlo methods(a bout SRIM)
3. Simulation of primary damage production by Monte Carlo methodsNE220 Lectures 14 - 15 17 - 22 February 2006Brian D. Wirth and D. R. Olander Nuclear Engineering Department1The TRIM codeReference: J.F. Ziegler, J.P. Biersack and U. Littmark, The Stopping and Range of Ions in Matter, Volume 1, Chapter 4, Pergamon Press (1985). http://www.srim.org ??TRIM (Transport of Ions in Matter) calculates all interactions of a projectile (ion or a neutral atom) with a solid consisting of stationary target atoms ??TRIM is part of the code package SRIM, which can be downloaded and run on a PC at the website www.srim.org ??The code is basically a Monte Carlo calculation. One projectile at a time is incident on a solid surface. The trajectories of the projectile and recoild target atoms are followed. The input information required for the calculation includes: ? projectile: Z1, m1, of initial energy E10 ??target: Z2, m2, thickness and density ? displacement energy of the atoms in the target material (Ed) ? surface atom binding energy (for sputtering calculations) ? binding (cohesive) energy of the target atoms, Eb2The TRIM codeThe output of the TRIM computation includes: ? The path of the projectile and all recoil target atoms ??The depth distribution of the atomic displacements (dpa) of the target atoms ??The final position of the projectile (I.e., range and range straggling) ? Projectile energy loss by ionization (electronic stopping); histories of all recoils ? Sputtering of the surface atoms ? Projectile reflection via the front surface or transmission through the rear surface Projectile - Solid Medium Interaction Process ??Only two-body collisions are considered ??The computation treats a three dimensional array of atoms in the solid, but does not consider crystallography. That is, the solid is assumed to be amorphous. ? Specific crystallographic effects (focusing or channeling) are not treated. ? The accumulation of damage (cascade overlap) is not treated. ? The target may consist of multiple layers, each containing several different elements.3TRIM: collision kinematics treatmentMethod of treating the mean free path between atomic collisions (projectile - target or target - target) ?? The usual mean free path (λ = (Nσ)-1) can not be used because the total cross section for most atomic collisions is infinite (e.g. the Rutherford cross section, Section 2, p26) ? Instead, a “free-flight distance”, L, is introduced. This is the distance between large-angle atomic collisions. Rather than trying to calculate the individual effect of many grazing collisions, only the angular deviation due to these collisions is tracked. When the angular deviation of a particle reaches an arbitrary specified (small) value, a large-angle collision is permitted. In addition, during the free-flight distance, electronic stopping is included to reduce the energy of the moving ion (atom).E’ previous collision L ionization atomic collision E = E’ - T cum - E ionizationE’ = projectile energy at start of the flight path Tcum = energy loss due to glancing atomic collisions during flight Eionization = ionization loss to electrons of medium during flight (Eq. (2.55)) E = projectile energy at point of large-angle nuclear collisions.4TRIM: collision kinematics treatment?? At the end of the free-flight path length, an impact parameter for the atomic collision (p) is randomly selected. The method for computing L and p will be discussed later. ??Knowing the energy of the projectile and the impact parameter, the center-of-mass (CM) scattering angle is computed by collision dynamics theory. The projectile or recoil atom loses energy as it proceeds through a series of such free flight path lengths and collisions. The process ceases when the energy of the moving atom is too low to cause further displacements (i.e., E & Ed) ??The detailed histories of recoils and the development of collision cascades are followed in the same manner as described above for the incident ion. Isolated cascade theory (e.g., Eq. (2.61)) is not used since the cascade is determined by following each particle individually.5TRIM: Universal Atomic Potential?? The projectile interacts with atoms of the medium according to interaction laws based on a composite (“Universal”) interatomic potential that fits a large number of projectile - target atom combinations over a wide range of energies:Z1 Z 2 e 2 ? r ? V(r) = Φ? ? ?a? r(3.1)Φ is a “universal” screening function that has been empirically determined by fitting exact interatomic potentials to Eq (3.1) for 521 element combinations (out of ~104)
over a wide energy range.6TRIM: Universal Atomic Potential? The curves are collapsed into the Universal Screening Function using the reduced separation distance r/a:? ? r ?? ?r? 4 Φ? ? = ∑ Ai exp??Bi ? ?? ? a ? i=1 ? ? a ??(3.2)where Ai and Bi are constants that are valid for all elements at all energies.The parameter a in Eqs (3.1) and (3.2) is an empirical screening length given by:a= 0.8854a Bohr0.23 Z1 + Z 0.23 2(3.3)where aBohr = 0.53 ? = Bohr radius of the hydrogen atomEq (3.2)7TRIM: Universal Atomic Potential? ??Substitution of Eq (3.1) into the classical scattering integral (Eq (2.33))θ= π?2∫∞ (p / a)dX = π?2 ∫ 1/2 ? 1 Φ (X) (p / a)2 ?1/2 r0 ? V(r) p 2 ? (r0 /a) 2 2 r ?1 ? ? ? X ?1 ? ? ? ? ? ? r2 ? X2 ? ? Ec ? X ε ? ∞pdr(3.4)yields the Center of Mass scattering angle as a function of the reduced impact parameter
(p/a), and the reduced relative kinetic energy of the collision:ε= Ec (Z1 Z 2 e 2 / a)(3.5)where the relative kinetic energy of the projectile or recoil is related to its kinetic energy in the lab frame by: Ec = m2 E m1 + m 2(3.6)The result is the Universal Scattering Integral8TRIM: Computational Algorithm1. Start with projectile of known energy E (at end of the free-flight path) 2. Pick the impact parameter p* 3. Use the universal scattering integral to determine the CM scattering angle (θ) a) The projectile scattering angle in the lab frame (Eq (2.11)) is:tan φ = sin θ cos θ + m1 / m 2(3.7)b) E’ = new energy of the projectile = E - T, with T from Eq (2.6) 4. T - E becomes E’ it is followed along with all of its secondary displacements until E & Ed 5. Find free-flight distance L* 6. At the end of the free-flight path of any energetic particle, the energy isE = E' ?Tcum ? E ionization(3.8)7. Follow the injected ion until its energy is too low to displace a lattice atom
Ei,min = Ed/Λ 8. Inject another ion at the surface 9. Stop when the statistics are “good enough” * The method of determining L and p is described below.9TRIM: Intercollision flight path and impact parameterHigh-energy projectile At high projectile energy, most deflections are small (i.e., less than 1° in the lab system). This is because the dominant contribution to the cross section is due to the Coulomb potential, which produces the forward-peaked Rutherford cross section. TRIM allows many of these small-angle atomic collisions to take place without calculating the details of each one. The large-angle collision is allowed to occur after the cumulative deflection of the projectile by the small-angle collisions reaches 5°. Such a path is schematically illustrated below:The free-flight length L is determined as follows. When the angular deflection is small, Eq. (3.7) reduces to:φ= θ 1 + m1 / m 2(3.9)The energy transferred to the target atom during the collision is:?θ? 1 T = ΛE(1 ? cos θ) = ΛE sin 2 ? ? ?2? 2(3.10)10TRIM: Intercollision flight path and impact parameterUsing Eq (3.9) and simplifying for low-angle collisions yields:? θ ? Λθ2 Λ ? m1 ? 2 T = Λ sin 2 ? ? ~ = ?1 + ? φ ?2? E 4 4 ? m2 ?2but,Λ ? m1 ? 1 4m1 m 2 (m1 + m 2 )2 m1 1 + = = ? ? 2 4 ? m2 ? 4 (m1 + m 2 )2 m2 m22Over many small-angle collisions, φ is interpreted as the cumulative deflection and T is the sum of the energy losses due to the small-angle atomic collisions that occur
over the free-flight path? m 2 Tcum ?1/2 φ cum ~ ? ? ? m1E ?(3.11)The energy transfer to atoms of the solid due to many glancing atomic collisions can Be related to a nuclear (I.e., atomic) stopping power:Tcum ~ dE L dx nuclTmax dE = N ∫ Tσ(E, T)dT dx nucl 0(3.12)The stopping power is defined by Eq (2.18):Where N is the density of the target atoms in the solid. The differential energy-transfer cross section is related to the impact parameter by the analog of Eq (2.35):σ(E, T)dT = 2 πpdp11TRIM: Intercollision flight path and impact parameterSo that Eq (3.10):p max ∞ ?θ? dE = N ∫ T2 πpdp = 2 πΛEN ∫ sin 2 ? ?pdp ?2? dx nucl 0 0(3.13)In the second integral, the upper limit has been set equal to pmax, which is the sum of the atomic radii of the projectile and target atoms. The interatomic potential is essentially
zero for larger separation distances. Substituting Eq (3.13) into (3.12) and then into (3.11) yields:φ2 cum = 2 πΛLN m 2 p max 2 ? θ ? ∫ sin ? ?pdp ?2? m1 0(3.14)In order to utilize the Universal Scattering Integral plotted on page 8, the variable integration is converted to p/a, where a is the universal screening length (Eq (3.3)).
the flight path L is given by: Setting φcum = 0.09 radians,?1 (0.09)2 (1 + m1 / m 2 )2 ?p max /a 2 ? θ ?? p ? ? p ?? L= ? ∫ sin ? ?? ?d? ?? ? 2 ?? a ? ? a ?? ? 0 8 πa 2 N ? ?(3.15)The integral is performed over the curve in the Figure on p. 8 appropriate to the energy of the collision. The code also checks if Eionization (=(dE/dx)eL) is & 5% of
E’. If not, L is reduced until it is.12TRIM: Intercollision flight path and impact parameterAfter travelling a distance L, the projectile energy is reduced by the sum of the nuclear stopping, Tcum, and the electronic stopping, Eionization. To determine the angular deflection and the energy transferred during the large-angle collision that occurs at the end of the free-flight path, the impact parameter must be specified. Basically, this is a method of mapping a random number between 0 and 1 onto the range of 0 to ∞. Let du = 2πpdpLN be the probability that the projectile finds a target atom between impact parameters p and p + dp in the interval L. Divide the flight path into j increments of length Δx = L/j. Over one of these increments, the probability of not finding a target atom with an impact parameter between 0 and p is 1 - πp2NΔx = 1 - πp2NL/j. Letting j become large, The probability v of not finding a target atom between 0 and p over the entire path length is:v = lim(1 ? πp2 NL / j) j = exp(?NLπp 2 )The combined probability = vdu of the projectile finding a target atom in (p,dp) and not between 0 and p over the flight path is:dw = exp(?πp2 LN)2 πLNpdp = e ?n dnwhere n = πp2NL. Integrating gives w = 1 - e-n, or n = -ln(1-w). Let w be a random number between 0 and 1; therefore 1 - w is also a random number between 0 and 1.13TRIM: Intercollision flight path and impact parameterExpressing n in terms of p yields the following formula for determining the impact parameter at the end of the flight path L: ? ? ln(1 ? w) ?1/2 (3.16) p =? ? ? πLN ? where L is given by Eq (3.15). With p known, the figure on p. 8 gives θ. The lab scattering angle φ is obtained from Eq (3.7) and T from Eq (3.10).
Low-Energy Projectile The free flight path given by Eq (3.15) and the impact parameter of Eq (3.16) are applied until L becomes equal to the interatomic spacing in the solid,L = N ?1/ 3(3.17)From this point on, the maximum impact parameter is set equal to one-half of the interatomic spacing, or pmax = N-1/3 /2. Since all impact parameters are equally
probable: 1 p = wpmax = wN ?1/ 3 (3.18) 2 Where w is a random number between 0 and 1.or (3.17) and (3.18) give L and p needed in steps 2 and 5 Equations (3.15) and (3.16) of the algorithm on page 9. 14Simple Estimate of Vacancy ProductionIn the standard TRIM program, the energy transferred to a target atom is analyzed further to yield such results as - ionization by recoiling atoms in a cascade - damage energy and number of vacancies produced in a collision cascade - damage energy and number of sub-threshold collisions in the cascade which transfer energies less than Ed, where knock-on (recoiling) atoms can not permanently escape their lattice site and their energy ends up in lattice vibrations (phonons). The recoils themselves are not individually followed in this Monte Carlo program (unless Full collsion cascades are followed). Their energy contributions to ionization and defect production are determined by standard defect production (cascade) theory. From this theory, the defect producing energy, Eν, is obtained from the transferred energy T of the recoil by taking into account electronic lossesEν = T 1 + k d g(εd )(3.19)where the electronic losses are governed by/ 3 ?1/2 k d = 0. m2and3/ 4 g(εd ) = εd + 0.40244 εd + 3. dwhere7/3 εd = 0.01014Z ? T 215Simple Estimate of Vacancy ProductionFrom the energy Eν, the number of displacements is calculated by the well known “modified Kinchin-Pease” (NRT) model**ν = 1, 0.8E ν ν= 2E d E d & E ν & 2.5E d E ν & 2.5E d(3.20)Thus, the three fractions of the recoil energy mentioned on the previous page, and the Number of vacancies per depth interval are calculated and provided as standard output
Of the TRIM program.** Kinchin and Pease, Rep Prog Phys, 18 (1955) 1. P. Sigmund, Rad Eff 1 (1969) 15. M.J. Norgett, M.T. Robinson and I.M. Torrens, Nucl. Eng. Design 33 (1974) 50.16Cascade (Recoil) Simulations by TRIMIn the previous discussion, we discussed the vacancy production by a modified KinchinPease model applied to every recoil atom which received a transferred energy, T & Ed, where Ed denotes the displacement energy of a lattice atoms. The number of vacancies is then stored at the depth position where the primary knock-on atom is created. In order to study the formation of vacancies, interstitials and replacement collisions in more detail, the individual recoils must be followed through a number of collisions and generations until their energy has dropped below Ed. This way, the spatial distribution of the defects is obtained, which often reveals transport of recoils (and energy) over large distances. Particularly for ion masses larger than or equal to the target atom mass, m1 & m2, the recoils may move farther than the incident ion. Such situations require simulations to treat, since semi-analytical transport theory procedures are not as precise. The “C”-key option of TRIM generates the file COLLISON.TXT, which follows the recoiling Atoms through many collision generations, until their energy has dropped below Ed. Any Knock-on atom receiving more energy than Ed is considered to leave its lattice position (Note, no crystallographic lattice is considered) and to create a vacancy. Thereafter a Second check is performed in order to determine whether the incident particle (the ion Itself or a previous recoil) is left with enou if its energy after the Collision is below Ed, then it is considered to remain within the recombination volume And finally combine with the vacancy, this way annihilating the previously created Vacancy and registering in TRIM as a “replacement” collision.17Cascade (Recoil) Simulations by TRIMBy this process, a large fraction of the vacancies may disappear again, mainly by target atom - target atom collisions at low energies. The ion - target atom collisions account only for a very small fraction (usually less than one percent) of all replacement collisions. If a moving particle (ion or recoil) is not ending up in a replacement collision before being slowed down to an energy less than Ed, then it is recorded as an “interstitial” atom.18TRIM/SRIM2003 examples: 1.3 MeV p+ in Ni19TRIM/SRIM2003 examples: 1.3 MeV p+ in Ni20TRIM/SRIM2003 examples: p+ and Ni in Ni5 MeV Ni21TRIM/SRIM2003 examples: 5 MeV Ni in Ni22TRIM/SRIM2003 examples: 5 MeV Ni in Ni23TRIM/SRIM2003 examples: alpha decay in U24TRIM/SRIM2003 examples: alpha decay in U25TRIM/SRIM2003 examples: 100 keV Fe PKA in FeBy comparison, Eq. 2.61 gives: v(100 keV) = 127026TRIM/SRIM2003 examples: 100 keV Fe PKA in Fe27TRIM/SRIM2003 examples: 100 keV Fe PKA in Fe28TRIM/SRIM2003 examples: 100 keV Fe PKA in Fe29TRIM/SRIM2003 examples: 100 keV Fe PKA in Fe30
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