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| In passing through matter, fast charged particles [[ionization|ionize]] the atoms or molecules which they encounter. Thus, the fast particles gradually lose [[energy]] in many small steps. '''Stopping power''' is defined as the [[conservative force]] acting on the particle during the interaction with materials (see [[Lennard-Jones potential]] and [[Yukawa potential]]). It is an important consideration in several scientific disciplines, including medicine and technology.<ref name=ICRU73>ICRU Report 73: Stopping of Ions heavier than Helium, Journal of the ICRU, 5 No. 1 (2005), Oxford Univ. Press ISBN 0-19-857012-0</ref>
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| [[File:Stopping H in Al.png|thumb|The stopping power of aluminium for [[proton]]s, plotted versus proton energy.]]
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| ==Definition and Bragg curve==
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| Both charged and uncharged particles lose energy while passing through matter. Positive ions are considered in most cases below.
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| The stopping power depends on the type and energy of the radiation and on the properties of the material it passes. For example the production of an [[ion]] pair (usually a positive ion and a (negative) electron) requires a fixed amount of energy (for example, 33.97 [[Electronvolt|eV]] in dry air<ref>{{cite book|title=Radiation Oncology Physics: A Handbook for Teachers and Students|year=2005|publisher=International Atomic Energy Agency|location=Vienna|isbn=92-0-107304-6|url=http://www-pub.iaea.org/mtcd/publications/pdf/pub1196_web.pdf|editor=Podgorsak, E. B.|accessdate=25 November 2012}}</ref>{{rp|305}}). The 'stopping power', referred to the material, is equal to the 'space gradient of residual [[kinetic energy]]', referred to the primary radiation:
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| : <math>F(E) = -\nabla E </math>
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| and so defined as the [[gradient]] of the radiation kinetic energy, because in the hypothesis of the conservation of [[mechanical energy]] the radiation kinetic energy is thought to be equal to the [[potential energy]] of the nuclear (for neutrons and uncharged [[hadron]]s and [[atom]]s) or electromagnetic (for [[photons]] and charged matter particles like [[leptons]] and [[ions]] ) field generated respectively by the material nuclei and by them and its electrons. The minus sign makes that the force always positive.[[File:Bragg Curve for Alphas in Air.png|thumb|Bragg curve of 5.49 MeV [[alpha particle]]s in air]] The conservative force usually increases toward the end of [[Range (particle radiation)|range]] and reaches a maximum, the [[Bragg peak]], shortly before the energy drops to zero. The curve that describes the force as function of the material depth is called the ''Bragg curve''. This is of great practical importance for [[radiation therapy]].
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| The equation above defines the '''linear stopping power''' which in the international system is expressed in [[newton (unit)|N]] but is usually indicated in other units like M[[Electronvolt|eV]]/mm or similar. If a substance is compared in gaseous and solid form, then the linear stopping powers of the two states are very different just because of the different density. One therefore often divides the force by the [[density]] of the material to obtain the '''mass stopping power''' which in the international system is expressed in [[meter|m]]<sup>4</sup>/[[second|s]] but is usually found in units like MeV/(mg/cm<sup>2</sup>) or similar. The mass stopping power then depends only very little on the density of the material.
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| The picture shows how the stopping power of 5.49 [[Electronvolt|MeV]] alpha particles increases while the particle traverses air, until it reaches the maximum. This particular energy corresponds to that of the alpha particle radiation from naturally [[radioactivity|radioactive]] gas [[radon]] (<sup>222</sup>Rn) which is present in the air in minute amounts wherever the ground contains granite.
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| The mean [[Range (particle radiation)|range]] can be calculated by [[Integral|integrating]] the reciprocal stopping power over energy:<ref name=icru16>{{cite book|last=International Commission on Radiation Units and Measurements|title=Linear Energy Transfer|year=1970|location=Washington D.C.|isbn=978-0913394090|url=http://epub.ub.uni-muenchen.de/8949/1/8949.pdf|accessdate=1 December 2012|id=ICRU report 16}}</ref>
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| <math>\Delta x=\int_{0}^{E_0}\frac{1}{F}\, dE</math>
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| where: | |
| :''E''<sub>0</sub> is the initial kinetic energy of the particle
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| :''Δx'' is the "continuous slowing down approximation (CSDA)" range and
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| :"F" is the stopping power.
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| The deposited energy can be obtained by integrating the stopping power over the entire path length of the ion while it moves in the material.
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| ==Electronic, nuclear and radiative stopping==
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| Electronic stopping refers to the slowing down of a projectile ion due to the inelastic collisions between bound electrons in the medium and the ion moving through it. The term inelastic is used to signify that energy is lost during the process (the collisions may result both in excitations of bound electrons of the medium, and in excitations of the electron cloud of the ion as well). Linear electronic stopping power is identical to [[Linear energy transfer|unrestricted linear energy transfer]].
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| Since the number of collisions an ion experiences with electrons is large, and since the charge state of the ion while traversing the medium may change frequently, it is very difficult to describe all possible interactions for all possible ion charge states. Instead, the electronic stopping power is often given as a simple function of energy <math>F_e (E)</math> which is an average taken over all energy loss processes for different charge states. It can be theoretically determined to an accuracy of a few % in the energy range above several hundred keV per [[nucleon]] from theoretical treatments, the best known being the [[Bethe formula]]. At energies lower than about 100 keV per nucleon, it becomes more difficult to determine the electronic stopping theoretically.<ref>P. Sigmund: [http://books.google.com/books?id=24st-gh4QHIC&pg=PP1 Stopping of heavy ions]. Springer Tracts in Modern Physics Vol. 204 (2004) ISBN 3-540-22273-1</ref>
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| [[File:Electronic nuclear stopping Al in Al.png|thumb|Electronic and nuclear stopping power for aluminum ions in aluminum, versus particle energy per nucleon. The maximum of the nuclear stopping curve typically occurs at energies of the order of 1 [[electron volt|keV]] per [[nucleon]].]]
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| Graphical presentations of experimental values of the electronic stopping power for many ions in many substances have been given by Paul.<ref>[http://www.exphys.uni-linz.ac.at/Stopping/ Stopping Power for Light Ions]</ref> The accuracy of various stopping tables has been determined using statistical comparisons.<ref>{{cite journal|doi=10.1016/j.nimb.2006.01.059|title=A comparison of recent stopping power tables for light and medium-heavy ions with experimental data, and applications to radiotherapy dosimetry|year=2006|last1=Paul|first1=H|journal=Nuclear Instruments and Methods in Physics Research Section B|volume=247|pages=166|bibcode = 2006NIMPB.247..166P|issue=2 }}</ref>
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| '''Nuclear stopping power''' refers to the elastic collisions between the projectile ion and atoms in the sample (the established designation "nuclear" may be confusing since nuclear stopping is not due to nuclear forces,<ref name="ICRU85">{{cite journal|last=International Commission on Radiation Units and Measurements|title=Fundamental Quantities and Units for Ionizing Radiation|journal=Journal of the ICRU|date=October 2011|edition=Revised|volume=11|issue=1|doi=10.1093/jicru/ndr012|url=http://www.engin.umich.edu/class/ners580/ners-bioe_481/lectures/pdfs/2011-04-ICRU_Report85a-QuantitiesUnits(revised).pdf|accessdate=14 December 2012|editor1-first=Stephen M.|editor1-last=Seltzer|id=ICRU report 85a}}</ref> but it is meant to note that this type of stopping involves the interaction of the ion with the ''nuclei'' in the target). If one knows the form of the repulsive potential energy <math>E (r)</math> between two atoms (see below), it is possible to calculate the nuclear stopping power <math>F_n (E)</math>. In the stopping power figure shown above for protons in aluminum, nuclear stopping is negligible except at the lowest energy. Nuclear stopping increases when the mass of the ion increases. In the figure shown here, nuclear stopping is larger than electronic stopping at low energy. For very light ions slowing down in heavy materials, the nuclear stopping is weaker than the electronic at all energies.
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| Especially in the field of radiation damage in detectors, the term "'''non-ionizing energy loss'''" (NIEL) is used as a term opposite to the [[linear energy transfer]] (LET), see e.g. Refs.<ref name="Huh02">{{cite journal|last=Huhtinen|first=Mika|title=: Simulation of non-ionising energy loss and defect formation in silicon|journal=Nucl. Instr. Meth. Phys. Res. B|year=2002|volume=491|issue=1-2|pages=194–215}}</ref><ref name="Bar95">{{cite journal|last=Barry|first=AL|coauthors=Houdayer, AJ; Hinrichsen, PF; Letourneau, WG; Vincent, J|title=The energy dependence of lifetime damage constants in GaAs LEDs for 1-500 MeV protons|journal=IEEE TRANSACTIONS ON NUCLEAR SCIENCE|year=1995|volume=42|issue=6|pages=2104–2107|bibcode = 1995ITNS...42.2104B |doi = 10.1109/23.489259 }}</ref><ref name="Lin01">{{cite journal|last=Lindström|first=G|title=Radiation hard silicon detectors - developments by the RD48 (ROSE) collaboration|journal=NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH SECTION A|year=2001|volume=466|pages=308–326}}</ref> Since per definition nuclear stopping power does not involve electronic excitations, NIEL and nuclear stopping can be considered to be the same quantity in the absence of nuclear reactions.
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| The total non-relativistic stopping power is therefore the sum of two terms: <math>F(E) = F_e (E) + F_n (E)</math>. Several semi-empirical stopping power formulas have been devised. The model given by Ziegler, Biersack and Littmark (the so-called "ZBL" stopping, see next chapter),<ref name=Z1985>J. F. Ziegler, J. P. Biersack, and U. Littmark. In The Stopping and Range of Ions in Matter, volume 1, New York, 1985. Pergamon. ISBN 0-08-022053-3</ref> implemented in different versions of the [[Stopping and Range of Ions in Matter|TRIM/SRIM]] codes,<ref name=srim/> is used most often today.
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| At extremely high ion energies,<ref name=ICRU73 /> one also has to consider radiative stopping power which is due to the emission of [[bremsstrahlung]] in the electric fields of the particles in the material traversed.<ref name=ICRU85 /> For electron projectiles, radiative stopping is always important. At high ion energies, there may also be energy losses due to nuclear reactions, but such processes are not normally described by stopping power.<ref name=ICRU85 />
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| Close to the surface of a solid target material, both nuclear and electronic stopping may lead to [[sputtering]].
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| ==The slowing-down process in solids==
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| [[File:Ion slowing.png|thumb|llustration of the slowing down of a single ion in a solid material.]]
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| In the beginning of the slowing-down process at high energies, the ion is slowed down mainly by electronic stopping, and it moves almost in a straight path. When the ion has slowed down sufficiently, the collisions with nuclei (the nuclear stopping) become more and more probable, finally dominating the slowing down. When atoms of the solid receive significant recoil energies when struck by the ion, they will be removed from their [[crystal structure|lattice]] positions, and produce a [[collision cascade|cascade of further collisions]] in the material. These
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| [[collision cascade]]s are the main cause of damage production during ion implantation in metals and semiconductors.
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| When the energies of all atoms in the system have fallen below the [[threshold displacement energy]], the production of new damage ceases, and the concept of nuclear stopping is no longer meaningful.
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| The total amount of energy deposited by the nuclear collisions to atoms in the materials is called the nuclear deposited energy.
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| The inset in the figure shows a typical range distribution of ions deposited in the solid. The case shown here might for instance be the slowing down of a 1 MeV silicon ion in silicon. The mean range for a 1 MeV ion is typically in the [[micrometer]] range.
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| ===Repulsive interatomic potentials===
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| At very small distances between the nuclei the repulsive interaction can be regarded as essentially Coulombic. At greater distances, the electron clouds screen the nuclei from each other. Thus the repulsive potential can be described by multiplying the Coulombic repulsion between nuclei with a screening function φ(r/a),
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| : <math> V(r) = { 1 \over 4 \pi \varepsilon_0} {Z_1Z_2 e^2 \over r} \varphi(r/a)</math> | |
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| where φ(r) → 1 when r → 0. Here <math>Z_1</math> and <math>Z_2</math> are the charges of the interacting nuclei, and ''r'' the distance between them; ''a'' is the so-called screening parameter.
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| A large number of different repulsive potentials and screening functions have been proposed over the years, some determined semi-empirically, others from theoretical calculations. A much used repulsive potential is the one given by Ziegler, Biersack and Littmark, the so-called ZBL repulsive potential. It has been constructed by fitting a universal screening function to theoretically obtained potentials calculated for a large variety of atom pairs.<ref name=Z1985/> The ZBL screening parameter and function have the forms
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| : <math> a = a_u = { 0.8854a_0 \over Z_1^{0.23} + Z_2^{0.23} }</math>
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| and
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| : <math>\varphi(x) = 0.1818e^{-3.2x} +0.5099e^{-0.9423x} +0.2802e^{-0.4029x} +0.02817e^{-0.2016x} </math>
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| where ''x = r/a<sub>u</sub>'', and ''a<sub>0</sub>'' is the Bohr atomic radius = 0.529 Å.
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| The standard deviation of the fit of the universal ZBL repulsive potential to the theoretically calculated pair-specific potentials it is fit to is 18% above 2 eV.<ref name=Z1985/>
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| Even more accurate repulsive potentials can be obtained from self-consistent total energy calculations using [[density-functional theory]] and the [[local-density approximation]]
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| (LDA) for electronic exchange and correlation.<ref>{{cite journal|doi=10.1016/S0168-583X(97)00447-3|title=Repulsive interatomic potentials calculated using Hartree-Fock and density-functional theory methods|year=1997|last1=Nordlund|first1=K|last2=Runeberg|first2=N|last3=Sundholm|first3=D|journal=Nuclear Instruments and Methods in Physics Research Section B|volume=132|pages=45|bibcode = 1997NIMPB.132...45N }}</ref>
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| ===Channeling===
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| {{Main|Channelling (physics)}}
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| In crystalline materials the ion may in some instances get "channeled", i.e., get focused into a channel between crystal planes where it experiences almost no collisions with nuclei. Also, the electronic stopping power may be weaker in the channel. Thus the nuclear and electronic stopping do not only depend on material type and density but also on its microscopic structure and cross-section.
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| ===Computer simulations of ion slowing down===
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| Computer simulation methods to calculate the motion of ions in a medium have been developed since the 1960s, and are now the dominant way of treating stopping power theoretically. The basic idea in them is to follow the movement of the ion in the medium by simulating the collisions with nuclei in the medium. The electronic stopping power is usually taken into account as a frictional force slowing down the ion.
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| Conventional methods used to calculate ion ranges are based on the [[binary collision approximation]] (BCA).<ref name="Rob74">{{cite journal|doi=10.1103/PhysRevB.9.5008|title=Computer simulation of atomic-displacement cascades in solids in the binary-collision approximation|year=1974|last1=Robinson|first1=Mark|last2=Torrens|first2=Ian|journal=Physical Review B|volume=9|pages=5008|bibcode = 1974PhRvB...9.5008R|issue=12 }}</ref> In these methods the movement of ions in the implanted sample is treated as a succession of individual collisions between the recoil ion and atoms in the sample. For each individual collision the classical scattering integral is solved by numerical integration.
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| The impact parameter ''p'' in the scattering integral is determined either from a stochastic distribution or in a way that takes into account the crystal structure of the sample.
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| The former method is suitable only in simulations of implantation into amorphous materials,
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| as it does not account for channeling.
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| The best known [[binary collision approximation|BCA]] simulation program is [[Stopping and Range of Ions in Matter|TRIM/SRIM]]
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| ([[Acronym and initialism|acronym]] for TRansport of Ions in Matter, in more recent versions | |
| called Stopping and Range of Ions in Matter), which is based on the ZBL electronic stopping and interatomic potential.<ref name=Z1985/><ref name=srim>[http://www.SRIM.org SRIM web site]</ref><ref>{{cite journal|doi=10.1016/0029-554X(80)90440-1|title=A Monte Carlo computer program for the transport of energetic ions in amorphous targets☆|year=1980|last1=Biersack|first1=J|last2=Haggmark|first2=L|journal=Nuclear Instruments and Methods|volume=174|pages=257|bibcode = 1980NucIM.174..257B }}</ref> It has a very easy-to-use user interface,
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| and has default parameters for all ions in all materials up to an ion energy of 1 GeV,
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| which has made it immensely popular. However, it doesn't take account of the crystal structure, which severely limits its usefulness in many cases. Several BCA programs overcome this difficulty; some fairly well known are MARLOWE,<ref name="Rob92">{{cite journal|doi=10.1016/0168-583X(92)95839-J|title=Computer simulation studies of high-energy collision cascades1|year=1992|last1=Robinson|first1=M|journal=Nuclear Instruments and Methods in Physics Research Section B|volume=67|pages=396|bibcode = 1992NIMPB..67..396R }}</ref> BCCRYS and crystal-TRIM.
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| Although the BCA methods have been successfully used in describing many physical processes, they have some obstacles for describing the slowing down process of energetic ions realistically. Basic assumption that collisions are binary results in severe problems when trying to take multiple interactions into account. Also, in simulating crystalline materials the selection process of the next colliding lattice atom and the impact parameter ''p'' always involve several parameters which may not have perfectly well defined values, which may affect the results 10-20% even for quite reasonable-seeming choices of the parameter values. The best reliability in BCA is obtained by including multiple collisions in the calculations, which is not easy to do correctly. However, at least MARLOWE does this.
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| A fundamentally more straightforward way to model multiple atomic collisions is provided by [[molecular dynamics]] (MD) simulations, in which the time evolution of a system of atoms is calculated by solving the equations of motion numerically. Special MD methods have been devised in which the number of interactions and atoms involved in MD simulations have been reduced in order to make them efficient enough for calculating ion ranges.<ref name="Nor95">{{cite journal|doi=10.1016/0927-0256(94)00085-Q|url=http://beam.acclab.helsinki.fi/~knordlun/mdh/mdh_captions.ps|title=Molecular dynamics simulation of ion ranges in the 1–100 keV energy range|year=1995|last1=Nordlund|first1=K|journal=Computational Materials Science|volume=3|pages=448|issue=4}}</ref><ref>{{cite journal|doi=10.1103/PhysRevE.57.7278|title=Efficient molecular dynamics scheme for the calculation of dopant profiles due to ion implantation|year=1998|last1=Beardmore|first1=Keith|last2=Grønbech-Jensen|first2=Niels|journal=Physical Review E|volume=57|pages=7278|arxiv = physics/9901054 |bibcode = 1998PhRvE..57.7278B|issue=6 }}</ref>
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| ==Minimum ionizing particle==
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| In physics, a minimum ionizing particle (or mip) is a particle whose mean energy loss rate through matter is close to the minimum.
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| When a fast charged particle passes through matter, it ionizes or excites the atoms or molecules that it encounters, losing energy in small steps. The mean rate at which it loses energy depends on the material, the kind of particle and also its momentum. In practical cases, most relativistic particles (e.g., cosmic-ray [[muon]]s) are minimum ionizing particles.
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| ==See also==
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| *[[Radiation length]]
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| *[[Attenuation length]]
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| * [[Collision cascade]]
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| * [[Radiation material science]]
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| *[[COSIRES conference]]
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| *[[REI conference]]
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| ==References==
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| {{reflist|2}}
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| ==Further reading==
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| *(Lindhard 1963) J. Lindhard, M. Scharff, and H. E. Shiøtt. Range concepts and heavy ion ranges. Mat. Fys. Medd. Dan. Vid. Selsk., 33(14):1, 1963.
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| *(Smith 1997) R. Smith (ed.), Atomic & ion collisions in solids and at surfaces: theory, simulation and applications, Cambridge University Press, Cambridge, UK, 1997,
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| ==External links==
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| *[http://bohr.inf.um.es/MELF-GOS.html Stopping power and energy loss straggling calculations in solids by MELF-GOS model]
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| *[http://www.nucleonica.net/wiki/index.php/Help:Range_%26_Stopping_Power A Web-based module for Range and Stopping Power in Nucleonica]
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| *[http://pdg.lbl.gov/2006/reviews/passagerpp.pdf Passage of charged particles through matter]
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| *[http://www.physics.nist.gov/PhysRefData/Star/Text/programs.html Stopping power for protons and alpha particles]
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| *[http://www.exphys.uni-linz.ac.at/Stopping/ Stopping Power: Graphs and Data]
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| *[http://www.phys.au.dk/~ahs/EBnotes.htm Penetration of charged particles through matter; lecture notes by E. Bonderup]
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| [[Category:Condensed matter physics]]
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| [[Category:Materials science]]
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| [[Category:Nuclear physics]]
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| [[Category:Radiation]]
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