Journal of the ICRU Vol 5 No 1 (2005) Report 73 Oxford University Press
2
DEFINITIONS
2.1
STOPPING Mean energy loss
The central quantity characterizing particle stopping is the stopping force or stopping power.2 For a point particle it is defined as the average loss of kinetic energy3 E per path length ‘, dE/d‘. The minus sign defines the stopping force as a positive quantity. The stopping force is related to the average change in momentum per path length according to dE dP ¼v , d‘ d‘
ð2:1Þ
where v is the projectile speed, P ¼ M1gn the momentum, M1 the projectile mass, 1 g ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 1 b2 b¼
v c
ð2:2Þ
Equation (2.4) in principle also applies to dressed atomic ions, i.e., ions carrying electrons for which the definition of the stopping force becomes ambiguous due to projectile excitation and charge exchange. However, currently accessible experimental and theoretical accuracy is below the level where terms of order m/M1 (m ¼ electron rest mass) would be significant. Therefore, the stopping force on a dressed ion may safely be related to the momentum change of the projectile nucleus, dE dP ¼ v : ð2:5Þ d‘ d‘ nucleus The definitions (2.4) and (2.5) allow the inclusion of contributions from all significant energy-loss channels. The stopping force is related to the velocity change in time ------ a quantity measured by some techniques ------ through dE dP dv ¼ ¼ M 1 g3 : d‘ dt dt
ð2:3Þ
and c the speed of light.4 For composite projectiles such as molecules and clusters for which energy can be transferred into internal degrees of freedom, the stopping force is defined by generalizing Eq. (2.1): dE X dP ¼ vi , ð2:4Þ d‘ d‘ i i where the sum extends over all constituents of the projectile.
ð2:6Þ
It is common practice to tabulate mass stopping forces
1dE , r d‘
ð2:7Þ
where r is the mass density of the target. A parameter of fundamental significance on the atomic or molecular scale is the stopping cross section S, defined by X S¼ wJ s J , ð2:8Þ J
2
The two terms will be used synonymously. While ‘stopping power’ is the offical nomenclature, ‘stopping force’ is more precise (Sigmund, 2000b). 3 Use of the symbol E for kinetic energy is just the first of a series of cases where notation in this report deviates from ICRU Reports 37 and 49. 4 The symbol g has at least four well-established functions in the context of heavy-ion penetration. In addition to Eq. (2.2), it characterizes the maximum energy transfer in an elastic binary collision, the effective-charge fraction of a dressed ion and Euler’s constant. Established notation will be maintained here, but proper specification will be added whenever ambiguities could arise.
where the sum extends over all energy-loss channels and where wJ and sJ denote the energy loss and pertinent cross section per target atom or molecule for the J’th channel. The stopping cross section is related to the stopping force through S¼
1dE , n d‘
ð2:9Þ
where n is the number of target atoms or molecules per volume. The relation to the mass stopping force
ª International Commission on Radiation Units and Measurements 2005
Downloaded from http://jicru.oxfordjournals.org/ at University of Alabama at Birmingham on December 3, 2012
2.1.1
doi:10.1093/jicru/ndi009
STOPPING OF IONS HEAVIER THAN HELIUM
reads 1 1dE , S¼ M2 r d‘
governing individual scattering events with a pronounced peak at small energy transfers and a tail at the high-energy end. Skewness decreases with increasing thickness, and eventually the spectrum reaches gaussian shape when (Bohr, 1948)
ð2:10Þ
where M2 is the mass of a target atom (or molecule) if the target consists of only one type of atom or molecule. For polyatomic and polymolecular targets, Eq. (2.10) remains valid if S and M2 are replaced by averages in accordance with the respective atomic or molecular abundances.
where wmax is the maximum energy loss in an individual event. At thicknesses for which the total energy loss amounts to a sizable fraction of the initial energy, the spectra skew again. As the rate of energy loss is governed by the speed, the range of thicknesses within which the gaussian approximation is valid widens for heavy ions because of increasing energy at constant speed. The opposite trend is observed when the projectile is an electron. There the gaussian limit of an energy-loss profile is hardly ever reached.
Energy-loss fluctuation
The energy loss DE at a given pathlength ‘ is a stochastic variable which obeys a statistical distribution F(DE, ‘) depending on ion----target combination, initial energy and travelled path length. The mean energy loss is connected to the stopping force by Z dE hDEi ¼ dð D EÞD E F ð D E; ‘Þ ¼ ‘, ð2:11Þ d‘
2.1.3
The impact parameter in a collision denotes the distance between the incoming (straight-line) trajectory and the target when the latter is initially at rest (Figure 2.1). While this is an inherently classical concept, it retains physical significance in the context of heavy-ion penetration. An impact parameter may refer to a target nucleus or a target electron. Unless stated otherwise, reference will be made here to the target nucleus. This is consistent with the so-called semiclassical picture where trajectories of nuclei participating in a collision are characterized by classical orbits, while electronic motion obeys the laws of quantum mechanics. Beam averages defined in Section 2.1.2 can be interpreted as integrations over the impact plane, i.e., a plane perpendicular to the beam direction with the impact parameter being the radial coordinate. The dependence on the impact parameter of the energy loss must be considered whenever the stopping medium is inhomogeneous or anisotropic or when finite geometric dimensions cannot be ignored. Prominent examples are the stopping of a beam under grazing incidence on a flat surface, or of a beam incident on a single crystal under channeling conditions (cf. Section 3.10). Angular scattering of heavy ions ------ being governed by interactions with
provided that ‘ is small enough so that the variation of dE/d‘ across the pathlength segment can be neglected. The notation h . . . i introduced in Eq. (2.11) denotes an average over a beam, i.e., a large number of trajectories. This includes all relevant parameters characterizing individual projectiles such as energy, charge and excitation state and position in space. The fluctuation in energy loss (‘energy-loss straggling’) after a given path length ‘ is described primarily by the variance D E V2 ¼ ð DE hDEiÞ2 : ð2:12Þ V2 is proportional to ‘ if individual energy-loss events are statistically independent. The straggling parameter W is then defined by W¼
1 dV2 , n d‘
Impact-parameter dependence
ð2:13Þ
and W may be expressed in atomic-scale parameters by X W¼ w2J sJ ð2:14Þ J
Despite the similarity to Eq. (2.8), the range of validity of Eq. (2.14) is more restricted. It does not describe charge-exchange straggling, and since it is based on Poisson statistics, limitations occur especially in crystals and quite generally in dense media (Sigmund, 1978, 1991). The question of the shape of an energy-loss spectrum deserves special attention. For very thin layers the spectrum must reflect the cross sections
p
Figure 2.1. Definition of impact parameter.
16
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2.1.2
ð2:15Þ
V wmax ,
DEFINITIONS target nuclei and the associated process of stopping by energy loss to recoil nuclei is another area for which knowledge of impact-parameter dependencies is vital.
2.2
In addition, the following range parameters (Lindhard et al., 1963b) are of interest (cf. Figure 2.2)
RANGE
The pathlength ‘ specifies the length of a segment of the trajectory measured along the path. The pathlength between two points 1 and 2 is related to the stopping force by 2
d‘ ¼
1
Z
E2 E1
dE ¼ d E=d‘
Z
E1
E2
dE , nSð EÞ
ð2:16Þ
provided that energy-loss fluctuations can be neglected. In the presence of small fluctuations, Eq. (2.16) is an approximate measure of the mean path length with the limits of integration specifying the end points of the path segment. In particular, the range or range along the path or csda (continuous-slowing-down approximation) range is given by R¼
Z
E0
0
dE , nSð EÞ
Range profiles as well as average ranges and variances can be associated with each of the range concepts defined above. These definitions override the approximate relationships (2.17) and (2.18) in the presence of significant straggling and /or angular scattering.
ð2:17Þ 2.3
where E0 is the initial energy (Figure 2.2). Eq. (2.17) approximates the mean range along the path when straggling is small. 2 The variance in range or range straggling VR may be estimated from the formula of Bohr (1948) V2R
¼
Z 0
E0
dE
nW ð EÞ ½nSð EÞ3
Radiation effects such as damage and ionization are characterized primarily by energy deposition profiles. Both one- and three-dimensional profiles are of interest, i.e., the energy deposited per depth or per volume by the ion and by all secondary particles such as recoiling target atoms and secondary electrons. It is necessary to distinguish between different modes of energy deposition (Lindhard et al., 1963a). Unlike range distributions which govern the statistical distribution of a single point, i.e., the end point of the ion trajectory, damage distributions characterize a collision or ionization cascade determined by the fate of a multitude of moving particles. Nevertheless, in the average over a large number of trajectories, damage and ionization profiles are governed by distribution functions analogous to and quantitatively not too different from the corresponding range distributions. Figure 2.3 shows the simplest estimate of an energy deposition profile, found by ignoring straggling and angular scattering. Integration of Eq. (2.9) specifies the projectile energy E ¼ E(‘) as a function of pathlength as the inverse of the relation
ð2:18Þ
,
again in the limit of low straggling.
R⊥
Rp x
R
RADIATION EFFECTS
R
‘ ¼ ‘ð EÞ ¼ Figure 2.2. Range concepts illustrated schematically.
Z
E0 E
17
0
dE : nSðE0 Þ
ð2:19Þ
Downloaded from http://jicru.oxfordjournals.org/ at University of Alabama at Birmingham on December 3, 2012
‘¼
Z
The vector range R, specifying the vector distance from the starting point to the end point of a trajectory, The projected range Rp, representing the component of the vector range in the initial direction of motion, The lateral range R? , representing the component of the vector range perpendicular to the initial direction of motion, and The penetration depth x, representing the component of the vector range along a given direction, e.g. the surface normal of a target with a plane surface. A useful dimensionless parameter Rp /R, the projected-range correction or detour factor, which is always < 1.
STOPPING OF IONS HEAVIER THAN HELIUM Table 2.1. Fundamental constants entering stopping parameters: numerical values (IUPAP 1987) and expressions in three systems of units.
The energy deposited in ionization per pathlength Fioniz(‘) may then be expressed as a function of pathlength by
Gaussian
a. u.
Bohr radius Rydberg energy Bohr velocity
a0
0.052917 nm
4pE0 h 2/me2
2/me2 h
1
R
13.6057 eV
e2/8pE0a0
e2/2a0
0.5
v0
c/137.036
e2/4pE0 h
e2/ h
1
2.5.1
Units
While the use of gaussian units is most common in the theoretical literature on particle stopping, the SI system of units is clearly preferrable from a user’s point of view. A third option, atomic units, is frequently encountered. This section serves to provide pertinent expressions for key parameters in these three systems of units and suitable conversion factors. Table 2.1 specifies expressions for three key parameters, Bohr radius, Rydberg energy and Bohr velocity.
ANGULAR SCATTERING
Angular scattering affects stopping measurements mainly in two ways: 1. Beam particles deflected away from the detecting device may give rise to distortion of the measured energy-loss spectrum and of all averages. 2. Differences between travelled path length and penetration depth through a layer give rise to a detour factor in energy loss and, more seriously, in range.
2.5.2
Examples
The use of different units is illustrated on the Coulomb factor in the standard expression for the electronic stopping cross section of a point charge which, in gaussian units, reads
While the second feature is a key ingredient in standard range theory, consideration of the first one plays a role in all analysis of stopping data. At this point, pertinent notation is introduced. The probability for angular deflection into a solid angle d2f is given by dP ¼ n‘K ðfÞd2 f,
SI
2.5 UNITS, FUNDAMENTAL CONSTANTS AND CONVERSION FACTORS
ð2:20Þ
where Sinoniz symbolizes the contribution of ionization processes to the total stopping cross section (Bragg curve). Similar relations may be written down for other radiation effects. 2.4
Value
total scattering angle a is then described by a distribution F(a, ‘)d2a which typically approaches the single-scattering profile Eq. (2.21) at large angles but takes on a gaussian-like shape around a ¼ 0. Complications arise from the fact that angular deflection and energy loss are correlated (cf. Section 3.14).
Figure 2.3. Beam energy (short-dashed line) and deposited energy (long-dashed line) versus penetration depth in the absence of energy-loss straggling (schematic).
Fioniz ð‘Þ ¼ nSioniz ð Eð‘ÞÞ,
Symbol
S¼
4pZ21 Z2 e4 L, mv2
ð2:22Þ
where L denotes the dimensionless stopping number (which would read L ¼ ln(2mv2/I) in case of the Bethe formula) and Z1 and Z2 denote the atomic numbers of projectile and target, respectively. The same relation in SI units reads
ð2:21Þ
for sufficiently small travelled pathlength ‘, where K(f) is the differential scattering cross section and f the deflection angle in the laboratory frame of reference. With dP increasing, multiple angular deflections become increasingly important. The distribution in
S¼
Z21 Z2 e4 L 4pE20 mv2
ð2:23Þ
but is rarely if ever encountered in the literature. In terms of fundamental constants (IUPAP, 18
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Depth
Name
DEFINITIONS 1987; NIST, 2001) listed in Table 2.1 the same relation reads v 2 0 S ¼ 4pZ21 Z2 L2Ra20 : ð2:24Þ v
and to the stopping cross section as dE , S in 1015 eVcm2 , S ¼ 1:6605A2 rd‘ d E=rd‘ in MeVcm2 =mg:
In atomic units this reduces to L S ¼ 4pZ21 Z2 2 : v
The stopping cross section relates to the stopping number as
ð2:25Þ
S ¼ 5:0991 · 104
The atomic unit of S is equivalent to5 27.2 eV · (0.0529 nm)2.
S in 1015 eVcm2 ,
ð2:30Þ
Conversion and the stopping force in atomic units is related to the mass stopping force as
In practical applications the kinetic energy per nucleon or per atomic mass unit E/A1, or specific energy is more convenient than the velocity variable.6 Mass stopping forces [Eq. (2.7)] will be reported in units of MeV cm2/mg and stopping cross sections in units of eVcm2 per atom or molecule. Pertinent relations for the most frequently occurring conversions are listed here. The specific energy is related to the projectile speed by ! E 1 ¼ uc2 ðg 1Þ ¼ 931:49 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 , A1 1 v2 =c2
dE dE ¼ 0:36749r , d‘ rd‘ r in g=cm3 , d E=rd‘ in MeV cm2 =mg:
dE dE ¼ 100r , d‘ rd‘
dE ¼ 3:0705 · rd‘
E A1
ð2:27Þ
2.5.4
¼ 0:02480 MeV:
ð2:33Þ
v ¼ v0
Notation
In the stopping literature, the symbol S may be found to denote both stopping cross section, stopping force and mass stopping force, and the symbol dE/dx may be found to denote either stopping force or mass stopping force.
L,
d E=rd‘ in MeVcm =mg,
ð2:32Þ
in atomic and modified SI units, respectively. Finally, the specific energy of an ion at the Bohr velocity v0 is given by
The mass stopping force relates to the stopping number as Z2 Z2 104 1 2 A2 b 2
d E=d‘ in keV=mm,
r in g=cm3 , d E=rd‘ in MeVcm2 =mg
where u is the atomic mass unit 1.6605 · 1027 kg. The inverse relation reads 1 1 þ E=2M1 c2 M1 v2 ¼ E : 2 2 ð1 þ E=M1 c2 Þ
ð2:31Þ
Alternatively,
ð2:26Þ
E=A1 in MeV,
Z21 Z2 L, b2
ð2:28Þ
5
Caution is indicated with respect to energy unit which is set to 2R here. This is consistent with the common practice of setting h ¼ m ¼ e ¼ 1. However, also the straight R may be encountered as the energy unit, and this is not always noted explicitly in the pertinent literature. 6 In general, no distinction will be made between energy per nucleon and energy per atomic mass unit because the numerical difference amounts to at most 0.25 % for stable isotopes of all elements from lithium upward.
19
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2.5.3
ð2:29Þ
2
DEFINITIONS
2.1
STOPPING Mean energy loss
The central quantity characterizing particle stopping is the stopping force or stopping power.2 For a point particle it is defined as the average loss of kinetic energy3 E per path length ‘, dE/d‘. The minus sign defines the stopping force as a positive quantity. The stopping force is related to the average change in momentum per path length according to dE dP ¼v , d‘ d‘
ð2:1Þ
where v is the projectile speed, P ¼ M1gn the momentum, M1 the projectile mass, 1 g ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 1 b2 b¼
v c
ð2:2Þ
Equation (2.4) in principle also applies to dressed atomic ions, i.e., ions carrying electrons for which the definition of the stopping force becomes ambiguous due to projectile excitation and charge exchange. However, currently accessible experimental and theoretical accuracy is below the level where terms of order m/M1 (m ¼ electron rest mass) would be significant. Therefore, the stopping force on a dressed ion may safely be related to the momentum change of the projectile nucleus, dE dP ¼ v : ð2:5Þ d‘ d‘ nucleus The definitions (2.4) and (2.5) allow the inclusion of contributions from all significant energy-loss channels. The stopping force is related to the velocity change in time ------ a quantity measured by some techniques ------ through dE dP dv ¼ ¼ M 1 g3 : d‘ dt dt
ð2:3Þ
and c the speed of light.4 For composite projectiles such as molecules and clusters for which energy can be transferred into internal degrees of freedom, the stopping force is defined by generalizing Eq. (2.1): dE X dP ¼ vi , ð2:4Þ d‘ d‘ i i where the sum extends over all constituents of the projectile.
ð2:6Þ
It is common practice to tabulate mass stopping forces
1dE , r d‘
ð2:7Þ
where r is the mass density of the target. A parameter of fundamental significance on the atomic or molecular scale is the stopping cross section S, defined by X S¼ wJ s J , ð2:8Þ J
2
The two terms will be used synonymously. While ‘stopping power’ is the offical nomenclature, ‘stopping force’ is more precise (Sigmund, 2000b). 3 Use of the symbol E for kinetic energy is just the first of a series of cases where notation in this report deviates from ICRU Reports 37 and 49. 4 The symbol g has at least four well-established functions in the context of heavy-ion penetration. In addition to Eq. (2.2), it characterizes the maximum energy transfer in an elastic binary collision, the effective-charge fraction of a dressed ion and Euler’s constant. Established notation will be maintained here, but proper specification will be added whenever ambiguities could arise.
where the sum extends over all energy-loss channels and where wJ and sJ denote the energy loss and pertinent cross section per target atom or molecule for the J’th channel. The stopping cross section is related to the stopping force through S¼
1dE , n d‘
ð2:9Þ
where n is the number of target atoms or molecules per volume. The relation to the mass stopping force
ª International Commission on Radiation Units and Measurements 2005
Downloaded from http://jicru.oxfordjournals.org/ at University of Alabama at Birmingham on December 3, 2012
2.1.1
doi:10.1093/jicru/ndi009
STOPPING OF IONS HEAVIER THAN HELIUM
reads 1 1dE , S¼ M2 r d‘
governing individual scattering events with a pronounced peak at small energy transfers and a tail at the high-energy end. Skewness decreases with increasing thickness, and eventually the spectrum reaches gaussian shape when (Bohr, 1948)
ð2:10Þ
where M2 is the mass of a target atom (or molecule) if the target consists of only one type of atom or molecule. For polyatomic and polymolecular targets, Eq. (2.10) remains valid if S and M2 are replaced by averages in accordance with the respective atomic or molecular abundances.
where wmax is the maximum energy loss in an individual event. At thicknesses for which the total energy loss amounts to a sizable fraction of the initial energy, the spectra skew again. As the rate of energy loss is governed by the speed, the range of thicknesses within which the gaussian approximation is valid widens for heavy ions because of increasing energy at constant speed. The opposite trend is observed when the projectile is an electron. There the gaussian limit of an energy-loss profile is hardly ever reached.
Energy-loss fluctuation
The energy loss DE at a given pathlength ‘ is a stochastic variable which obeys a statistical distribution F(DE, ‘) depending on ion----target combination, initial energy and travelled path length. The mean energy loss is connected to the stopping force by Z dE hDEi ¼ dð D EÞD E F ð D E; ‘Þ ¼ ‘, ð2:11Þ d‘
2.1.3
The impact parameter in a collision denotes the distance between the incoming (straight-line) trajectory and the target when the latter is initially at rest (Figure 2.1). While this is an inherently classical concept, it retains physical significance in the context of heavy-ion penetration. An impact parameter may refer to a target nucleus or a target electron. Unless stated otherwise, reference will be made here to the target nucleus. This is consistent with the so-called semiclassical picture where trajectories of nuclei participating in a collision are characterized by classical orbits, while electronic motion obeys the laws of quantum mechanics. Beam averages defined in Section 2.1.2 can be interpreted as integrations over the impact plane, i.e., a plane perpendicular to the beam direction with the impact parameter being the radial coordinate. The dependence on the impact parameter of the energy loss must be considered whenever the stopping medium is inhomogeneous or anisotropic or when finite geometric dimensions cannot be ignored. Prominent examples are the stopping of a beam under grazing incidence on a flat surface, or of a beam incident on a single crystal under channeling conditions (cf. Section 3.10). Angular scattering of heavy ions ------ being governed by interactions with
provided that ‘ is small enough so that the variation of dE/d‘ across the pathlength segment can be neglected. The notation h . . . i introduced in Eq. (2.11) denotes an average over a beam, i.e., a large number of trajectories. This includes all relevant parameters characterizing individual projectiles such as energy, charge and excitation state and position in space. The fluctuation in energy loss (‘energy-loss straggling’) after a given path length ‘ is described primarily by the variance D E V2 ¼ ð DE hDEiÞ2 : ð2:12Þ V2 is proportional to ‘ if individual energy-loss events are statistically independent. The straggling parameter W is then defined by W¼
1 dV2 , n d‘
Impact-parameter dependence
ð2:13Þ
and W may be expressed in atomic-scale parameters by X W¼ w2J sJ ð2:14Þ J
Despite the similarity to Eq. (2.8), the range of validity of Eq. (2.14) is more restricted. It does not describe charge-exchange straggling, and since it is based on Poisson statistics, limitations occur especially in crystals and quite generally in dense media (Sigmund, 1978, 1991). The question of the shape of an energy-loss spectrum deserves special attention. For very thin layers the spectrum must reflect the cross sections
p
Figure 2.1. Definition of impact parameter.
16
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2.1.2
ð2:15Þ
V wmax ,
DEFINITIONS target nuclei and the associated process of stopping by energy loss to recoil nuclei is another area for which knowledge of impact-parameter dependencies is vital.
2.2
In addition, the following range parameters (Lindhard et al., 1963b) are of interest (cf. Figure 2.2)
RANGE
The pathlength ‘ specifies the length of a segment of the trajectory measured along the path. The pathlength between two points 1 and 2 is related to the stopping force by 2
d‘ ¼
1
Z
E2 E1
dE ¼ d E=d‘
Z
E1
E2
dE , nSð EÞ
ð2:16Þ
provided that energy-loss fluctuations can be neglected. In the presence of small fluctuations, Eq. (2.16) is an approximate measure of the mean path length with the limits of integration specifying the end points of the path segment. In particular, the range or range along the path or csda (continuous-slowing-down approximation) range is given by R¼
Z
E0
0
dE , nSð EÞ
Range profiles as well as average ranges and variances can be associated with each of the range concepts defined above. These definitions override the approximate relationships (2.17) and (2.18) in the presence of significant straggling and /or angular scattering.
ð2:17Þ 2.3
where E0 is the initial energy (Figure 2.2). Eq. (2.17) approximates the mean range along the path when straggling is small. 2 The variance in range or range straggling VR may be estimated from the formula of Bohr (1948) V2R
¼
Z 0
E0
dE
nW ð EÞ ½nSð EÞ3
Radiation effects such as damage and ionization are characterized primarily by energy deposition profiles. Both one- and three-dimensional profiles are of interest, i.e., the energy deposited per depth or per volume by the ion and by all secondary particles such as recoiling target atoms and secondary electrons. It is necessary to distinguish between different modes of energy deposition (Lindhard et al., 1963a). Unlike range distributions which govern the statistical distribution of a single point, i.e., the end point of the ion trajectory, damage distributions characterize a collision or ionization cascade determined by the fate of a multitude of moving particles. Nevertheless, in the average over a large number of trajectories, damage and ionization profiles are governed by distribution functions analogous to and quantitatively not too different from the corresponding range distributions. Figure 2.3 shows the simplest estimate of an energy deposition profile, found by ignoring straggling and angular scattering. Integration of Eq. (2.9) specifies the projectile energy E ¼ E(‘) as a function of pathlength as the inverse of the relation
ð2:18Þ
,
again in the limit of low straggling.
R⊥
Rp x
R
RADIATION EFFECTS
R
‘ ¼ ‘ð EÞ ¼ Figure 2.2. Range concepts illustrated schematically.
Z
E0 E
17
0
dE : nSðE0 Þ
ð2:19Þ
Downloaded from http://jicru.oxfordjournals.org/ at University of Alabama at Birmingham on December 3, 2012
‘¼
Z
The vector range R, specifying the vector distance from the starting point to the end point of a trajectory, The projected range Rp, representing the component of the vector range in the initial direction of motion, The lateral range R? , representing the component of the vector range perpendicular to the initial direction of motion, and The penetration depth x, representing the component of the vector range along a given direction, e.g. the surface normal of a target with a plane surface. A useful dimensionless parameter Rp /R, the projected-range correction or detour factor, which is always < 1.
STOPPING OF IONS HEAVIER THAN HELIUM Table 2.1. Fundamental constants entering stopping parameters: numerical values (IUPAP 1987) and expressions in three systems of units.
The energy deposited in ionization per pathlength Fioniz(‘) may then be expressed as a function of pathlength by
Gaussian
a. u.
Bohr radius Rydberg energy Bohr velocity
a0
0.052917 nm
4pE0 h 2/me2
2/me2 h
1
R
13.6057 eV
e2/8pE0a0
e2/2a0
0.5
v0
c/137.036
e2/4pE0 h
e2/ h
1
2.5.1
Units
While the use of gaussian units is most common in the theoretical literature on particle stopping, the SI system of units is clearly preferrable from a user’s point of view. A third option, atomic units, is frequently encountered. This section serves to provide pertinent expressions for key parameters in these three systems of units and suitable conversion factors. Table 2.1 specifies expressions for three key parameters, Bohr radius, Rydberg energy and Bohr velocity.
ANGULAR SCATTERING
Angular scattering affects stopping measurements mainly in two ways: 1. Beam particles deflected away from the detecting device may give rise to distortion of the measured energy-loss spectrum and of all averages. 2. Differences between travelled path length and penetration depth through a layer give rise to a detour factor in energy loss and, more seriously, in range.
2.5.2
Examples
The use of different units is illustrated on the Coulomb factor in the standard expression for the electronic stopping cross section of a point charge which, in gaussian units, reads
While the second feature is a key ingredient in standard range theory, consideration of the first one plays a role in all analysis of stopping data. At this point, pertinent notation is introduced. The probability for angular deflection into a solid angle d2f is given by dP ¼ n‘K ðfÞd2 f,
SI
2.5 UNITS, FUNDAMENTAL CONSTANTS AND CONVERSION FACTORS
ð2:20Þ
where Sinoniz symbolizes the contribution of ionization processes to the total stopping cross section (Bragg curve). Similar relations may be written down for other radiation effects. 2.4
Value
total scattering angle a is then described by a distribution F(a, ‘)d2a which typically approaches the single-scattering profile Eq. (2.21) at large angles but takes on a gaussian-like shape around a ¼ 0. Complications arise from the fact that angular deflection and energy loss are correlated (cf. Section 3.14).
Figure 2.3. Beam energy (short-dashed line) and deposited energy (long-dashed line) versus penetration depth in the absence of energy-loss straggling (schematic).
Fioniz ð‘Þ ¼ nSioniz ð Eð‘ÞÞ,
Symbol
S¼
4pZ21 Z2 e4 L, mv2
ð2:22Þ
where L denotes the dimensionless stopping number (which would read L ¼ ln(2mv2/I) in case of the Bethe formula) and Z1 and Z2 denote the atomic numbers of projectile and target, respectively. The same relation in SI units reads
ð2:21Þ
for sufficiently small travelled pathlength ‘, where K(f) is the differential scattering cross section and f the deflection angle in the laboratory frame of reference. With dP increasing, multiple angular deflections become increasingly important. The distribution in
S¼
Z21 Z2 e4 L 4pE20 mv2
ð2:23Þ
but is rarely if ever encountered in the literature. In terms of fundamental constants (IUPAP, 18
Downloaded from http://jicru.oxfordjournals.org/ at University of Alabama at Birmingham on December 3, 2012
Depth
Name
DEFINITIONS 1987; NIST, 2001) listed in Table 2.1 the same relation reads v 2 0 S ¼ 4pZ21 Z2 L2Ra20 : ð2:24Þ v
and to the stopping cross section as dE , S in 1015 eVcm2 , S ¼ 1:6605A2 rd‘ d E=rd‘ in MeVcm2 =mg:
In atomic units this reduces to L S ¼ 4pZ21 Z2 2 : v
The stopping cross section relates to the stopping number as
ð2:25Þ
S ¼ 5:0991 · 104
The atomic unit of S is equivalent to5 27.2 eV · (0.0529 nm)2.
S in 1015 eVcm2 ,
ð2:30Þ
Conversion and the stopping force in atomic units is related to the mass stopping force as
In practical applications the kinetic energy per nucleon or per atomic mass unit E/A1, or specific energy is more convenient than the velocity variable.6 Mass stopping forces [Eq. (2.7)] will be reported in units of MeV cm2/mg and stopping cross sections in units of eVcm2 per atom or molecule. Pertinent relations for the most frequently occurring conversions are listed here. The specific energy is related to the projectile speed by ! E 1 ¼ uc2 ðg 1Þ ¼ 931:49 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 , A1 1 v2 =c2
dE dE ¼ 0:36749r , d‘ rd‘ r in g=cm3 , d E=rd‘ in MeV cm2 =mg:
dE dE ¼ 100r , d‘ rd‘
dE ¼ 3:0705 · rd‘
E A1
ð2:27Þ
2.5.4
¼ 0:02480 MeV:
ð2:33Þ
v ¼ v0
Notation
In the stopping literature, the symbol S may be found to denote both stopping cross section, stopping force and mass stopping force, and the symbol dE/dx may be found to denote either stopping force or mass stopping force.
L,
d E=rd‘ in MeVcm =mg,
ð2:32Þ
in atomic and modified SI units, respectively. Finally, the specific energy of an ion at the Bohr velocity v0 is given by
The mass stopping force relates to the stopping number as Z2 Z2 104 1 2 A2 b 2
d E=d‘ in keV=mm,
r in g=cm3 , d E=rd‘ in MeVcm2 =mg
where u is the atomic mass unit 1.6605 · 1027 kg. The inverse relation reads 1 1 þ E=2M1 c2 M1 v2 ¼ E : 2 2 ð1 þ E=M1 c2 Þ
ð2:31Þ
Alternatively,
ð2:26Þ
E=A1 in MeV,
Z21 Z2 L, b2
ð2:28Þ
5
Caution is indicated with respect to energy unit which is set to 2R here. This is consistent with the common practice of setting h ¼ m ¼ e ¼ 1. However, also the straight R may be encountered as the energy unit, and this is not always noted explicitly in the pertinent literature. 6 In general, no distinction will be made between energy per nucleon and energy per atomic mass unit because the numerical difference amounts to at most 0.25 % for stable isotopes of all elements from lithium upward.
19
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2.5.3
ð2:29Þ
