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Scaled representation of mean energy losses of heavy charged particles in matter
Volume 121, number 6
PHYSICS LETTERS A
4 May 1987
SCALED REPRESENTATION OF MEAN ENERGY LOSSES OF HEAVY CHARGED PARTICLES IN MAUER S.A. GERASIMOV and V.M. KOROL Department ofNuclear Physics, Institute ofPhysics, Rostov State University, Rostov-on-Don 344006, USSR Received 2 January 1987; accepted for publication 26 February 1987
A scaling law for a simple description of the mean energy of transmitted heavy charged particles in matter as a function oftarget thickness and atomic number is presented. The method of scaling is obtained using the Thomas—Fermi transformation of variables.
Our aim here is to establish a scaling law for the description ofthe mean energy oftransmitted charged particles as a function of target thickness and atomic number. Such an analytic relation is useful in the study of thin films and surfaces. Only a few appropriate measurements on transmission of protons and a-particles in various targets have been reported in the literature [1—3].The thickness dependence of the mean energy loss is unknown for most elements and compounds. The Bethe formula is valid only for a limited range of energy losses. The use of the Bethe formula to describe the mean energy loss in thick films is useless because of a poor agreement between theory and experiment at large depth in the target. This is demonstrated in fig. 1. Here the energy T of transmitted heavy charged particles as a function of target thicknessx, the initial energy T0 and the atomic number Z is the solution ofthe equation r K dK
J i~k u,
(1)
=
x
where
2 /Zm
IC~~ = 2T0h
4,
(2)
1e
u = 8~t Zfpxh 4/AmA m
4 Z, (3) 1 me and K= 2Th 2/Zm 4, p is the density of mass, A ~ the atomic weight,1ern is the mass of an electron, m is the mass ofthe projectile, mA = 1.66 x 1 O_24 g, and1 Z 1e is the projectile charge.
Any corrections which extend the Bethe formula
1.0
I
0.8
o~
a
0.2
0 0
0.1
0.2 U
0.3
OA
Fig. 1. Thickness dependence of the mean energy of transmitted protons. The initial energy ofprotons is 1.55 MeV. The solid line is given by the Bethe—Bloch formula. Experimental points (squares) from ref. [2].
have been applied but unfortunately such treatments require excessive labor in this field. The present work is an attempt to establish a semi-empirical relation based on the principle of scaling. One starting point ofsuch a consideration, a useful one here, is the scaling property ofthe Thomas—Fermi theory [4], which means that any physical relation describing a statistical atom contains the atomic number Z only in the forms 1”3, pZ —2/3,
EZ~4”3,
(4)
rZ
where r is the radial coordinate, p is the momentum ofan atomic electron, and E is the electron energy. 293
Volume 121, number 6
PHYSICS LETTERS A
The differential cross section da/dAE for the exchange of the energy AE by an incident charged particle to an atom must be transformed as Z3dg/db.E. It is connected with the fact that the melastic scattering cross section da may be treated as an area (—~r2Z213) and in the high-energy formulation must be proportional to the number of atomic electrons Z. In this case the transformation “energy—’energy x Z canbe applied for both the energy transfer iS.E and the energy ofthe projectile T. - ‘~“
4 May 1987
only. Relations of the Thomson—Whiddington type [5] are special cases of this assumption. It is a wellknown factthat the range—energy dependence can be described by a semi-empirical relation of the power type. Therefore, for a given atom: TIT
7
—
0
(
0
where a is a parameter which must be defined, and ~, is some function of x/Tg. Ifwe now apply the principle of scaling we should replace T 0 by (5) and x by (6). Any other parameters must be considered as constants. Thus, for any atom
On the other hand, since the probability ofthe energy exchange is proportional to dc and the number of atoms per unit lengthof the specimen, one can condude that a macroscopic scale x (range of charged particle in matter, thickness of target,5’3. depth in mateIt should be rial) mustbe transformed into xp/AZ borne in mind that the probabilityis invariant in this principle of the scaling. Before going on, it should be noted that processes of stopping of charged particles are defined by the magnitude ofthe velocity and the inelastic scatteringcross section is proportional to the square ofthe projectile charge Z 1. Thus, mT/m 413 (5) 1Z is the universal variable for the charged particle energy and
instead of (5) and (6), respectively. In these notations the scaled argument of the dependence (7) is
Z~pmx/m
h/t
5~’3
(6)
1AZ is the one for a macroscopic coordinate. This principle of scaling means that any result obtained for the given atom has a universal character so that the conversion to another atomof canvariables. be achievedOf by the simple transformations course, possibilities of this scaling are limited because the Thomas—Fermi theory is valid only for strongly bound atomic electrons playing the dominant role in an inelastic scattering with large AE. Therefore, this principle of scaling is a high-energy formulation. A simple and good way to estimate the mean energy loss of the charged particle in matter is to describe the relative energy change T/T 0 in a scatterer. For a given target whose atomic number is Z, the relative energy change depends on the initial energy T0 and the thickness of the scatterer x. Evidently, for any Z, T/T0— 1 as x—~.0and T/T0—~.0as x-+R, where R is the maximum range of the charged particle in matter. In order to satisfy these conditions one may suppose that the relative energy change depends on x/R 294
2
~
(4a_ 5)13
Z1 m1 a~ a (8) AT0argument ofthe relationship (7). Let is the m universal us define dimensionless variables 2/3
/
2
t = 1 (3E ~
2
—
\.
~l T 4 ) e~’m 413 1Z 713 Z~pxh4
2 (3it’\ 3 ~ 4)
9
AmAmIm?Z513’
0°~.
( )
(10)
Looking back at the scaling one may remark that the relation (7) must contain the atomic number only in the following 3 ~ = 1 2form 3 (11)
z~’
‘
‘
‘
Finally, since x—~0 as Z—4 a n/4 n> 5 —
~,
then a> 5/4. Hence (12)
It should be noted here that the conditions (11) and (12) are approximate because the range—energy relation is described semi-empirically. It is clear that the atomic number dependence of the relative energy change cannot be sharp. Therefore, a test for the first few n is sufficient. As an illustration, fig. 2 shows the universal thickness dependence of T/T0 calculated for different mitial energies T0. Experimental data have been taken from refs. [1] and [2]. Calculations were made for all n in the range 5 ~ n ~ 10. From the point of view ofthe scaling, appropriate results were obtained only
Volume 121, number 6
PHYSICS LETTERS A
4 May 1987 I
1.0
1.0
~ 0.5
0.5
0
0
0.1
Q
0.2
h/t~’3
Fig. 2. Universal thickness dependence of the relative energy change of a-particles in aluminum (solid circles) and protons in aluminum (open circles), copper (triangles) and gold (squares). Experimental points for a-particles from ref. [1], others from ref. [2]. The initial energies ofprotons and a-particles arelarger than 1 MeV. The solid curve is the dependence (13) for a= 4.82 and b=2/3.
for a= 7/4. Deviations from the scaling at all other a are large. A more exact value of a is 20/12 (n~21 /12). The results presented in fig. 2 correspond to this case. Thus, it turns out that this scaling procedure is useful for a universal description of the mean energy loss T 0 T as a function of the atomic number and other parameters. Approximately, this scaled dependence can be expressed by —
t/t0 = (1 — ah/t~)”, h ~ ta/a.
(13)
One can conclude that ab= 1 if the independence condition of dT/dx on T0 is used here. The dashed line in fig. 2 corresponds to this case when a = 5 and b= 3/5. However, this condition is essential for small x only. It is connected with effects of multiple scattering at large x in a material. Therefore one can propose the more exact dependence (13) presented in fig. 2 by the solid line. Now, a = 4.82 and b = 2/3. Thus, we have written a new scaling law for the description ofthe mean energy losses of charged partides in matter. This kind of scaling gives satisfactory results in a wide region of target thickness, atomic numbers and initial energies of heavy charged particles. Indeed, there is the Lindhard—Scharff scaling [6,7]. Practically, it is a scaling of the Bethe formula. One can see from (2) and (3) that in the case ofthe Lindhard—Scharffscaling mT/m 1Z is the scaled variable of energy T and Z~mpx/m1 AZ is one of
I
0
0.1
I
I
0.2
u/Kg’30.3
0.~
Fig. 3. Thickness dependence of the relative energy change in the Lindhard—Scharff representation of variables. The experimental data are the same as in fig. 2. The full, dashed and dash-dotted curves are given by the formula (13) with a = 4.82 and b= 2/3 for aluminum, copper and gold, respectively.
macroscopic size. Both methods of scaling are identical in the description of mean energy losses if a = 2. As was mentioned above, the deviations from the scaling are large in this case. Nevertheless, let us apply the Lindhard—Scharff scaling to the description of mean energy losses of heavy charged particles in matter. We should replace T0 by mT0/m1Z and x by Z~mpx/m 1AZ in eq. (7) if the scaling procedure is used here. Taking into account that Z1 and m1 dependences of both transformations ofvariables are identical and considering the relation (7) for protons and a-particles one can conclude from fig. 2 that a=5/3. Therefore the relationship 3)
K/K0 u/K~’ must =f( be universal in the Lindhard—Scharff approach of the scaling. Fig. 3 demonstrates large deviations from the scaling in this case. Basic distinctions between the two methods of scaling must be mentioned in conclusion. From the mathematical point of view the Thomas—Fermi scaling principle can be obtained using the Bethe—Bloch theory with ln y replaced by y ‘~. This approximation is valid within the range 10 ~ y~ 100. From the physical point ofview the range of validity of the Thomas—Fermi scaling is wider since the close collision contribution to inelastic energy losses is taken into account. The collective distant collision 295
Volume 121, number 6
PHYSICS LETTERS A
contribution is written approximately Thomas—Fermi scaling procedure.
in the
References [1] D.A. Sykes andSj. Harris, NucI. Ins~r.rn.Methods 94 (1971) 39. [2] M.D. Bedri, SJ. Harris and H.J. Parish, Rad. Eff. 27 (1976) 183.
296
4 May 1987
[3] P.D. Bourland, W.K. Chu and D. Powers, Phys. Rev. B 3 (1971) 3625. [4] D.A. Kirzhnits, Yu.E. Lozovik and G.V. Shpatakovskaya, Usp. Fiz. Nauk 117 (1975) 3. [5] H. Bichel and C. Tschalaer, NucI. Data Tables A 3 (1967) 343. [61 J. Lindhard and M. Scbarfl~K. Dan. Vidensk. Seisk. Mat. Fys. Medd. 27 (1953) No. 15. [7] T.E. Everhart and P.H. Hoff, J. Appi. Phys. 42 (1971) 5837.
PHYSICS LETTERS A
4 May 1987
SCALED REPRESENTATION OF MEAN ENERGY LOSSES OF HEAVY CHARGED PARTICLES IN MAUER S.A. GERASIMOV and V.M. KOROL Department ofNuclear Physics, Institute ofPhysics, Rostov State University, Rostov-on-Don 344006, USSR Received 2 January 1987; accepted for publication 26 February 1987
A scaling law for a simple description of the mean energy of transmitted heavy charged particles in matter as a function oftarget thickness and atomic number is presented. The method of scaling is obtained using the Thomas—Fermi transformation of variables.
Our aim here is to establish a scaling law for the description ofthe mean energy oftransmitted charged particles as a function of target thickness and atomic number. Such an analytic relation is useful in the study of thin films and surfaces. Only a few appropriate measurements on transmission of protons and a-particles in various targets have been reported in the literature [1—3].The thickness dependence of the mean energy loss is unknown for most elements and compounds. The Bethe formula is valid only for a limited range of energy losses. The use of the Bethe formula to describe the mean energy loss in thick films is useless because of a poor agreement between theory and experiment at large depth in the target. This is demonstrated in fig. 1. Here the energy T of transmitted heavy charged particles as a function of target thicknessx, the initial energy T0 and the atomic number Z is the solution ofthe equation r K dK
J i~k u,
(1)
=
x
where
2 /Zm
IC~~ = 2T0h
4,
(2)
1e
u = 8~t Zfpxh 4/AmA m
4 Z, (3) 1 me and K= 2Th 2/Zm 4, p is the density of mass, A ~ the atomic weight,1ern is the mass of an electron, m is the mass ofthe projectile, mA = 1.66 x 1 O_24 g, and1 Z 1e is the projectile charge.
Any corrections which extend the Bethe formula
1.0
I
0.8
o~
a
0.2
0 0
0.1
0.2 U
0.3
OA
Fig. 1. Thickness dependence of the mean energy of transmitted protons. The initial energy ofprotons is 1.55 MeV. The solid line is given by the Bethe—Bloch formula. Experimental points (squares) from ref. [2].
have been applied but unfortunately such treatments require excessive labor in this field. The present work is an attempt to establish a semi-empirical relation based on the principle of scaling. One starting point ofsuch a consideration, a useful one here, is the scaling property ofthe Thomas—Fermi theory [4], which means that any physical relation describing a statistical atom contains the atomic number Z only in the forms 1”3, pZ —2/3,
EZ~4”3,
(4)
rZ
where r is the radial coordinate, p is the momentum ofan atomic electron, and E is the electron energy. 293
Volume 121, number 6
PHYSICS LETTERS A
The differential cross section da/dAE for the exchange of the energy AE by an incident charged particle to an atom must be transformed as Z3dg/db.E. It is connected with the fact that the melastic scattering cross section da may be treated as an area (—~r2Z213) and in the high-energy formulation must be proportional to the number of atomic electrons Z. In this case the transformation “energy—’energy x Z canbe applied for both the energy transfer iS.E and the energy ofthe projectile T. - ‘~“
4 May 1987
only. Relations of the Thomson—Whiddington type [5] are special cases of this assumption. It is a wellknown factthat the range—energy dependence can be described by a semi-empirical relation of the power type. Therefore, for a given atom: TIT
7
—
0
(
0
where a is a parameter which must be defined, and ~, is some function of x/Tg. Ifwe now apply the principle of scaling we should replace T 0 by (5) and x by (6). Any other parameters must be considered as constants. Thus, for any atom
On the other hand, since the probability ofthe energy exchange is proportional to dc and the number of atoms per unit lengthof the specimen, one can condude that a macroscopic scale x (range of charged particle in matter, thickness of target,5’3. depth in mateIt should be rial) mustbe transformed into xp/AZ borne in mind that the probabilityis invariant in this principle of the scaling. Before going on, it should be noted that processes of stopping of charged particles are defined by the magnitude ofthe velocity and the inelastic scatteringcross section is proportional to the square ofthe projectile charge Z 1. Thus, mT/m 413 (5) 1Z is the universal variable for the charged particle energy and
instead of (5) and (6), respectively. In these notations the scaled argument of the dependence (7) is
Z~pmx/m
h/t
5~’3
(6)
1AZ is the one for a macroscopic coordinate. This principle of scaling means that any result obtained for the given atom has a universal character so that the conversion to another atomof canvariables. be achievedOf by the simple transformations course, possibilities of this scaling are limited because the Thomas—Fermi theory is valid only for strongly bound atomic electrons playing the dominant role in an inelastic scattering with large AE. Therefore, this principle of scaling is a high-energy formulation. A simple and good way to estimate the mean energy loss of the charged particle in matter is to describe the relative energy change T/T 0 in a scatterer. For a given target whose atomic number is Z, the relative energy change depends on the initial energy T0 and the thickness of the scatterer x. Evidently, for any Z, T/T0— 1 as x—~.0and T/T0—~.0as x-+R, where R is the maximum range of the charged particle in matter. In order to satisfy these conditions one may suppose that the relative energy change depends on x/R 294
2
~
(4a_ 5)13
Z1 m1 a~ a (8) AT0argument ofthe relationship (7). Let is the m universal us define dimensionless variables 2/3
/
2
t = 1 (3E ~
2
—
\.
~l T 4 ) e~’m 413 1Z 713 Z~pxh4
2 (3it’\ 3 ~ 4)
9
AmAmIm?Z513’
0°~.
( )
(10)
Looking back at the scaling one may remark that the relation (7) must contain the atomic number only in the following 3 ~ = 1 2form 3 (11)
z~’
‘
‘
‘
Finally, since x—~0 as Z—4 a n/4 n> 5 —
~,
then a> 5/4. Hence (12)
It should be noted here that the conditions (11) and (12) are approximate because the range—energy relation is described semi-empirically. It is clear that the atomic number dependence of the relative energy change cannot be sharp. Therefore, a test for the first few n is sufficient. As an illustration, fig. 2 shows the universal thickness dependence of T/T0 calculated for different mitial energies T0. Experimental data have been taken from refs. [1] and [2]. Calculations were made for all n in the range 5 ~ n ~ 10. From the point of view ofthe scaling, appropriate results were obtained only
Volume 121, number 6
PHYSICS LETTERS A
4 May 1987 I
1.0
1.0
~ 0.5
0.5
0
0
0.1
Q
0.2
h/t~’3
Fig. 2. Universal thickness dependence of the relative energy change of a-particles in aluminum (solid circles) and protons in aluminum (open circles), copper (triangles) and gold (squares). Experimental points for a-particles from ref. [1], others from ref. [2]. The initial energies ofprotons and a-particles arelarger than 1 MeV. The solid curve is the dependence (13) for a= 4.82 and b=2/3.
for a= 7/4. Deviations from the scaling at all other a are large. A more exact value of a is 20/12 (n~21 /12). The results presented in fig. 2 correspond to this case. Thus, it turns out that this scaling procedure is useful for a universal description of the mean energy loss T 0 T as a function of the atomic number and other parameters. Approximately, this scaled dependence can be expressed by —
t/t0 = (1 — ah/t~)”, h ~ ta/a.
(13)
One can conclude that ab= 1 if the independence condition of dT/dx on T0 is used here. The dashed line in fig. 2 corresponds to this case when a = 5 and b= 3/5. However, this condition is essential for small x only. It is connected with effects of multiple scattering at large x in a material. Therefore one can propose the more exact dependence (13) presented in fig. 2 by the solid line. Now, a = 4.82 and b = 2/3. Thus, we have written a new scaling law for the description ofthe mean energy losses of charged partides in matter. This kind of scaling gives satisfactory results in a wide region of target thickness, atomic numbers and initial energies of heavy charged particles. Indeed, there is the Lindhard—Scharff scaling [6,7]. Practically, it is a scaling of the Bethe formula. One can see from (2) and (3) that in the case ofthe Lindhard—Scharffscaling mT/m 1Z is the scaled variable of energy T and Z~mpx/m1 AZ is one of
I
0
0.1
I
I
0.2
u/Kg’30.3
0.~
Fig. 3. Thickness dependence of the relative energy change in the Lindhard—Scharff representation of variables. The experimental data are the same as in fig. 2. The full, dashed and dash-dotted curves are given by the formula (13) with a = 4.82 and b= 2/3 for aluminum, copper and gold, respectively.
macroscopic size. Both methods of scaling are identical in the description of mean energy losses if a = 2. As was mentioned above, the deviations from the scaling are large in this case. Nevertheless, let us apply the Lindhard—Scharff scaling to the description of mean energy losses of heavy charged particles in matter. We should replace T0 by mT0/m1Z and x by Z~mpx/m 1AZ in eq. (7) if the scaling procedure is used here. Taking into account that Z1 and m1 dependences of both transformations ofvariables are identical and considering the relation (7) for protons and a-particles one can conclude from fig. 2 that a=5/3. Therefore the relationship 3)
K/K0 u/K~’ must =f( be universal in the Lindhard—Scharff approach of the scaling. Fig. 3 demonstrates large deviations from the scaling in this case. Basic distinctions between the two methods of scaling must be mentioned in conclusion. From the mathematical point of view the Thomas—Fermi scaling principle can be obtained using the Bethe—Bloch theory with ln y replaced by y ‘~. This approximation is valid within the range 10 ~ y~ 100. From the physical point ofview the range of validity of the Thomas—Fermi scaling is wider since the close collision contribution to inelastic energy losses is taken into account. The collective distant collision 295
Volume 121, number 6
PHYSICS LETTERS A
contribution is written approximately Thomas—Fermi scaling procedure.
in the
References [1] D.A. Sykes andSj. Harris, NucI. Ins~r.rn.Methods 94 (1971) 39. [2] M.D. Bedri, SJ. Harris and H.J. Parish, Rad. Eff. 27 (1976) 183.
296
4 May 1987
[3] P.D. Bourland, W.K. Chu and D. Powers, Phys. Rev. B 3 (1971) 3625. [4] D.A. Kirzhnits, Yu.E. Lozovik and G.V. Shpatakovskaya, Usp. Fiz. Nauk 117 (1975) 3. [5] H. Bichel and C. Tschalaer, NucI. Data Tables A 3 (1967) 343. [61 J. Lindhard and M. Scbarfl~K. Dan. Vidensk. Seisk. Mat. Fys. Medd. 27 (1953) No. 15. [7] T.E. Everhart and P.H. Hoff, J. Appi. Phys. 42 (1971) 5837.