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NOV 1 3 1964
Effects of Shell Corrections to Stopping power
in Theoretical Dose Studies
J. E. TURNER
Health Physics Division, Oak Ridge National Laboratory
Oak Ridge, Tennessee
MASTER
Abstract.--The theoretical evaluation of absorbed radiation dose
in tissue from external sources is based on the absorbed energy
and rate of linear energy transfer (LET) calculated from the
Bcthe stopping-power formula. For the chemical elements in
tissue, it is known experimentally that shell corrections to the
Bethe formula are needed for incident particle energies in the
range 0 to 15 MeV. Shell correction curves that agree with
experimental data for protons with a number of metals, ranging
in atomic number from Be to Pb, have been developed. This work
is applied to the light elements present in soft tissue. Shell
correction curves, estimations of the mean excitation energies
a:d the results of several theoretical studies based on this work
are presented.
1.
Introduction
The theoretical evaluation of the RBE dost iri tissue frian suurces os
iorizing radiation is often made on the basis of the amount of e. ergy
assorbed per unit mass in the irradiated system and the spatial and
Twinpor si distributions of the absorbed energy. To many experiments in
: viuse rate is not varied, only the spatial distribution and the
Jo. umount of energy absorbed per unit mass are of concern. Measure-
Lorem: ci these quantities can be difficult or impossible, and one
.requently relies on physical theory to aid in assessing dose-effect
relationships. Both the absorbed energy and its spatial distribution are
then determined by the stopping power of the irradiated medium (soft
tissue, bone, etc.). If the incident radiation consists of charged
particles, Ithen one uses directly the stopping power of the medium for
i radiation. For neutrons and gamma rays, one can consider the scopping
Juwe, of the medium for the charged secondary particles, e.g., recoil
protons or other nuclei and electrons. In any case, stopping power is the
key physical quantity of interest in theoretical dose studies. This
quantity is also of prime importance in dos imeter design and interpretation
of dos imeter response.
This paper describes the results of some recent developments in the
theory of shell corrections to stopping power as applied to dos imetry, both
from the standpoint of dose-effect studies and from the scandpoint of
dos imeter design and response. The theory of shell corrections is reviewed
.. Section 2 of this paper and some numerical results are given in Section
- To be definite, the numerical results are calculated for the slowing
down of protons in the energy range < 15 MeV in three elements of importance
LII
(1) Research sponsored by the U. S. Atomic Energy Commission under contract
with Union Carbide Corporation.
in dos imetry: nitrogen, argon, and xenon.
2. Stopping Power Theory
For a heavy charged particle the stopping power, .de/dx, of a
uniform isotropic medium characterized by N atoms/ cm3 with atomic number
z is given by Bethe's formula3
i24.
-
DE
dx
42° e NZ
- Arz“e*NZ (in -- 2m vezi op
my 111 - B?
NIO
)
(1)
Here z is the charge of the incident particle (z = 1 for protons); e and
m are the electron charge and mass; v is the speed of the incident
particle; B = V/C, where c is the speed of light; I is the mean excitation
energy of the target medium; and c/Z is a term representing shell corrections
as explained below. Density corrections are not important in the appli-
cations discussed here and have been omitted in Eq. (1). The quantity 1
is the energy-independent parameter given by
ini - IVE, - Efni
where E -E is the excitation energy of an atom of the target above its
ground state energy and f, is the corresponding oscillator strength of the
acom.
Under the assumption that the speed of the incident particle is large
compared with orbital speeds v. of electrons in the ground states of the
carget atoms, ive., when vive » 1, the direct quantum mechanical calcu.
lation of stopping power gives En (1) with c/2 = 0. This assumption, now-
ever, is often not valid since v can never be greater than c. For example,
in heavy elements the inner electron speeds are already comparable to c,
and one cannot obtain v >> Va in the light elements, such as those in
tissue, the condition v >> v. is fulfilled experimentally at high energy,
althought it is not fulfilled at low incident particle energy. It has been
pointed out,4 for example, that the K-shell ionization potential for oxygen
is about 540 volts, and the condition v» va is not fulfilled for alpha
particles with energies less than about 4 Mev. Thus shell corrections to
s topping power are necessary in heavy elements at all incident particle
velocities and in light elements at the lower velocities.
To apply stopping power theory in studies of dos ime try, it is
des i rable to know in detail all of the quantities appearing in Eq. (1).
in practice, each of them can be determined by direct experimental measure-
ment except | and c/2. An experiment that measures a value (- dE/ dx) of
stopping power determines, in effect, the value of the quantity
my
2mv²
1 - B
Inl + - • in
2
24
4Trz e NZ
(3)
dx
exp
Note that since I is independent of incident particle energy, the experi-
mentally determined quantities on the right hand side of 13), for various
energies, furnish directly the energy dependence of the shell correction
C/Z. An uncertainty in the quantity ini acts as an additive constant to
the shell correction.
in order to determine 1 and c/Z separately, further work must be
done. Although I is a well-defined parameter of stopping power theory,
its theoretical evaluation is difficult and has not been carried out except
for the simplest a toms and molecules. For the Thomas-Fermi model, Blochs
showed that I is proportional to the atomic number 2. The ratio is
approximately 1/2 ~ 10-15 ev, decreasing somewhat at high Z, where the
Thomas-Fermi model is most applicable. Thus mean excitation energies are
known at least approximately for the heavier elements from theoretical
considerations. Since it is the logarithm of l that is related to the
experimental stopping power in (3) above, exact values of I are often not
critical in applications.
Since the innermost electrons in an atom have the highest velocities,
the condition v >> V fails first for K-electrons as a fast charged
particle slows down from high speed. Separate calculation of the contri-
bution of K-elections to stopping power leads to a K-shell correction Cu/z.
Walske carried out such a calculation for both Ko and L-shell electrons6
using hydrogenic wave functions. In recent years it has become apparent
that still higher shell corrections are needed. In addition to being
laborious, a direct extension of Walske's work to the M- and higher shells
would be of doubtful validity because higher shell electrons are not in a
hydrogenic "1/pll potential. In a modified procedure, the dependence of
shell corrections on energy found by Walske has been employed by Bichsel,
us ing adjustable scaling parameters, to es ĉimate higher shell corrections
in such a way that agreement with empirical stopping power and range data
is obtained. 8
Fano has given a method of expanding complete expressions for the
shell corrections of atoms into powers of 1/v2.9 The basic stopping power
expression can be written?
(4)
my
where l - 9/2m, with a representing the magnitude of the momentum trans-
ferred from the incident particle to an atomic electron, and where
le 16 is
given in terms of the oscillator strength fn by IF 1 - Qf/1E,-E). The
integration in (4) has to be carried out over all values of Q and E.-E
cons is tent with the conservation of energy. It can be shown that
evaluation of the expression (4) leads to the stopping power given by
Eq. (1) with the total shell correction for an atom given to order 17 v4
approximately by
cm2f(T) +15%), * , Ry“ z PVO)
(5)
(mv2,2
Here T reri esents the total kinetic energy of the electrons in the atom,
Ry is the. Rydberg energy (13.6 eV), and plo) is the electronic dens ity at
the position of the nucleus of the atom. The symbols i la denote the
expectation value of the enclosed quantity in the ground state or the a tom.
An important feature of expression (5) is that the shell corrections
expressed in this form depend only upon well defined properties of the
ground state of an atom, independently of any model used in the calcu-
lation.". The kinetic energy, for example, can be evaluated from data given
by Gombás-> or from Hartree-Fock or Thomas-Fermi calculations. The data of
Gombás are approxima ted by (T), ~ 244 Ry, which can be applied to give the 1/vc
correction in Eq. (5) as a function of 2. The quantities in the numerator
of the 1/v4 term in (5) arise mostly from contributions by the two
K-elections. The screened hydrogenic approximation for the two K-electrons13
gives (T) ~ 1012 - 0.3)4 Ry? and p(0) - ? (2 - 0.3). A full account of
this work is given in Ref. 10.
3. Some Numerical Results
in this section we show numerical results for the effects of shell
corrections in three problems: (1) the determination of mean excitation
energies, l, from high-energy experiments; (2) the contributions of shell
corrections to stopping power as a function of incident particle energy;
and (3) the determination of mean excitation energies from low-energy
experiments. Results are given specifically for protons in nitrogen, argon,
and xenon.
(1). Determination of 1 from High-Energy Experiments. It is sometimes
assumed that shell corrections vanish in the high-energy limit. However,
as pointed out above, even at arbitrarily high energy the condition v >> v.
cannot be fulfilled if v~. Shell corrections in the high-energy limit
vec, or B = 1, were evaluated in Ref. 10 by means of Eq. (5) above for a
number of elements throughout the periodic system. 14 The results are shown
in Fig. 1. The differences A1 in the mean, excitation energies of atoms
introduced by assuming (c/Z)e, • 0 are shown in Table 1. The differences
are given by
"
E
.
A1 ladj
1
(6)
COX22D
Element
Values of Al- leds - 1 for Several Elements
Table 1
OvNoo
Al (ev)
m
a
yor
improv
e
...
......
......
.....
.
.
.
.
.
..
.
..
.
.
.
where
in ladi - In 1 + (C/2)B1
(7)
The quantity lad defined in this way has been called the adjusted mean
excitation energy," which corresponds to the assumption that shell
corrections vanish in the high-energy limit. For the purpose of esti-
nating All in Table 1, the value 1 - 163 eV was used for aluminum 15 and
the approximation I ~ 10 Z was used for the heavier elements. Because of
the logarithmic dependence of 1 and lode in (7), the difference Al is
insensitive to the particular value of 1 over a wide range. The table
shows that in the interpretation of experimental data at high energies,
the assumption of vanishing shell corrections can be of significance only
in the heavier elements.
in dosimeters, xenon is a relatively heavy gas inat is sometimes
employed. Table | indicates that the total shell correction for this
element at B = 1 is equivalent to about 10 eV in the value of 1 appearing
in Eq. (1). For the determination of 1 values for other elements of
interest in dos imetry, e.g., carbon, nitrogen, and oxygen, shell. corrections
play an insignificant role in high-energy experiments, such as those of
Zrelov and Stoletov, 16 Bakker and Segre, 17 and Thompson. 18
(2). Shell Corrections as a Function of Energy.-As described in
Ref. 7, shell correction curves were drawn for five metals (Be, Al, Cu,
Ag, and Pb) to fit smoothed values of proton stopping power and range given
in the form of tables and curves by Bichsel8 and by Whaling.19 A plot was
made of the right hand side of Eq. (3) as a function of incident proton
energy for each metal. Since I docs not depend on energy, the variation
in these curves show's directly the variation of the shell correction c/z
as a function of energy. The position of each curve on the c/Z axis was
adjusted to give the high-energy limit (C/Z) - shown in Fig. I for that
element.
.
.
-
.
.
Similar curves for protons in nitrogen, argon, and xenon were drawn
to fit smoothly when interpolated between the curves given in Ref. 7 and
to have the high-energy limit shown in Fig. 1. The curves were also spot
checked for agreement with other data. The results are shown in Fig. 2.
The abscissa x in this figure is defined by
X
(8)
where v. is the orbital electron velocity in the ground state of hydrogen,
and is the same parameter employed by Lindhard and Scharff. For the three.
elements in the figure, x is given as a function of incident proton energy
in Fig. 3. The error bars in Fig. 2 ind i cate the magnitude of an
uncertainty of approximately + 1 per cent in the proton s topping power for
:,
-
-
-
-
..
an element at the two values of x where they are drawn. From this one
sees the actual extent to which shell corrections contribute to the stopping
power of even the light elements in tissue at the lower energies. The
curves for carbon and oxygen are not shown, but lle close to the nitrogen
curve.21
(3). Determination of 1 from Low-Energy Experiments.-The curves in
Fig. 2 have been used to estimate mean excitation energies for nitrogen,
argon, and xenon from the experimental data of Brolley and Ribe.23 These
experiments were carried out in part with 4.43-MeV protons and with
8.86-MeV deuterons, which have a velocity equal to that of 4.43-MeV protons.
in these experiments shell corrections .contribute an important amount to
the stopping power. Brolley and Ribe give the experimental stopping power
values in Table II. To evaluate 1 from these data we have from Eq. (3)
Inl + { = 9.167 - 32,8 z (-)
in which I is in ev, (-dE/dx)ex is in MeV/ g/cm², and A is the atomic weight
of the element. Using the curves of Fig. 2 at the appropriate values of x
for 4.43-MeV protons, we obtain the values of 1 given in Table 11.
The value of 1 for argon agrees with an independent calculation using
Brolley and Ribe's data and the K- and L-shell corrections of Walske. It
is also in agreement with the recent experimental value 1 = 190 + 17 eV
reported by Martin and Northcliffe<5 for heavy ions slowing down in argon.
The nitrogen value found here is greater by 4 eV than that found from the
Brolley-Ribe data in Ref. 24, which is not significant. Xenon apparently
was not calculated in this reference. The value found in Table Il is not
consistent with the value 1 ~ 10 Z for elements in the region of the atomic
number of xenon. 15 However, Brolley and Ribe point out that their measured
xenon stopping power is too low compared with other gases in the experiment.
As noted in the table, the value for xenon from this experiment is probably
too low by about 65 ev.
-
-
-
--
The above examples show the effect of shell corrections on the value
of 1. Another important consideration in theoretical dose studies is the
direct influence of shell corrections on the stopping power itself. Figure
4 shows the fraction by which the stopping power of nitrogen for protons
is reduced due to shell corrections. These values are also typical of carbon
and oxygen. The contribution for protons has a maximum of about 5 per cent
in the energy region of several hundred kev. On the low-energy side of the
maximum the shell corrections pass through zero in Fig. 2 and then approach
large negative values. Beyond this region, electron capture-and-loss phenomena
set in and the theory is no longer applicable. On the high-energy side of the
maximum the contribution drops to less than 1 per cent at several Mev. Since
the range in tissue of a 3-MeV proton, for example, is only ~ 0.2 mm, shell
corrections will be of interest in theoretical dose studies when dose is
investigate in volumos with dimensions of this order of magnitude, in
.
.
..
1...
Table 11
Experimental Stopping Powers (Ref. 23)
Element
I (ev)
-
- dE/ dx
(ev-cm² x 10-15,
-
.
-
.
-
1.76 + 0.04
3.72 + 0.08
8.06 + 0.18
89
189
485a
avalue probably too low by abryt 65 av.
in historiastasize viesti**...**
-
V
resultater
TV
i
a
the
m
in
de
...
PE-
these cases, neglecting shell corrections would increase the stopping power
by the order of several per cent. Since the stopping power is identical
with the LET (rate of linear energy transfer), the assignment of a value of
RBE (relative biological effectiveness) would be affected somewhat also.
4. Conclusion
Shell corrections are relevant in theoretical studies in dosimetry
from two standpoints. First, their evaluation is needed in order to
determine accurately mean excitation energies (l values) used in dos i meter
design. Unless experiments to determine ! are carried out at high energies --
and then only for the lightest elemen cs--C/Z will contribute measurably to
the stopping power. The logarithmic dependence of dE/ dx on 1 tends to
decrease the importance of shell corrections because experiment measures,
in effect, the quantity Ini + C/Z. Second, shell corrections can reduce by
several per cent or more at low energies the stopping power of light elements.
This could have a practical effect in special studies of low-energy dos imetry,
such as with protons and alpha particles with energies ~ 1 Mev, in which one
is interested in doses and RBE in volumes of tissue with dimensions comparable
to the ranges of ihes e particles.
1
:;
.
L
Y
er
10
References
(1) Research sporsored by the U. $. Atomic Energy Commission under
contract with Union Carbide Corporation.
(2) "Radiation Quantities and Units," linternational Commission on
Radiological Units and Measurements (ICRU) Report 10a, National Bureau
of Standards Handbcok 84, Washington, D. C., 1962.
(3) H. A. Bethe, Ann. d. Phys. 5, 325 (1930); "Quantenmechanik der
Ein-und Zwei-Elektronenprobleme," "Handbuch der Physik," Springer,
Berlin, 1933, Vol. 24/1, pp. 491 ff.
(4) H. A. Bethe and J. Ashkin, "Passage of Radiations through Mat ter,"
"Experimental Nuclear Physics," E. Segre, Ed., John Wiley and Sons, Inc.,
New York, 1953, p. 1696
(5) F. Bloch, Zeit. für Phys. 81, 363 (1933).
(6) M. C. Walske, Phys. Rev. 88, 1283 (1952); Phys. Rev. 101, 940 (1956).
(7) V. Fano, "Penetration of Protons, Alpha Particles, and Mesons,"
"Annual Reviews of Nuclear Science," E. Segre, Ed., Annual Reviews, Inc.,
Palo Alto, 1963, Vol. 13, p. !.
(8) H. Bichsel, 'Passage of Charged Particles through Matter," "Handbook
of Physics," Second Edition, McGraw-Hill Book Co., Inc., New York, N. Y.,
1963, pp. 8-20.
.
.
.
.... .
11.
(9) CF. Section 4.4 of Ref. 7. The method treats the problem non-
relativistically.
.
- terr orism:
(10) U. Fano and J. E. Turner, "Contributions to the Theory of Shell
Corrections," "Studies on the Penetration of Charged particles in Matter,"
Report No. 4, National Academy of Sciences - National Research Council
(11) Cf. also G. Placzak, Phys. Rev. 86, 377 (1952).
(12) P. Gombás, "Statistische Behandlung des Atoms," "Encyclopedia of
Physics," S. Flügge, Ed., Springer, Berlin, 1956, Vol. 36/2, p. 183.
(13) H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and
TWO-Electron Atoms," Academic Press, Inc., New York, N. Y., 1957, p. 163.
(14) The numerical evaluations in Ref. 10 were actually carried out with
a slightly modified form of Eq. (5) in which the effect of electron-electron
correlations in a toms was calculated explicitly for collisions corresponding
to low momentum transfer (low u). The effect is largest in this case, but
amounts only to less than ~ 3% of the kinetic energy.
(15) "Studies on the Penetration of Charged particles in Matter,"
National Academy of Sciences-Natlonal Research Council Publication 1133,
U. Fano, Ed., Washington, D. C., 1964.
(16) V. P. Zrelov and G. D. Stoletov, J. Exp. Theor. Phys. (USSR) 36,
658 (1959). English translation JETP 9, 461 (1959).
(17) C. J. Bakker and E. Segre, Phys. Rev. 81, 489 (1951).
(18) T. J. Thompson, "Effect of Chenii cal Structure in Stopping Power
for High-Energy Pro tons," University of California Radiation Laboratory
Report UCRL-1910 (1952).
(19) W. Whaling, "'The Energy loss of Charged Particles in Matter,"
"Encyclopedia of Physics," S. Flugge, Ed., Springer, Berlin, 1958,
vol. 34/2, p. 193.
(20) J. Lindhard and M. Scharff, Kal. Danske Videnskab. Selskab.,
Mat.-Fys. Medd. 27, No. 15 (1953).
(21) Note that the stopping power of a tomic hydrogen can be calculated
completely from theory (Ct. Refs. 22 and 5).
(22) L. M. Brown, Phys. Rev. Z2, 297 (1950).
(23) J. E. Brolley, Jr., and F. L. Ribe, Phys. Rev. 28, 1112 (1955).
(24) "Stopping Powers for Use with Cavity Chambers," National Bureau
of Standards Handbook 79, Washington, D. C., 1961.
(25) F. W. Martin and L. C. Northcliffe, Phys. Rev. 128, 1166 (1962).
,
UNCLASSIFIED
ORNL- DWG. 64-9731
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To
20-30-40-50-60-70-80-
90
100
ATOMIC NUMBER Z
Fig. 1.--Estimated proton sholl corrections at B - | as a function of
atomic number.
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UNCLASSIFIED
ORNL- DWG. 64-9730
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5
10
50
50
100
100
500
500
__
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Fig. 2.--Proton shell corrections for nitrogen, argon, and xenon as a
function of x = v1/v2.
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UNCLASSIFIED
ORNL-DWG. 64-9728

NITROGEN
ARGON
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7
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0.5
1
5
10
50
100
500 1000
PROTON ENERGY (Mev)
Fig. 3.--x as a function of energy for protons in nitrogen, argon, and
xenon.
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UNCLASSIFIED
ORNL - DWG. 64-9729

PERCENT
0.5
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PROTON ENERGY (Mev)
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Fig. 4.--Per cent reduction in stopping power due to shell corrections
for protons in nitrogen.
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-LEGAL NOTICE

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