NBSIR 83-2785

Density Effect for the Ionization

Loss of Charged Particles in

Various Substances

Department of Physics

Brookhaven National Laboratory

Upton, New York 1 1 973

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Radiation Research

Washington, DC 20234

November 1983

Prepared rcr: U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Office of Standard Reference Data

Washington, DC 20234

U.S. Department of Energy

Washington, DC 20545

Office of Naval Research

Arlington, Virginia 22217

NBSIR 83-2785

DENSITY EFFECT FOR THE IONIZATION

LOSS OF CHARGED PARTICLES IN

VARIOUS SUBSTANCES

R. M. Sternheimer

Department of Physics

Brookhaven National Laboratory

Upton, New York 11973

M. J. Berger and S. M. Seltzer

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Radiation Research

Washington, DC 20234

November 1983

Prepared for: U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Office of Standard Reference Data

Washington, DC 20234

U.S. Department of Energy

Washington, DC 20545

Office of Naval Research

Arlington, Virginia 22217

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Director

DENSITY EFFECT FOR THE IONIZATION LOSS

OF CHARGED PARTICLES IN VARIOUS SUBSTANCES*

R. M. Sternheimer

Department of Physics

Brookhaven National Laboratory

Upton, New York 11973

M. J. Berger and S. M. Seltzer

Center for Radiation Research

National Bureau of Standards

Washington, D.C. 20234

ABSTRACT

The density-effect correction 5(6) for the ionization energy loss

of charged particles has been evaluated for a total of 278 substances

including 98 cases of elements of the Periodic Table (12 gases and 86

condensed materials, including liquid hydrogen and graphite of three

different densities) and including also 180 chemical compounds and

substances of biological interest (13 gases and 167 liquid or solid

substances). In the calculations, up-to-date values of the mean excitation potential I and of the atomic absorption edges hv. were employed as input data for the general equations for o( 6) previously derived by

Sternheimer

A version of this report with a shortened text but the same tables will

appear in Atomic Data and Nuclear Data Tables.
1 . Introduction

The density-effect correction 6 for the ionization loss of charged

particles 1-12 has been evaluated previously for a large number of

substances. 5 - 12 The last previous extensive effort in this direction was

made in the paper of Sternheimer, Seltzer, and Berger 12 in which the density

effect was evaluated for a total of 72 substances (34 metallic elements, 26

compounds, 11 gases and liquid hydrogen). In Ref. 12, the basic equations of

Sternheimer (Refs. 3 and 5) were used in order to evaluate the density effect,

employing up-to-date values of the mean excitation potential 13 - 14 I, and of

the atomic absorption edges 15 hv_.

.

In the present work, the results of Ref. 12 have been extended to a total

of 278 substances including 98 cases of elements of the Periodic Table (12

gases and 86 condensed materials including liquid hydrogen and graphite of

three different densities) and including also 180 chemical compounds and substances of biological interest (13 gases and 167 liquid or solid compounds).

The essential advance of the present calculations over those previously

carried out in Refs. 5-12 consists in the development and implementation of a

computer algorithm which carries out in a single operation the numerical

evaluation of the density effect and the fitting of the numerical results by

an approximation formula.

2. Numerical Evaluation of the Density Effect

The calculations of 6(6) are based on the following equations derived by

Sternheimer 3 ’ 5 in 1945 and 1952:

6(6) = I f in [U. 2 + t

2 )/r 2

] - l

2

{ 1 - 6

2

) , (1)

i =1

1 1 1

2
where 3 = v/c is the particle velocity divided by the velocity of light, and l

is the solution of the equation:

1_

3

2

1 =

n f

I

^

i =1 v. 2 + l

2

i

( 2 )

In Eq. (2), v. is defined by:

v . = v . p/

v

i p (3)

th

where hv^ is the absorption edge for the i oscillator of the dispersion

model. The quantity hv^ is the plasma energy of the electrons of the

substance considered as free electrons, and is given by 16

hv = 28.816 (

p

0 Z/A) 1/2 eV , r (4)

where p 0 is the density of the medium (in g/cm 3 ), Z is the atomic number and

A is the atomic weight. In the case of a compound or molecular gas, Z/A is to

be replaced by the ratio of the total number of electrons to the effective

molecular weight or the sum of atomic weights of the constituent atoms:

EZ./IA.. As in Ref. 12, a separate dispersion oscillator is used for each

subshell of the atom considered, e.g., K, U, Ljt, and L T jy for neon. The

quantity p in Eq. (3) is the adjustment factor which was introduced by

Sternheimer 5 in 1952 and which is designed to give agreement of the oscillator

energies 3 hv.p (or rather hv ».) with the observed mean excitation cotential I.

i p r

Specifically, in Eq. (1), the constants JT are defined by:

l. = (v 2 + ~f.) 1/2 for v. >0 i i o i i

oJ

(5)
£

n = f

n 1/2 for v

n = 0 (conduction electrons in a metal). (6)

In Eq. (5), the factor 2/3 takes into account the Lorentz-Lorenz correction

[see Ref. 5., Eqs. ( 48) -(52)3 in the expression for the polarizability a(v);

note that this factor does not enter for the case of conduction electrons for

which i

= f 1/2 as qiven above.

The mean excitation potential I of the medium is given by

Jtn I = J f . £n(hv i.) (7)

4 i

v p v

K '

By making use of Eq. (3) for v. , we obtain the following expression, which is

used to determine the value of the Sternheimer adjustment factor p:

n - 1 o 1/2

£n I = y f. Zn[(hv.p) 2 +^-f. (hv) 2

] + f Jin (h v f 1/2 ). (8)

. i LX

i

1 3v p' J n

v p n

’ v '

For a conductor, f is taken as n /Z, where n is the effective number of

n c c conduction electrons per atom of the substance. Note that for a compound

(insulator) or for a gas, n_ = 0, the sum in Eq. (8) extends from i = 1 to n,

and the last term on the right-hand side of Eq. (8) is not present. The

values of p thus determined from the experimental values of I and hv^ lie

generally in the range 1.5 - 2.5. Physically the meaning of p is that it

takes into account the fact that for the excitations of an inner shell with

absorption edge hv., the contribution of the excitation (ionization) to continuum states involves energies which are larger than hv.

.

A very approximate

estimate of p was made in Ref. 12 [Eq. (11)], with the result that p is of the

order of e 1/2 = 1.649.

4
In Eqs. (1), (2), and (5)-(8), f. is the oscillator strength for the i th

oscillator, which was taken as n./Z for the inner (non-conduction) electrons;

here n. is the number of electrons for the subshell considered, e.g., n^ = 4

for the L T T

subshel 1 . In the case of a metal, n was taken to be the lowest

III c k

chemical valence of the element considered. The values of the absorption

edges hv. for the various subshells of all elements were obtained from the

compilation of Carlson. 15 The values of I were obtained from two recent

papers of Berger and Seltzer. 13 ’ 14

In Fig. 1, we have plotted the values of the Sternheimer adjustment

factor p as a function of Z. The solid curve has been drawn through the p

values for metals as obtained by means of Eq. (8). The p values for the 12

gases are shown separately as crosses. It can be seen that except for the

four gases 0

2 , N2 , F, and Ne, the crosses lie very close to the curve determined by the p values for condensed substances. The most striking feature of

the curve of Fig. 1 is the existence of successive maxima and minima as a

function of Z. The maxima and minima reflect the existence of similar

features in the curve of I/Z _vs_. Z, as presented in Fig. 1 of Ref. 12, but in

the present case, i.e., for p, these fluctuations are much more pronounced.

They can be related to the electronic shell structure of the atoms considered.

k

An alternative prescription would be to use as the effective number of con- duction electrons the number of electrons participating in plasma excitations

in metals. The latter number can be deduced from optical data and from

measured electron energy-loss spectra. Effective numbers of plasma electrons

have been deduced from the experimental literature by Raether 17 for 27 metals

and by Isaacson 18 for 47 metals; see also Mann and Brandt, 19 and Ziegler,

Biersack and Littmark. 20 We have made some numerical tests, and have found,

for example, that the use of results of Raether or Isaacson would change the

density-effect correction such that the electron stopping power in gold would

differ by less than 0.3% and that in copper by less than 0.25%, compared to

the values obtained when the number of conduction electrons is deduced from

the lowest valence state.

5
Thus the maxima at Z = 11, Z = 30, Z = 47, and Z = 70, correspond approximately to the filling of the 2p 6

, 3d 10

, 4d 10, and 4f 14 shells, respectively.

In addition, the pronounced shoulder in the neighborhood of Z = 80 can be

correlated with the completion of the 5d 10 shell in this region of the

Periodic Table.

On the other hand, the minima of p at Z = 20, Z = 39, Z = 57, and Z = 89

correspond approximately to the filling of the ns

2 shell in the alkaline

earths Ca (Z = 20), Sr (Z = 38), Ba (Z = 56), and Ra (Z = 88), respectively.

We would like to note that these alkaline earths correspond to the closing of

the successive supershells 21 of the Periodic Table, where a supershell is

defined as the set of all shells nil with the same value of the quantum number

k = n + i. Thus both the curves l/l vs. Z and p

_vs . Z give additional support

to the k ordering of atomic structure. 22

3. Fitting Formula

Using the procedures described above, numerical values of 6(3) were

calculated for each material at many points on a logarithmically spaced energy

grid. The energy variable used was T/m 0 c

2

, where T is the kinetic energy and

m

Q c

2 is the particle rest energy. The grid values were chosen to be

T

(i)

/m

o c2 = 100*000, 80,000, 60,000, 50,000, 40,000, 30,000, 20,000, 15,000,

10,000, and so on, down to T^/m0 c

2 = 0.01. The numerical values of 6 were

fitted to the formula proposed by Sternheimer 5 in 1952, namely:

5(X) = 4.6052X + a(X

x - X) m

+ C, (X

0 < X < Xx

) (9)

5(X) = 4.6052X + C, (X > X

L

) ( 10 )

6
where X = log 10 (p/m 0 c) = 1og 10 (By) = 1/2 log 10 [(T/m 0 c 2

) (T/m 0 c

2 + 2)], with

p the momentum of the incident particle and y = (1 - g

2 )- 1/2 . X

Q is the value

of X below which 5(X) is zero for the case of an insulator or gas, and the

value of X below which 6(X) for a metal (conductor) is small, i.e.,

<5(X) < 0.14. X

x is the value of X above which 5(X) has essentially attained

its asymptotic value (to within 0.015). In Eqs. (9) and (10), a and m are

adjustable parameters which will be determined below, and C is given by:

C = -2 in ( I/h v ) - 1 , (11)

where I is the mean excitation potential of the substance for use in the

Bethe-Bloch stopping-power formula. 23 ’ 24 In the present paper, we will

frequently use the notation C for -C = |C|.

4. Determination of the Parameters in the Fitting Formula

The experience of Sternheimer 5 - 10 in fitting 6(0) indicates that X

x of

Eq. (9) can be taken as that value of X for which the deviation of 6(X) from

its asymptotic value [Eq. (10)] is of the order of 0.01, and in particular

does not exceed 0.015.

For each grid value X^, the computer program calculates the values of 6

and 6 , the asymptotic value defined by Eq. (10). We define 6, as follows:

aS ±

s 6 - 6 as

Furthermore we define X as follows [see Ref. 10, Eq. (8)]:

a

X = C/4.6052 . d

( 12 )

( 13 )

7
Incidentally, the physical significance of the difference is clearly shown

(for the case of neon gas at normal temperature and pressure) in Fig. 1 of

Ref. 10.

4.1 Non-Conductors

We first consider the case of non-conducting materials for which 5(0) = 0

at low velocities 0 < 0 O

, where 0 O is the velocity for which l

2 = 0 according

to Eq. (2). We then have X 0 = log 10 (3 0 Yo)» where Yo = (1 - & 0 2 )~ 1/2 . After

X

Q has been thus determined it is necessary to determine X

: in Eqs. (9) and

(10). Now the numerical values of to be denoted by S

L num [see Eq. (12)]

are approximated by the monomial expression a(

x - X) m

, as shown by Eq. (9).

We will denote the fitted values of a(X

x - X) m

at the mesh points by

Thus we have

6

1 )fn (X) = a ( X j - X) m

. (14)

The values of X

x

, a and m must be so chosen as to minimize the maximum

deviations

A6 i = 6 i,fit ,num

(15)

We have one condition relating a, m, X

0

, and X

x

, namely that 5(X

0

) = 0. From

Eq. (9) we obtain directly:

4.6052X 0 + a(X

j

- X 0 ) m -5 = 0 ( 16 )

where C = - C. Upon solving for a, and using Eq. (13), one finds that

8
a

= 4.6052(X

a

- X

0

) (h - x

0

) m

(17)

The remaining task is to determine the best values of X

1 and m. For- each

insulator or gas, nine separate calculations were run with X

L determined by

the condition that 6

1 (X1

) [see Eq. (12)] has the values 0.0015, 0.002, 0.003,

0.004, 0.005, 0.006, 0.008, 0.010, and 0.015, respecti vely . This procedure

directly limits the maximum inaccuracy introduced by neglecting the numerical

value of ^

= 6 - S as for X > X

x

. The resulting errors are certainly

tolerable because when the stopping number is ~ 20, an error of 0.015 in 6(X)

introduces a relative error of only 0.015/20 = 0.00075 = 0.075%.

The equation for a and the above procedure for determining a reasonable

range of values of X

x leave only the exponent m undetermined. In the previous

fits in Refs. 5, 7-10, and 12, it was found that it is best to require an

exact fit of Eq. (9) to the numerical value of 6. at one additional point

l,num

in the range X

0 < X < X ls preferably for an X value near the value of X

a defined by Eq. (13). This intermediate X value for which the additional fit

was made will be denoted by X 2

. Trial values of X 2 were chosen to be the ten

grid points X ^

immediately below, and the ten grid points X^. ^ immediately

above X^ defined by Eq. (13), subject to the condition that X

0 < X 2

< X

L

.

We can now solve for m as follows. For a given value of X 2 , we have:

a(X x - X

2 ) = «

1 (X 2

)

. (IB)

In view of the definition of 6 (

Q

) and the requirement of an exact fit at

X = Xq

,

we have also:

m

a (

i - Xq) = Sj(Xq) (19)

9
Dividing Eq. (19) by Eq. (18),

( 20 )

and therefore:

Tog 10 [ 6

1 (X0 )/ 61 (X 2 )] m = log 10 [(X 1 -X

0 )/(X 1 -X 2 )]

’ (21)

With m thus determined and for the given values of X

: and X

0 , a can now be

obtained from Eq. (17).

The following computer algorithm was used for selecting the parameters a

and m:

1. For each trial combination X! and X 2 , a and m were calculated

according to Eqs. (17) and (21).

2. These trial values of a and m were used to evaluate 6 , r ., according

1 , f 1

1

3

to Eq. (9) at each grid-point X^ between X

Q and Xx

, and the maximum

difference A = 1 5. - 6. | for the trial was noted,

max

1 l,fit l,num'

It should be noted that 6 is a monotonical ly increasing function of X. This

condition is satisfied only when the fitting parameter m is smaller than a maximum value m which -- for insulators and gases -- is given by 11 max

3 3 x

i - x

o m = y 7

- . In 26 of the 278 cases considered, the fitting procedure

m 3.x a A

q d

u resulted in a value of m somewhat larqer than m , with the result that

/ \

IT! clX (for compounds) the value of 6 from Eq. (9) was slightly negative in a narrow energy region near threshold. These values of m were nevertheless

accepted because the resulting error was negligible, the absolute value of 5 in this region being smaller than ~ 0 . 02 .

10
3. This procedure was repeated in 180 trials, i.e., using the 9 choices

of Xj and 20 choices of X

2 discussed earlier. The values of X

Q

, X

x

,

a and m finally selected were those from the trial giving the

smallest value of A . Values of these parameters will be given in max

r 3

Tables I and II.

4.2 Conductors

We now proceed to a discussion of the density effect for metallic

conductors. In this case, 5(0) does not vanish for arbitrarily small

velocities, as already discussed by Sternheimer in Ref. 7. The basic reason

is that for substances with conduction electrons, Eq. (2) contains a term with

v

p = 0, and this leads to the result that i

2 > 0 for any nonvanishing 3 2

.

Therefore a suitable value of X 0 must be chosen for which 6(X 0

) is small, but

not zero. X

Q cannot be made too small al gebraical ly (e.q., very negative),

since this would spoil the overall fit to Eq. (9) at larger values of X. It

has been our general experience in obtaining the fits published in Ref. 12

that Xq must generally be chosen such that 6(X 0 ) is close to 0.1 in all cases

(see Table I of Ref. 12). In view of this observation, and in order to widen

the choice of parameters so as to obtain the smallest values of A , the max computer program was run for each of the 180 aforementioned choices with an

additional choice of five values of X 0 , such that the calculated values of

6 (

0

) were 0.06, 0.08, 0.10, 0.12, and 0.14, respect i vely. Thus a total of

180 x 5 = 900 possible fits were run for each metallic substance, and again

that fit was chosen which qives the smallest value of Amax

11
For the case of metals, the equations for a and m are slightly changed

because 6(X 0 ) is not zero. The appropriate equations have been derived in

Ref. 11 and are as follows: We define X by

a, o

c + s(x

0

) X a,6

= 4.6052 ( 22 )

In terms of X . the modified equations for a and m are given by:

a, 6 max

4. 6052 (X . - X.)

a = £1

(*l

- X

0

) m

(23)

X

l " Xq

'max

=

X - X

0 (24)

Obviously, for insulators [4(X n ) = 0], , reduces to X. as defined above u 0 a [Eq. (13)].

For metals we have found that the density effect $ for X below X

0 can be

approximated sati sf actori ly by the formula

«(X) « 6(X 0 ) x io

2(x - x o), X < x„ . (25)

The error in 6 incurred by the use of Eq. (25) is always smaller than the

uncertainty ^maxfor the fit above X

0

.

Before we proceed to a detailed explanation of Tables I and II, we note

that in some cases, for the 72 substances considered by us in Ref. 12,

although the same values of the mean excitation potential I were used, the new

values of a and m are nevertheless appreciably different. For example, for

12
borosilicate glass (Pyrex) we have a

x = 0.2988 and m x = 2.805 in the fit of

Ref. 12 (with X

Q = 0.1479, X

x = 2.5) and we have a 2

= 0.08270 and m

2

= 3.5224

(with Xg = 0.1479, X l = 2.9933) in the present fit. For gold, we found

a

x = 0.1533 and m1 = 2.881 (with XQ = 0.0966, s(X

Q

) = 0.0912; Xj = 3.5) in

Ref. 12, and a 2 = 0.09756 and m 2 = 3.1101 (with X 0 = 0.2021, <5(X 0 ) = 0.14;

X

x = 3.6979) in the present work. Even though the parameters a and m are

individually quite sensitive to the choices of X 0 , X x , X

2 , (X x

) (and 6

0 (X 0 )

in the case of metals), the variations of a and m are correlated so that the

fitted values are quite similar.

We note that the compositions for the various substances, in particular

for the organic compounds and the biological substances, are not listed in

Table II. For those compositions, the reader is referred to the recent paper

of Seltzer and Berger. 13

5. Example of the Use of Tables I and II

The density-effect correction 5 is to be used in the Bethe stopping-power

formula

1_ dE

P 0 dX

0.153536

S 2 j

{F( e) - 2 An I «} (26)

In this expression, - —41 is the mean energy loss per unit pathlength, in

Pg U A MeV/(g cm -2

). The term 2 C/1 is the shell correction, which is generally

negligible at energies at which the density-effect correction 6 is

significant. For heavy charged particles (muons, pions, protons, ...)

Examination of our data indicates that the correlation is such that — = n^ m 2“rn i^

s where n has a value in the ranqe 4 to 8.

a2

13
F( 0) = 2 in (27)

2m 0 c

2

e

2

1 - 0

2

and for electrons

F(0) = in m

Q c

2 T 0

2

2(1 - 0

2

) (2/TF - 1 + 0 2

) Zn2 + 1 - 0

2 + |

(1 - /17b*). (28)

As an example we consider the case of aluminum. We find X

Q = 0.1708,

Xj_ = 3.0127, 6(X

0

) = 0.12, fi^Xj) = 0.0015, a = 0.08024, m = 3.6345,

C = 4.2395. As a result, from Eqs. (9) and (10), <5(X) is given by:

6(X) = 0.12[10

2(X '°' 1708)

] (X < 0.1708)

3 6345

<5( X) = 4.6052X + 0. 08024(3. 0127-X) - 4.2395 (0.1708 < X < 3.0127)

6(X) = 4.6052X - 4.2395 (X > 3.0127).

We note that X

0 = 0.1708 corresponds to a momentum p/m Q c = = 1.482,

or a kinetic energy (in units m 0 c

2

) T/m 0 c

2 = 0.788.

6. Acknowl edgments

The work carried out at Brookhaven National Laboratory (RMS) was supported

by the U.S. Department of Energy under Contract No. DE-AC02-76CH00016. The

work carried out at the National Bureau of Standards (MJB and SMS) was

supported by the NBS Office of Standard Reference Data, by the Department of

Energy (Office of Health and Environmental Research), and by the Office of

Naval Research.

14
REFERENCES

1. E. Fermi, Phys. Rev. 57, 485 (1940).

2. G. C. Wick, Nuovo Cimento _1, 302 (1943).

3. R. M. Sternheimer, Thesis, Univ. of Chicaqo (1946).

4. 0. Halpern and H. Hall, Phys. Rev. _73, 477 (1948).

5. R. M. Sternheimer, Phys. Rev. 88, 851 (1952).

6. R. M. Sternheimer, Phys Rev. 91, 256 (1953); 93, 351 (1954); 93, 1434

(1954). _ _ _

7. R. M. Sternheimer, Phys. Rev. 103 , 511 (1956).

8. R. M. Sternheimer, Phys. Rev. 145 , 247 (1966).

9. R. M. Sternheimer, Phys. Rev. 164 , 349 (1967).

10. R. M. Sternheimer and R. F. Peierls, Phys. Rev. B_3, 3681 (1971).

11. R. M. Sternheimer, Phys. Rev. B 24 , 6288 (1981).

12. R. M. Sternheimer, S. M. Seltzer, and M. J. Berqer, Phys. Rev. B26, 6067

(1982); B27, 6971 (1983) (E). —

13. S. M. Seltzer and M. J. Berqer, Int. J. Appl. Radiat. Isot. 33, 1189

(1982).

14. M. J. Berqer and S. M. Seltzer, "Stopping Powers and Ranqes of Electrons

and Positrons" (2nd Ed.), National Bureau of Standards Report No.

NBSIR 82-2550A (1983).

15. T. A. Carlson, Photoelectron and Auger Spectroscopy (Plenum Press, New

York, 1975), Appendix 1.

16. U. Fano, Ann. Rev. Nucl. Sci. 13, 1 (1963).

17. H. Raether, Excitation of Plasmons and Interband Transitions by Electrons

( Spri nger-Ver I ag, Berlin, Heidelberg, New York, 1980).

18. D. J. Isaacson. "Positron Trappings of Defects in Solids," PhD.

Disertation, New York University (1981).

19. A. Mann and W. Brandt, Phys. Rev. B 24_, 4999 (1981).

20. J. F. Ziegler, 0. P. Biersack, and U. Littmark, Proc. of U.S. -Japan

Seminar on Charged Particle Penetration Phenomena, Oak Ridge National

Laboratory Report C0NF-8201 31 (1982), p. 88.

15
21 . R. M. Sternheimer, Phys. Rev. A15, 1817 (1977).

22. R. M. Sternheimer, Trans. New York Acad. Sci., Series II, _40, 190 (1980).

23. H. A. Bethe, Ann. Phys. (Leipzig) _5, 325 (1930).

24. F. Bloch, Z. Phys. 81, 363 (1933).

25. ICRU Report 10b, "Physical aspects of irradiation. Recommendations of

the International Commission on Radiological Units and Measurements,"

published as National Bureau of Standards Handbook 85 (1964).

26. "Report of the task group on reference man," International Commission on Radiological Protection, Pergamon Press, New York ( 1 y

/

b )

.

16
Definition of Parameters in Tables I and II

Z Atomic number

Z/A Ratio of atomic number to atomic weight

I Mean excitation energy (in eV)

p 0 Density (in g/cm 3)

hVp Plasma energy (in units eV) [Eq. (4)]

p Sternheimer adjustment factor for the atomic excitation energies

[Eqs. (3) and (8)].

-C

m

a

,

5

o

[Eq. (11)].

Parameters in fitting formulas [Eqs. (9) and (10)].

Density-effect value used as fitting parameter in Eq. (25).

A Upper bound for the error inherent in fittinq procedure. The

max y absolute value of the difference between the fitted and the numerical

value of 5 is at all enerqies 3 smaller than Amax

The composition of the compounds and mixtures in Table II,

fractions by weight of the atomic constituents, can be found in

Berger. 13 The designation (ICRU) indicates tissue compositions

International Commission on Radiation Units and Measurements, 25

designation (ICRP) indicates tissue compositions adopted by the

Commission on Radiological Protection. 26

in terms of

Seltzer and

adopted by the

and the

International

Table

I.

Density-Effect

Parameters

for

Elemental

Substances

X

*3

g=

s^ sT CM <T* sr 00 cm cm cm 0 cm cm to 0000000

00 00 so — *— 0 00 to to CO 0 CM •- c* 0 0 0 •- •— •- 0

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U.s. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report NoJ 3. Publication Date

4. TITLE AND SUBTITLE

DENSITY EFFECT FOR THE IONIZATION LOSS

OF CHARGED PARTICLES IN VARIOUS SUBSTANCES

5. AUTHOR(S)

R. M. Sternheimer, M. J. Berger and S. M. Seltzer

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

10. SUPPLEMENTARY NOTES

| 1 Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

The density-effect correction 5(B) for the ionization energy loss

of charged particles has been evaluated for a total of 278 substances

including 98 cases of elements of the Periodic Table (12 gases and 86

condensed materials, including liquid hydrogen and graphite of three

different densities) and including also 180 chemical compounds and

substances of biological interest (13 gases and 167 liquid or solid

substances). In the calculations, up-to-date values of the mean excitation potential I and of the atomic absorption edges hy. were employed as

input data for the general equations for 5(B) previously derived by

Sternheimer

.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

charged particles; compounds; density effect; elements; mean excitation

energies; stopping power.

13- AVAILABILITY

i I Unlimited

! ] For Official Distribution. Do Not Release to NTIS

i | Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

lZ Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

15. price

USCOMM-DC S043-PS0