Collision stopping power

EGSnrc ``inherits'' the treatment of restricted collision stopping powers from EGS4, i.e. uses the formulas recommended by Seltzer and Berger [61] which are based on the Bethe-Bloch theory [62,63,64]. The standard treatment (see also ICRU report 37 [50] which was used as the source of the formulas below) assumes that there is a certain value for energy transfer to atomic electrons, $T_{\rm med}$, that is (i) large compared to the binding energies (ii) corresponds to an impact parameter that is large compared to the atomic dimensions. Collision processes that are associated with energy loss $T'$ less than $T_{\rm med}$ are treated according to the theory of Bethe, the main result of which is that

$$L_{\rm coll}^\pm(T,T' < T_{\rm med}) = {2 \pi r_0^2 m n \over \be... ... \beta^2 T_{\rm med} \over (1 - \beta^2) I^2 \right) - \beta^2 - \delta \right]$$ (4.6.1)

where $\beta$ is again the electron velocity in units of the speed of light, $I$ is the mean ionization energy and $\delta$ the density effect correction that takes into account the polarization of the medium due to the electron field. As $T_{\rm med}$ is defined being large compared to the binding energies of the atom, collision processes with energy transfer larger than $T_{\rm med}$ can be treated using the Møller [53] (electrons) or Bhabha [54] (positron) cross section (see also section 2.4.3), i.e.

$$L_{\rm coll}^\pm(T,T' > T_{\rm med}) = \int\limits_{T_{\rm med}}^{T_c} {\rm d}T' T' {{\rm d} \sigma_{\rm inel}^\pm \over {\rm d}T'}$$ (4.6.2)

Using some additional approximations, $T_{\rm med}$ drops out in the sum of $L_{\rm coll}^\pm(E,T' < T_{\rm med})$ and $L_{\rm coll}^\pm(E,T' > T_{\rm med})$ and one obtains

$$L_{\rm coll}^\pm(T,T_c) = {2 \pi r_0^2 m n \over \beta^2} \left[ \ln {T^2 \over I^2} + \ln(1 + \tau/2) + G^\pm(\tau) - \delta \right]$$ (4.6.3)

where $\tau = T/m$ and the functions $G^\pm$ are different for electrons and positrons due differences in the Møller and Bhabha cross sections and are given by

$$\begin{split}G^-(\tau) & = -1 - \beta^2 + \ln\left[ \eta (1 -... ...ver 3} \eta^3 - {y^3 \tau^3 \over 4} \eta^4 \right] \end{split}$$ (4.6.4)

where $\eta = T_c/T$ and $y$ is defined in Eq. (4.3.10).

From the above discussion and from the general discussion of a Class II condensed history implementation in section 2.4.1 it is clear that the formalism used to treat inelastic collisions with atomic electrons is only applicable if

$$T_c \gg ~~ .$$ (4.6.5)

This imposes a rather severe limitation on the use of Class II condensed history codes in high-$Z$ materials (the $K$-shell binding energy for lead is, for instance, 88 keV). We are therefore currently investigating a more realistic approach for situations when the condition (4.6.5) is not satisfied, but its implementation into EGSnrc is left for the next release of the system. One should probably also mention that the many successful studies carried out with the EGS4 system indicate that the implications of violating the requirement (4.6.5) are perhaps less severe than one might expect from purely theoretical arguments.

The only non-trivial parameters of the restricted stopping power formula are the mean ionization energy $I$ and the density effect correction $\delta$. The default mean ionization energies for elements used in PEGS4, along with atomic numbers, weights, chemical symbols, and mass densities are summarized in Table 2. Mean ionization energies for compounds are derived from

$$\ln I = \sum p_i Z_i \ln I(Z_i)$$ (4.6.6)

where $p_i$ is the stoichiometric index of the $i$'th element which has atomic number $Z_i$ and a mean ionization energy $I(Z_i)$, unless the material belongs to a set of pre-defined materials to be found in Table 2.13.2 of the EGS4 manual [12] or listed in the BLOCK DATA section of pegs4.mortran.

The density effects correction has been treated extensively in the literature. The default PEGS4 approach is based on the formulation of Sternheimer and Peierls[65] which basically parameterizes the density effect in terms of 6 parameters (AFACT, SK, X0, X1, CBAR, and IEV). This same parameterized approach was used for the calculations by Berger and Seltzer [66] and by Sternheimer, Berger and Seltzer [67] who fitted the parameters to the density effect as calculated for the ICRU for increasingly larger numbers of materials. The power of PEGS4 is that it will generate a density effect for any arbitrary material, if need be with no recourse to the fitted parameters. In this case it will use the Sternheimer and Peierls[65] general formula. There is also an option in PEGS4 which allows the 6 parameters to be read in directly (using the ISSB=1 option in PEGS4, see section 6.2 page ). However, these parameterizations are only fits to the actual density effect data. An option was added to PEGS4 in 1989 which allowed the density effect data to be used directly[68] and the EGSnrc distribution includes a huge data base of all the density effect values calculated by Seltzer and Berger for ICRU Report 37[50]. To turn this option on the user must set the flag EPSTFL to 1 in the PEGS4 input file. See section 6.1.2 (page ) for more details. In this case the density effect correction is calculated by interpolation from pre-computed values stored in a separate file. Selecting this option has the additional effect that the mean ionization energy of the material is taken directly from the density effect correction file. It should be noted that in general the differences in collision stopping powers between using the default Sternheimer and Peierls density effect, or the fitted parameters or the direct ICRU 37 values, are small, of the order of a few percent at most. These differences are only likely of importance for very detailed work.

Table 2: Default atomic numbers, symbols, atomic weights, mass densities, and I values for elements in PEGS4.

Z	Symbol	Atomic	Density	I(eV)
		weight	(g/cm$^3$)
1	H	1.00797	0.0808	19.2
2	HE	4.00260	0.1900	41.8
3	LI	6.93900	0.5340	40.0
4	BE	9.01220	1.8500	63.7
5	B	10.81100	2.5000	76.0
6	C	12.01115	2.2600	78.0
7	N	14.00670	1.1400	82.0
8	O	15.99940	1.5680	95.0
9	F	18.99840	1.5000	115.0
10	NE	20.18300	1.0000	137.0
11	NA	22.98980	0.9712	149.0
12	MG	24.31200	1.7400	156.0
13	AL	26.98150	2.7020	166.0
14	SI	28.08800	2.4000	173.0
15	P	30.97380	1.8200	173.0
16	S	32.06400	2.0700	180.0
17	CL	35.45300	2.2000	174.0
18	AR	39.94800	1.6500	188.0
19	K	39.10200	0.8600	190.0
20	CA	40.08000	1.5500	191.0
21	SC	44.95600	3.0200	216.0
22	TI	47.90000	4.5400	233.0
23	V	50.94200	5.8700	245.0
24	CR	51.99800	7.1400	257.0
25	MN	54.93800	7.3000	272.0
26	FE	55.84700	7.8600	286.0
27	CO	58.93320	8.7100	297.0
28	NI	58.71000	8.9000	311.0
29	CU	63.54000	8.9333	322.0
30	ZN	65.37000	7.1400	330.0
31	GA	69.72000	5.9100	334.0
32	GE	72.59000	5.3600	350.0
33	AS	74.92160	5.7300	347.0
34	SE	78.96000	4.8000	348.0
35	BR	79.80800	4.2000	357.0
36	KR	83.80000	3.4000	352.0
37	RB	85.47000	1.5300	363.0
38	SR	87.62000	2.6000	366.0
39	Y	88.90500	4.4700	379.0
40	ZR	91.22000	6.4000	393.0
41	NB	92.90600	8.5700	417.0
42	MO	95.94000	9.0100	424.0
43	TC	99.00000	11.5000	428.0
44	RU	101.07000	12.2000	441.0
45	RH	102.90500	12.5000	449.0
46	PD	106.40000	12.0000	470.0
47	AG	107.87000	10.5000	470.0
48	CD	112.40000	8.6500	469.0
49	IN	114.82000	7.3000	488.0
50	SN	118.69000	7.3100	488.0
51	SB	121.75000	6.6840	487.0
52	TE	127.60000	6.2400	485.0
53	I	126.90440	4.9300	491.0
54	XE	131.30000	2.7000	482.0
55	CS	132.90500	1.8730	488.0
56	BA	137.34000	3.5000	491.0
57	LA	138.91000	6.1500	501.0
58	CE	140.12000	6.9000	523.0
59	PR	140.90700	6.7690	535.0
60	ND	144.24001	7.0070	546.0
61	PM	147.00000	1.0000	560.0
62	SM	150.35001	7.5400	574.0
63	EU	151.98000	5.1700	580.0
64	GD	157.25000	7.8700	591.0
65	TB	158.92400	8.2500	614.0
66	DY	162.50000	8.5600	628.0
67	HO	164.92999	8.8000	650.0
68	ER	167.25999	9.0600	658.0
69	TM	168.93401	9.3200	674.0
70	YB	173.03999	6.9600	684.0
71	LU	174.97000	9.8500	694.0
72	HF	178.49001	11.4000	705.0
73	TA	180.94800	16.6000	718.0
74	W	183.85001	19.3000	727.0
75	RE	186.20000	20.5300	736.0
76	OS	190.20000	22.4800	746.0
77	IR	192.20000	22.4200	757.0
78	PT	195.08000	21.4500	790.0
79	AU	196.98700	19.3000	790.0
80	HG	200.59000	14.1900	800.0
81	TL	204.37000	11.8500	810.0
82	PB	207.19000	11.3400	823.0
83	BI	208.98000	9.7800	823.0
84	PO	210.00000	9.3000	830.0
85	AT	210.00000	1.0000	825.0
86	RN	222.00000	4.0000	794.0
87	FR	223.00000	1.0000	827.0
88	RA	226.00000	5.0000	826.0
89	AC	227.00000	1.0000	841.0
90	TH	232.03600	11.0000	847.0
91	PA	231.00000	15.3700	878.0
92	U	238.03000	18.9000	890.0
93	NP	237.00000	20.5000	902.0
94	PU	242.00000	19.7370	921.0
95	AM	243.00000	11.7000	934.0
96	CM	247.00000	7.0000	939.0
97	BK	247.00000	1.0000	952.0
98	CF	248.00000	1.0000	966.0
99	ES	254.00000	1.0000	980.0
100	FM	253.00000	1.0000	994.0