Statistical Analyses, and rejected Data.

For several years, we have been carrying out statistical comparisons between experimental data of electronic stopping power and various tables and programs. The purpose is to determine the reliability of the tables and programs. But during these analyses we also find experimental data sets that appear unreliable und must be excluded from the analysis (see sect. E below).

A. Method.

For a certain range of Z₁, for a certain range of target atomic numbers Z₂, and for every data point in a certain range of specific energy E/A₁ (where Z₁ is the projectile’s atomic number, E its energy and A₁ its mass number), we calculate the normalized difference . Here, is the mass stopping power, ρ is the density, is the linear stopping power, and x is the path length. In every range of specific energy, we then determine the mean normalized difference and its standard deviation , using our program „Judge“ [1]. Here, signifies an unweighted average, S_exp is an experimental stopping power value taken from our collection, and S_table the corresponding value from a particular stopping power table or program. A small |Δ| usually signifies good agreement between table and data, and the standard deviation σ is related to the accuracy of the experimental data. If |Δ| is small, as we frequently find, σ may be taken as a measure of the accuracy of the table, as determined from experiment. The number of data points is also given in the tables, as an indication of the size of the data base.

The averages are unweighted, except for the data that were excluded from the analysis, i.e., given weight zero, since they were found in conflict with respect to other data of the same Z₁-Z₂-combination (where Z₂ is the atomic number of the target element). Solids and gases (i.e., substances that solid or gaseous, resp., at normal temperature and pressure) are treated separately.

B. Statistical Analysis for Protons and Alphas in Elements.

The following statistical analyses are taken from reference [2]. The data are compared to tables by Andersen and Ziegler [3]; Ziegler [4]; Janni [5]; Ziegler, Biersack and Littmark [6]; ICRU Report 49 [7]; and SRIM 2003 [8]. The latter two give the best agreement.

Table B1. Mean normalized difference Δ ± σ (in %) for H ions in 17 solid elements (these are the solid elements covered by ICRU 49)
E/A1 (MeV)	0.001 - 0.01	0.01 - 0.1	0.1 - 1.0	1 - 10	10 - 100	0.001 - 100
No. of points	207	1272	2393	1156	196	5224
AZ 77	5.5 ± 12	-1.2 ± 12	-3.4 ± 8.3	-1.1 ± 3.9	-0.7 ± 0.6	-1.9 ± 8.9
J 82	11.7 ± 12	2.1 ± 11	-1.1 ± 7.3	-0.9 ± 3.7	-0.2 ± 0.5	0.2 ± 8.4
ZBL 85	-7.0 ± 24	-1.2 ± 12	-3.0 ± 7.8	-0.3 ± 4.2	0.3 ± 2.1	-2.0 ± 9.5
ICRU 49	5.8 ± 12	0.8 ± 11	-0.7 ± 7.1	-0.2 ± 4.1	0.0 ± 0.5	0.1 ± 7.9
SRIM 2003	4.8 ± 13	0.6 ± 11	-0.9 ± 6.8	-0.6 ± 3.8	-0.1 ± 0.6	-0.2 ± 7.7

Table B2. Mean normalized difference Δ ± σ (in %) for H ions in all elemental gases except F, Cl, Rn
E/A1 (MeV)	0.001 - 0.01	0.01 - 0.1	0.1 - 1.0	1 - 10	10 - 100	0.001 - 100
No. of points	116	329	535	303	11	1294
AZ 77	-1.2 ± 6.5	-1.1 ± 5.1	-1.8 ± 4.2	-0.3 ± 2.0	-0.1 ± 0.3	-1.2 ± 4.3
J 82	-1.1 ± 9.4	-0.1 ± 4.6	0.5 ± 3.9	0.9 ± 3.2	3.2 ± 0.6	0.4 ± 4.7
ZBL 85	23 ± 13	22 ± 11	0.4 ± 6.8	-1.1 ± 1.7	-1.0 ± 0.5	7.6 ± 13
ICRU	-0.7 ± 6.5	-1.1 ± 5.0	-1.2 ± 3.7	-0.8 ± 1.6	-0.2 ± 0.5	-1.0 ± 4.1
SRIM 2003	1.7 ± 4.9	-0.1 ± 4.7	-0.4 ± 3.6	-0.2 ± 1.6	0.2 ± 0.3	-0.1 ± 3.8

Table B3. Mean normalized difference Δ ± σ (in %) for He ions in 16 elemental solids (These are the solid elements covered by ICRU 49)
E/A1 (MeV)	0 - 0.01	0.01 - 0.1	0.1 - 1.0	1 - 10	10 - 100	0 - 100
No. of points	94	942	1610	332	11	2989
Z 77	6.1 ± 25	4.8 ± 8.4	0.5 ± 5.6	0.1 ± 3.3	0.5 ± 1.0	2.0 ± 8.1
ZBL 85	19 ± 24	3.5 ± 8.1	0.7 ± 5.8	-0.5 ± 3.5	0.8 ± 2.4	2.0 ± 8.3
ICRU	4.9 ± 24	2.6 ± 7.9	0.2 ± 5.7	0.5 ± 3.4	0.9 ± 0.9	1.1 ± 7.6
SRIM 2003	10.2 ± 21	3.5 ± 7.8	0.5 ± 5.4	-0.1 ± 3.3	0.2 ± 0.9	1.7 ± 7.3

Table B4. Mean normalized difference Δ ± σ (in %) for He ions in all elemental gases except F, Cl, Rn
E/A1 (MeV)	0 - 0.01	0.01 - 0.1	0.1 - 1.0	1 - 10	0 - 10
No. of points	5	181	669	205	1060
Z 77	-0.5 ± 6.0	-1.6 ± 3.6	1.0 ± 3.3	1.6 ± 2.2	0.7 ± 3.3
ZBL 85	7.2 ± 13	2.6 ± 5.7	3.2 ± 4.3	-0.7 ± 1.5	2.4 ± 4.6
ICRU	0.5 ± 6.8	-1.4 ± 3.5	0.3 ± 3.6	0.5 ± 1.2	0.1 ± 3.3
SRIM 2003	-5.4 ± 6.1	-0.1 ± 3.2	0.3 ± 3.2	-0.2 ± 1.1	0.1 ± 3.0

Remarkably, the experimental accuracy for measurements on gases is here, on the average, twice as good as for solids.

C. Statistical Analysis for protons and alphas in Compounds.

The following comparisons with the tables from ICRU Report 49 [7] and SRIM 2003 [8] are taken from [14].

Table C1. Mean normalized difference Δ ± σ (in %) for He ions in ethylene
E/A1 (MeV)	0 – 0.03	0.03 – 0.3	0.3 – 3.0	0 – 3.0
No. of points	8	53	66	127
ICRU 49	-5.4 ± 5.1	-1.0 ± 2.4	2.7 ± 1.8	0.6 ± 3.3
SRIM 2003_26, CAB corrected	-23 ± 8.1	-3.8 ± 6.5	1.9 ± 1.4	-2.1 ± 7.7

In Table C1, there are eight different measurements for the same substance, in good agreement with each other. The large |Δ| for SRIM at low energy is evident, indicating that SRIM is too high there.

Table C2. Mean normalized difference Δ ± σ (in %) for H or He ions in about 150 compounds (CO and dimethyl sulfite omitted), compared to SRIM (CAB corrected)
Ions	Targets	E/A1 (MeV)	0 – 0.03	0.03 – 0.3	0.3 – 3.0	3 – 30	0 – 30
H	condensed	No. of pts.	62	441	817	172	1492
		Δ ± σ	-3.2 ± 15	-0.3 ± 7.5	1.5 ± 6.3	-0.3 ± 3.8	0.6 ± 7.1
	gaseous	No. of pts.	11	556	334	12	913
		Δ ± σ	2.7 ± 4.6	-0.8 ± 4.2	-0.1 ± 3.3	-0.8 ± 2.2	-0.5 ± 3.9
He	condensed	No. of pts.	61	542	1268	7	1878
		Δ ± σ	-3.3 ± 9.9	0.9 ± 6.7	-0.8 ± 4.1	-1.2 ± 3.3	-0.4 ± 5.3
	gaseous	No. of pts.	73	1111	1496	0	2680
		Δ ± σ	-16 ± 11	-1.4 ± 6.5	1.0 ± 2.8		-0.4 ± 5.7

Here, the results for H ions are rather similar to those for elements shown above. For low energy He ions in gases, there is again a large negative value Δ as in Table C1 above.

Table C3. Mean normalized difference Δ ± σ (in %) for H and He ions in 23 compounds covered by ICRU 49
E/A1 (MeV)	0 – 0.03	0.03 – 0.3	0.3 – 3.0	3 – 30	0 – 30
No. of points	116	1036	1237	135	2524
ICRU Rep. 49	0.2 ± 8.9	1.4 ± 5.9	1.3 ± 5.2	1.0 ± 4.4	1.3 ± 5.7
SRIM 2003_26, CAB corrected	-7.8 ± 12	-1.0 ± 6.4	0.4 ± 5.6	-0.6 ± 4.0	-0.6 ± 6.6

Here, the ICRU table is clearly better than SRIM.

Table C4. Mean normalized difference Δ ± σ (in %) for H ions in 20 gaseous hydrocarbon compounds, with respect to two SRIM calculations
E/A1 (MeV)	0 – 0.03	0.03 – 0.3	0.3 – 3.0	3 – 30	0 – 30
No. of points	0	371	190	4	565
SRIM 2003, Bragg		3.0 ± 4.4	3.1 ± 2.5	-0.1 ± 1.0	3.0 ± 3.9
SRIM 2003, CAB, g		-1.1 ± 4.4	-0.7 ± 3.2	0.2 ± 1.3	-1.0 ± 4.0

Table C4 shows the positive effect of the CAB correction (which is very hard to discern generally): the corrections decrease Δ by 4 % and bring SRIM very close to the data.

D. Statistical Analysis for ions from ₃Li to ₁₈Ar.
The following comparisons with the tables MSTAR [9, 10], SRIM 2003 [8], and ICRU Report 73 [11] have been taken from [12]. Comparisons with additional tables can be found in [11]. Separate results for the various ions (as compared to MSTAR) can be found in [10].

Table D1. Mean normalized difference Δ ± σ (in %) for ions from 3Li to 18Ar in the elemental solids covered by ICRU 73.
E/A1 (MeV)	0.025 - 0.1	0.1- 1	1 - 10	10 - 100	100-1000	0.025-1000
No. of points	1399	3452	1262	175	11	6299
MSTAR v.3, mode b	2.5 ± 9.9	0.1 ± 7.3	0.8 ± 5.5	0.1 ± 2.2	0.7 ± 1.4	0.8 ± 7.6
SRIM 2003.26	1.3 ± 9.7	-0.9 ± 7.0	-0.3 ± 5.6	-1.6 ± 2.9	-0.1 ± 1.6	-0.3 ± 7.4
ICRU 73	-11.4 ± 20	-6.8 ± 12	-3.0 ± 6.6	-0.8 ± 3.0	-0.8 ± 1.9	-6.9 ± 13

Table D2. Mean normalized difference Δ ± σ (in %) for ions from 3Li to 18Ar in aluminum oxide, kapton polyimide, polycarbonate (makrolon), polyethylene, polyethylene terephthalate (mylar), polypropylene, polyvinyl chloride, silicon dioxide, toluene, and water (liquid)
E/A1 (MeV)	0.025 – 0.1	0.1 – 1	1- 10	10 - 100	0.025-100
No. of points	133	586	368	13	1100
MSTAR v. 3, mode b	6.6 ± 10.4	1.6 ± 6.3	5.2 ± 4.0	0.0 ± 1.3	3.4 ± 6.6
SRIM 2003.26	-0.8 ± 8.3	-0.1 ± 5.2	-0.4 ± 5.0	-2.3 ± 1.7	-0.3 ± 5.6
ICRU 73	-11 ± 12	-2.1 ± 7.4	-1.0 ± 5.1	-0.5 ± 1.4	-2.8 ± 8.1

Table D3. Mean normalized difference Δ ± σ (in %) for ions from 3Li to 18Ar in all gases covered by MSTAR and ICRU 73 for which we have data.
E/A1 (MeV)	0.025 – 0.1	0.1 – 1	1- 10	10 - 100	0.025-100
No. of points	167	190	551	189	1097
MSTAR v. 3, mode b	-2.5 ± 10.3	-2.2 ± 13	0.2 ± 3.8	0.7 ± 2.4	-0.5 ± 7.3
SRIM2003.26	3.0 ± 10.1	-7.7 ± 12	-0.4 ± 5.2	-2.2 ± 3.9	-1.4 ± 8.1
ICRU 73	-50 ± 28	-2.9 ± 16	-2.0 ± 10.5	-0.1 ± 3.8	-9.1 ± 23

Evidently, MSTAR and SRIM describe the data about equally well. For the ICRU table, the agreement at low energy is generally worse.

E. Rejected or omitted data.

These data were rejected because of obvious discrepancies with other data for the same Z₁ – Z₂ – combination.

Table E1. Rejected proton and alpha data from [2], with later additions that include also some compounds. June, 2008
Z1	Target name/File no.	Reason for rejection (or omission)	Ref.
1	Ag.003	low compared to many others	Wa49
	Ag.011, Au.024, Cu.010	low	No75
	Au.053, Pd.003	wrongly rejected before June, 2008	Vs00
	C.018, C.019	5 - 10% high compared to others	Op75
	Ce.002, Yb.003	much lower than Kn80 (“obviously incorrect” acc. to Kn80)	Si72
	Cu.031	very low	Gt62
	D2Oc.001	Temporarily rejected (low compared to tables)	Ad77
	H.008, He.006	low compared to many others	Cr42
	He.011, He.012	Uncertainty about threshold effect	Gl91, RG01
	LiF.003, 004	Temporarily rejected awaiting new Bauer data	Mö04
	N.017	solid gas	Bö82a
	Nb.002	low compared to Si84, Bi86	Bh73
	Si.001	very low	Ar69
	Si.014	low	Gm76
	Ta.008	low compared to Lu79, Si84, etc.	Si72
	Ti.004, Ti.005	high compared to Or71	Gt62
	Ti.006	high compared to Or71	Ar69
	Al2O3.007	strange results with very large stated errors	Rt72
	GaSb.001	25 % error	Hl74
	LiF.001	60% too high according to P. Bauer	Ed97
	SiC.001	Data for O and Al ions low w.r.t. Zha03b	Js04
	ZnTe.002	Low compared to ZnTe.001; uncertain density required for conversion from linear stopping power	BL74
2	[Cr.06,Cu.18, Mo.08, Ni.22]	Based on ranges (5 - 100 keV). The stopping values go down to 0.01 keV, but these are not really measured. Rather, they are extrapolated down from 100 keV using the shape of SRIM 95 stopping. Data not rejected, but replaced by reevaluated values from 5 to 100 keV.	Sp98
	Ag.26	low compared to Gt62, Th81	No75
	Ag.24	very high compared to Gt62, Th81	Te57
	Air.04, CO2.05, He.08	Data differ markedly from other similar data	Hb72
	Au.26, C.14	high compared to many others	Pe81
	Au.33	low compared to Bl80, Th80, Kr82	No75
	H2Ov.01	Apparently replaced by Pl80	Pl78
	Ne.06	Too steep compared to others	Fu99
	Ta2O5.01, SiO2.04	Off by large factors	SB76
	ZnTe.01	Uncertain density, and discrepancy with ZnTe.02 (PH77)	BL74
	Targets CO and Dimethyl sulfite were omitted from statistical analysis because of very large Bragg corrections in SRIM; the large Bragg correction for SF6 was set to zero.

Table E2. Rejected or omitted heavy ion data from [1], with later additions to the original list. 20 Oct 2010
Ion	Target.File-number	Reason for rejection (or omission)	Ref.
238U	Air.1, He.3, Kr.4	Differentiated range-energy curve; strange shape; large stated errors	Bez75
63Cu	H2.2, N2.2	Two single points from new ITEP setup; large stated errors	Fer06
58Ni	Cu.3	Data unusually low	Ay81b
40Ar	Au.8	low by a factor 2 – 3 compared to Sc82 (and Wr79)	Nd77
32S	Au.2	high by a factor 2 – 3 w.r.t. Sd75, Fs76, Am68 (error of Bt66: 25%)	Bt66
32S	Ag.2	low by a factor 1.5 – 2 w.r.t. Fs76 (error of Bt66: 25%)	Bt66
32S	Ni.2	in analogy, to avoid large discrepancies	Bt66
28Si	Au.2	In analogy to other Nd77 data (see Table B of [1])	Nd77
24Mg	Ag.3,Au.2,Cu.1,Fe.2,Mo.1,Pt.1, Ti.1,W.1	omitted (see p. 308 of [1])	At90
24Mg	Co.1,Hf.1,Nb.1,Pd.1, Re.1,V.1	In analogy, although not covered by MSTAR (8 Jul 03)	At90
24Mg	Ni.3	in analogy, to avoid large discrepancies, see p. 13/3	At90
24Mg	Ta.1	in analogy, p. 17/5	At90
26Mg	Ge.1, Si.1	omitted (see text by Paul I, p. 308)	At91
26Mg	Ta.2	very similar to 24MgTa.1 (At90)	Ku91
20Ne	Al.5, Al.8	high energy points too low compared to Po61, Sha73 and Ang00	Tp62
20Ne	Au.3	In analogy to other Nd77 data (see Table B of [10]	Nd77
16O	Ag.14	high compared to BG65, Sk86, Am68, Wr72	Bt66
16O	Au.11	high w.r.t. Ku88, BG65, Sk90, Am68	Sd74
16O	Au.15	too steep, in part too high w.r.t. Wr79, Po60, Ab93, Sa92	Nd77
14N	Au.11	too steep, in part too high w.r.t. Wr79, Sa91, Sc82, Po61, Ld85	Nd77
14N	CH4.1	In analogy to some other Tp62 data	Tp62
15N	Ar.6, He.5	high compared to And69 data for Ar, Ef75 for N2, Rl60 for O2, and Tp62 for air and Ar targets (p. 169 of [13])	Pr93
14N	He.1, Kr.1, Ne.2, Xe.1
12C	Au.9	In analogy to other Nd77 data	Nd77
12C	W.1	Too high as seen by statistical analysis (Judge)	Ant91
11B	Al.2, Al.3	low compared to Rä91,Zh98a	Tp62
11B	CH4.1	In analogy to some other Tp62 data	Tp62
7Li	Ag.6	too low compared to Se90, Sa84b, Li86	Tp62
7Li	Cu.5	high; apparently replaced by Me80 (which is in good agreement with An80)	Me79
7Li	Air.3, Ar.4, H2.3, He.4	Low compared to other comparable data, especially to An78	All56
7Li	CH4.1	In analogy to some other Tp62 data	Tp62
7Li	W.1	Too high as seen by statistical analysis (Judge)	Ant91
2<Z1<27	Si	Data shown on figures for Li, B, C, N, O, Si, P are all low compared to others.	Whl02b

REFERENCES.

[1] H. Paul and A. Schinner, "An empirical approach to the stopping power of solids and gases for ions from ₃Li to ₁₈Ar, Nucl. Instr. Meth. Phys. Res. B 179 (2001) 299

[2] H. Paul and A. Schinner, “Judging the reliability of stopping power tables and programs for protons and alpha particles using statistical methods”, Nucl. Instr. Methods B 227 (2005) 461

[3] H.H. Andersen and J.F. Ziegler, The Stopping and Ranges of Ions in Matter, Vol. 3, Pergamon, New York, 1977

[4] J.F. Ziegler, Helium: Stopping Power and Ranges in all Elemental Matter, The Stopping and Ranges of Ions in Matter, Vol. 4, Pergamone, New York, 1977

[5] J.F. Janni, Atomic Data Nucl. Data Tables 27 (1982) 147

[6] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Ranges of Ions in Matter, Vol. 1, Pergamon, New York, 1985

[7] ICRU Report 49, International Commission on Radiation Units and Measurements, Bethesda, MD, USA, 1993

[8] SRIM 2003, obtained from http://www.srim.org. The more recent program SRIM 2006 yields the same stopping powers

[9] A. Schinner and H. Paul, Program MSTAR v. 3 (2003), see this internet site

[10] H. Paul and A. Schinner, “Empirical stopping power tables for ions from ₃Li to ₁₈Ar and from 0.001 to 1000 MeVnucleon in solids and gases”, Atomic Data Nucl. Data Tables 85 (2003) 377

[11] ICRU Report 73, International Commission on Radiation Units and Measurements, J. ICRU 5 (1) (2005)

[12] H. Paul, "A comparison of recent stopping power tables for light and medium-heavy ions with experimental data, and applications to radiotherapy dosimetry", Nucl. Instrum. Methods B 247 (2006) 166

[13] H. Paul and A. Schinner, "An empirical approach to the stopping power of solids and gases for ions from ₃Li to ₁₈Ar, Part II, Nucl. Instr. Meth. Phys. Res. B 195 (2002) 166

[14] H. Paul and A. Schinner, "Statistical analysis of stopping data for protons and alphas in compounds", Nucl. Instrum. Methods B 249 (2006) 1