Method

The mean energy loss of the heavy ions can be expressed as: $\Delta E\; =\; Q2S$_p( {TA}) ρt

where

Q

charge of the ion;

$S$_p( {TA})

energy loss by the a proton with kinetic energy $T$_p= T/A ;

T,A

kinetic energy and mass number of the ion;

$\rho$

density of the medium;

t

step length

This formula is used to calculate the ionisation loss for all the charged hadrons.

In the case of ions, the charge of the ion makes the problem more difficult. Electron-exchange processes with the atoms change the charge as the ion traverses the medium. The main features of this process can be summarised as follows:

• Q loses the dependence on the initial ion charge after a very short initial step, usually $\ll 1\mu$ for condensed media;
• Q does depend on the velocity of the ion;
• even in the case of constant velocity, Q fluctuates around a mean value;
• Q features a small dependence on the medium, especially for small values of T.

 

T/A	He ions	Pb ions
MeV	error ()	error ()
10	3	1
5	4	3
3	10	4

Table: Comparison of the measured and calculated $\Delta$ E for 10 MeV/A ions in Pb

If we want to calculate the energy loss of the ion, we need a good parametrisation for the quantity $Q=Q$_eff . We give here a relatively simple parametrisation [,]:

$Q$eff	$=$	$Z$I[ 1- ( 1.034 - 0.1777 e-0.08114 ZI) e-v]
$v$	$=$	$121.4139\; \{\beta Z$I2/3}+ 0.0378sin( 190.7165 {βZI2/3})

where $\beta$ is the velocity and $Z$_I the atomic number of the ion (i.e. the charge of the bare nucleus).

 

1|c|T (MeV)	6c|417 $\mu$ g $cm-2$	6c|110 $\mu$ g $cm-2$
	4c|MC (keV)	2c|data (keV)	4c|MC (keV)	2c|data (keV)
	$\Delta$ E	dif	FWHM	dif	$\Delta$ E	FWHM	$\Delta$ E	dif	FWHM	dif	$\Delta$ E	FWHM
4.88							785	8	59	-23	725	77
9.85	3600	14	138	-33	3150	206	816	8	67	-26	756	91
19.79	3090	3	142	-35	2990	218	719	3	70	-32	699	103
29.27	2660	1	141	-31	2630	204
29.75							620	4	69	-26	598	93
39.70	2320	1	138	-28	2300	191	550	4	68	-24	528	90

Table: Comparison of the measured and calculated $\Delta$ E and FWHM for O ions in Al; errors are in percent.

It can be seen that () neglects the (small) medium dependence of $Q$_eff . For very high energies ($\beta \to 1$ ) $Q$_eff→Z_I . For very low energies ($T/A\; \sim$ few keV) the formula breaks down. $Q$_eff can even become negative for keV and $Z$_I> 20 . However this is not a serious source of error when calculating $\Delta E$ , since in this case the range of the ion is very small, and it can almost be said that it stops immediately.

The calculation of the energy loss straggling (fluctuations) differs from that of normal charged hadrons. For the charged hadrons the fluctuations come from the statistical nature of the projectile-atom interactions; for the ions there is another process which broadens the energy loss distribution: the fluctuation of the charge. For heavier ions this process dominates the energy loss straggling for $T/A\; \le 10$ MeV.

Figure: Stopping powers in Carbon

The heavy ions are in the Gaussian regime (see [PHYS332]) of the collisional fluctuations even in the case of very thin absorbers. If $T/A$ is not too high, the $\sigma 2$ of the distribution: $\sigma$_coll²= D {Q_eff²ZA}ρ t  m[ 1 + {T_AM_u}+ {12}( {T_AM_u}) ²]

where

D

$0.307\; \{MeV\; cm2g\}$ ;

$T$_A

$\{TA\}$ ;

$M$_u

atomic mass unit;

Z

atomic number of the medium;

A

mass number of the medium.

 

	9c|Pb ions in gas (energies in MeV)
	3c|$N$2	3c|Ar	3c|Xe
	2cMonteCarlo	1c|data	2cMonteCarlo	1c|data	2cMonteCarlo	1c|data
t (cm)	$\Delta$ E	FWHM	FWHM	$\Delta$ E	FWHM	FWHM	$\Delta$ E	FWHM	FWHM
0.2	25.9	1.15	1.1	25.4	1.30	1.4	51.1	2.26	2.5
0.4	53.6	1.62	1.5	52.4	1.84	1.9	107.0	3.22	3.3
0.8	112.0	2.25	2.0	111.0	2.59	2.6	236.0	4.51
1.2	162.0	2.37	2.0	168.0	2.95	2.8

Table: Comparison of the measured and calculated and FWHM for 1.4 MeV/A Pb ions in gas.

Analysing the experimental straggling data it is possible to find that the electron-exchange charge fluctuations can be described by a Gaussian with width: $\sigma$_ch²= D {Q_eff²ZA}ρ t  m {C2}( 1 - {Q_effZ_I})

where the parameter $C\; \sim 2.5$ has been derived from the experimental straggling data.

If $Q$_eff→Z_I , which is the case for high energy heavy ions and for few MeV/A He ions, then $\sigma$_ch²→0 .

Comparing equations () and () it can be seen that for heavy ions and for $T/A\; \ll M$_u $\sigma$_ch²> σ_coll² . The total energy loss fluctuation can be described by a Gaussian distribution with: $\sigma 2=\; \sigma$_ch²+ σ_coll²

The mean energy loss and energy loss fluctuation calculation is performed in the routine GTHION, making use of the proton energy loss tables.

Note: The Gaussian fluctuation gives too broad a distribution for high energy in the case of thin absorbers. A correction has been introduced in GTHION which cures this discrepancy. In the absence of high energy straggling data for ions, the correction has been tuned using high energy $\pi$ energy loss data, where the $\pi$ has been tracked by GTHION.