
906 Charged Particle and Photon Interactions with Matter
31.4.3.1 interaction of slow ions with matter
The interaction mechanism for recoil ions of a few tens of keV energy with matter is considerably
different from that of fast ions described in Section 31.2. The nuclear stopping, S
nc
, which can be
ignored in fast ion collisions, becomes comparable to or larger than the electronic stopping, S
el
. The
total stopping, S
T
, can be expressed as the sum of the two. Apart from the difculty in obtaining
values for S
nc
and S
el
for slow collisions, a simple relation, S
T
= S
nc
+ S
el
, produces another complex-
ity. The interaction of slow ions with matter has been extensively studied by Lindhard etal. (1963).
The nuclear process follows the usual procedure of a screened Rutherford scattering. The electronic
interaction is based on the Thomas–Fermi model. S
e
is expressed as (dε/dρ)
e
= kε
1/2
, where ε and ρ
are the dimensionless reduced energy and reduced range. When the projectile and the target are the
same element (Z
1
= Z
2
; indices 1 and 2 stand for the projectile and the target atom, respectively), k
is
given by the atomic number Z and mass A, as follows:
k Z A=
−
0 133
2
2 3
2
1 2
.
/ /
(31.18)
The secondary ions will enter into the collision processes again, and this will go on. After the
cascade processes for stopping collisions take place, most of the ion energy is spent in the atomic
motion, ν, and wasted as heat in ordinary detectors. Only the energy η is utilized in electronic exci-
tation and contributes to the ionization or scintillation. The ratio η/ε is called the nuclear quenching
factor, q
nc
, or the Lindhard factor. The asymptotic equation for ν (=ε − η) for recoil ions in a homo-
nuclear
medium (Z
1
= Z
2
) is given as follows (Lindhard etal., 1963):
ν
ε
ε
=
+ ⋅1 k g( )
(31.19)
An
approximate expression for g(ε) is given by (Lewin and Smith, 1996)
g( ) .
. .
ε ε ε ε= + +3 0 7
0 15 0 6
(31.20)
Once a value for q
nc
is obtained, it can be used for the study of the track structure and quenching.
The electronic LET (LET
el
= −dη/dR), that is, the electronic energy deposited per unit length along
the ion track, can be an important parameter to consider the radiation effects due to slow recoil
ions (Hitachi, 2005, 2008). The value of LET
el
obtained for recoil ions is close to the LET value for
α-particles. This makes α-particles a good measure for decay times, the S/T ratio, etc., for recoil
ions,
as discussed in Section 31.2.
The
excitation density in the recoil ion track can be as high as that in an α-track; therefore,
scintillation quenching, q
el
, is expected, as discussed in Section 31.2. The total quenching factor,
q
T
, is expressed as q
nc
× q
el
. The track structure can be considered by using LET
el
and ion velocity.
Then, the value of q
el
is estimated as discussed in Section 31.2. We have T = q
nc
E in Equation 31.9.
Most of the δ rays produced by recoil ions do not have sufcient energy to escape from the core
and form an undifferentiated core, that is, T
c
/T = 1 for recoil ions. The quenching factors calculated
for recoil ions in LAr and LXe (Hitachi, 2005) are shown in Figure 31.25 together with reported
experimental values (Arneodo etal., 2000; Bernabei et al. 2002; Akimov etal., 2002; Aprile etal.,
2005; Chepel etal., 2006).
Practically, the recoil ion-to-γ ratio, RN/γ, is a good parameter for the design and development of
a detector for dark matter searches, and it can be expressed as (Hitachi, 2005)
RN q q
Lγ
γ
=
nc el
⋅
(31.21)