Biomedical Engineering Reference
In-Depth Information
a
b
Fig. 15.2 (a)
Experimental and fitted OELF of liquid water. Continuous and dashed lines represent
the extended Drude model fitting to the IXSS data (squares) [
16
] and REF measurements (circles)
[
15
], respectively.
(b)
At high energy transfers we obtain the OELF from the FFAST database
of NIST for the water molecule (triangles) [
40
], and from the x-ray scattering factors of H and
O (crosses) [
41
]; the solid line represents the extended Drude model fitting to the IXSS data, to
which we have added the contribution from the oxygen K-shell electrons through their GOS after
the K-shell binding energy .E
K
D
540 eV/. See the text for more details
15.3.2
Extension algorithms at
k ¤ 0
of the valence excitation
spectrum
In order to use Eqs.
15.2
and
15.3
for calculations of the stopping power or the
energy-loss straggling, the ELF must be known for arbitrary momentum-transfer
(i.e. k
¤
0). However, experimental data for the ELF of liquid water at k
¤
0 is
only available from the Sendai group [
42
,
43
] in the range 0:19
3:59 a.u.
Current theories overcome the problem of describing the ELF over the complete
Bethe surface by extrapolating the optical data through a suitable extension algo-
rithm. In the simplest case, an extension algorithm is a dispersion relation, i.e. an
analytic expression of energy-transfer as a single-valued function of momentum-
transfer. In what follows we present a summary of the most used extension
algorithms for the calculation of the OELF at finite k.
k
15.3.2.1
Methods based on the Drude dielectric function
A widely used methodology for the extension algorithm of the ELF to the
whole energy-momentum region .k; !/, was proposed by Ritchie and Howie [
17
].
These authors suggested to incorporate non-zero momentum-transfers through the
k
-dependence of the parameters W
i
.k/ and
i
.k/:
