Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 20

excitations is nonlinear. Here, nonlinearity of the FPT can turn into an
advantage through the possibility of diminishing the influence of noise on the
reconstructed data.
Before proceeding to detailed applications of the FPT, some basic def
initions with a brief, but thorough discussion of pertinent aspects of the
quantummechanical origin of resonances will be given next.

1.2 The quantum-mechanical concept of resonances in
scattering and spectroscopy
When two nuclei collide with each other, certain particle transformations
known as nuclear transmutations can occur. For instance, in the case of
scattering of proton ( 1 H) with carbon ( 1 6 C) or aluminum ( 2 13 Al), the follow
ing nuclear reactions termed radiative capture are possible with emission of
photons (γ)

1 H + 1 6 C−→ 1 7 N ∗ −→ 1 7 N + γ (1.1)
1 H + 2 13 Al−→ 2 14 Si ∗ −→ 2 14 Si + γ. (1.2)
In the collisional channel 1 H + 1 6 C−→ 1 7 N ∗ from process (1.1), or in the
similar part 1 H + 2 13 Al−→ 2 14 Si ∗ from (1.2), the socalled compound nucleus
of nitrogen ( 1 7 N ∗ ) or silicon ( 2 14 Si ∗ ) can be created in an excited state as
formally denoted with the star superscript. Such nuclear excited states are
intrinsically unstable and, therefore, they will decay through radiative de
excitations like

1

N + γ or 2 14 Si ∗ −→ 2 14 Si + γ, as per (1.1) or (1.2),

13
7

N ∗ −→ 1 7

respectively.
It is a wellestablished fact that at a fixed incident energy E, cross sections
Q(E) for resonant scattering of particles are adequately described by the
one and manylevel BreitWigner (BW) formulae [90]. This latter expression
appears as either a single Lorentzian

∝ Γ 1 /4
(E−E 1 ) 2 + Γ 1 /4

Q(E)≈Q 1 (1.3)

or a linear combination of any number of such Lorentzians

Q(E)≈Q K ∝ K

Γ k /4
(E−E k ) 2 + Γ k /4

(1.4)

k=1

where E k is the resonance energy and Γ k is the resonance width 2 . In its

2 Throughout this topic, we will use the units in which the reduced Planck constant is equal
to unity ( ~ = 1) such that the energy and frequency are permitted to be interchangeably
employed as synonyms, E = ~ ω = ω.