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Y. Alhassid
4 Experimental Methods
The measurement of level densities is a challenging task. There are several methods
but all have systematic uncertainties and are limited to certain energy regimes:
• Level counting at low excitation energies. This requires the knowledge of a
complete set of measured energy levels [9].
• Neutron and proton resonance data [10] provide an estimate of the level density
at the neutron or proton threshold energy. The measured resonance level spacing
(usually s wave and sometimes also p wave) provides the level density at certain
values of the spin/parity determined by the selection rules. The conversion to
total densities requires a model for the spin distribution, and often a spin-cutoff
model with rigid-body moment of inertia is used.
• Particle evaporation spectra [11], which depend on the level density through the
Hauser–Feshbach formalism [1]. This method requires the knowledge of particle
transmission coefficients, which can be calculated from optical potential models.
• The “Oslo method” which uses the measured particle and γ -ray coincidence
matrix [12]. The extraction of level densities in this method requires the
knowledge of level counting data at low energies and neutron resonance data.
Progress has often been achieved by combining several of these methods.
5 Empirical Models
Several phenomenological models have been introduced to describe level densities
in the presence of correlations.
5.1 Back-Shifted Fermi Gas Formula
Pairing correlations and shell effects are empirically taken into account in Bethe’s
formula by shifting the ground-state energy by a back-shift parameter
ρ(E x ) =
√
π
12
a
−1/4 (E x − )
−5/4 e
2
√
a(E x −) .
(18)
This back-shifted Bethe formula for the state density includes two parameters a and
that can be treated as adjustable parameters. They can for example be determined
from level counting data at low excitation energies and neutron resonance data [13,
14].
Y. Alhassid
4 Experimental Methods
The measurement of level densities is a challenging task. There are several methods
but all have systematic uncertainties and are limited to certain energy regimes:
• Level counting at low excitation energies. This requires the knowledge of a
complete set of measured energy levels [9].
• Neutron and proton resonance data [10] provide an estimate of the level density
at the neutron or proton threshold energy. The measured resonance level spacing
(usually s wave and sometimes also p wave) provides the level density at certain
values of the spin/parity determined by the selection rules. The conversion to
total densities requires a model for the spin distribution, and often a spin-cutoff
model with rigid-body moment of inertia is used.
• Particle evaporation spectra [11], which depend on the level density through the
Hauser–Feshbach formalism [1]. This method requires the knowledge of particle
transmission coefficients, which can be calculated from optical potential models.
• The “Oslo method” which uses the measured particle and γ -ray coincidence
matrix [12]. The extraction of level densities in this method requires the
knowledge of level counting data at low energies and neutron resonance data.
Progress has often been achieved by combining several of these methods.
5 Empirical Models
Several phenomenological models have been introduced to describe level densities
in the presence of correlations.
5.1 Back-Shifted Fermi Gas Formula
Pairing correlations and shell effects are empirically taken into account in Bethe’s
formula by shifting the ground-state energy by a back-shift parameter
ρ(E x ) =
√
π
12
a
−1/4 (E x − )
−5/4 e
2
√
a(E x −) .
(18)
This back-shifted Bethe formula for the state density includes two parameters a and
that can be treated as adjustable parameters. They can for example be determined
from level counting data at low excitation energies and neutron resonance data [13,
14].