PUBLISHING HOUSE
OF THE ROMANIAN ACADEMY
PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A,
Volume 3, Number 1-2, 2002.
THEORY OF THRESHOLD PHENOMENA
Cornel HAŢEGAN
Romanian Academy
E-mail : hategan@ifin.nipne.ro
Theory of Threshold Phenomena in Quantum Scattering is developed in terms of Reduced Scattering
Matrix. Relationships of different types of threshold anomalies both to nuclear reaction mechanisms
and to nuclear reaction models are established. Magnitude of threshold effect is related to spectroscopic factor of zero-energy neutron state.
1. INTRODUCTION
Heisenberg, (Nobel Prize for Physics 1932, see
[1]), has introduced the concept of Scattering- (S-)
Matrix, [2], in order to describe the dynamics of
quantum scattering. The only singularities of Scattering Matrix, as function of energy, are Resonances and Threshold Effects; they both were firstly
approached by Wigner (Nobel Prize for Physics
1963, see [1]). Wigner has predicted the Threshold
Cusp, [3]; this effect was thought as anomaly in
excitation function of an reaction, resulting from
opening of a new neutron channel. The Cusp denominates a discontinuity, coincident with a neutron channel threshold, in cross-section energy derivative of an open competing channel.
The Wigner Cusp should be an universal
threshold effect, appearing at every new neutron
channel. However, extensive experimental studies
along many decades have proved that the nuclear
threshold effects are rather rare and apparently
diverse; this diversity resulting in developement of
several distinct theoretical threshold models. The
experimental
and
theoretical
problems,
accumulated in the field during time, do impose an
unitary approach to threshold phenomena: a theory
explaining the diversity and rare evidences of
threshold effects, also containing, as limit cases,
previous theoretical threshold models.
2. CUSP THEORY
The threshold effects originate in conservation
of the flux. If a new channel opens, a redistribution
of the flux in old open channels appears, i.e. a
modification of their reaction cross-sections; such
changes of cross-sections of open channels, due to
opening of a new one, are called threshold effects.
The magnitude of a threshold effect depends on
amount of flux absorbed by the threshold channel.
If the threshold channel has no (coulombian and
centrifugal) barriers, i.e. it is a s-wave neutron one,
then absorption of flux happens suddenly and this
results in a specific threshold effect named
Threshold Cusp. Since the flux in neutron threshold channel is proportional to (neutron) wave
number (square root of energy), it results the crosssection threshold cusp has infinite energy derivative at zero (threshold) energy; in fact from here
the name of Cusp comes.
The concept of Cusp was introduced by
Wigner, in fifties; decade later, quantitative
treatments of the cusp were developed by Breit [4],
in R-Matrix Theory, and by Baz [5], in S-Matrix
Theory. The Cusp Theory assumes the Scattering
Matrix elements, near neutron channel threshold,
depend on neutron energy En , or its wave number
kn , only via monotone penetration factors, (N, M,
A - constant matrices),
Snn  1  N nnkn ; San  M ankn1 / 2
0
Sab  Sab
 Aabkn
(kn  0)
The unitary S 0 - Scattering Matrix describes the
N open channels a, b  1,2 ... N  either at neutron
threshold energy or, more general, (the open
channels system) in absence of threshold channel n
(= N + 1). The Cusp term of Scattering Matrix for
Cornel HAŢEGAN
open channels is related to successive transitions to
and from threshold channel
Sab 
1
1
San Snb  M an M nbkn (kn  0)
2
2
The limits of Cusp Theory are: it is valid (1) only
in zero-energy limit ( k n  0 ), (2) only for
electrical-neutral threshold channels, and (3) only
if resonances are not in threshold vicinity.
The
cross-section
cusp,
0
*
 ab ~ ReS ab S ab  ~ k n , is zero in threshold
limit and has a vertical slope (infinite energy
derivative ~ 1 / E n1 / 2 ). Another experimental
characteristic of cusp refers to energy dependance
of cross-section  an for reactions populating
threshold channel,
constant
 an / E n1 / 2 ~
E n  0 . The Cusp Theory predicts a threshold
effect at opening of every s-wave neutron channel
but does not estimate the magnitude of this effect.
One can prove however that the threshold cusp
component of total cross-section is bounded,

2

 ab  ab  S nn  1 ; it is negligeable in zero
energy limit, S nn  1 . An enhacement of the cusp
could result from a virtual state in neutron
threshold channel, [6]. The above results were,
until seventies, major contributions to theory of
threshold phenomena, see [7].
3. ON EXPERIMENTAL EVIDENCES FOR
THRESHOLD EFFECTS
Extensive experimental studies on threshold
effects in nuclear reactions were done, along many
decades, since formulating the problem; in spite of
these searches, only three groups of threshold
effects were observed and each group is restricted
to rather few cases, see [8].
A first group of threshold effects was observed
with light nuclei, mainly in proton elastic
scattering. They were analyzed theoretically not as
a genuine s-wave Wigner cusp but rather within
generalizations of Cusp Theory, see [8].
A second group of anomal effects, evinced in
deuteron stripping reactions on medium (A~90)
mass nuclei, was related to neutron analogue
channel,[9]. Initially, it was described by a protonneutron charge exchange model, [10]; then Lane,
[11], proposed a model based on interplay of pwave neutron single particle resonance with
threshold. The Cusp Theory cannot account for this
threshold anomaly, [12, 13, 14].
2
A third group, the Isotopic Threshold Effect,
was observed in proton reactions on light-medium
mirror nuclei (A~30), [15]. Alternative evidences
for this effect were found in a (d,p) reaction, [16],
as well as in some other experimental data on
proton isospin coupled reactions, [17].
The Nuclear Reactions Physics of seventies was
confronted with several problems related to
threshold phenomena, [18] : (1) the threshold
effects are rather rare, occuring under some
restrictive conditions, (2) the observed threshold
effects are apparently peculiar, differing each from
other in many aspects, (3) special reaction models
were devised to describe different effects or even
same threshold anomaly. A theory of threshold
phenomena has to fulfill corresponding
requirements: (1) to understand necessary
conditions for experimental observation of a
threshold effect, (2) to provide unitary description
of different types of threshold effects, (3) to
include previous theoretical threshold models as
limit cases.
4. REDUCED SCATTERING MATRIX AND
THRESHOLD PHENOMENA
An alternative approach to problem of threshold
phenomena is here presented, by incorporating it in
a broader class of multichannel problems studying
the effect of an unobserved (invisible or
eliminated) channel on the observed (or retained)
open channels; such problems could be formally
considered as scattering problems in truncated
space of retained channels This class of
multichannel problems comprises not only the
threshold phenomena [19], but also the
quasiresonant scattering, [20], the atomic quantum
defect, [21], or even classical electromagnetic
scattering problems, [22].
The usual approaches to multichannel scattering
problems in truncated space of channels are
"Reduced R- or K- Matrix", [23], and "Effective
Hamiltonian" (Projector Method), [24]; the last
method has been applied for studying effective
interactions in Nuclear Physics, as e.g. Nuclear
Optical Potential Anomaly, [25]. Since the
Scattering Matrix is primary object of Scattering
Theory, (rather than Hamiltonian and R- or KMatrix), the concept of "effective" or "reduced"
operator should be necessarilly extended to SMatrix.
Consider the multichannel system of N open
(retained) channels, decoupled from the
Theory of Threshold Phenomena
3
unobserved (eliminated) threshold channel n. The
independent open channels bare system is
described by unitary Scattering Matrix,
0
. By coupling threshold channel n to
S N0  S ab
open ones N, via S na matrix elements, one obtains
the Reduced Scattering Matrix S N  S ab for
group of retained channels, [17]; it includes both
bare Scattering Matrix, S N0 , and the effect, S , of
eliminated channel1.
0
0
Sab  Sab
 Sab  Sab
 San (1  Snn )1 Snb
The Reduced Scattering Matrix does include, as
limit cases, the Cusp Theory S nn  1 [5], and its
generalization, [26]; it does cover also the KMatrix Theory of Charge Exchange Coupling,
[25].
The Reduced S - Matrix is obtained by using
different parametrizations of the Scattering Matrix;
depending on (K - or R - Matrix) parametrization,
the logarithmic derivative of eliminated channel
appears either implicitly or explicitly in formula.
(The real K - and R - Matrix functions yield an
unitary S - Matrix; the unitarity, i.e. the
conservation of the flux, is a vital condition for
every approach to threshold phenomena.) The
Reduced Scattering Matrix was generalized, [28],
in order to separate the resonant and threshold
structures in K - and Level- Matrix formulae of the
S - Matrix.
The Reduced Scattering Matrix takes into
account both the real and virtual transitions to and
from eliminated channel; the corresponding
formula for eliminated closed channel was derived
either as extension below threshold, [19], or by the
Quantum Defect Method, [21].
The method and results, "not altogether
expected" (Annals of Physics), do response to
main questions of the field. The formal merits of
this method are: the Reduced S - Matrix formula is
valid both near and far away from threshold, it is
valid both for potential and resonant scattering
near threshold and it is valid even for threshold
channels with barriers, as e.g. p-wave or
coulombian ones. The physical merit of this
approach is it does fix the relation between
threshold effects , S , and reaction mechanisms in
threshold channel, via S nn matrix element, (see
chapter on Threshold Effects and Reaction
Mechanisms).
1
Paraphrasing Dirac, ("Pretty Mathematics", International
Journal of Theoretical Physics, 21, p. 603, 1982), a basic result has to be displayed by a simple and aesthetical formula
The reduced S - Matrix approach to threshold
effects yields "closure relations" for summ of
cross-section threshold effects (  ) in all
reactions induced from given channel, say a, both
below (-) and above (+) threshold
N
 
( )
ab
( )
 2 Re S aa
()
ab
()
 2 Re S aa
 | S an |2
b 1
N
 
b 1
The cross-section threshold effects as interference
terms could have different signs; consequently, the
"closure relations" do not enforce strong
constraints on threshold effects magnitude  ab .
However if all  ab terms have same sign, then a
large threshold effect in one channel results in
small threshold effects in other open competing
channels, (see chapter on Threshold Effects and
Reaction Models).
5. THRESHOLD EFFECTS AND REACTION
MECHANISMS
a. Cusp and Potential Scattering
Consider, firstly, potential non-resonant
scattering both in threshold and open channels. In
this limit, the Reduced S - Matrix formula does
S nn  1 ,
restrict
to
cusp-one,
S ab  1 / 2S an S nb ~ k n  0 . It can be proved
that S ab is a monotone increasing function of
energy, provided S - Matrix is dependent on
energy only via penetration factors. The cusp
magnitude is vanishing at threshold; if far away
from threshold S ab is a small quantity , it results
the threshold cusp is negligeable in all energy
range, [19]. This unexpected result was, later on,
confirmed by alternative theoretical methods, [7].
b. S-Wave Anomaly and Resonant Scattering
A compound nucleus resonance (  ) located in
vicinity of a neutron threshold induces a nonnegligeable threshold effect for s-waves; the
threshold channel partial width n has to be
dominant term in the total width  . The
compound nucleus resonant scattering is necessary
condition for an enhanced s-wave threshold effect:
(1) energy coincidence of neutron threshold with a
resonance in compound nucleus E   E thr   ,
and (2) a large partial width n ~  for
Cornel HAŢEGAN
resonance decay in threshold channel, [19], [8].
c. P-Wave Threshold Anomaly and QuasiResonant Scattering
A compound nucleus resonance cannot result in
an observable p- wave threshold effect. The pwave threshold effect is related, [29], to: (1) a
neutron single particle resonance coincident with
threshold, [11],and (2) the direct interaction
coupling of observed open channels to the
threshold one, [30]. These two conditions define
the Quasi-Resonant Scattering (Coupled Channel
Resonance), [20].
6. THRESHOLD EFFECTS AND
SPECTROSCOPIC FACTORS
The flux transfer involved in a cusp effect is
essentially determined by penetration factors of
threshold channel S - Matrix elements. The flux
dynamics for the other threshold anomalies is
related to pole residues (resonance partial decay
widths) of threshold channel S - Matrix elements.
The decay widths are determined not only by
penetration factors but also by spectroscopic
reduced widths; large reduced width ( Wigner unit)
of threshold neutron resonance does act as
amplifier of flux transfer to and from neutron
threshold channel. The magnitude of s-wave and of
p-wave threshold effects is proportional,
respectively, to neutron resonance reduced width,
[19], and to neutron strength function, [31], i.e. to
spectroscopic factor of neutron threshold state.
7. THRESHOLD EFFECTS AND REACTION
MODELS
The reaction models are approximative
algorithms for calculating Scattering Matrix,based
on physical assumptions concerning reaction
mechanisms. Some nuclear reaction models, as
Optical Model or Distorded Waves Born
Approximation, do not take into account the
opening of a new reaction channel; such models
generate S - Matrix elements which are smoothly
varying even across-thresholds. Other nuclear
reaction models, as Channel Coupling Models,
Weak Coupling Models, Bohr Models, HauserFeshbach Statistical Model, do take into account
the opening of a new reaction channel. The subject
of this chapter is problem of threshold effects
within these reaction models; they are studied by
4
means of "closure relations" and by using only
some general assumptions. The cross-section
threshold effects in the Weak Coupling, Bohr and
Hauser-Feshbach models are negative and bounded
as magnitude.
a. Weak Coupling Models
The Weak Coupling Models are defined by the
condition that the transition matrix T (= S - 1)
elements have independent scattering phase-shifts
for different channels.
Tab  exp i a  | Tab | exp ib 
The phases of inelastic transition matrix elements
are determined by elastic scattering phase-shifts
(  a and  b ) in associated initial and final
channels. The cross-section threshold effects in
Weak Coupling Models are negative,  ab  0 .
b. Bohr Models
The Bohr Models are defined by the condition
the transition matrix elements
Tab   a .b
are split into two factors, which depend on input
and exit channels, irrespective on number of
involved channels. Examples of Bohr Models are
the isolated resonance or a summ of isolated (noninterfering)
resonances.
The
cross-section
threshold effects in Bohr Models are negative,
 ab  0 .
c. Hauser-Feshbach Statistical Model
The Hauser-Feshbach Statistical Model assumes
the Scattering Matrix has direct terms, generated
by Optical Model, only in elastic channels, (< > energy average),
S ab  S aa   S abfl
 S aa  S aa (OM );  S abfl  0
The non-diagonal S - Matrix elements of this
model are purely fluctuating (Bohr hypothesis of
random phases); they define the energy averaged
compound
nucleus
cross-section,
 ab 
S abfl
1/ 2
 a  b .
nucleus cross-section  ab
The
compound
is calculated in terms
of transmission coefficients, Ta  1  S aa OM  ,
which are smoothly varying even acrossthresholds. There is no threshold effect at negative
2
Theory of Threshold Phenomena
5
energy,
 ab   0 ; there is a drop of cross-
6.
section above n- threshold,  ab   0 .
7.
The Theory of Threshold Phenomena does
reproduce quantitatively the Hauser - Feshbach
Statistical Model results; the derivation is done by
means of "closure relations" and by assuming Bohr
hypothesis of random phases for open channels. If
threshold channel is both fluctuant and direct
coupled to open chaannels, then a negative
threshold effect appears below zero-energy too.
The all cross-section threshold effects within
Hauser-Feshbach, Bohr and Weak Coupling
Models are negative. According to these models, a
large threshold effect in one open channel implies
small threshold effects in other open competing
channels; this limit provides a "hydrodinamical
picture" for threshold effects.
8.
9.
10.
11.
12.
13.
14.
15.
16.
8. CONCLUSIONS
The Theory of Threshold Phenomena, based on
Reduced Scattering Matrix, does establish relationships between different types of threshold effects
and nuclear reaction mechanisms: the cusp and
non-resonant potential scattering, s-wave threshold
anomaly and compound nucleus resonant scattering, p-wave anomaly and quasi-resonant scattering.
A threshold anomaly related to resonant or quasiresonant scattering is enhanced provided the neutron threshold state has large spectroscopic amplitude. The Theory contains, as limit cases, Cusp
Theories and also results of different nuclear reactions models as Charge Exchange, Weak Coupling,
Bohr and Hauser-Feshbach models.
This work is partially based on communications
on Multichannel Phenomena in Nuclear Reactions,
reported to The Section of Physical Sciences of
The Romanian Academy (October 1992) and to a
Humboldt Colloquium (September 1997).
The author acknowledges help of H. Comisel
and R.A. Ionescu in preparing this manuscript.
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Recievied : July 13,2001