Table Of ContentPROGRESS IN NUCLEAR PHYSICS Volume 12
EDITORS: D. M. BRINK AND J. H. MULVEY
Part 2
TOTAL CROSS-SECTION
MEASUREMENTS
by
G. GIACOMELLI
PERGAMON PRESS
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78 G. GIACOMELLI
5.5. The AN and ΣΝ systems 139
5.6. The antinucleon-nucleon system 139
5.7. Comparison of the high-energy behaviours 143
5.8. Dispersion relations 151
5.9. Absorption cross-sections off nuclei 155
6. CONCLUDING REMARKS 160
ACKNOWLEDGEMENTS 162
REFERENCES 163
APPENDIX A: Interpolated values of total cross-sections 166
APPENDIX B: A listing of all measured total cross-sections 167
1. Introduction
In the field of elementary particle physics, the total cross-section
measurement is, at the moment, the simplest and the most precise cross-
section measurement. When a new energy range becomes available at a
new accelerator, it is one of the first measurements to be performed and, as
time goes by, the measurements are repeated with greater and greater
accuracy. This procedure has in the past been highly rewarding, since the
increased precision allowed the detection of new structures.
Most of the systematic total cross-section measurements have been
performed using the transmission method, in "good geometry". Recent
measurements at Brookhaven,(1_4) CERN,(5) Rutherford*6,7) and
Serpukhov(8) achieved statistical precisions of better than ±01% with
absolute scale errors of about 0*3—1 Ό%. However, the results of dif-
ferent experiments in the same energy range, although having the same
behaviour with energy, are often in systematic disagreement with one
another, well outside their quoted statistical and sometimes systematic
errors.
The amount of total cross-section data available from the literature is
rather large (about 2200 data points). The largest amount is for the
πΝ system ; in decreasing order of available data follow the pN, KN, pJV,
and K+N systems. Very few data are available for the AN and ΣΝ
systems.
TOTAL CROSS-SECTION MEASUREMENTS 79
In this review we shall discuss total cross-section measurements from
both the technical and the physics points of view. We shall be concerned
mainly with elementary hadron collisions.
Phenomenologically one may speak of two energy regions :
(i) the resonance region up to about 5 GeV/c laboratory momentum ;
(ii) the high-energy region for momenta larger than 5 GeV/c.
In the resonance region, the total cross-sections are characterized by
the presence of structures, a number of bumps usually interpreted as
resonances. The heights of the bumps decrease with increasing energy,
and around 5 GeV the resonance region joins smoothly to the high-
energy one. In the high-energy region the total cross-sections are slowly
varying functions of the energy and do not exhibit any appreciable
structure. In the first region, the cross-sections depend strongly on the
type of colliding particles and on their isospin, whilst in the second region
these dependences tend to disappear.
The π+ρ,Κ+ρ, and pp are pure isospin states. In the other cases one has
a mixture of two isospin states. The determination of the pure isotopic
spin cross-sections requires the measurement of two cross-sections,
which involves changing either the incident or the target particle. For
pions it is easy to measure both n+p and n~p total cross-sections, and
hence to derive the total cross-sections σ± and σ^ in the pure isospin
states. For the other cases, the simplest solution is to measure the cross-
sections off protons and off neutrons. Unfortunately, the best neutron
target is a bound neutron-proton state (the deuteron), so that in this case
the problems of nuclear physics in the deuteron severely limit the analysis
of the data.
A total cross-section measurement in the resonance region may be
thought of as a method for detecting the existence of new resonances(.9)
The word "resonance" is used here in a rather loose way: it may be
applied to any structure in any cross-section. Resonances discovered to
date have been found by using two different methods, those of formation
and production. In a formation experiment, the resonance is produced as
a compound state when a target nucléon is bombarded with a particle of
appropriate energy (e.g. up -► N* -► πΝ). In a production experiment,
the resonance is produced together with other particles (e.g. πρ -+ πΝ*
and subsequent N* decay).
In a formation experiment, the search for resonances is performed by
measuring a partial cross-section or the total cross-section as a function
80 G. GIACOMELLI
of the momentum of the incoming particle. At the resonance energy, the
cross-sections will show an enhancement, in the best case a peak, sometimes
a shoulder.
In general, the measurement of an appropriate partial cross-section
as a function of energy will be more sensitive to the presence of resonances
than will the measurement of the total cross-section ; the enhancements
will stand out more clearly. On the other hand, the relatively poor intrinsic
sensitivity of the total cross-sections is largely compensated by the sim-
plicity of the measurements and by the high precisions obtainable. In
the past, most of the baryon resonances have, in fact, been discovered by
measuring total cross-sections.
Low-mass resonances are, as a rule, easy to detect, because they usually
produce large effects. Higher-mass resonances, however, show up as
broad and non-prominent structures, often overlapping with one another,
so that the measurement of many closely spaced points with high accuracy
is required. Errors in the absolute values can be tolerated only if they
are essentially energy-independent. Most of the measurements performed
with the main aim of "bump-hunting" are, in fact, characterized by small
energy-independent errors (say 0-1%) and a relatively large-scale factor
(say 1-2%).
If a structure is found in a total cross-section measurement, the infor-
mation that it yields includes the mass M, the width at half height Γ,
the height a , and the isotopic spin I.
R
The total cross-section method of resonance-hunting must be considered
as a "coarse spectrometer", with a not-so-good energy resolution. In
fact "spectrometers" with higher resolutions (for instance, the phase-
shift analysis) have shown that some of the peaks found in total cross-
section measurements are in reality due to two or more nearby resonances.
Total cross-section measurements do not provide enough information
to establish conclusively that a peak is a resonant state, i.e. a state with
definite quantum numbers. In fact, a structure could also come from a
threshold effect, such as the opening up of a new important channel or
other kinematical effects. On the other hand, the separation between a
resonance and a threshold effect is tenuous, since threshold effects are
often also resonances/9-11)
Measurements of total cross-sections at high energies should in principle
yield information on the asymptotic region/4,8) What is of interest is the
energy dependence of the cross-sections, as well as the relations amongst
TOTAL CROSS-SECTION MEASUREMENTS 81
the different cross-sections. In the asymptotic region the effects of symmetry-
breaking interactions should be minimal, and the properties of the under-
lying symmetries revealed. The most conventional theories predict the
following: (a) the total cross-sections decrease slowly with energy to a
constant non-zero value; (b) the total cross-sections of the particles
belonging to the same isospin multiplet become equal ; (c) the total cross-
sections of particle and antiparticle become equal ; (d) the ratios between
various total cross-sections are governed by some internal symmetry.
Recent measurements(8) cast some doubts on these statements.
At present energies we are not yet in the asymptotic region. The avail-
able experimental information indicates that we are approaching
"Asymptopia", but also that we are far away from it.
The total cross-section data are also useful for a variety of purposes.
The imaginary part of the forward elastic scattering amplitude is
obtained from the total cross-sections, via the optical theorem. Dispersion
relations allow the calculation of the real part of the forward elastic
scattering amplitude in terms of integrals over total cross-sections. They
also allow the calculation of the low-energy parameters such as the S-wave
scattering lengths and the coupling constants.
In Section 2 we shall discuss the experimental methods used for
measuring total cross-sections. The computation of the pure isospin
cross-section is taken up in Section 3, where the deuteron problem is
discussed in detail. Section 4 will be devoted to a discussion o fthe experi-
mental errors. Sections 2,3, and 4 are rather technical. Section 5 is devoted
to the discussion of the experimental data ; dispersion relations and other
specialized subjects are also briefly discussed.
In the concluding remarks we shall try to summarize the present status
of the total cross-section art and knowledge, and point out a number of
open problems.
The survey of the literature was stopped at the end of 1968. An effort
has been made to have this report as complete as possible; I apologize
for probable omissions and incorrect reporting of the data. The available
total cross-section data were collected on punched cards and are listed in
the appendices.
2. Experimental Methods
The transmission method, employing many circular transmission
counters, is the simplest and most widely used method for systematic
TOTAL CROSS-SECTION MEASUREMENTS 81
the different cross-sections. In the asymptotic region the effects of symmetry-
breaking interactions should be minimal, and the properties of the under-
lying symmetries revealed. The most conventional theories predict the
following: (a) the total cross-sections decrease slowly with energy to a
constant non-zero value; (b) the total cross-sections of the particles
belonging to the same isospin multiplet become equal ; (c) the total cross-
sections of particle and antiparticle become equal ; (d) the ratios between
various total cross-sections are governed by some internal symmetry.
Recent measurements(8) cast some doubts on these statements.
At present energies we are not yet in the asymptotic region. The avail-
able experimental information indicates that we are approaching
"Asymptopia", but also that we are far away from it.
The total cross-section data are also useful for a variety of purposes.
The imaginary part of the forward elastic scattering amplitude is
obtained from the total cross-sections, via the optical theorem. Dispersion
relations allow the calculation of the real part of the forward elastic
scattering amplitude in terms of integrals over total cross-sections. They
also allow the calculation of the low-energy parameters such as the S-wave
scattering lengths and the coupling constants.
In Section 2 we shall discuss the experimental methods used for
measuring total cross-sections. The computation of the pure isospin
cross-section is taken up in Section 3, where the deuteron problem is
discussed in detail. Section 4 will be devoted to a discussion o fthe experi-
mental errors. Sections 2,3, and 4 are rather technical. Section 5 is devoted
to the discussion of the experimental data ; dispersion relations and other
specialized subjects are also briefly discussed.
In the concluding remarks we shall try to summarize the present status
of the total cross-section art and knowledge, and point out a number of
open problems.
The survey of the literature was stopped at the end of 1968. An effort
has been made to have this report as complete as possible; I apologize
for probable omissions and incorrect reporting of the data. The available
total cross-section data were collected on punched cards and are listed in
the appendices.
2. Experimental Methods
The transmission method, employing many circular transmission
counters, is the simplest and most widely used method for systematic
82 G. GIACOMELU
measurements of the total cross-sections of charged particles. Other
methods use bubble chambers/12) spark chambers(5) and counter
hodoscopes.(4)
The measurement of the total cross-sections of neutral particles is a
more difficult task, because neutral beams are not monoenergetic and
neutral particles are more difficult to detect.
In this section we shall discuss extensively the transmission measure-
ments of charged particles, and describe the other methods briefly.
2.1. The transmission method for charged particles
As examples of this method we shall take the recent measurements
performed at the Brookhaven,(1_3) Rutherford*6,7) and Serpukhov(8)
laboratories. Whilst the transmission method is in principle very simple,
it may become rather complicated technically when one is trying to achieve
the highest precisions.
2.1.1. Principle of a transmission measurement
The classical geometry used for a total cross-section measurement
(Fig. 1) consists of: (a) a system of counters to monitor the incoming beam ;
DUMMY
H2 BEAM
BEAM TARGETS TRANSMISSION
MONITOR COUNTERS
■I.
FIG. 1. Schematic layout of a transmission measurement. The beam is defined and
counted by the beam monitor ; it is absorbed in a target, say hydrogen, which can
be replaced by a dummy one for a background measurement. One transmission
counter would suffice for an ideal pencil beam ; in practice, one is forced to use a
whole array of counters.
(b) the targets where the beam is absorbed ; and (c) a series of transmission
counters which detect the transmitted particles.
Ideally one would like a beam of negligible transverse dimensions (a
pencil beam), a target and one transmission counter, again of negligible
TOTAL CROSS-SECTION MEASUREMENTS 83
transverse dimensions. Since actual beams have finite dimensions, one
is forced to use large transmission counters, which therefore do not measure
the total cross-section but a slightly smaller value. The total cross-section
is then obtained by extrapolation to zero solid angle. The extrapolation
procedure is one of the most delicate problems of the transmission
method.
The partial cross-section σ seen by the ith transmission counter is
{
given by
«-fat®
(1)
where JV is the Avogadro number, A is the atomic weight, pi is the target
thickness in g cm-2, R is the fraction of beam reaching the ith trans-
oi
mission counter when the dummy target is in place, and R the same
t
fraction for the full target.
The assumptions that are implicit in deriving eq. (1) are: (a) that the
dummy target is exactly equivalent to the real targets when they are
empty; (b) that the whole system is stable over the time required to
alternate from a full target to the dummy one ; (c) that there should not
be electronics effects arising from the different full and empty counting
rates in the transmission counters; and (d) that the energy loss suffered
by the incident particles in the full target does not change the absorption
probability in the region between the target and the transmission counters.
Each of the transmission counters measures the total cross-section
minus the cross-section for having a charged secondary in the solid angle
subtended by the ith transmission counter :
"v^w
(2)
V dQ
Using the square of the momentum transfer t as a variable, and assuming
that the differential cross-section has elastic and inelastic contributions
which are roughly exponential in i, one has, after removing Coulomb
scattering effects as discussed in the section on the corrections:*
* At high energies and small angles t Ä (PiatAab)2· Therefore t is proportional to the solid
angle Ω subtended by the ith transmission counter. For many-prong events there may not
be a simple definition of t.
84 G. GIACOMELLI
σ(ί)^σ, , - {aebt + cedt)dt
0
+ <3)
-«--[!K') 3 (·*"')}
where a = (da/dt) for the elastic part, and c is the same quantity for
t=0
the inelastic one ; b and d are the slopes of the elastic and inelastic contri-
butions. Equation (3) must be regarded as an approximation valid over a
limited ί-range [say 0-005 < |i| < 0-06 (GeV/c)2] and when the contribu-
tion of many-prong events is not too large.
Most of the experiments have been analysed using either a linear
or quadratic approximation of eq. (3) :
Linear
approximation: σ(ή ~ a — (a + c)\t\ = A + Bt (3a)
tot
Quadratic . , ,, ,
^ . . φ) x ~ - fa + c)\t\-
σ
approximation : '
A + Bt + Ct2. (3b)
The linear approximation implies a constant differential cross-section ;
its use is justified if the slopes b and d are small and i fthe f-range covered
by the transmission counters is small. The linear approximation is
expected to give a systematic underestimate of a ; the underestimate will
m
increase slowly with momentum if the transmission counters are kept
fixed, that is if the r-range involved in the extrapolation increases with
momentum.
The quadratic extrapolation is the form used by most experimenters.
Figure 2 shows a plot of the partial cross-sections as function of t for
a Brookhaven experiment at 3 GeV/c.(3) At higher energies and larger
f-ranges higher order terms in both (3) and (3a) have been considered (:8)
σ(ή = exp (A + Bt + Ct2). (3c)
The above procedure is not adequate for scattering off nuclei because
the coherent diffraction scattering has a very steep slope/3'13) The pro-
cedure is therefore that of computing the coherent elastic scattering, of
subtracting its contribution from each partial cross-section, and of then
