Determination of Low-Energy Parameters of Neutron–Proton Scattering
on the Basis of Modern Experimental Data from Partial-Wave Analyses
V. A. Babenko∗ and N. M. Petrov
Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine,
Metrologicheskaya ul. 14b, 03143 Kiev, Ukraine
The triplet and singlet low-energy parameters in the effective-range expansion for neutron–
proton scattering are determined by using the latest experimental data on respective phase
shifts from the SAID nucleon–nucleon database. The results differ markedly from the analogous
parameters obtained on the basis of the phase shifts of the Nijmegen group and contradict the
parameter values that are presently used as experimental ones. The values found with the
aid of the phase shifts from the SAID nucleon–nucleon database for the total cross section
for the scattering of zero-energy neutrons by protons, σ0 = 20.426 b, and the neutron–proton
coherent scattering length, f = −3.755 fm, agree perfectly with the experimental cross-section
values obtained by Houk, σ0 = 20.436 ± 0.023 b, and experimental scattering-length values
obtained by Houk andWilson, f = −3.756±0.009 fm, but they contradict cross-section values of
σ0 = 20.491±0.014 b according to Dilg and coherent-scattering-length values of f = −3.7409±
0.0011 fm according to Koester and Nistler.
PACS: 13.75.Cs, 21.30.-x, 25.40.Dn
DOI: 10.1134/S1063778807040072
1. Along with the deuteron parameters, the low-energy parameters in the effective-range
expansion for neutron–proton scattering,
k cot δ = −
rk2 + v2k
4 + v3k
6 + v4k
8 + . . . , (1)
are fundamental quantities that play a key role in studying strong nucleon–nucleon interaction.
∗E-mail: pet@online.com.ua
http://arxiv.org/abs/0704.1024v1
These parameters are of great importance for constructing various realistic nuclear-force models,
which, in turn, form a basis for studying the structure of nuclei and various nuclear processes.
For this reason, it is highly desirable to determine reliably and accurately the parameters in
the effective-range expansion, including the scattering length a, the effective range r, the shape
parameter v2, and higher order parameters vn.
Although low-energy parameters for neutron–proton scattering have been determined and
studied since the early 1950s, even the experimental values of such parameters as the scattering
length a and the effective range r are ambiguous to date. As for the shape parameter v2, even
its sign is unknown at the present time. The theoretical value of this parameter depends greatly
on the nuclear-force model used: as we go over from one model to another, the parameter v2
in the triplet state changes within a broad interval, from −0.95 [1, 2] to 1.371 fm3 [3], whence
it follows that the shape parameter is a very subtle and sensitive feature of nucleon–nucleon
interaction.
We would like to note that not only does the shape parameter v2 depend on the form of
interaction, but it is also strongly dependent on the scattering length a and the effective range r.
In particular, a change of only a few tenths of a percent in the scattering length a may lead to a
severalfold change in the shape parameter v2 [4]. The shape parameters vn of order higher than
that of v2 have been still more poorly determined and are more sensitive to details of nucleon–
nucleon interaction. The aforesaid highlights once again the importance of reliably determining
the scattering length a and the effective range r, the more so as these are quantities that are
most frequently used as inputs in constructing various models of nucleon–nucleon interaction.
2. It is well known [5] that the neutron–proton system may occur either in the triplet (the
total spin is S = 1) or the singlet (the total spin is S = 0) spin state. In determining the
scattering lengths a and the effective ranges r in the triplet (t) and singlet (s) spin states,
one employs the experimental dependence of the total (spin-averaged) cross section for the
scattering of slow neutrons by free protons and data characterizing the scattering of zero-
energy neutrons by para-hydrogen. In order to determine the triplet and singlet scattering
lengths (at and as, respectively), use is usually made of equations that relate these quantities
to the total cross section for the scattering of zero-energy neutrons by protons,
σ0 = π
, (2)
and to the coherent scattering length,
(3at + as) . (3)
In this case, the cross section σ0 is determined from the results of experiments that study
slow-neutron scattering on protons bound in various molecules (H2, H2O, C6H6, CH3OH),
corrections associated with neutron capture by a proton and with effects of proton binding
in molecules being subsequently eliminated. The elimination of binding-effect corrections is a
nontrivial many-body problem, since, in addition to proton and neutron motion, it is necessary
to take into account the motion of the molecular residue. A number of significant simplifications
and approximations are made in solving this problem [6]. A compendium of experimental results
from [7–13] on the total cross section for the scattering of zero-energy neutrons by free protons,
σ0, is given in Table 1.
Two values of the total cross section σ0 are recommended at the present time. These are
the value obtained by Houk (1971) [12],
σ0 = 20.436(23) b, (4)
and the value obtained by Dilg (1975) [13],
σ0 = 20.491(14) b. (5)
Since these two values of σ0 are inconsistent, their weighted-mean value
σ0 = 20.476(12) b (6)
can also be used in determining the scattering lengths.
It should be noted that the total cross section σ0 has not been measured since 1975.
The coherent scattering length f , which is determined by relation (3), is found either from
experiments where slow neutrons are scattered by pure para-hydrogen [8, 14, 15] or by crystals
[16] or — and this is a more precise method — from experiments where neutrons are reflected
by a liquid mirror and where use is made of a number of pure hydrocarbons [9, 10, 17–22]. Also,
a method for determining the coherent scattering length by means of neutron interferometry
from experiments to study neutron scattering on molecular hydrogen was proposed in [23]. The
values found by various authors for the neutron–proton coherent scattering length f are quoted
in Table 2, whence it can be seen that the value of this quantity is even more ambiguous than
the value of σ0.
In determining the scattering lengths in the triplet and the singlet state (at and as, respec-
tively), one employs most frequently, at the present time, the coherent-length value obtained
by Koester and Nistler [22],
f = −3.7409(11) fm, (7)
and the coherent-length value presented in the compilation of Dumbrajs et al. [24],
f = −3.738(1) fm. (8)
Recent experiments aimed at determining the neutron–proton coherent scattering length by
means of neutron interferometry [23], which were mentioned above, yielded the value
f = −3.7384(20) fm. (9)
Within the experimental errors, the value in (9) agrees with the result of Koester and Nistler
in (7) and with the value in (8), which was used by Dumbrajs et al. [24].
Table 3 presents values obtained in a number of previous studies [9, 10, 13, 18, 21, 22,
24–28] for the scattering lengths and effective ranges in the triplet and singlet spin states. All
of them have been used as experimental values. The values of the triplet (at) and singlet (as)
scattering lengths from Table 3 were obtained on the basis of formulas (2) and (3) by using
various values for the total cross section σ0 and the neutron–proton coherent scattering length
The values of the triplet effective range rt in Table 3 were determined primarily in an
approximation that does not depend on the form of interaction; that is,
rt ≡ ρ (−εd, 0) = 2R
, (10)
where ρ (−εd, 0) is the mixed effective radius of the deuteron;
R = 1/α (11)
is a parameter that characterizes the spatial dimensions of the deuteron; and α is the deuteron
wave number, which is related to the deuteron binding energy εd by the equation
εd = h̄
2α2/mN . (12)
In a number of studies [24, 26], the triplet effective range was determined in accordance
with the formula
rt = ρ (−εd, 0) + δrt , (13)
where the correction δrt is a model-dependent quantity. According to the estimates obtained by
Noyes on the basis of the dispersion relations [26], the correction δrt arising owing to one-pion
exchange is
δrt = −0.013 fm . (14)
According to other estimates [24], this correction is
δrt ≃ −0.001 fm , (15)
which is an order of magnitude smaller in absolute value than the estimate in (14). In the
latter case, the effective range rt is therefore nearly coincident with the mixed effective radius
ρ (−εd, 0).
The singlet effective range rs is usually determined on the basis of an analysis of the total
cross section for neutron–proton scattering, σ (E), in the low-energy region at fixed values of
the parameters at, as, and rt. The values found in this way for the singlet effective range
rs appear to be even more ambiguous than the values of the triplet effective range. As can
be seen from Table 3, the scattering-length and effective-range values used as experimental
ones change within rather broad ranges. The scatter of these values is due first of all to the
fact that different experimental values of the cross section for the scattering of zero-energy
neutrons by free protons, σ0, and of the neutron–proton coherent scattering length f are used
to determine these quantities. The ambiguity in determining the singlet effective range rs is also
associated with an insufficient accuracy of the experimental total cross sections for neutron–
proton scattering at energies below 5MeV. The values found by different authors for the singlet
effective range rs change within a broad range, from 2.42 [9] to 2.81 fm [13].
Thus, the accuracy of the experiments performed in the 1950s–1970s is insufficient for
unambiguously determining the low-energy parameters of neutron–proton scattering. At the
same time, these parameters play an important role in the theory of few-nucleon systems, which
is based on nucleon–nucleon interaction. As was shown in [29, 30], the binding energies of the
3H and 4He nuclei depend greatly on the singlet effective range rs, increasing as rs becomes
smaller. By way of example, we indicate that, as rs decreases by 0.1 fm, the binding energies
of the 3H and 4He nuclei increase by 0.3 and 1.5MeV, respectively. We note that the decrease
of 0.01 fm in the triplet scattering length at also leads to the increase of 0.025MeV in the
triton binding energy [31, 32]. At the same time, it is well known that, in calculations with
realistic nucleon–nucleon potentials, the binding energies of few-nucleon systems prove to be
underestimated. In such calculations, the 3H binding energy is as a rule underestimated by
1MeV. A reliable and precise determination of the low-energy parameters of neutron–proton
scattering and their use in calculating the binding energies of systems that contain three or
more nucleons may contribute to solving the problem of underestimating the binding energies
of few-nucleon systems without introducing three-particle forces, quark degrees of freedom,
and other concepts that would require revising basic points in the traditional theory of nuclear
forces, which relies on pair nucleon–nucleon interaction.
To conclude this section, we present, for low-energy parameters, values that are currently
used as experimental ones. Most frequently, the present-day literature quotes two sets of low-
energy parameters. These are the set from [24],
at = 5.424(4) fm, rt = 1.759(5) fm;
as = −23.748(10) fm, rs = 2.75(5) fm,
which is matched with the experimental value (5) of the total cross section at zero energy due
to Dilg [13] and with the value in (8) for the neutron–proton coherent scattering length from
[24], and the set from [28],
at = 5.419(7) fm, rt = 1.753(8) fm;
as = −23.740(20) fm, rs = 2.77(5) fm,
which corresponds to the weighted-mean value (6) of the cross sections presented by Houk [12]
and Dilg [13] and to the value in (7) for the coherent length due to Koester and Nistler [22].
It should be noted that the experiments performed in the 1950–1970s were the main source
of information used to deduce the values in (16) and (17) for the low-energy parameters of
neutron–proton scattering.
3. In recent years, the accuracy of experimental data on nucleon–nucleon scattering has
been improved considerably; moreover, methods of their partial-wave analysis, which make it
possible to describe the results of scattering experiments in terms of phase shifts, have also been
refined [33, 34]. Owing to this, the triplet and singlet low-energy parameters of neutron–proton
scattering can be determined independently of one another by using the 3S1- and
1S0-state
phase shifts [4, 35]. The results of the partial-wave analysis performed by the GWU group [33]
(data from the well-known SAID nucleon–nucleon database) and by the Nijmegen group [34] are
presently the most precise and most widely used data on the phase shifts for nucleon–nucleon
scattering. The most popular modern realistic nucleon–nucleon potentials constructed within
the last decade, which include the Nijm-I, Nijm-II, Reid93 [36], Argonne V18 [37], CD-Bonn [28,
38], and Moscow [39] potentials, are based on fits to data of the Nijmegen group [34]. However,
it should be noted that the partial-wave analysis of the Nijmegen group is a result of processing
and averaging experimental data on nucleon–nucleon scattering over a period from 1955 to
1992, but this analysis provides an insufficiently accurate description of modern experimental
data on nucleon–nucleon scattering. Despite the proximity of the phase shifts for neutron–
proton scattering that were obtained by the GWU and Nijmegen groups, the corresponding
values of the low-energy parameters in the effective-range expansion are markedly different [4],
this difference being not only quantitative but also qualitative.
Using the approximation of the effective-range function k cot δ at low energies by polyno-
mials and Padé approximants within the least squares method, we calculated the triplet and
singlet low-energy parameters of neutron–proton scattering for the experimental data on the
GWU [33] and Nijmegen [34] phase shifts. The results obtained for the low-energy parameters
in the present study by employing the data from the partial-wave analysis of the GWU group,
at = 5.4030 fm, rt = 1.7494 fm, v2t = 0.163 fm
as = −23.719 fm, rs = 2.626 fm, v2s = −0.005 fm
differ significantly from the parameter values
at = 5.420 fm, rt = 1.753 fm, v2t = 0.040 fm
as = −23.739 fm, rs = 2.678 fm, v2s = −0.48 fm
which were obtained on the basis of the data from the partial-wave analysis of the Nijmegen
group. The triplet low-energy parameters calculated here for the phase shifts of the Nijmegen
group are virtually coincident with the analogous parameters obtained previously in [35]. Un-
fortunately, the data presented by the Nijmegen group do not contain the singlet low-energy
parameters of neutron–proton scattering. The value of the singlet shape parameter v2s for the
Nijmegen phase shifts was calculated in [1], and it is in agreement with our value.
Using expressions (18) and (19) for the scattering lengths and relying on formulas (2) and
(3), we find for the cross section σ0 and for the coherent scattering length f that
σ0 = 20.426 b , f = −3.755 fm (20)
in the case of the GWU phase shifts and that
σ0 = 20.473 b , f = −3.7395 fm (21)
in the case of the Nijmegen phase shifts.
The values in (21) are in good agreement with the weighted mean of the cross sections
obtained by Houk and Dilg, σ0 = 20.476(12) b, and with the coherent-scattering-length value
of f = −3.7409(11) fm according to Koester and Nistler [22]. It should be emphasized, however,
that this agreement is not accidental; it is directly related to the fact that, in the partial-wave
analysis of the Nijmegen group, the cross-section values obtained by Houk [12] and Dilg [13]
and the coherent-scattering-length value obtained by Koester and Nistler [22] were used as
input experimental parameters. It is precisely the reason why all of the experimental low-
energy parameters in (17), with the exception of the singlet effective range, agree within the
experimental error with the corresponding parameters in (19), which were calculated on the
basis of the Nijmegen phase shifts.
The singlet-effective-range value of rs = 2.678 fm, which was calculated for the phase shifts
obtained by the Nijmegen group, is much smaller than the experimental value of rs = 2.77(5) fm,
which was quoted by Dilg in [13]. In this connection, it should be noted that, in [13], the
singlet effective range rs was determined from experimental data on the total cross section
for neutron–proton scattering at energies below 5MeV at the scattering-length values fixed
at at = 5.423(4) fm and as = −23.749 fm and the triplet-effective-range value fixed at rt =
1.760(5) fm, but, as was indicated above, this method for determining the effective range is
highly unreliable (see Table 3). A determination of the singlet effective range rs directly from
the singlet phase shift irrespective of the triplet parameters is more correct and consistent,
which reduces substantially the uncertainty in this quantity.
For the sake of comparison, the low-energy parameters for neutron–proton scattering that
correspond to the GWU (GWU PWA) and Nijmegen (Nijm PWA) phase shifts are given in Ta-
ble 4, along with the values of these parameters for a number of the realistic potentials (Argonne
V18 [37], CD-Bonn [28, 38], and Moscow [39] potentials) whose parameters were matched with
the Nijmegen nucleon–nucleon database. Also quoted there are the experimental values of the
low-energy parameters. Table 4 shows that the values of the low-energy parameters obtained
for the Nijmegen phase shifts are in perfect agreement with the corresponding parameters for
the potentials fitted to the Nijmegen nucleon–nucleon database.
A significant distinction between the values of the triplet low-energy parameters for the
GWU and Nijmegen data was discussed in detail in our previous article [4]. Here, we only
indicate that the difference of the triplet scattering lengths by 0.3% is in fact a more important
circumstance than the fourfold distinction between the values of the triplet shape parameters.
This is because many important features of the neutron–proton system — such as the asymp-
totic deuteron normalization factor AS and the root-mean-square radius rd of the deuteron
— are highly sensitive to variations in the triplet scattering length [40]. We also note that,
although the triplet effective ranges obtained from experimental data of the two main groups
are close to each other, the values of the difference δrt of the effective range rt and the mixed
effective radius ρ (−εd, 0) for the GWU [33] and Nijmegen [34] phase shifts differ significantly.
For example, the correction δrt for the phase shifts of the GWU group is positive, taking the
value
δrt = 0.0163 fm . (22)
For the phase shifts of the Nijmegen group, this correction is negative and, in absolute value, is
an order of magnitude smaller than the correction in (22): δrt = −0.001 fm. The value of the
singlet effective range for the phase shifts of the GWU group also differs from its counterpart
for the Nijmegen phase shifts (by about 2%), and the corresponding difference of the singlet
shape parameters is formidable, reaching two orders of magnitude.
In contrast to the partial-wave analysis of the Nijmegen group, the partial-wave analysis of
the GWU group does not employ the values of the cross section σ0 and the coherent scattering
length f as input parameters. The theoretical values of σ0 = 20.426 b and f = −3.755 fm,
which we obtained here for the cross section in question and for the neutron–proton coherent
scattering length from data of the partial-wave analysis performed by the GWU group, are in
perfect agreement with the experimental cross-section value of σ0 = 20.436(23) b according to
Houk [12], and the experimental coherent-scattering-length value obtained by Houk and Wilson
[9, 10],
f = −3.756(9) fm , (23)
but they contradict the cross-section value of σ0 = 20.491(14) b according to Dilg [13] and the
coherent-scattering-length value of f = −3.7384(20) fm, which was obtained recently by the
neutron-interferometry method in [23]. Thus, we see that a reliable experimental determination
of the total cross section for neutron–proton scattering at zero energy, σ0, and of the coherent
scattering length, f , is now quite a pressing problem. Precise values of these quantities would
make it possible to determine unambiguously the triplet and singlet scattering lengths and to
solve the problem of choosing a correct set of the low-energy parameters and phase shifts among
currently recommended experimental values.
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Table 1. Total cross section for neutron scattering on a proton at zero energy
No. References σ0 , b
1 Melkonian [7] (1949) 20.36(10)
2 Stewart and Squires [8] (1953) 20.41(14)
3 Houk and Wilson [9] (1967) 20.37(2)
4 Houk and Wilson [10] (1968) 20.442(23)
5 Neill et al. [11] (1968) 20.366(76)
6 Houk [12] (1971) 20.436(23)
7 Dilg [13] (1975) 20.491(14)
Table 2. Amplitude for coherent neutron–proton scattering
No. References f , fm
1 Shull et al. [16] (1948) −3.900(100)
2 Hughes et al. [17] (1950) −3.75(3)
3 Burgy et al. [18] (1951) −3.78(2)
4 Stewart and Squires [8] (1955) −3.80(5)
5 Dickinson et al. [19] (1962) −3.740(20)
6 Koester [20] (1967) −3.719(2)
7 Houk and Wilson [9, 10] (1967, 1968) −3.756(9)
8 Koester and Nistler [21] (1971) −3.740(3)
9 Koester and Nistler [22] (1975) −3.7409(11)
10 Callerame et al. [15] (1975) −3.733(4)
11 Schoen et al. [23] (2003) −3.7384(20)
Table 3. Low-energy parameters of neutron–proton scattering from various studies
No. References at , fm as , fm rt , fm rs , fm
1 Burgy et al. [18] (1951) 5.377(21) −23.690(55) 1.704(28) −
2 Noyes [25] (1963) 5.396(11) −23.678(28) 1.727(14) 2.51(11)
5.392(6) −23.689(13) 1.724(7) 2.42(9)
3 Houk and Wilson [9] (1967) 5.399(11) −23.680(28) 1.732(12) 2.48(11)
5.411(4) −23.671(12) 1.747(4) 2.59(8)
4 Houk and Wilson [10] (1968) 5.405(6) −23.728(13) 1.738(7) 2.56(10)
5 Koester and Nistler [21] (1971) 5.414(5) −23.719(13) − −
6 Noyes [26] (1972) 5.413(5) −23.719(13) 1.735 2.66
5.423(5) −23.712(13) 1.748(6) 2.75(10)
7 Lomon and Wilson [27] (1974) 5.414(5) −23.719(13) 1.750(5) 2.76(5)
2.77(5)
8 Dilg [13] (1975) 5.423(4) −23.749(9) 1.760(5) 2.81(5)
2.78(5)
9 Koester and Nistler [22] (1975) 5.424(3) −23.749(8) 1.760(5) 2.81(5)
10 Dumbrajs et al. [24] (1983) 5.424(4) −23.748(10) 1.759(5) 2.75(5)
11 Machleidt [28] (2001) 5.419(7) −23.740(20) 1.753(8) 2.77(5)
Table 4. Low-energy parameters of neutron–proton scattering that were obtained on the basis
of the present-day data of the partial-wave analysis and modern realistic models of nucleon–
nucleon interaction
No. Model at , fm as , fm rt , fm rs , fm σ0 , b f , fm
1 GWU PWA 5.4030 −23.719 1.7494 2.626 20.426 −3.755
2 Nijm PWA 5.420 −23.739 1.753 2.678 20.473 −3.7395
3 Argonne V18 5.419 −23.732 1.753 2.697 20.461 −3.7375
4 CD Bonn 5.4199 −23.738 1.751 2.671 20.471 −3.7392
5 Moscow 5.422 −23.740 1.754 2.66 20.476 −3.7370
6 Expt. [10, 12] 5.405(6) −23.728(13) 1.738(7) 2.56(10) 20.436(23) −3.756(9)
7 Expt. [24] 5.424(4) −23.748(10) 1.759(5) 2.75(5) 20.491(14) −3.738(1)
8 Expt. [28] 5.419(7) −23.740(20) 1.753(8) 2.77(5) 20.476(12) −3.7409(11)

  The triplet and singlet low-energy parameters in the effective-range
expansion for neutron--proton scattering are determined by using the latest
experimental data on respective phase shifts from the SAID nucleon--nucleon
database. The results differ markedly from the analogous parameters obtained on
the basis of the phase shifts of the Nijmegen group and contradict the
parameter values that are presently used as experimental ones. The values found
with the aid of the phase shifts from the SAID nucleon--nucleon database for
the total cross section for the scattering of zero-energy neutrons by protons,
$\sigma_{0}=20.426 $b, and the neutron--proton coherent scattering length,
$f=-3.755 $fm, agree perfectly with the experimental cross-section values
obtained by Houk, $\sigma_{0}=20.436\pm 0.023 $b, and experimental
scattering-length values obtained by Houk and Wilson, $f=-3.756\pm 0.009 $fm,
but they contradict cross-section values of $\sigma_{0}=20.491\pm 0.014 $b
according to Dilg and coherent-scattering-length values of $f=-3.7409\pm 0.0011
$fm according to Koester and Nistler.

Determination of Low-Energy Parameters of Neutron–Proton Scattering
on the Basis of Modern Experimental Data from Partial-Wave Analyses
V. A. Babenko∗ and N. M. Petrov
Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine,
Metrologicheskaya ul. 14b, 03143 Kiev, Ukraine
The triplet and singlet low-energy parameters in the effective-range expansion for neutron–
proton scattering are determined by using the latest experimental data on respective phase
shifts from the SAID nucleon–nucleon database. The results differ markedly from the analogous
parameters obtained on the basis of the phase shifts of the Nijmegen group and contradict the
parameter values that are presently used as experimental ones. The values found with the
aid of the phase shifts from the SAID nucleon–nucleon database for the total cross section
for the scattering of zero-energy neutrons by protons, σ0 = 20.426 b, and the neutron–proton
coherent scattering length, f = −3.755 fm, agree perfectly with the experimental cross-section
values obtained by Houk, σ0 = 20.436 ± 0.023 b, and experimental scattering-length values
obtained by Houk andWilson, f = −3.756±0.009 fm, but they contradict cross-section values of
σ0 = 20.491±0.014 b according to Dilg and coherent-scattering-length values of f = −3.7409±
0.0011 fm according to Koester and Nistler.
PACS: 13.75.Cs, 21.30.-x, 25.40.Dn
DOI: 10.1134/S1063778807040072
1. Along with the deuteron parameters, the low-energy parameters in the effective-range
expansion for neutron–proton scattering,
k cot δ = −
rk2 + v2k
4 + v3k
6 + v4k
8 + . . . , (1)
are fundamental quantities that play a key role in studying strong nucleon–nucleon interaction.
∗E-mail: pet@online.com.ua
http://arxiv.org/abs/0704.1024v1
These parameters are of great importance for constructing various realistic nuclear-force models,
which, in turn, form a basis for studying the structure of nuclei and various nuclear processes.
For this reason, it is highly desirable to determine reliably and accurately the parameters in
the effective-range expansion, including the scattering length a, the effective range r, the shape
parameter v2, and higher order parameters vn.
Although low-energy parameters for neutron–proton scattering have been determined and
studied since the early 1950s, even the experimental values of such parameters as the scattering
length a and the effective range r are ambiguous to date. As for the shape parameter v2, even
its sign is unknown at the present time. The theoretical value of this parameter depends greatly
on the nuclear-force model used: as we go over from one model to another, the parameter v2
in the triplet state changes within a broad interval, from −0.95 [1, 2] to 1.371 fm3 [3], whence
it follows that the shape parameter is a very subtle and sensitive feature of nucleon–nucleon
interaction.
We would like to note that not only does the shape parameter v2 depend on the form of
interaction, but it is also strongly dependent on the scattering length a and the effective range r.
In particular, a change of only a few tenths of a percent in the scattering length a may lead to a
severalfold change in the shape parameter v2 [4]. The shape parameters vn of order higher than
that of v2 have been still more poorly determined and are more sensitive to details of nucleon–
nucleon interaction. The aforesaid highlights once again the importance of reliably determining
the scattering length a and the effective range r, the more so as these are quantities that are
most frequently used as inputs in constructing various models of nucleon–nucleon interaction.
2. It is well known [5] that the neutron–proton system may occur either in the triplet (the
total spin is S = 1) or the singlet (the total spin is S = 0) spin state. In determining the
scattering lengths a and the effective ranges r in the triplet (t) and singlet (s) spin states,
one employs the experimental dependence of the total (spin-averaged) cross section for the
scattering of slow neutrons by free protons and data characterizing the scattering of zero-
energy neutrons by para-hydrogen. In order to determine the triplet and singlet scattering
lengths (at and as, respectively), use is usually made of equations that relate these quantities
to the total cross section for the scattering of zero-energy neutrons by protons,
σ0 = π
, (2)
and to the coherent scattering length,
(3at + as) . (3)
In this case, the cross section σ0 is determined from the results of experiments that study
slow-neutron scattering on protons bound in various molecules (H2, H2O, C6H6, CH3OH),
corrections associated with neutron capture by a proton and with effects of proton binding
in molecules being subsequently eliminated. The elimination of binding-effect corrections is a
nontrivial many-body problem, since, in addition to proton and neutron motion, it is necessary
to take into account the motion of the molecular residue. A number of significant simplifications
and approximations are made in solving this problem [6]. A compendium of experimental results
from [7–13] on the total cross section for the scattering of zero-energy neutrons by free protons,
σ0, is given in Table 1.
Two values of the total cross section σ0 are recommended at the present time. These are
the value obtained by Houk (1971) [12],
σ0 = 20.436(23) b, (4)
and the value obtained by Dilg (1975) [13],
σ0 = 20.491(14) b. (5)
Since these two values of σ0 are inconsistent, their weighted-mean value
σ0 = 20.476(12) b (6)
can also be used in determining the scattering lengths.
It should be noted that the total cross section σ0 has not been measured since 1975.
The coherent scattering length f , which is determined by relation (3), is found either from
experiments where slow neutrons are scattered by pure para-hydrogen [8, 14, 15] or by crystals
[16] or — and this is a more precise method — from experiments where neutrons are reflected
by a liquid mirror and where use is made of a number of pure hydrocarbons [9, 10, 17–22]. Also,
a method for determining the coherent scattering length by means of neutron interferometry
from experiments to study neutron scattering on molecular hydrogen was proposed in [23]. The
values found by various authors for the neutron–proton coherent scattering length f are quoted
in Table 2, whence it can be seen that the value of this quantity is even more ambiguous than
the value of σ0.
In determining the scattering lengths in the triplet and the singlet state (at and as, respec-
tively), one employs most frequently, at the present time, the coherent-length value obtained
by Koester and Nistler [22],
f = −3.7409(11) fm, (7)
and the coherent-length value presented in the compilation of Dumbrajs et al. [24],
f = −3.738(1) fm. (8)
Recent experiments aimed at determining the neutron–proton coherent scattering length by
means of neutron interferometry [23], which were mentioned above, yielded the value
f = −3.7384(20) fm. (9)
Within the experimental errors, the value in (9) agrees with the result of Koester and Nistler
in (7) and with the value in (8), which was used by Dumbrajs et al. [24].
Table 3 presents values obtained in a number of previous studies [9, 10, 13, 18, 21, 22,
24–28] for the scattering lengths and effective ranges in the triplet and singlet spin states. All
of them have been used as experimental values. The values of the triplet (at) and singlet (as)
scattering lengths from Table 3 were obtained on the basis of formulas (2) and (3) by using
various values for the total cross section σ0 and the neutron–proton coherent scattering length
The values of the triplet effective range rt in Table 3 were determined primarily in an
approximation that does not depend on the form of interaction; that is,
rt ≡ ρ (−εd, 0) = 2R
, (10)
where ρ (−εd, 0) is the mixed effective radius of the deuteron;
R = 1/α (11)
is a parameter that characterizes the spatial dimensions of the deuteron; and α is the deuteron
wave number, which is related to the deuteron binding energy εd by the equation
εd = h̄
2α2/mN . (12)
In a number of studies [24, 26], the triplet effective range was determined in accordance
with the formula
rt = ρ (−εd, 0) + δrt , (13)
where the correction δrt is a model-dependent quantity. According to the estimates obtained by
Noyes on the basis of the dispersion relations [26], the correction δrt arising owing to one-pion
exchange is
δrt = −0.013 fm . (14)
According to other estimates [24], this correction is
δrt ≃ −0.001 fm , (15)
which is an order of magnitude smaller in absolute value than the estimate in (14). In the
latter case, the effective range rt is therefore nearly coincident with the mixed effective radius
ρ (−εd, 0).
The singlet effective range rs is usually determined on the basis of an analysis of the total
cross section for neutron–proton scattering, σ (E), in the low-energy region at fixed values of
the parameters at, as, and rt. The values found in this way for the singlet effective range
rs appear to be even more ambiguous than the values of the triplet effective range. As can
be seen from Table 3, the scattering-length and effective-range values used as experimental
ones change within rather broad ranges. The scatter of these values is due first of all to the
fact that different experimental values of the cross section for the scattering of zero-energy
neutrons by free protons, σ0, and of the neutron–proton coherent scattering length f are used
to determine these quantities. The ambiguity in determining the singlet effective range rs is also
associated with an insufficient accuracy of the experimental total cross sections for neutron–
proton scattering at energies below 5MeV. The values found by different authors for the singlet
effective range rs change within a broad range, from 2.42 [9] to 2.81 fm [13].
Thus, the accuracy of the experiments performed in the 1950s–1970s is insufficient for
unambiguously determining the low-energy parameters of neutron–proton scattering. At the
same time, these parameters play an important role in the theory of few-nucleon systems, which
is based on nucleon–nucleon interaction. As was shown in [29, 30], the binding energies of the
3H and 4He nuclei depend greatly on the singlet effective range rs, increasing as rs becomes
smaller. By way of example, we indicate that, as rs decreases by 0.1 fm, the binding energies
of the 3H and 4He nuclei increase by 0.3 and 1.5MeV, respectively. We note that the decrease
of 0.01 fm in the triplet scattering length at also leads to the increase of 0.025MeV in the
triton binding energy [31, 32]. At the same time, it is well known that, in calculations with
realistic nucleon–nucleon potentials, the binding energies of few-nucleon systems prove to be
underestimated. In such calculations, the 3H binding energy is as a rule underestimated by
1MeV. A reliable and precise determination of the low-energy parameters of neutron–proton
scattering and their use in calculating the binding energies of systems that contain three or
more nucleons may contribute to solving the problem of underestimating the binding energies
of few-nucleon systems without introducing three-particle forces, quark degrees of freedom,
and other concepts that would require revising basic points in the traditional theory of nuclear
forces, which relies on pair nucleon–nucleon interaction.
To conclude this section, we present, for low-energy parameters, values that are currently
used as experimental ones. Most frequently, the present-day literature quotes two sets of low-
energy parameters. These are the set from [24],
at = 5.424(4) fm, rt = 1.759(5) fm;
as = −23.748(10) fm, rs = 2.75(5) fm,
which is matched with the experimental value (5) of the total cross section at zero energy due
to Dilg [13] and with the value in (8) for the neutron–proton coherent scattering length from
[24], and the set from [28],
at = 5.419(7) fm, rt = 1.753(8) fm;
as = −23.740(20) fm, rs = 2.77(5) fm,
which corresponds to the weighted-mean value (6) of the cross sections presented by Houk [12]
and Dilg [13] and to the value in (7) for the coherent length due to Koester and Nistler [22].
It should be noted that the experiments performed in the 1950–1970s were the main source
of information used to deduce the values in (16) and (17) for the low-energy parameters of
neutron–proton scattering.
3. In recent years, the accuracy of experimental data on nucleon–nucleon scattering has
been improved considerably; moreover, methods of their partial-wave analysis, which make it
possible to describe the results of scattering experiments in terms of phase shifts, have also been
refined [33, 34]. Owing to this, the triplet and singlet low-energy parameters of neutron–proton
scattering can be determined independently of one another by using the 3S1- and
1S0-state
phase shifts [4, 35]. The results of the partial-wave analysis performed by the GWU group [33]
(data from the well-known SAID nucleon–nucleon database) and by the Nijmegen group [34] are
presently the most precise and most widely used data on the phase shifts for nucleon–nucleon
scattering. The most popular modern realistic nucleon–nucleon potentials constructed within
the last decade, which include the Nijm-I, Nijm-II, Reid93 [36], Argonne V18 [37], CD-Bonn [28,
38], and Moscow [39] potentials, are based on fits to data of the Nijmegen group [34]. However,
it should be noted that the partial-wave analysis of the Nijmegen group is a result of processing
and averaging experimental data on nucleon–nucleon scattering over a period from 1955 to
1992, but this analysis provides an insufficiently accurate description of modern experimental
data on nucleon–nucleon scattering. Despite the proximity of the phase shifts for neutron–
proton scattering that were obtained by the GWU and Nijmegen groups, the corresponding
values of the low-energy parameters in the effective-range expansion are markedly different [4],
this difference being not only quantitative but also qualitative.
Using the approximation of the effective-range function k cot δ at low energies by polyno-
mials and Padé approximants within the least squares method, we calculated the triplet and
singlet low-energy parameters of neutron–proton scattering for the experimental data on the
GWU [33] and Nijmegen [34] phase shifts. The results obtained for the low-energy parameters
in the present study by employing the data from the partial-wave analysis of the GWU group,
at = 5.4030 fm, rt = 1.7494 fm, v2t = 0.163 fm
as = −23.719 fm, rs = 2.626 fm, v2s = −0.005 fm
differ significantly from the parameter values
at = 5.420 fm, rt = 1.753 fm, v2t = 0.040 fm
as = −23.739 fm, rs = 2.678 fm, v2s = −0.48 fm
which were obtained on the basis of the data from the partial-wave analysis of the Nijmegen
group. The triplet low-energy parameters calculated here for the phase shifts of the Nijmegen
group are virtually coincident with the analogous parameters obtained previously in [35]. Un-
fortunately, the data presented by the Nijmegen group do not contain the singlet low-energy
parameters of neutron–proton scattering. The value of the singlet shape parameter v2s for the
Nijmegen phase shifts was calculated in [1], and it is in agreement with our value.
Using expressions (18) and (19) for the scattering lengths and relying on formulas (2) and
(3), we find for the cross section σ0 and for the coherent scattering length f that
σ0 = 20.426 b , f = −3.755 fm (20)
in the case of the GWU phase shifts and that
σ0 = 20.473 b , f = −3.7395 fm (21)
in the case of the Nijmegen phase shifts.
The values in (21) are in good agreement with the weighted mean of the cross sections
obtained by Houk and Dilg, σ0 = 20.476(12) b, and with the coherent-scattering-length value
of f = −3.7409(11) fm according to Koester and Nistler [22]. It should be emphasized, however,
that this agreement is not accidental; it is directly related to the fact that, in the partial-wave
analysis of the Nijmegen group, the cross-section values obtained by Houk [12] and Dilg [13]
and the coherent-scattering-length value obtained by Koester and Nistler [22] were used as
input experimental parameters. It is precisely the reason why all of the experimental low-
energy parameters in (17), with the exception of the singlet effective range, agree within the
experimental error with the corresponding parameters in (19), which were calculated on the
basis of the Nijmegen phase shifts.
The singlet-effective-range value of rs = 2.678 fm, which was calculated for the phase shifts
obtained by the Nijmegen group, is much smaller than the experimental value of rs = 2.77(5) fm,
which was quoted by Dilg in [13]. In this connection, it should be noted that, in [13], the
singlet effective range rs was determined from experimental data on the total cross section
for neutron–proton scattering at energies below 5MeV at the scattering-length values fixed
at at = 5.423(4) fm and as = −23.749 fm and the triplet-effective-range value fixed at rt =
1.760(5) fm, but, as was indicated above, this method for determining the effective range is
highly unreliable (see Table 3). A determination of the singlet effective range rs directly from
the singlet phase shift irrespective of the triplet parameters is more correct and consistent,
which reduces substantially the uncertainty in this quantity.
For the sake of comparison, the low-energy parameters for neutron–proton scattering that
correspond to the GWU (GWU PWA) and Nijmegen (Nijm PWA) phase shifts are given in Ta-
ble 4, along with the values of these parameters for a number of the realistic potentials (Argonne
V18 [37], CD-Bonn [28, 38], and Moscow [39] potentials) whose parameters were matched with
the Nijmegen nucleon–nucleon database. Also quoted there are the experimental values of the
low-energy parameters. Table 4 shows that the values of the low-energy parameters obtained
for the Nijmegen phase shifts are in perfect agreement with the corresponding parameters for
the potentials fitted to the Nijmegen nucleon–nucleon database.
A significant distinction between the values of the triplet low-energy parameters for the
GWU and Nijmegen data was discussed in detail in our previous article [4]. Here, we only
indicate that the difference of the triplet scattering lengths by 0.3% is in fact a more important
circumstance than the fourfold distinction between the values of the triplet shape parameters.
This is because many important features of the neutron–proton system — such as the asymp-
totic deuteron normalization factor AS and the root-mean-square radius rd of the deuteron
— are highly sensitive to variations in the triplet scattering length [40]. We also note that,
although the triplet effective ranges obtained from experimental data of the two main groups
are close to each other, the values of the difference δrt of the effective range rt and the mixed
effective radius ρ (−εd, 0) for the GWU [33] and Nijmegen [34] phase shifts differ significantly.
For example, the correction δrt for the phase shifts of the GWU group is positive, taking the
value
δrt = 0.0163 fm . (22)
For the phase shifts of the Nijmegen group, this correction is negative and, in absolute value, is
an order of magnitude smaller than the correction in (22): δrt = −0.001 fm. The value of the
singlet effective range for the phase shifts of the GWU group also differs from its counterpart
for the Nijmegen phase shifts (by about 2%), and the corresponding difference of the singlet
shape parameters is formidable, reaching two orders of magnitude.
In contrast to the partial-wave analysis of the Nijmegen group, the partial-wave analysis of
the GWU group does not employ the values of the cross section σ0 and the coherent scattering
length f as input parameters. The theoretical values of σ0 = 20.426 b and f = −3.755 fm,
which we obtained here for the cross section in question and for the neutron–proton coherent
scattering length from data of the partial-wave analysis performed by the GWU group, are in
perfect agreement with the experimental cross-section value of σ0 = 20.436(23) b according to
Houk [12], and the experimental coherent-scattering-length value obtained by Houk and Wilson
[9, 10],
f = −3.756(9) fm , (23)
but they contradict the cross-section value of σ0 = 20.491(14) b according to Dilg [13] and the
coherent-scattering-length value of f = −3.7384(20) fm, which was obtained recently by the
neutron-interferometry method in [23]. Thus, we see that a reliable experimental determination
of the total cross section for neutron–proton scattering at zero energy, σ0, and of the coherent
scattering length, f , is now quite a pressing problem. Precise values of these quantities would
make it possible to determine unambiguously the triplet and singlet scattering lengths and to
solve the problem of choosing a correct set of the low-energy parameters and phase shifts among
currently recommended experimental values.
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Table 1. Total cross section for neutron scattering on a proton at zero energy
No. References σ0 , b
1 Melkonian [7] (1949) 20.36(10)
2 Stewart and Squires [8] (1953) 20.41(14)
3 Houk and Wilson [9] (1967) 20.37(2)
4 Houk and Wilson [10] (1968) 20.442(23)
5 Neill et al. [11] (1968) 20.366(76)
6 Houk [12] (1971) 20.436(23)
7 Dilg [13] (1975) 20.491(14)
Table 2. Amplitude for coherent neutron–proton scattering
No. References f , fm
1 Shull et al. [16] (1948) −3.900(100)
2 Hughes et al. [17] (1950) −3.75(3)
3 Burgy et al. [18] (1951) −3.78(2)
4 Stewart and Squires [8] (1955) −3.80(5)
5 Dickinson et al. [19] (1962) −3.740(20)
6 Koester [20] (1967) −3.719(2)
7 Houk and Wilson [9, 10] (1967, 1968) −3.756(9)
8 Koester and Nistler [21] (1971) −3.740(3)
9 Koester and Nistler [22] (1975) −3.7409(11)
10 Callerame et al. [15] (1975) −3.733(4)
11 Schoen et al. [23] (2003) −3.7384(20)
Table 3. Low-energy parameters of neutron–proton scattering from various studies
No. References at , fm as , fm rt , fm rs , fm
1 Burgy et al. [18] (1951) 5.377(21) −23.690(55) 1.704(28) −
2 Noyes [25] (1963) 5.396(11) −23.678(28) 1.727(14) 2.51(11)
5.392(6) −23.689(13) 1.724(7) 2.42(9)
3 Houk and Wilson [9] (1967) 5.399(11) −23.680(28) 1.732(12) 2.48(11)
5.411(4) −23.671(12) 1.747(4) 2.59(8)
4 Houk and Wilson [10] (1968) 5.405(6) −23.728(13) 1.738(7) 2.56(10)
5 Koester and Nistler [21] (1971) 5.414(5) −23.719(13) − −
6 Noyes [26] (1972) 5.413(5) −23.719(13) 1.735 2.66
5.423(5) −23.712(13) 1.748(6) 2.75(10)
7 Lomon and Wilson [27] (1974) 5.414(5) −23.719(13) 1.750(5) 2.76(5)
2.77(5)
8 Dilg [13] (1975) 5.423(4) −23.749(9) 1.760(5) 2.81(5)
2.78(5)
9 Koester and Nistler [22] (1975) 5.424(3) −23.749(8) 1.760(5) 2.81(5)
10 Dumbrajs et al. [24] (1983) 5.424(4) −23.748(10) 1.759(5) 2.75(5)
11 Machleidt [28] (2001) 5.419(7) −23.740(20) 1.753(8) 2.77(5)
Table 4. Low-energy parameters of neutron–proton scattering that were obtained on the basis
of the present-day data of the partial-wave analysis and modern realistic models of nucleon–
nucleon interaction
No. Model at , fm as , fm rt , fm rs , fm σ0 , b f , fm
1 GWU PWA 5.4030 −23.719 1.7494 2.626 20.426 −3.755
2 Nijm PWA 5.420 −23.739 1.753 2.678 20.473 −3.7395
3 Argonne V18 5.419 −23.732 1.753 2.697 20.461 −3.7375
4 CD Bonn 5.4199 −23.738 1.751 2.671 20.471 −3.7392
5 Moscow 5.422 −23.740 1.754 2.66 20.476 −3.7370
6 Expt. [10, 12] 5.405(6) −23.728(13) 1.738(7) 2.56(10) 20.436(23) −3.756(9)
7 Expt. [24] 5.424(4) −23.748(10) 1.759(5) 2.75(5) 20.491(14) −3.738(1)
8 Expt. [28] 5.419(7) −23.740(20) 1.753(8) 2.77(5) 20.476(12) −3.7409(11)
