# Laboratory electron screening in nuclear resonant reactions

C. Iliadis<EMAIL_ADDRESS>Department of Physics & Astronomy, University of
North Carolina at Chapel Hill, NC 27599-3255, USA  Triangle Universities
Nuclear Laboratory (TUNL), Duke University, Durham, North Carolina 27708, USA

###### Abstract

Both nonresonant and resonance reaction data are subject to laboratory
electron screening effects. For nonresonant reactions, such effects are well
documented and the measured cross sections can be corrected to find the
unscreened ones. Frequently, the procedure and expression to calculate
laboratory electron screening factors for nonresonant reactions are also
applied to isolated narrow resonances, without much theoretical support or
experimental evidence.

A simple model is applied to estimate electron screening factors, lengths, and
potentials for narrow resonances. The corrections to the measured data result
in an enhancement of the unscreened resonance strengths by less than 0.2%,
contrary to published narrow-resonance screening correction factors, which
predict a reduction of the unscreened strengths by up to 25%. Unless it can be
proven otherwise, it is recommended that measured strengths of isolated narrow
resonances not be corrected for laboratory electron screening.

The prospects of investigating laboratory electron screening effects by
measuring almost negligible differences in resonance strengths are not
promising. Instead, the difference of the resonance energy for the unscreened
and screened situation may be measurable. As an example, the case of the
$E_{r}$ $=$ $956$-keV resonance in the 27Al(p,$\gamma$)28Si reaction is
discussed. It is also demonstrated that the claim of a previously reported
detection of a resonance near $800$ keV in the 176Lu(p,n)176Hf reaction is
incorrect.

##  I Introduction

Nonresonant charged-particle nuclear reaction measurements at low bombarding
energies are impacted by the presence of electrons in the vicinity of the
interacting nuclei. These electrons, either bound to individual target or
projectile atoms, or freely moving in the conduction band in the case of a
metal, give rise to an attractive potential that effectively reduces the width
of the overall potential barrier to be penetrated by the projectile.
Therefore, the astrophysical $S$ factor extracted from a nonresonant cross
section measured in the laboratory is expected to be larger compared to the
$S$ factor that would have been obtained in the absence of electrons,
especially at the lowest bombarding energies. This effect has been observed in
several experiments (see, e.g., Ref. [1]). It is important to correct the
measured cross section for such laboratory electron screening effects, and,
thereby, determine the cross section applicable to bare interacting nuclei.
The latter quantity can then be used, together with a prescription of stellar
electron screening, to calculate thermonuclear reaction rates, which are an
essential ingredient for models of stellar evolution and explosion. The
electron screening correction factors differ for the laboratory and stellar
environment. The focus of the present work is on the former. The latter have
been calculated, e.g., by Refs. [2, 3].

Many authors (see e.g., Refs. [4, 5], and references therein) have pointed out
that the magnitude of the laboratory electron screening corrections extracted
from low-energy nonresonant cross section data are larger than what is
predicted from theory. Sophisticated theoretical models have been applied to
the problem, but significant inconsistencies between theory and experiment
remain (for a review, see Ref. [5]).

The aim of the present work is not to provide more accurate predictions for
the nonresonant laboratory electron screening corrections, but to investigate
the correction pertaining to isolated narrow resonances. Assenbaum et al. [6]
were first to suggest that the electron screening correction factors obtained
for nonresonant reactions can be applied equally to narrow resonances. They
also predicted that electron screening effects would result in a shift of the
resonance energy compared to the case of unscreened nuclei. As will be
discussed below, their first claim turns out to be incorrect, while the second
one is confirmed in the present work. Measuring such shifts of the resonance
energy may allow for a detailed study of the interplay between atomic and
nuclear processes.

The effects of atomic electrons on nuclear resonance scattering have been
studied many times before [7, 8, 9, 10]. However, a review of such effects in
nuclear resonance reactions has not been given in any detail. For this reason,
in the literature, the correction factors obtained for nonresonant reactions
are also applied to narrow resonances (see, e.g., Refs. [11, 12, 13, 14]).
Such corrections always result in a bare (unscreened) resonance strength that
is lower, by up to 25%, depending on the reaction, compared to the measured
(screened) strength. However, it is neither obvious why the same laboratory
screening correction factors should be applied to both nonresonant and narrow-
resonance reaction data, nor whether there are compelling reasons to correct
the latter data for laboratory screening effects at all.

In Secs. II and III, laboratory electron screening effects for nonresonant
reactions and narrow resonances, respectively, will be reviewed. Screening
energies and lengths are presented in Sec. IV. Results are provided in Sec. V
and future measurements are discussed in Sec. VI. A concluding summary is
given in Sec. VII.

##  II Electron screening in nonresonant reactions

The nonresonant cross section, $\sigma(E)$, at a center-of-mass energy, $E$,
can be parameterized as [15]

$\sigma(E)\equiv\frac{1}{E}S(E)e^{-2\pi\eta(E)}$ (1)

where the astrophysical $S$ factor is frequently a function that varies slowly
with energy; $\eta(E)$ denotes the Sommerfeld parameter, $\eta$ $\equiv$
($Z_{0}Z_{1}e^{2}/\hbar)\sqrt{\mu/(2E)}$; $Z_{0}$, $Z_{1}$, $e$, and $\mu$ are
the charges of the interacting nuclei, elementary charge, and reduced mass,
respectively. The energy-dependent Gamow factor, $e^{-2\pi\eta}$, describes
the $s$-wave transmission through the Coulomb barrier.

The situation is depicted in Fig. 1. The unscreened Coulomb barrier,
$V_{C}(r)$, is shown as the blue curve. A negative screening potential,
$U_{e}$, is represented by the green line. It is depicted here as a constant
potential, which is the usual assumption made in the literature for
nonresonant reactions. The magnitude of $U_{e}$ is highly exaggerated for
illustrative purposes. The screened Coulomb potential, $V_{C}(r)$ $+$ $U_{e}$,
i.e., the sum of the blue and green lines, is shown as the red curve. When a
particle is incident on the unscreened barrier at a center-of-mass energy, $E$
(gray arrow at right), it needs to tunnel through a distance $R_{u}$ $-$
$R_{n}$ to initiate the reaction, where $R_{u}$ and $R_{n}$ denote the
classical turning point for the unscreened barrier and the nuclear radius,
respectively. A particle of energy $E$ incident on the screened barrier will
tunnel through a shorter distance of $R_{s}$ $-$ $R_{n}$, where $R_{s}$ is the
classical turning point of the screened barrier. The increase in the measured
nonresonant cross section is described by the ratio of transmission
probabilities, $T^{\prime}$ and $T$, through the screened and unscreened
barriers, respectively, at energy, $E$,

$f_{nr}\equiv\frac{\sigma_{\mathrm{screen}}}{\sigma_{\mathrm{unscreen}}}=\frac{T^{\prime}(E)}{T(E)}$
(2)

The transmission coefficient in the Wentzel-Kramers-Brillouin (WKB)
approximation for the unscreened Coulomb barrier is given by [16]

$T(E)\approx\exp\left(-\frac{\sqrt{8\mu}}{\hbar}\int_{R_{n}}^{R_{u}}\sqrt{V_{C}(r)-E}\,dr\right)$
(3)

where $\mu$ is the reduced mass, and $V_{C}(r)$ is the (unscreened) Coulomb
potential. The outer turning point is given by $R_{u}$ $=$
$Z_{0}Z_{1}e^{2}/(4\pi\epsilon_{0}E)$, with $\epsilon_{0}$ denoting the vacuum
permittivity. For a particle approaching the screened barrier at energy $E$,
we can write

$T^{\prime}(E)\approx\exp\left(-\frac{\sqrt{8\mu}}{\hbar}\int_{R_{n}}^{R_{u}}\sqrt{V_{C}(r)+U_{e}-E}\,dr\right)$
(4)

It can be seen that Eq. (4) is equivalent to the transmission of the
unscreened barrier at an energy of $E_{\mathrm{eff}}$ $=$ $E$ $+$ $|U_{e}|$,
i.e., $T^{\prime}(E)$ $=$ $T(E_{\mathrm{eff}})$, as indicated by the blue
arrow in Fig. 1. This is the reason why usually the transmission coefficients,
$T^{\prime}(E)$ and $T(E)$, are not computed numerically. Instead, they are
approximated by the Gamow factors, $T(E)$ $\approx$
$\mathrm{exp}(-2\pi\eta(E))$ and $T^{\prime}(E)$ $\approx$
$\mathrm{exp}(-2\pi\eta(E_{\mathrm{eff}}))$, so that the nonresonant electron
screening correction factor becomes

$f_{nr}\approx\frac{e^{-2\pi\eta(E_{\mathrm{eff}})}}{e^{-2\pi\eta(E)}}\approx
e^{\pi\eta(E)\frac{|U_{e}|}{E}}$ (5)

In the last step, it is assumed that the energy of the incident particle is
large compared to the screening energy, i.e., $E$ $\gg$ $|U_{e}|$. The
electron screening potential, $U_{e}$, is assumed to be independent of energy.
The factor, $f_{nr}$, amounts to unity at higher energies, where $E$ $\gg$
$|U_{e}|$, and increases as the energy decreases. Therefore, its magnitude is
$f_{nr}$ $\geq$ $1$.

Figure 1:  Schematic representation (not to scale) of electron screening for a
nonresonant charged-particle nuclear reaction in the laboratory, showing the
unscreened Coulomb potential (blue curve), constant negative screening
potential, $U_{e}$ (green line), screened Coulomb potential (red curve), total
energy, $E$ (gray arrows), and effective energy, $E_{\mathrm{eff}}$ $=$ $E$
$+$ $|U_{e}|$ (blue arrow); $R_{n}$, $R_{s}$, and $R_{u}$ denote the nuclear
radius, and the classical turning points at energy $E$ for the screened and
unscreened barrier, respectively. The actual reaction in the laboratory is
represented by the second gray arrow (on the left) extending to the red curve.
Notice that $R_{s}$ is also equal to the classical turning point for the
unscreened barrier (blue curve) at the effective energy, $E_{\mathrm{eff}}$
(blue arrow). No screening potential is shown inside the nucleus, because it
is irrelevant for the derivation of $f_{nr}$ in Eq. (5).

Equation (5) has been applied in Refs. [17, 6] and is the commonly adopted
formalism for nonresonant cross sections. As can be seen from the above
derivation, the incident particle does not actually gain total energy, as is
sometimes stated. Instead, the energy shift, from $E$ to $E_{\mathrm{eff}}$,
facilitates the convenient calculation of $f_{nr}$ by using the Gamow factors
at these two energies (see also Ref. [18]), without the need of computing the
ratio of transmission coefficients at energy $E$ numerically. Also, sometimes
a pre-factor containing the ratio of energies and $S$ factors at $E$ and
$E_{\mathrm{eff}}$ is included in Eq. (5). This is incorrect since the
reaction takes place at energy $E$, not at $E_{\mathrm{eff}}$.

The electron screening potential for nonresonant reactions can be estimated
with a suitable model representing the electronic configuration of the target
and projectile. For example, for gaseous targets and low bombarding energies,
the adiabatic (limit) approximation is frequently used [6]. It assumes that
the electron velocities are much larger than the relative motion between the
target and projectile nuclei. This implies that the electron cloud instantly
adjusts to the ground state of a molecule-like system consisting of the two
approaching nuclei with respective charges of $Z_{0}$ and $Z_{1}$. The
(negative) screening potential, $U_{\mathrm{ad}}$, can then be approximated by
the difference in electron binding energies,

$U_{\mathrm{ad}}\approx B_{e}(Z_{0})+B_{e}(Z_{1})-B_{e}(Z_{0}+Z_{1})$ (6)

where $B_{e}(Z_{0})$, $B_{e}(Z_{1})$, and $B_{e}(Z_{0}+Z_{1})$ denote the
(positive) total electron binding energies in the atoms with charges of
$Z_{0}$, $Z_{1}$, and $Z_{0}+Z_{1}$, respectively (see Eq. (5) in Ref. [6]).

As already pointed out in Sec. I, the values of $|U_{e}|$ extracted from low-
energy cross section data are, in most cases, significantly larger than those
calculated using the adiabatic approximation, $|U_{\mathrm{ad}}|$, by about a
factor of two. A tabulated comparison between values can be found, e.g., in
Ref. [4].

##  III Electron screening for narrow resonances

For an isolated narrow resonance, what is usually measured is not directly the
cross section, but the integrated cross section over the energy region of the
resonance. This quantity is referred to as the resonance strength and can be
extracted in the laboratory from the measured thick-target resonance yield
curve [15]. The resonance strength, $\omega\gamma$, is defined by

$\omega\gamma\equiv\omega\frac{\Gamma_{a}\Gamma_{b}}{\Gamma}$ (7)

where $\Gamma_{a}$, $\Gamma_{b}$, and $\Gamma$ $=$ $\Gamma_{a}$ $+$
$\Gamma_{b}$ $+$ … denote the energy-dependent partial widths of the incoming
channel and the outgoing channel, respectively, and the total resonance width;
$\omega$ $\equiv$ $(2J+1)/[(2j_{p}+1)(2j_{t}+1)]$ is the statistical spin
factor, with $J$, $j_{p}$, and $j_{t}$ representing the spins of the
resonance, projectile, and target, respectively. The general form of the
resonance electron screening correction factor can then be written as

$f_{r}\equiv\frac{\omega\gamma_{\mathrm{screen}}}{\omega\gamma_{\mathrm{unscreen}}}=\frac{\Gamma_{a}^{\prime}}{\Gamma_{a}}\frac{\Gamma_{b}^{\prime}}{\Gamma_{b}}\frac{\Gamma}{\Gamma^{\prime}}$
(8)

where the primed and unprimed quantities refer to the screened and unscreened
widths, respectively. The meaning of a “narrow resonance” in the present
context will be defined at the end of this section.

In resonant charged-particle reactions at sufficiently low bombarding
energies, which are of main interest in nuclear astrophysics measurements, the
entrance channel width is much smaller than the exit channel width, i.e.,
$\Gamma_{a}$ $\ll$ $\Gamma_{b}$. In this case, Eq. (8) reduces to

$f_{r}=\frac{\Gamma_{a}^{\prime}}{\Gamma_{a}}=\frac{P^{\prime}}{P}\approx\frac{T^{\prime}}{T}$
(9)

Here, it is assumed that the main energy dependence of the particle partial
width, $\Gamma_{a}$, arises from the penetration factor, $P_{\ell}$ (see,
e.g., Ref. [15]), and the latter quantity is approximated by the barrier
transmission coefficient, $T$.111The definition of the transmission
coefficient usually contains the ratio of wave numbers to the left and right
of the barrier, whereas the penetration factor does not [19, 20]. However, the
wave numbers are implicitly included in the WKB wave function normalizations
[16]. Therefore, the energy dependencies of the transmission coefficient and
the penetration factor for the same value of the orbital angular momentum
should be nearly equal.

In the opposite case, $\Gamma_{a}$ $\gg$ $\Gamma_{b}$, the resonance electron
screening correction factor reduces to $f_{r}$ $\approx$
$\Gamma_{b}^{\prime}/\Gamma_{b}$. If such a resonance decays by emission of a
$\gamma$ ray or neutron, electron screening will only impact the value of
$f_{r}$ through the weak energy dependence of $\Gamma_{b}$, with the result
that $f_{r}$ $\approx$ $1$. If the emitted particle is charged (e.g., a proton
or $\alpha$ particle), its transmission through the screened barrier must be
considered in addition (see Eq. (8)).

Figure 2 presents the situation for a resonance with $\Gamma_{a}$ $\ll$
$\Gamma_{b}$, which is of primary interest in the present work. The unscreened
Coulomb barrier is shown as the blue curve. The outer turning point for a
particle approaching this barrier at the resonance energy, $E_{r}$,
corresponding to a resonance level (blue horizontal line) inside the nucleus
at the same energy, is denoted by $R_{u}$. The energy $E_{r}$ is a property of
the compound nucleus only. Whereas outside the nuclear radius a constant
screening potential was assumed for the discussion in Sec. II and Fig. 1, this
restriction will now be relaxed by adopting a negative screening potential,
$V_{\mathrm{screen}}(r)$, that varies with distance (depicted in green in Fig.
2). At large radial distances, $r\rightarrow\infty$, the screening potential
will approach zero, $V_{\mathrm{screen}}(r)\rightarrow 0$ (see also Sec. IV).
Furthermore, inside the nucleus, the screening potential, $U_{e}$, is assumed
to be constant (green horizontal line).222If we simplify the problem and
assume that the K-shell electrons (see Sec. V) form a uniformly charged sphere
surrounding the target nucleus, then the screening potential will be nearly
constant over the much smaller nuclear region. A constant screening potential
inside the nucleus was also assumed, e.g., in Refs. [21, 22].

Figure 2:  Schematic representation (not to scale) of electron screening for a
resonance in the laboratory, showing the unscreened Coulomb potential (blue
curve), negative screening potential, $V_{\mathrm{screen}}$ (green), screened
Coulomb potential (red curve), resonance energy, $E_{r}$ (blue arrow), and
shifted energy, $E_{r}^{\prime}$ $=$ $E_{r}$ $-$ $|U_{e}|$ (gray arrow);
$R_{n}$ denotes the nuclear radius, $R_{u}$ is the classical turning point at
energy $E_{r}$ for the unscreened barrier, and $R_{s}$ is the turning point at
energy $E_{r}$ $-$ $|U_{e}|$ for the screened barrier. The actual reaction in
the laboratory is represented by the gray arrow and the red curve. Notice that
the tunneling distance, $R_{s}$ $-$ $R_{n}$, through the screened barrier at
energy $E_{r}$ $-$ $|U_{e}|$ is larger than the distance $R_{u}$ $-$ $R_{n}$
through the unscreened barrier at $E_{r}$. If the screening potential,
$V_{\mathrm{screen}}(r)$, would be constant, the tunneling distances would be
the same and no change in either the transmission coefficient or resonance
strength would be expected.

A laboratory measurement of an isolated narrow resonance is impacted by
electron screening in two ways: (i) outside the nucleus, the sum of the
unscreened Coulomb potential (blue line) and screening potential (green line)
gives rise to the screened Coulomb potential, shown in red; (ii) the
attractive screening potential performs work on the projectile approaching the
target atom, and, therefore, the energy at which the narrow resonance will be
excited in the laboratory becomes $E_{r}^{\prime}$ $=$ $E_{r}$ $-$ $|U_{e}|$,
where $E_{r}^{\prime}$ $<$ $E_{r}$ (see the gray arrow in Fig. 2). Or,
expressed differently, the virtual level inside the compound nucleus (red
horizontal line) is lowered by an amount of $|U_{e}|$.

The transmission coefficent for the unscreened barrier is given by Eq. (3),
where the center-of-mass resonance energy, $E_{r}$, replaces the energy, $E$.
But, unlike the nonresonant case in Sec. II, the transmission coefficient in
the presence of electrons is given by

$T^{\prime}\approx\exp\left(-\frac{\sqrt{8\mu}}{\hbar}\int_{R_{n}}^{R_{s}}\sqrt{V_{C}(r)+V_{\mathrm{screen}}(r)-(E_{r}+U_{e})}\,dr\right)$
(10)

where the outer turning point for the screened case, $R_{s}$, is obtained from
$V_{C}(R_{s})$ $+$ $V_{\mathrm{screen}}(R_{s})$ $=$ $E_{r}$ $+$ $U_{e}$. It
can be seen that, for the special case of a constant screening potential over
the region of the outer turning point, i.e., $V_{\mathrm{screen}}(r)$ $=$
$U_{e}$ $=$ const, the two effects discussed above, (i) and (ii), cancel each
other exactly. Consequently, the two turning points for the screened and
unscreened case, $R_{s}$ and $R_{u}$, would coincide and Eq. (10) reduces to
Eq. (3). In other words, the electron screening correction factor, $f_{r}$,
would become unity. This also means, contrary to the claim in Ref. [6], that
it is incorrect to apply the screening factor for nonresonant reactions,
$f_{nr}$ in Eq. (2), to the measured strength of an isolated narrow resonance,
because this procedure disregards the shift down in resonance energy from
$E_{r}$ to $E_{r}$ $-$ $|U_{e}|$ in the calculation of the transmission
coefficient. The possibility of measuring this resonance energy shift will be
addressed in Sec. VI.

When $V_{\mathrm{screen}}(r)$ is not constant, but declines outside the
nuclear radius toward zero, the transmission coefficient for the screened
Coulomb barrier is, in fact, smaller than the transmission through the
unscreened barrier. This can be seen in Fig. 2, where the distance the
particle needs to tunnel through the screened barrier, $R_{s}$ $-$ $R_{n}$, at
$E_{r}$ $-$ $|U_{e}|$ is larger than the distance for tunneling through the
unscreened barrier, $R_{u}$ $-$ $R_{n}$, at the energy $E_{r}$. Therefore, the
unscreened resonance strength is generally larger than the screened value,
which is the opposite of the assumption generally made in the literature for
the laboratory screening correction for a narrow resonance (see Sec. I). In
other words, unlike the correction factor for nonresonant cross sections,
$f_{nr}$ $\geq$ $1$, the magnitude of the narrow-resonance correction factor
is $f_{r}$ $\leq$ $1$, as long as the screening potential,
$V_{\mathrm{screen}}(r)$, is negative. It is assumed here that the screening
potential, $U_{e}$, is constant inside the nucleus and can simply be
subtracted from the unscreened resonance energy. It follows from the above
discussion that the important quantity in this context is not only the
magnitude of the electron screening potential, but also its rate of decline
over the tunneling region.

Arguments similar to the above had been presented earlier in connection with
electron screening in $\alpha$-particle radioactivity [21, 23] and screening
effects for narrow resonances in astrophysical plasmas [3, 22]. The shift in
the energy of the virtual resonance level, caused by electron screening, is
frequently disregarded in the literature (see, e.g., Refs. [17, 24, 25]),
leading to incorrect predictions.

In the present context, a “narrow resonance” is defined by $\Gamma$ $\ll$
$|U_{e}|$, i.e., its total width must be small compared to the shift in the
resonance energy, $U_{e}$ $=$ $E_{r}^{\prime}$ $-$ $E_{r}$, caused by electron
screening. As discussed above, for this condition the reaction occurs at the
screened energy, $E_{r}^{\prime}$, instead of the unscreened one, $E_{r}$. For
a broad resonance, i.e., $\Gamma$ $\gtrsim$ $|U_{e}|$, the reaction can
proceed over an extended range of incident energies, including the unscreened
resonance energy, and the electron screening correction factor must be
computed numerically from an expression more complicated than Eq. (9).

##  IV Screening lengths and screening potentials

A simple model is used in this work to estimate numerical values for the
screening effects on the measured strength of a narrow resonance. The
resonance screening factor, $f_{r}$, is found by numerically integrating Eqs.
(3) and (10). A Yukawa-type expression is adopted for the screened Coulomb
potential outside the nuclear radius,

$V_{C}(r)+V_{\mathrm{screen}}(r)=\frac{e^{2}}{4\pi\epsilon_{0}}\frac{Z_{0}Z_{1}}{r}~{}e^{-r/L}\,\,,r\geq
R_{n}$ (11)

where $L$ represents the electron screening length scale. The exponential
factor damps the overall potential to nearly zero after a few screening
lengths. For $r$ $\ll$ $L$, and keeping only the linear term in the expansion
of the exponential factor, Eq. (11) reduces to

$V_{C}(r)+V_{\mathrm{screen}}(r)\approx\frac{e^{2}}{4\pi\epsilon_{0}}\frac{Z_{0}Z_{1}}{r}-\frac{e^{2}}{4\pi\epsilon_{0}}\frac{Z_{0}Z_{1}}{L}\,\,,r\ll
L$ (12)

Therefore, and following Refs. [21, 22], the constant screening potential
inside the nucleus, $U_{e}$, can be approximated by

$U_{e}=-\frac{e^{2}}{4\pi\epsilon_{0}}\frac{Z_{0}Z_{1}}{L}$ (13)

For the nuclear radius, a value of

$R_{n}=1.2(A_{0}^{1/3}+A_{1}^{1/3})~{}\mathrm{fm}$ (14)

will be assumed, where $A_{0}$ and $A_{1}$ are the integer mass numbers of the
projectile and target, respectively.

The last task before the electron screening factor for narrow resonances,
$f_{r}$, can be computed numerically is to specify the electron screening
length, $L$. The smaller the screening length scale, the larger the screening
energy inside the nucleus, $U_{e}$, and its rate of decline outside the
nuclear radius, and the larger the screening correction factor, $f_{r}$, will
become. The screening length will depend on the atoms under consideration and
the environment in which the nuclear reaction takes place.

A dominant contribution to the electron screening is provided by the (inner)
core electrons, especially the K electrons. Their contribution will be
estimated by approximating their screening length with the radius of the K
shell,

$L_{KS}=r_{K}$ (15)

The latter values were calculated by Ref. [26] using the electron localization
function (ELF), together with Hartree-Fock wave functions of the neutral
atoms. Typical values of $r_{K}$ range from $0.58a_{0}$ for carbon to
$0.094a_{0}$ for iron, where $a_{0}$ $=$ $5.29\times 10^{4}$ fm denotes the
Bohr radius.

When the target atoms either form a metal lattice or are embedded in a metal
backing, the screening effect of the (free) conduction-band electrons must be
considered in addition. An approximation of their screening length can be
obtained from the Thomas-Fermi model of a metal [27], which predicts333The
numerical value of $3.7\times 10^{-10}$ provided in Eq. (3) of Ref. [21] is
incorrect and should be replaced by $6.1\times 10^{-9}$.

$L_{TF}=\sqrt{\frac{2\epsilon_{0}E_{F}}{3\rho e^{2}}}=6.1\times
10^{4}\sqrt{\frac{E_{F}~{}[\mathrm{eV}]}{\rho~{}[10^{22}~{}\mathrm{cm^{-3}}]}}~{}\mathrm{fm}$
(16)

where $E_{F}$ denotes the Fermi energy and $\rho$ is the electron density.
Typical values for metals are $E_{F}$ $\approx$ $10$ eV and $\rho$ $\approx$
$10\times 10^{22}$ cm-3 [27], giving a shielding length of $L_{TF}$ $\approx$
$6.10\times 10^{4}$ fm.

A number of authors (see, e.g., Refs. [28, 29, 25]) have computed screening
lengths using the Debye-Hückel model, which yields444The numerical value of
$2.18\times 10^{-8}$ provided in Eq. (4) of Ref. [21] is incorrect and should
be replaced by $2.18\times 10^{-11}$.

$L_{DH}=\sqrt{\frac{\epsilon_{0}k_{B}T}{\rho e^{2}}}=6.9\times
10^{2}\sqrt{\frac{T~{}[\mathrm{K}]}{\rho~{}[10^{22}~{}\mathrm{cm^{-3}}]}}~{}\mathrm{fm}$
(17)

where $k_{B}$ and $T$ denote the Boltzmann constant and temperature,
respectively. This model gives much smaller screening lengths, resulting in a
stronger electron screening effect. Equation (17) is useful for a plasma [2],
but this formulation does not apply to metals at room temperature, as pointed
out, e.g., by Refs. [18, 30]. For doped semiconductors or electrolytes, the
Debye-Hückel model results in modified expressions [27] compared to Eq. (17).

Here, only the dominant contributions to the electron screening, according to
Eqs. (15) and (16), are considered. For a metal target and low bombarding
energies, the velocity of the incident projectile is much smaller than the
Fermi velocity of the electrons, and, therefore, the electron screening effect
is caused by the static polarization of both the surrounding bound and
conduction electrons. When applicable, the effects of K-shell and conduction
electrons will be combined by adopting a shielding length of $L^{-1}$ $=$
$r_{K}^{-1}$ $+$ $L_{TF}^{-1}$, which assumes that the total screening
potential is given by the sum of the individual contributions. Numerical
results will be presented in the next section.

##  V Results and discussion

Table 1 gives the main results, including a comparison with values from the
literature. Six narrow resonances are listed in the reactions
17O(p,$\alpha$)14N, 18O(p,$\gamma$)19F, 22Ne(p,$\gamma$)23Na,
25Mg(p,$\gamma$)26Al, and 27Al(p,$\gamma$)28Si. All of these fulfill the
conditions $\Gamma_{a}$ $\ll$ $\Gamma_{b}$ and $\Gamma$ $\lesssim$ $100$ eV
(see Sec. III). The target compositions are given in column 4. They range from
wide-gap semiconductor material (Ta2O5), gas (22Ne), to metal (Mg, Al). The
screening lengths of the K-shell electrons in the neutral target atoms,
$r_{K}$, which are listed in column 5, were assumed to be approximately equal
to the K-shell radii found in Tab. 1 of Ref. [26]. For the two metals, the
screening lengths, $L_{TF}$, calculated from the Thomas-Fermi model according
to Eq. (16), are given in column 6. The outer turning point radii, $R_{s}$, of
the screened Coulomb potential, calculated from Eq. (10), are listed in column
7. A comparison of length scales indicates that the screening lengths, $r_{K}$
and $L_{TF}$, are much larger than the outer turning-point radii, $R_{s}$.
Consequently, any screening correction factors are expected to be small.
Column 8 provides values for the constant screening potential, $U_{e}$ (see
Eq. (13)), inside the compound nucleus, which are approximately equal to the
energy difference between the unscreened resonance energy, $E_{r}$, and the
screened one. Values of $U_{e}$ range from $-0.5$ to $-2.0$ keV. They are
similar to the adiabatic approximation estimates obtained from Eq. (6), which
are given in column 9.

The present estimates of the screening correction factors for narrow
resonances, $f_{r}$, calculated according to Eqs. (9) $-$ (16), are listed in
column 10. As can be seen, the values of $f_{r}$ are unity within $0.2$%.
Also, the results predict that the screened resonance strengths are slightly
smaller than the unscreened ones, consistent with the discussion in Sec. III.

In comparison, screening “enhancement” factors for narrow resonances from the
literature, $f_{\mathrm{Lit}}$, calculated from Eqs. (5) and (6), are given in
column 11. These factors yield screened resonance strengths that exceed the
unscreened values by 7% to 25%, depending on the reaction. Again, it must be
emphasized that it is not appropriate to calculate electron screening factors
for narrow resonances using Eq. (5), which applies to nonresonant cross
sections and disregards the shift in the resonance energy, as explained in
Sec. III. Notice, that the (incorrect) literature “enhancement” factors are
significant, even when the measured resonance strength uncertainties are taken
into account.

A number of tests were performed to investigate the sensitivity of the present
results to parameter variations. Changing the nuclear radius parameter in Eq.
(14) from $1.2$ fm to either $0.5$ fm or $2.0$ fm did not impact the numerical
values of $f_{r}$ noticeably. The inclusion of a centrifugal term,
$\hbar^{2}\ell(\ell+1)/(2\mu r^{2})$, in Eqs. (3) and (10), and varying the
orbital angular momentum, $\ell$, between $0$ and $3$, did not change any of
the results either. Increasing the screening lengths adopted here (i.e., the
values of $r_{K}$ and $L_{TF}$ listed in columns 5 and 6, respectively, of
Tab. 1) will result in values of $f_{r}$ that are even closer to unity. When
the screening lengths are reduced by a factor of two, the electron screening
correction factors, $f_{r}$, are unity within 1%. These changes are negligibly
small, contrary to the correction factors reported in the literature for
narrow resonances (column 11).

The simple procedure for calculating narrow-resonance screening factors
presented here has a number of shortcomings. A static, time-independent model
has been adopted, although a dynamical approach would be more appropriate. A
constant screening potential is assumed inside the compound nucleus, see Eq.
(13), which oversimplifies the actual situation. Similar arguments apply to
approximating the screened potential by the damped, Yukawa-type, expression of
Eq. (11). The numerical results are impacted slightly by the adopted values of
the screening lengths for the K-shell and conduction electrons, for which
rough estimates have been employed here. It is worthwhile to address these
issues in the future using more sophisticated models, e.g., similar to those
developed for the related case of $\alpha$-particle radioactivity [31, 23,
30].

Table 1: Electron screening factors, $f_{r}$, and related quantities, for reported measured narrow resonances. Reaction | E${}_{r}^{c.m.}$ (keV) |  $\Gamma$ (eV) | Target |  $r_{K}$ (fm)111Electron screening length of K-shell electrons in the neutral target atom; see Tab. 1 of Ref. [26]. |  $L_{TF}$ (fm)222Electron screening length of the Thomas-Fermi model for metals; see Eq. (16). |  $R_{s}$ (fm)333Outer turning point of screened potential; see Eq. (10). |  $U_{e}$ (keV)444Constant screening potential inside the compound nucleus, which is approximately equal to the energy shift down from the unscreened resonance energy to the screened one; see Eq. (13). |  $U_{\mathrm{ad}}$ (keV)555Adiabatic approximation estimate of the screening potential; see Eq. (6). |  $f_{r}$666Present estimate of the screening correction factor for narrow resonances; see Eq. (9). | Literature
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  |  | $f_{\mathrm{Lit}}$ | Ref.
17O(p,$\alpha$)14N | 64.5 | 130$\pm$5888From Refs. [35, 36], using $\Gamma$ $\approx$ $\Gamma_{\alpha}$. | Ta2O5121212Wide-gap semiconductor. | 21160 |  | 178.6 |  $-$0.54 |  $-$0.68 | 0.9996 | 1.15 | [12]
18O(p,$\gamma$)19F | 90.0 | 121$\pm$5999From R matrix fit of Ref. [33] (see their Table 4). | Ta2O5121212Wide-gap semiconductor. | 21160 |  | 128.0 |  $-$0.54 |  $-$0.68 | 0.9998 | 1.10 | [32]
18O(p,$\alpha$)15N | 90.0 | 121$\pm$5999From R matrix fit of Ref. [33] (see their Table 4). | Ta2O5121212Wide-gap semiconductor. | 21160 |  | 128.0 |  $-$0.54 |  $-$0.68 | 0.9998 | 1.09 | [33, 34]
22Ne(p,$\gamma$)23Na | 149.4 |  $<60$101010Upper limit using $\Gamma$ $\approx$ $\Gamma_{\gamma}$, with $\Gamma_{\gamma}$ estimated using the Recommended Upper Limits (RUL) [37] for the primary transitions observed by Refs. [11] and [38] for 22Ne $+$ $p$ and 25Mg $+$ $p$, respectively. | Ne gas | 15870 |  | 96.4 |  $-$0.91 |  $-$0.91 | 0.9998 | 1.07 | [11]
25Mg(p,$\gamma$)26Al | 92.2 |  $<30$101010Upper limit using $\Gamma$ $\approx$ $\Gamma_{\gamma}$, with $\Gamma_{\gamma}$ estimated using the Recommended Upper Limits (RUL) [37] for the primary transitions observed by Refs. [11] and [38] for 22Ne $+$ $p$ and 25Mg $+$ $p$, respectively. | Mg metal | 12484 | 55315 | 187.4 |  $-$1.7 |  $-$1.2 | 0.9976 | 1.25 | [13]
27Al(p,$\gamma$)28Si | 956 | 70$\pm$14111111From Ref. [39]. | Al metal | 11310 | 49044 | 19.6 |  $-$2.0 |  $-$1.3 | 0.9999 |  |
176Lu(p,n)176Hf | 805777This resonance could not have been observed by Ref. [25] because its strength would be far below the present-day detection limit; see Sec. VI. |  | Lu metal |  |  |  |  |  |  |  | [25]

##  VI Resonance energy shifts caused by electron screening

Experimental studies of electron screening effects in resonant reactions face
a number of obstacles.

First, electron screening is expected to impact a resonance strength in a
charged-particle induced reaction only when the entrance channel width is
significantly smaller than the exit-channel one, $\Gamma_{a}$ $\ll$
$\Gamma_{b}$555For the condition $\Gamma_{a}$ $\gg$ $\Gamma_{b}$ (or
$\omega\gamma$ $\approx$ $\omega\Gamma_{b}$), and assuming that the resonance
decays by emission of a neutron or $\gamma$ ray, electron screening will
impact the exit channel width, $\Gamma_{b}$, only through the small change in
the decay energy (Sec. III). In this case, the value of $f_{r}$ will be close
to unity for an exothermic reaction. (see Sec. III). Second, even when the
condition $\Gamma_{a}$ $\ll$ $\Gamma_{b}$ is fulfilled, the ratio of screened
versus unscreened resonance strengths, $f_{r}$, will be close to unity (see
Table 1) because the effects of the screened Coulomb potential and the shift
in the resonance energy compensate each other largely (see Sec. IV).
Consequently, electron screening will not significantly impact the values of
measured resonance strengths, which are frequently extracted from the plateau
height of thick-target yield curves [15].

Because of these difficulties, it is worthwhile to consider, instead, to
measure the shift in the resonance energy, $E_{r}^{\prime}$ $-$ $E_{r}$,
caused by electron screening. Such a shift is expected to occur, in principle,
in a charged-particle resonance reaction regardless of the relative magnitudes
of the entrance and exit channel partial widths ($\Gamma_{a}$ and
$\Gamma_{b}$).

A shift in resonance energy, presumably caused by electron screening, had been
reported by Kettner et al. [25]. They measured the thick-target yield curve of
a 176Lu(p,n)176Hf resonance at 805 keV (center-of-mass energy), using three
different target-backing combinations (Lu2O3 insulator, Lu metal, and PdLu
alloy). No other information on this resonance is available in the literature.
They observed an energy difference in the leading edge of the yield curves
between the metal (and alloy) and the insulator target. By assuming that the
insulator target exhibits insignificant screening, the observed energy shift
down by $-32\pm 2$ keV (Tab. 1) was interpreted as the electron screening
potential for the metal (and alloy) target. Huke et al. [18] discussed the
energy shift reported by Ref. [25], but attributed it instead to differences
in target preparation and resulting stopping powers. The Wigner limit for an
$s$-wave proton partial width in the 176Lu(p,n)176Hf reaction at $805$ keV in
the center of mass corresponds to a value of $\approx$ $10^{-16}$ eV, which is
far below the present-day experimental detection limit. Therefore, the claim
of Ref. [25] to have detected a resonance at $805$ keV in the 176Lu(p,n)176Hf
reaction, which is still being discussed in the recent literature [40, 41,
42], is incorrect.

No unambiguous evidence has so far been published demonstrating the existence
of a shift in resonance energy caused by laboratory electron screening. Such
an energy shift could be detected by comparing a resonance energy measured in
the laboratory with the corresponding unscreened value. The latter corresponds
to the resonance energy that would be obtained in the absence of all electrons
surrounding the interacting nuclei. It can be determined from

$E_{r}=E_{x}-Q_{\mathrm{nu}}$ (18)

where $E_{r}$ is the unscreened resonance energy in the center-of-mass system
(same as Sec. III), which is a property of the compound nucleus only; $E_{x}$
denotes the excitation energy of the compound nucleus, and $Q_{\mathrm{nu}}$
represents the $Q$ value computed from nuclear (as opposed to atomic) masses
[43]. This value can be compared to the resonance energy that is obtained from
a laboratory measurement, by using the relativistic expression

$E_{r}^{\prime}=\sqrt{2m_{0}c^{2}E_{r}^{lab}+[(m_{0}+m_{1})c^{2}]^{2}}-(m_{0}+m_{1})c^{2}$
(19)

where $E_{r}^{\prime}$ and $E_{r}^{lab}$ denote the center-of-mass energy of
the resonance in the presence of electrons (same as in Sec. III), and the
measured resonance energy in the laboratory reference frame, respectively;
$m_{0}$ and $m_{1}$ represent the masses of the target and projectile. The
energy shift caused by electron screening contributes to the measured
difference, $E_{r}^{\prime}$ $-$ $E_{r}$.

This procedure requires a careful assessment of all input quantities. The
candidate resonance needs to be narrow (i.e., $\Gamma$ $\lesssim$ $100$ eV),
and the target well characterized and free of surface contamination. The
energy spread of the incident beam must be small (i.e., no more than a few
hundreds of electron volts). The excitation energy, $E_{x}$, in Eq. (18) needs
to be precisely measured, preferably by $\gamma$-ray spectrometry. The
laboratory resonance energy, $E_{r}^{lab}$, in Eq. (19) must be measured
precisely using methods that do not depend on the energies of other
(calibration) resonances. Finally, additional effects caused by the presence
of atomic electrons in the target need to be accounted for, e.g., the
excitation and ionization of bound electrons in the atom in which the nuclear
reaction is taking place [10, 44], and the Lewis effect [45].

As an example, let us consider the resonance in the 27Al(p,$\gamma$)28Si
reaction near a center-of-mass energy of $956$-keV ($J^{\pi}$ $=$ $3^{+}$;
$\Gamma$ $=$ $70\pm 14$ eV [39]). The corresponding excitation energy, which
was determined from the measured $\gamma$-ray energies of the primary decays,
is reported as $E_{x}$ $=$ $12541.31\pm 0.14$ keV [46]. The nuclear $Q$ value
amounts to $Q_{\mathrm{nu}}$ $=$ $11583.63\pm 0.05$ keV [47]. Consequently,
this yields an unscreened resonance energy of $E_{r}$ $=$ $957.68\pm 0.15$
keV, according to Eq. (18). The laboratory value of the resonance energy is
reported as $E_{r}^{lab}$ $=$ $991.756\pm 0.017$ keV [48]. In that experiment,
an aluminum metal target was used and the energy was determined relative to a
Josephson-derived 1-V standard. Also, the reported value includes corrections
caused by the ionization of atomic electrons (corresponding to an energy shift
of $24\pm 12$ eV). The above laboratory resonance energy results in a screened
resonance energy in the center-of-mass system of $E_{r}^{\prime}$ $=$
$956.032\pm 0.016$ keV, according to Eq. (19). The energy difference,
$E_{r}^{\prime}$ $-$ $E_{r}$, amounts to $-1.65\pm 0.15$ keV. This result is
near the screening energy of $U_{e}$ $=$ $-2.0$ keV (Table 1), which was
estimated using the simple model of the present work, based on a Yukawa-type
screening potential and screening lengths for electrons in the K shell and the
conduction band (Sec. IV). It is also close to the value of $U_{\mathrm{ad}}$
$=$ $-1.3$ keV that is found from the adiabatic approximation (see Eq. (6)).
Although these two estimates of the screening potential roughly agree with the
energy difference, $E_{r}^{\prime}$ $-$ $E_{r}$, estimated above for the
$E_{r}$ $=$ $956$-keV resonance in the 27Al(p,$\gamma$)28Si reaction, further
studies will be needed to confirm this claim.

##  VII Summary

The present work addressed the estimation of laboratory electron screening
correction factors for isolated narrow resonances. Such corrections are
frequently performed in the literature with the same procedure and expression
used to correct laboratory nonresonant cross sections. It was pointed out that
electron screening affects nonresonant cross sections and resonance strengths
differently, and that it is not appropriate to correct measured resonance
strengths using the same procedure and expression employed for the correction
of measured nonresonant cross sections. The reported literature screening
factors applied to narrow resonances result in unscreened resonance strengths
that are smaller, by 7% to 25% depending on the reaction, than the measured
(screened) ones. On the contrary, the present work demonstrated that
unscreened resonance strengths are equal to the measured ones within $0.2$%.
This small correction is of no practical importance. Unless demonstrated
otherwise, measured resonance strengths do not need to be corrected for
laboratory electron screening effects.

Since electron screening has a negligible impact on the strengths of narrow
resonances, any attempts to study such effects by measuring the thick-target
yield are futile. Instead, and regardless of the relative magnitudes of the
entrance and exit channel partial widths, it may be more promising to detect
the shift in the resonance energy down from the unscreened value (i.e.,
obtained in the absence of any electrons) to the screened one (i.e., measured
in the laboratory). Although no unambiguous evidence has been published so far
demonstrating such an energy shift, it is pointed out this effect is likely
present in the data for the $E_{r}$ $=$ $956$-keV resonance in the
27Al(p,$\gamma$)28Si reaction. It is also demonstrated that the claim of a
previously reported detection [25] of a resonance in the 176Lu(p,n)176Hf
reaction is incorrect.

###### Acknowledgements.

The comments of Alain, Coc, Robert Janssens, Yosuke Kanai, Richard Longland,
Caleb Marshall, and Thanassis Psaltis are highly appreciated. This work is
supported by the DOE, Office of Science, Office of Nuclear Physics, under
grants DE-FG02-97ER41041 (UNC) and DE-FG02-97ER41033 (TUNL).

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