#  Determination of hydrogen density in solids by transmission ERDA under
nuclear-elastically enhanced recoiling of H by 8 and 9 MeV He

Hiroshi Kudo1 ***Corresponding author E-mail address<EMAIL_ADDRESS>,
Masanori Kurosawa2, Hiroshi Naramoto1, Masao Sataka1, Satoshi Ishii1, Kimikazu
Sasa1,3 and Shigeo Tomita3

1Research Facility Center for Science and Technology, University of Tsukuba,
1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan
2Faculty of Life and Environmental Sciences, University of Tsukuba, 1-1-1
Tennodai, Tsukuba, Ibaraki 305-8572, Japan
3Faculty of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai,
Tsukuba, Ibaraki 305-8571, Japan

ABSTRACT

Hydrogen number densities in thin self-supporting samples of polyphenylene
sulfide and muscovite have been determined by nuclear-elastic recoil detection
analysis of transmission layout. The analysis procedure is based only on the
database of stopping power and recoil cross section for material analysis,
without using any reference sample of known H content. For the polyphenylene
sulfide sample, the determined value of $(2.87\pm 0.26)\times 10^{22}\,{\rm
H}/$cm3 is in good agreement with the calculated value of $3.01\times
10^{22}\,{\rm H}/$cm3. For the muscovite sample, the H density originating
each from bound water and absorbed water is uniform over the entire thickness
of the sample. The determined density $(9.43\pm 0.75)\times 10^{21}$ H$/$cm3
of the muscovite agrees excellently with the value of $9.36\times 10^{21}$
H$/$cm3 obtained from other quantitative analyses typically applied for
minerals. The present results demonstrate the capability of accurate
determination of H contents in materials and minerals by transmission ERDA.

##  I Introduction

Knowledge about behavior of hydrogen in solids is of scientific and also of
practical importance since hydrogen can seriously affect physical and,
especially, mechanical properties of materials [1]. Also, hydrogen in solids
attracts geological interest since geological samples typically contain H in
the form of water (H2O) or hydroxyls (OH) [2] [3] [4]. For example, it is
certainly helpful to discuss the origin of the lunar water and the earth’s sea
via hydrogen analyses of glasses and minerals in lunar rocks [5] [6].

For analysis of hydrogen in solids, possible experimental methods, in
particular, of non-destructive type are extremely restricted because analysis
techniques based on electronic transitions of atoms are hardly applicable in
this case. A notable and unique feature of elastic recoil detection analysis
(ERDA) using a fast ion beam is non-destructive detection of light atoms in a
solid sample [7]. For analysis of hydrogen in thin self-supporting samples, a
useful method is ERDA of transmission layout (T-ERDA), in which recoil H
resulting from hard collisions with He or heavier projectiles is energy-
analyzed in the direction close to 0∘ after passing through the sample. For
example, Tirira and coworkers carried out profiling of H in thin self-
supporting samples by T-ERDA using 2–3 MeV He [8] [9]. At about the same time,
elastic recoil coincidence spectroscopy (ERCS), which is a variation of
T-ERDA, has been developed [10]. Later, ERCS using proton–proton scattering
was applied to observe the fluid inclusion in minerals [11], localized H
distribution in diamond [12], and H density in clinopyroxene crystals [13]. In
the studies mentioned above, the workers obtained higher spatial resolutions
than for the typical case of neutron tomography [14], which is a competitive
analysis technique with ERDA.

Previously, we observed 3D distribution of hydrogen bubbles in a H-introduced
Al foil by T-ERDA [15]. Actually, such 3D analysis of hydrogen can be carried
out by taking advantage of the enhanced recoil cross section as large as
$\sim$2 b/sr due to the nuclear elastic interaction of 8–10 MeV 4He with H
[16]. This allows sufficient count rate of recoil H needed for the 3D
observation by employing the collimated 10–100 pA He2+ beam of the $\mu$m
size.

In our developing stage of nuclear-elastic T-ERDA, the main factors
determining the spatial and depth resolutions typically in the $\mu$m range
were studied using test samples [17]. Further fundamental studies are needed
to establish this method as a standard technique of hydrogen analysis. For
this reason, we have carried out T-ERDA using samples of PPS [polyphenylene
sulfide (C6H4S)n] and natural muscovite. The former is a synthetic resin with
well-known atomic composition, while the latter sample has been highly
characterized by other analysis approaches, as will be noted in §II. Rigorous
analysis of hydrogen content in these samples is of practical importance since
these are often used as reference samples for hydrogen analysis by ERDA [18]
[19]. The main interest of the present work is the reproducibility of the
known densities of hydrogen in the samples without relying on any reference
sample to determine the absolute H densities. Therefore, the present analysis
method is based fully on the open database of stopping power and recoil cross
section for material analysis using ion beams.

##  II  Experiments

In the present experiments, the measured samples are stacked PPS–Al layers and
muscovite, which are on the Al foil of 50 $\mu$m thickness to prevent the
incident He from entering the particle detector. The PPS–Al stacked samples
are those used in the previous work [17], which are abbreviated as PPS/Al(50),
Al(10)/PPS/Al(50), and Al(20)/PPS/Al(50), where the numbers in the parentheses
denote the thicknesses in $\mu$m. The thicknesses of the PPS film and the Al
foils were determined by measuring the weight and area, assuming the densities
of 2.70 and 1.35 g/cm3 for Al and PPS, respectively. The nominal purity of the
Al foils is 99.99%. The measurement reproduced the nominal Al thicknesses of
10, 20, and 50 $\mu$m within the error of $\pm$2%. The measured PPS thickness
is $1.35\pm 0.05\,\mu$m, which is thin enough to be used as a localized H
marker. Furthermore, the PPS film was characterized by Rutherford
backscattering spectroscopy (RBS) using 2 MeV He, which showed no discernible
inhomogeneity of the PPS film, according to the composition analysis of the
RBS spectra (S and C) applying the commonly used SIMNRA code.

The muscovite sample was prepared by cleaving the bulk muscovite which was
collected from a granitic pegmatite rock at Ishikawa, Fukushima Prefecture,
Japan. The precise thickness of the muscovite sample on the order of 10 $\mu$m
can be determined afterward from the analysis of the T-ERDA spectrum. The
density of the muscovite 2.83 g/cm3 was measured with a high precision
electronic densimeter. The chemical composition of the muscovite was
determined with scanning electron microscope coupled with energy dispersive
X-ray spectroscopy (SEM-EDS), while the water content of the muscovite was
determined by Karl Fischer titration. The analysis results are summarized in
table 1, which has well reproduced the published data [20].

Table 1: Measured chemical composition and thereby obtained atomic composition of the muscovite collected from Ishikawa. The water content H2O($-$) extracted by heating up to 200∘C is considered as absorbed water molecules, while H2O($+$) extracted by heating from 200 to 1000∘C should originate from water combined as hydroxyl in the crystal structure. For this reason, the sum of the chemical composition excluding H2O($-$) is normalized to 100 wt%. The notations H($-$) and H($+$) are used similarly.  Chem. com. | wt% |  Atom. com. | at% | mass%
---|---|---|---|---
SiO2 | 46.72 |  Si | 14.70 | 21.75
Al2O3 | 33.73 |  Al | 12.51 | 17.78
K2O | 10.93 |  K | 4.39 | 9.04
TiO2 | 0.07 |  Ti | 0.02 | 0.04
FeO | 3.07 |  Fe | 0.81 | 2.38
MnO | 0.09 |  Mn | 0.02 | 0.07
MgO | 0.21 |  Mg | 0.10 | 0.13
Na2O | 0.65 |  Na | 0.40 | 0.48
H2O ($+$) | 4.53 |  H ($+$) | 9.51 | 0.50
H2O ($-$) | 0.40 |  H ($-$) | 0.84 | 0.04
|  |  O | 56.70 | 47.79

Figure 1 shows the experimental setup similar to the previous case [17],
except for the slightly narrow acceptance angle of the surface-barrier silicon
detector (SSD), $\varphi_{0}=3.78^{\circ}$. The energy resolution of the
detector was 33 keV, which is sufficient for the present T-ERDA. The
experiments were carried out under the pressure of $2\times 10^{-5}$ Pa. The
incident He energies of 8 and 9 MeV were chosen for effective use of the
extremely enhanced He-H differential recoil cross section in the forward
direction. The estimated angular spread of the incident He2+ beam from the
UTTAC 6MV tandem accelerator is $\sim$0.2∘, which is much less than
$\varphi_{0}$. The He2+ beam was focused to $\sim$30$\times 30\,\mu$m2 on the
samples. The energy-measuring system for H was calibrated with 5.48 MeV
$\alpha$ particles emitted from 241Am.

The beam current was adjusted within the range of 150–300 pA, depending on the
sample, to obtain the count rate of 200 counts/s or less for recoil H. This
assures negligible counting loss of the H signals in the present experiments.
The beam current was frequently monitored during the measurements by the
Faraday cup attached to the sample holder, shown in figure 1. The integrated
beam current (incident charge) for 1 minute was measured repeatedly with a
current integrator, from which the uncertainty of the incident charge $\pm 5$%
was obtained. The integrated beam current for 1 minute was also measured
before and after accumulating every hydrogen spectrum for 10 minutes. This
procedure was repeated two or three times and, after confirming the
reproducibility, the two or three hydrogen spectra were summed up to obtain
the final data.

Under fixed beam conditions, we have repeated to measure H spectra several
times at several fixed beam positions on the samples and confirmed that there
is only negligible difference among these spectra. This indicates not only
negligible loss of H from the samples by the He irradiation, but also
uniformity of atomic composition and thickness of the samples.

##  III  Results and discussion

According to the previous study [17], the present T-ERDA can be discussed
using the straight-path approximation for the trajectories of He and recoil H.
Also, the enhanced energy loss for the acceptance angle of
$\varphi_{0}=3.78^{\circ}$ relative to $0^{\circ}$ is negligibly small since
the path length is enhanced only by $1/\cos{\varphi_{0}}-1=0.22$%. It is
notable that the angular spread of recoil H after passing through the present
sample causes negligible change in the H yield. This is because the
differential recoil cross section changes slowly with recoil angle in the
narrow forward direction under consideration, hence the H yield is unchanged
as a whole even when the recoil directions of recoil H are smeared and mixed
each other by the multiple scattering suffered in the sample. In a
representative case, for example, the angular spread of 5.12 MeV H (recoil H
by 8 MeV He) after passing through an Al foil of 50 $\mu$m is $\pm
1.2^{\circ}$ [17], which corresponds to only $\pm 1$% change in the value of
differential recoil cross section at the recoil angle of 3.78∘ [16]. It is of
essential importance to note that the analysis of observed energy spectra of
recoil H requires to know, in advance, the stopping powers for incident He and
recoil H, which are evaluated from the density and the atomic composition of
the sample. This situation might seem logically inconsistent in the
determination of H density. In most cases, however, hydrogen in the sample
makes only negligible contribution to the stopping powers, and therefore, we
can determine yet unknown H density in the sample. The following analysis of
the experimental data is based on the physical backgrounds mentioned above.

###  III.1 PPS

Figure 2 shows the energy spectra of recoil H measured using 8 and 9 MeV He2+
for the stacked PPS–Al samples. The absence of PPS/Al(50) data for 9 MeV He is
because the incident He can pass through the sample and certainly damage the
detector. The sharp peaks in the spectra originate from recoil H in the PPS
film. The peak energies for 8 MeV He2+ agree with the previously observed
values [17]. These peak energies can be reproduced with a discrepancy of only
0.6% by the evaluations of energy losses in PPS and Al, in which the straight
path approximation and the stopping powers by SRIM are employed. On the other
hand, the similar evaluations for 9 MeV underestimate the peak energies by
1.2%, corresponding to $\sim$40 keV, which is not negligible in comparison
with the energy resolution of the detector 33 keV, as noted earlier.
Nevertheless, such discrepancies with the peak energies do not affect the
present analysis of the experimental data.

Since the PPS thickness (1.35 $\mu$m) cannot be resolved with the depth
resolution, 1.5–1.6 $\mu$m, of the present T-ERDA [17], the areas under the
peaks are considered to determine the number density of H in PPS. In this
case, the slight and approximately uniform background yield originating
probably from He–Al nuclear reactions must be subtracted from every spectrum.
The data for the five different experimental conditions, each providing the
number density of H independently, are useful to confirm the consistency and
reliability of the present analysis procedure.

The integrated H yield under the peak $Y_{\rm peak}$ is related to $n$, the
number density of H, by

$\displaystyle Y_{\rm peak}=Nnt\times\int\frac{d\sigma}{d\Omega}\,d\Omega\;,$
(1)

where $N$ is the number of incident He, $t=1.35\,\mu$m is the thickness of the
PPS sample, $d\sigma/d\Omega$ is the He–H differential recoil cross section in
the laboratory frame, which depends on the He energy at recoil ${\cal E}_{\rm
r}$ and the solid angle $\Omega$, and the integral is performed for the solid
angle of acceptance of the detector. ${\cal E}_{\rm r}$ can be calculated
using the stopping powers for He. Equation (1) leads to

$\displaystyle n=\frac{Y_{\rm peak}}{Nt\,\sigma({\cal E}_{\rm
r},\varphi_{0})}\;,$ (2)

where the cross section $\sigma({\cal E}_{\rm r},\varphi_{0})$ is expressed as

$\displaystyle\sigma({\cal E}_{\rm
r},\varphi_{0})=\int\frac{d\sigma}{d\Omega}\,d\Omega=\int_{0}^{\varphi_{0}}\frac{d\sigma}{d\Omega}\,2\pi\sin{\varphi}\,d\varphi\;.$
(3)

where $\varphi$ is the polar angle with respect to the He beam axis, and
$\varphi_{0}=3.78^{\circ}$, as noted in §II. Figure 3 shows $\sigma({\cal
E}_{\rm r},\varphi)$ for the parameter ranges of interest in the present
analysis, which was calculated using the numerical table of differential
recoil cross sections given by IBANDL [16]. At ${\cal E}_{\rm r}=9$ MeV,
$\sigma({\cal E}_{\rm r},\varphi)$ is maximal, implying that the highest
efficiency for collecting H signals is obtained for the present setup of
$\varphi=\varphi_{0}\,(=3.78^{\circ})$. Nevertheless, the background yield
increases with increasing the He energy and therefore the He energy must be
chosen by considering the balance between the efficiency and sensitivity
required for hydrogen analysis of a given sample. Comparing the values of
$\sigma({\cal E}_{\rm r},\varphi)$ for $\phi=3.78^{\circ}$ and 4∘, we notice
the appreciable difference by $\sim$10% only for the small angular change of
0.22∘. This is the reason that special care was taken for the acceptance angle
of the detector in the experimental setup.

Table 2 shows the determined values of the H density of PPS, for which the
experimental data used are in the ranges of $\Delta Y=$(1.4–2.4)$\times
10^{4}$ and $N=$(1.2–2.8)$\times 10^{11}$. For the case of 9 MeV He on the
Al(10)/PPS/Al(50) sample, for example, the density of $2.68\times 10^{22}$
H/cm3 results from $\Delta Y=$2.311$\times 10^{4}$ after subtracting the
background yield of $1.55\times 10^{3}$, $N=2.588\times 10^{11}$, and ${\cal
E}_{\rm r}=7.77$ MeV which corresponds to $\sigma=24.65$ mb (see figure 3).
Note that the energy loss of He and recoil H in the PPS sample is negligibly
small in the determination of the H density, as is obvious from the small
energy differences in the values of ${\cal E}_{\rm r}$, shown in table 2.

Table 2: Number density of H in PPS determined under different measurement conditions. The He energy at recoil ${\cal E}_{\rm r}$ is shown in terms of the central energy with $\pm$difference resulting from the finite thickness of PPS. The estimated uncertainty in the values of H density is $\pm$8%. The rightmost column shows the ratio of the determined density to the calculated density (3.01$\times$$10^{22}{\rm H}/$cm3) from the atomic composition of PPS.  Sample | He energy | He energy | H density | Ratio to |
---|---|---|---|---|---
|  ${\cal E}_{\rm in}$(MeV) |  ${\cal E}_{\rm r}$(MeV) | ($10^{22}{\rm H}/$cm3) | 3.01$\times$$10^{22}{\rm H}/$cm3 |
PPS$/$Al(50) | 8 | 7.95$\pm$0.05 | 2.64 | 0.88 |
Al(10)$/$PPS$/$Al(50) | 8 | 6.66$\pm$0.06 | 3.17 | 1.05 |
Al(10)$/$PPS$/$Al(50) | 9 | 7.77$\pm$0.05 | 2.68 | 0.89 |
Al(20)$/$PPS$/$Al(50) | 8 | 5.21$\pm$0.07 | 2.94 | 0.98 |
Al(20)$/$PPS$/$Al(50) | 9 | 6.44$\pm$0.06 | 2.90 | 0.96 |

The uncertainty of the determined values of H density stems mainly from four
sources: (i) the $\pm 5$% error in $N$ associated with the integrated current,
as is mentioned in §II, (ii) a possible error of $\pm 4$% in $\sigma({\cal
E}_{\rm r},\varphi)$ resulting from the $\pm 0.07^{\circ}$ uncertainty of the
acceptance angle of the detector, (iii) a possible error of $\pm 4$%
associated with the measured value of $t$, and (iv) the statistical error of
$\Delta Y\simeq 1$%. Because of no interaction among these four sources, the
total uncertainty is estimated as $\pm(5^{2}+4^{2}+4^{2}+1^{2})^{1/2}\simeq\pm
8$% by assuming the convolution characteristic of Gaussian distributions. It
should be noted that in the literatures there is a discrepancy of $\sim$$\pm
5$% between the experimental and theoretical recoil cross sections for the
energy and angular ranges of interest [22][23]. Even if this discrepancy is
taken into account, only minor change arises in the above estimate, i.e., from
8 to 9%.

The number density of H in PPS calculated from the nominal density of 1.35
g/cm3 and the atomic composition of (C6H4S)n, using Avogadro’s constant, is
$3.01\times 10^{22}\,{\rm H}/$cm3. The ratios of the determined value to the
calculated value, shown in table 2, are approximately equal to 1. Averaging
the five determined values, we may conclude $(2.87\pm 0.26)\times
10^{22}\,{\rm H}/$cm3 for the H density of PPS. This value agrees well and
consistently with the calculated one, which demonstrates the validity of the
present analysis.

###  III.2 Muscovite

Figure 4 shows T-ERDA spectra of recoil H from the muscovite sample, which
were measured with 8 and 9 MeV He2+. The counts of the spectrum yield for 9
MeV have been multiplied by 1.103 to compare the two spectra for the equal
number of incident He2+, $1.05\times 10^{12}$ for the 8 MeV case. The narrower
and higher spectrum of recoil H for 9 MeV results from the reduced energy loss
of He and H in the sample and from the increased recoil cross sections,
compared with the 8 MeV case.

For evaluation of the stopping powers of the muscovite, its density 2.83 g/cm3
and the atomic composition shown in table 1 are the input parameters of the
SRIM code [21]. Compared with pure muscovite of 2.9 g/cm3, which consists of
K, Al, Si, O, and H [ KAl2 AlSi3 O10 (OH)2 ], there are several substituted
atoms in the muscovite sample. However, such difference, including the
different H contents, hardly affect the stopping powers. Actually, the
stopping powers of the muscovite sample for He and H in the energy ranges
under consideration are less than those of pure muscovite by only 0.2%.

In the straight-path approximation, the measured H energy $E$ can be converted
to the depth $z$ where the recoil occurs in the muscovite of thickness $L_{\rm
mus}$. We start with writing down the He energy at recoil, which was
introduced in §III.1, i.e.,

$\displaystyle{\cal E}_{\rm r}={\cal E_{\rm in}}-\int_{0}^{z}S_{\rm mus-
He}\,dz\>,$ (4)

where ${\cal E_{\rm in}}=8$ or 9 MeV is the kinetic energy of incident He in
the present case, and the notation $S_{\rm mus-He}$ indicates the stopping
power of muscovite for He. Since recoil H passes through the Al foil of
thickness $L_{\rm al}=50\,\mu$m before entering the detector, the relation
between $E$ and $z$ is written in terms of the integral along $z$ which is
also the beam axis of incident He, i.e.,

$\displaystyle E=\alpha\,{\cal E}_{\rm r}-\int_{z}^{L_{\rm mus}}S_{\rm
mus-H}\,dz-\int_{L_{\rm mus}}^{L_{\rm mus}+L_{\rm al}}S_{\rm al-H}\,dz\>,$ (5)

where $\alpha=4M_{\rm H}M_{\rm He}/(M_{\rm H}+M_{\rm He})^{2}=0.64$ with
$M_{\rm H}$ and $M_{\rm He}$ being the atomic masses of H and He,
respectively, and the stopping powers are written with the same notation as in
equation (4). The stopping powers in equations (4) and (5) depend on the
kinetic energy of He or H, and therefore, on $z$, hence they increase as $z$
increases (with losing the kinetic energies) in the energy ranges of He and H
under consideration. The first term on the right side of equation (5)
represents the energy of recoil H induced at $z$, while the second and the
third represent the energy loss of H in the outgoing path.

For a given value of $L_{\rm mus}$, $E$ decreases with increasing $z$ from
$z=0$ because of the larger stopping power for He than for H. It is obvious
that $E$ at $z=0$ decreases with increasing $L_{\rm mus}$ and, accordingly,
$L_{\rm mus}$ can be determined from such a condition that the calculated
value of $E$ at $z=0$ reproduces the high-energy end of the measured spectrum.
Similarly, the low-energy end of the measured spectrum can also be used to
determine $L_{\rm mus}$, which allows to reconfirm the value of $L_{\rm mus}$
obtained from the high-energy end. Actually, the high- and low-energy ends are
given by the H energies at half-heights of the right and left steps of the
spectra shown in figure 4. This procedure is illustrated in figure 5, thereby
we have determined $L_{\rm mus}=12.5\pm 0.5\,\mu$m using the values of
stopping powers by SRIM, The depth scales in figure 2 have been obtained
accordingly. These depth scales and the smeared high- and low-energy ends of
the spectra indicate that the depth resolution of the present T-ERDA is
$\sim$2 $\mu$m.

The depth-dependent number density of H in the muscovite sample $n(z)$ can be
obtained from the spectra in figure 4. When the $z$ axis is divided by a short
interval of $\Delta z$, the H yield contributed from the thickness range of
$z-\Delta z/2$ to $z+\Delta z/2$ is expressed as

$\displaystyle\Delta Y(z)=N\,n(z)\,\Delta
z\int\frac{d\sigma}{d\Omega}\,d\Omega\;,$ (6)

where $N$ is the number of incident He, $d\sigma/d\Omega$ is the He–H
differential recoil cross section in the laboratory frame, which depends on
the He energy at recoil ${\cal E}_{\rm r}$ and the solid angle $\Omega$, and
the integral is performed for the solid angle of acceptance of the detector.
Notably, equation (6) is on the basis of the fact that the counting efficiency
of the detector for recoil H is effectively 100%, which has been assured,
although indirectly, from the analysis results of PPS in §III.1. From equation
(6), we obtain

$\displaystyle n(z)=\frac{\Delta Y(z)}{N\,\sigma({\cal E}_{\rm
r},\varphi_{0})\,\Delta z}\;,$ (7)

where the recoil cross section $\sigma({\cal E}_{\rm r},\varphi)$ is expressed
as

$\displaystyle\sigma({\cal E}_{\rm
r},\varphi)=\int\frac{d\sigma}{d\Omega}\,d\Omega=\int_{0}^{\varphi}\frac{d\sigma}{d\Omega}\,2\pi\sin{\varphi}\,d\varphi\;.$
(8)

It is straightforward that $n(z)$ for $\Delta z=1\,\mu$m can be determined
using equation (7) and $\sigma({\cal E}_{\rm r},\varphi)$ shown in figure 3.
In this case, $\Delta Y$ for every 1 $\mu$m depth is obtained from the spectra
in figure 4 after subtracting the slightly recognized background. Figure 6
shows $n(z)$ determined in this manner. For example, the density of
$n(z)=9.52\times 10^{21}$ H$/$cm3 at $z=6.5\,\mu$m for 8 MeV results from
$\Delta Y=20,179$ counts for $6\leq z\leq 7\,\mu$m, $N=1.054\times 10^{12}$,
and $\sigma({\cal E}_{\rm r},\varphi_{0})=20.10$ mb.

We see in figure 6 that the results of $n(z)$ for 8 and 9 MeV He agree
consistently with each other at the depth from 1.5 to 11 $\mu$m. Outside this
depth range, the slopes near the high- and low-energy ends of the spectra
should lead to unreliable results. We see the uniformity of $n(z)$ at $1.5\leq
z\leq 11\,\mu$m, hence the average value ${\bar{n}}=(9.43\pm 0.75)\times
10^{21}$ H$/$cm3 is obtained. The uncertainty of ${\bar{n}}$ has been
estimated as $\pm 8$% in the similar manner to the case of PPS, except that
the possible error of the PPS thickness is replaced by that of the muscovite
thickness, $0.5/12.5=4$%, as noted earlier. In figure 6, the value of
${\bar{n}}$ agrees well with the number density of $9.36\times 10^{21}$
H$/$cm3 obtained from the data in table 1, using the density of the muscovite
2.83 g/cm3 and Avogadro’s constant. Furthermore. the value of ${\bar{n}}$
cannot be accounted for with the density of $8.61\times 10^{21}$ H$/$cm3 which
is obtained for the bound water only. Therefore, we may conclude that the
absorbed water is distributed uniformly over the entire thickness of the
muscovite sample with the present depth resolution of $\sim$2 $\mu$m. This is
unique information obtained by the nondestructive T-ERDA. These analysis
results demonstrate the capability of accurate determination of hydrogen
contents in materials and minerals by the present T-ERDA system.

##  IV  Summary and Conclusion

Hydrogen number densities in PPS and muscovite have been determined by T-ERDA
without using reference samples. For PPS, the determined H density of
$(2.87\pm 0.26)\times 10^{22}\,{\rm H}/$cm3 is in good agreement with the
calculated density of $3.01\times 10^{22}\,{\rm H}/$cm3. For muscovite, the H
density is uniform over the entire thickness of the sample with the depth
resolution of $\sim$2 $\mu$m, and the average value $(9.43\pm 0.75)\times
10^{21}$ H$/$cm3 agrees excellently with $9.36\times 10^{21}$ H$/$cm3 which is
obtained from the measured chemical composition of the muscovite sample,
listed in table 1. This also implies that the H density originating from
absorbed water is uniform in the muscovite, like that from bound water
constituting the crystal structure of muscovite. These analysis results assure
reliability of T-ERDA under the condition of nuclear elastically enhanced
recoiling.

In the present T-ERDA, the sensitivity to H content is restricted by the
background yield, noted in §III.1. Obviously, the background yield can be
suppressed by replacing the present stopping foil (50 $\mu$m thick Al) by a
foil of heavier element with higher Coulomb barrier. Such a refinement in the
experimental setup might be of technical importance when the T-ERDA system is
used for analysis of not only small amount of hydrogen, but also trace amount
of deuterium introduced in a foil sample.

ACKNOWLEDGEMENTS
The authors thank the technical staff of UTTAC for operating the 6MV tandem
accelerator for supporting the experiments. We are grateful to Toray KP Films
Inc. for providing the PPS films, and to Toyo Aluminium K. K. for providing a
set of Al foils of different thicknesses.

## References

  * [1] Condon J B and Schober T, 1993 J. Nucl. Mater. 207 1–24
  * [2] Sweeney R J, Prozesky V M and Springhorn K A 1997 Geochim. Cosmochim. Acta 61 101–113
  * [3] Raepsaet C, Bureau H, Khodja H, Aubaud C and Carraro A 2008 Nucl. Instrum. Meth. Phys. Res. B 266 1333–1337
  * [4] Bureau H, Raepsaet C, Khodja H, Carraro A and Aubaud C 2009 Geochim. Cosmochim. Acta 73 3311–3322
  * [5] Stephant A, Anand M, Ashcroft H O, Zhao X, Hu S, Korotev R L, Strekopytov S, Greenwood R C, Humphreys-Williams E, Liu Y, Tang G, Li Q and Franchi I A 2019 Geochim. Cosmochim. Acta 266 163–183
  * [6] Lin Y and van Westrenen W 2021 Global Planet. Change 197 103393
  * [7] Lanford W A, Nuclear Reactions for Hydrogen Analysis. In: Tesmer J R, Nastasi M, Barbour J C, Maggiore C J and Mayer J W, editors. Handbook of Modern Ion Beam Materials Analysis, Materials Research Society, 1995
  * [8] Tirira J, Trocellier P and Frontier J P 1990 Nucl. Instrum. Meth. Phys. Res. B 45 147–150
  * [9] Tirira J, Trocellier P, Frontier J P, Massiot Ph, Costantini J M and Mori V, 1990 Nucl. Instrum. Meth. Phys. Res. B 50 135–139
  * [10] Hofsäss H C, Parikh N R, Swanson M L and Chu W-K 1990 Nucl. Instrum. Meth. Phys. Res. B 45 151–156
  * [11] Furuno K, Komatsubara T, Sasa K, Oshima H, Yamato Y, Ishii S, Kimura H and Kurosawa M 2003 Nucl. Instrum. Meth. Phys. Res. B 210 459–463
  * [12] Reichart P, Datzmann G, Hauptner A, Hertenberger R, Wild C and Dollinger G 2004 Science 306 1537–1540
  * [13] Weis F A, Ros L, Reichart P, Skogby H, Kristiansson P and Dollinger G 2018 Phys. Chem. Miner. 45 669–678
  * [14] Griesche A, Dabah E, Kannengiesser T, Kardjilov N, Hilger A and Manke I 2014 Acta Materialia 78 14–22
  * [15] Yamazaki A, Sasa K, Tomita S, Ishii S, Naramoto H, Sataka M, Kudo H, Itoh G and Ohkubo M 2019 AIP Advances 9 105111
  * [16] IBANDL (Ion Beam Analysis Nuclear Data Library). https://www-nds.iaea.org/exfor/ibandl.htm
  * [17] Kudo H, Naramoto H, Sataka M, Ishii S, Sasa K and Tomita S 2021 J. Phys. Condensed Matter 33 465901
  * [18] Ros L, Kristiansson P, Borysiuk M, Abdel N, Elfman M, Nilsson E J C and Pallon J 2015 Nucl. Instrum. Meth. Phys. Res. B 348 273–277
  * [19] Kinomura A, Nakao S, Suzuki K, Kuzuya Y, Nakajima M and Yasuda K 2021 Nucl. Instrum. Meth. Phys. Res. B 502 54–58
  * [20] Udagawa S, Urabe K, Ikawa H, Katoo K and Hasu H 1973 J. Clay Sci. Soc. Jp. (in Japanese) 13 131–138
  * [21] Ziegler J F and Biersack J P, SRIM2013, The Stopping and Range of Ions in Matter http://www.srim.org/
  * [22] Pusa P, Rauhala E, Gurbich A and Nurmela A 2004 Nucl. Instrum. Meth. Phys. Res. B 222 686–689
  * [23] Hupin G, Quaglioni S and Navrátil P 2014 Phys. Rev. 90 061601(R)

Figure 1: Schematic diagram of the experimental setup for T-ERDA.

Figure 2: T-ERDA spectra for incidence of 8 and 9 MeV He on the three PPS
samples, shown for the same number of incident He2+, $2\times 10^{11}$. Note
that H energy of 6.683 keV corresponds to 1 channel width on the horizontal
axis.

Figure 3:  $\sigma({\cal E}_{\rm r},\varphi)$ calculated using the numerical
table of differential recoil cross sections given by IBANDL [16].

Figure 4: T-ERDA spectra of recoil H from the muscovite sample, shown for the
same number $1.05\times 10^{12}$ of incident 8 and 9 MeV He2+. Note that H
energy of 6.617 keV corresponds to 1 channel width on the horizontal axis.

Figure 5: Calculated high- and low-energy ends (denoted as HE and LE,
respectively) of the H-energy spectrum as a function of the thickness of
muscovite. The black dots on the curves indicate the HE and LE values of the
measured spectra shown in figure 4, from which the most probable thickness of
$12.5\,\mu$m (the green dashed arrow) has been determined.

Figure 6: Number density of H in the muscovite sample, determined by T-ERDA
using 8 and 9 MeV He2+. The uncertainty of $\pm$8% in the determined values is
shown only for the 8 MeV case to avoid a confusing figure. The dashed
horizontal lines indicate the calculated densities of $9.36\times 10^{21}$
H$/$cm3 (upper) and $8.61\times 10^{21}$ H$/$cm3 (lower), with and without the
absorbed water in the muscovite, respectively.