Elastic scattering in the proton-nitrogen and proton-oxygen systems near the Coulomb barrier

Measurement and parameterization of proton elastic scattering cross-sections for nitrogen
I. Bogdanović Radović*, Z. Siketić and M. Jakšić

Ruđer Bošković Institute, P.O. Box 180, 10000 Zagreb, Croatia
A.F. Gurbich

Institute of Physics and Power Engineering, Bondarenko sq. 1, 249033 Obninsk, Russian Federation

The cross-sections for the elastic scattering of protons from natural nitrogen at non-Rutherford scattering energies were measured at three laboratory scattering angles: 118°, 150° and 165°. The experimental data were parameterized in the framework of nuclear physics models. A benchmark experiment was performed in order to prove that the excitation functions obtained in the present work can be used to adequately simulate the yield from a thick target containing nitrogen.

^* Author to whom correspondence should be addressed. Electronic mail: [e-mail]
I. INTRODUCTION
For materials analysis, Rutherford Backscattering Spectroscopy is an established analytical technique. It can provide information about thickness, elemental composition or stochiometry of the analyzed material. For light element detection, analytical use of protons instead of alpha particles is more and more common due to a larger probing depth and enhanced scattering cross-sections coming from the nuclear component of the interaction above the Coulomb barrier. The presence of this non-Rutherford component in the scattering cross-section implies the need of detailed measurements of excitation functions at different scattering angles. A significant amount of work has been done in the last few decades in order to achieve a reliable database for the scattering of both protons and alpha particles from light elements. This work is summarized in the IBANDL database.¹ The database contains most of the available experimental nuclear cross-sections relevant to Ion Beam Analysis. For some projectile-nucleus scattering systems evaluated cross sections are also available.² The evaluation consists in the elaboration of the most accurate available cross-sections through the incorporation of the all relevant experimental data in the framework of nuclear physics theory.³ The advantage of evaluated cross-sections is that excitation functions for analytical purposes can be calculated for any scattering angle or energy, with a reliability exceeding that of any individual measurement.

Nitrogen is often present as a main constituent in bulk targets or as an important constituent in thin films. Analysis with protons of samples containing nitrogen is typically done at energies above the Coulomb barrier where the cross-section is no longer Rutherford. Therefore, for a quantitative analysis and depth profiling of nitrogen it is important to know the exact cross-sections. The motivation for the present work was to supply new data for the ¹⁴N(p,p)¹⁴N cross-sections, especially at energies above 3 MeV.

Up to now, scattering of protons from nitrogen has been reported in numerous publications for angles important for backscattering analysis (>120°) mainly at energies not exceeding 3 MeV.^4-14 In works published almost half a century ago, authors have used gaseous N target and CsI(Tl) crystal detectors to study states in compound nucleus ¹⁵O. ^4-9 In more recent works measurements have been done with nitrogen-containing thin films deposited on the carbon substrate, similar to that used in the present work. ^10-14 For detection, silicon surface barrier detectors were applied at scattering angles important for the backscattering analysis. Previously acquired data were used to produce the evaluated cross-sections in the energy range up to 3.4 MeV.¹⁵ The motivation for the present work was lack of the reliable cross-section data for the elastic scattering of protons from nitrogen up to 5 MeV.

II. EXPERIMENTAL METHODS

1. 
   The apparatus
   

Measurements were performed using proton beams from the 6.0 MV Tandem Van de Graaff accelerator at the Ruđer Bošković Institute in Zagreb. The beam was extracted from a sputtering ion source using a TiH₂ cathode. The energy calibration of the accelerator was done using narrow resonance in ²⁷Al(p,γ)²⁸Si at 991.88 keV and neutron threshold reaction ⁷Li(p,n)⁷Be at 1880.6 keV. Final energy spread of the beam was calculated to be 0.1% of the incident beam energy.

The proton beam, delimited by horizontal and vertical slits to a spot of 2×2 mm² impinged on the sample at normal incidence. To detect backscattered protons from the target, three surface barrier detectors with a 2.5 msr solid angle each were positioned at 118°, 150° and 165°. The energy resolution of the detectors used was 12 keV for protons in the measured energy range. The excitation function of the ¹⁴N(p,p)¹⁴N was measured between 2.4 and 5.0 MeV with minimum step of 10 keV where the cross-section varied rapidly and 25 keV elsewhere.
B. Samples
It is quite difficult to select solid target appropriate for the N cross section measurements. As we wanted to have a N signal well separated from signals coming from other elements, a thin AlN film deposited by reactive sputtering on a vitreous graphite substrate was used. On the top, a 6 nm Au layer was evaporated for normalization purposes, assuming that cross-sections for backscattering of protons from Au are Rutherford in the entire energy range studied in this work. In addition, the normalization to gold eliminates further corrections due to the dead time, knowledge of absolute solid angle value and errors due to imperfect charge collection. Target stability was examined regularly by measuring N and Au intensity ratios at 3 MeV. It was found that this ratio was within 2.7% during the measurements.
III. DATA ANALYSIS
The average differential cross-section for backscattering of protons from N at scattering angle  is given by:
, (1)
where E is incident proton energy, E_Au is energy loss in Au layerE_AlN is energy loss in AlN film, A_N is area under the N peak, Q the number of incident H ions, () the detector solid angle and N_N the number of nitrogen atoms per unit area (at/cm²), and where the detected counts are actually given by an integral of the differential cross-section over the energy thickness of the film. Very sharp resonances cannot be resolved by this method.

The differential cross section for backscattering of protons from Au for an angle θ is given by the following equation:

, (2)
where A_Au is area under the Au peak, and N_Au the number of Au atoms per unit area. Normalization against Au signal leads to:
(3)
To calculate the cross sections using Eq. (3) it is necessary to know the N_Au/N_N ratio. Therefore, the thin film was carefully investigated by 2 MeV ⁴He backscattering since for this beam all the cross-sections are pure Rutherford to determine the stochiometry, layer thickness, homogeneity and presence of impurities that could affect the results. It was found that N_N=(639±30)x10¹⁵ at/cm² and N_Au=(38.6±0.6)x10¹⁵ at/cm² which gives N_Au/N_N ratio equal to 0.0604±0.0030.

The energy loss of protons in the Au and AlN layers was calculated using stopping power data incorporated in the program SRIM 2003.¹⁶ The energy loss of protons in the Au layer varied between 0.5 and 0.3 keV and in the AlN layer between 5.5 and 3.2 keV for the minimum and maximum projectile energy, respectively. Because of its small value, the correction in Eq. (3) can be neglected.

IV. PARAMETERIZATION
The experimental data obtained were parameterized in the framework of the R-matrix theory with optical model phases being used instead of hard spheres ones. The initial values for the resonance parameters were taken from the paper by M.L.West et al.⁹ Because the substitution of optical model phases for hard sphere ones influenced the interference between potential and resonance scattering, the shape of the resonances changed and so some of the resonances were assigned spins and parities different from those assumed by West et al. The final list of the resonance parameters is presented in Table 1. The natural boundary conditions were set at the resonance energies listed in Table 2. The parameters of spin and orbital momentum mixing are presented in Table 1 following Nelson et al. definitions.¹⁹ The channel spin mixing ratio is defined as

, (4)

where s_> is the higher channel spin and  is the total width of the decay channel. The l-mixing ratio is defined as

(5)

with corresponding mixing angles _s = atan (_s) introduced for convenience. The range of  is from 0 to 1, and the range of  is from -90 to +90.

With two s values and two l values contributing there are a maximum of four elastic scattering partial widths _s,l . These widths may be expressed in terms of three mixing ratios , _<, _> and the total elastic scattering width  by the following relations:

(6)
The potential parameters were found by fitting the non-resonance background in the excitation functions. In order to accurately reproduce shape resonances the real potential well was split on the l-number. The final set of the optical parameters is presented in Table 3 (in usual optical model notation). The real potential has energy dependence V=V₀-bE, where V₀ values are listed in the Table 3 and b=2.865. The imaginary potential at energies less than 4 MeV was assumed to be zero. Fig.1 demonstrates behavior of the theoretical cross-section on scattering angle and energy.
V. RESULTS AND DISCUSSION
Experimentally determined proton backscattering cross-sections from N together with theoretical calculations and all other available experimental data are plotted for laboratory angles of 118°, 150° and 165° in Fig. 2. In the entire investigated energy region the cross-sections are significantly higher then Rutherford. The energies to which the reported cross sections are assigned correspond to incident proton energies minus half the energy loss in the AlN layer. The uncertainty in the measured cross-sections is calculated to be less then 6% for energies below 4.6 MeV and between 6% and 8% for energies from 4.6 and 5 MeV for 165° and 150°. For 118° the relative uncertainty was below 5% for all energies. The following factors were taken into consideration for estimating the uncertainty: the counting statistics of the peak areas and the systematic error in determining the N_Au/N_N ratio. Uncertainties due to dead time, solid angle and improper charge measurement are eliminated with the normalization to backscattering protons from gold. Errors of the detector angular settings were estimated to be negligible. Numerical values of the cross sections together with the corresponding uncertainties are available on request from the authors. They will be also uploaded to the IBANDL database.¹ The agreement between calculated and measured cross sections is very good for larger scattering angles (165° and 150°) in the entire energy region where calculations were performed. Calculations predict only one strong and narrow resonance at 3196 keV with FWHM of 12 keV. The experimental points are shifted about 5 keV compared to theoretical predictions toward the lower energies. For 118° and below 3.3 MeV theoretical calculations predict for about 15% higher cross sections. Except narrow resonance at 3196 keV another strong but significantly broader resonance is measured at 3870 keV with cross sections value 85 times larger then Rutherford. From Fig. 2 is visible that excitation functions are characterized with broad regions where cross section is almost constant and considerably larger than Rutherford. Those plateaus with enhanced cross sections can be used for analysis of smaller amounts of nitrogen in heavier matrices for backscattering analysis.

For comparison with our results, other available experimental data are shown at the same figure. For the investigated energy region, our measurements can be compared with data from several references^4-8,14. Cross sections at 118° are compared with data from Ref. 4 for 121.1°. In the region from 2.4 up to 3.25 MeV present data are for about 10% lower than data from Ref. 4. Above 3.25 MeV agreement is better. Data from West et al.⁹ are given for 124° and are for around 10% higher than present data. Around 150° there are three data sets that can be compared with our results. Those from Ref. 7 for 153,4° are slightly higher, but data from Ref. 8 for 155.2° are in fair agreement with the present ones. Recently measured data from Jiang et al.¹⁴ are for about 30% lower then other measurements in the region between 2.5 and 3.2 MeV. Results for 165° can be compared with data for 159.5° from Bashkin et al.⁵, 167.2° from Olness et al.⁶ and 165° from Lambert and Durand⁸. Again, in the energy range from 2.6 to 3.5 MeV, our data are up to 10% lower than those data but above 3.5 MeV agreement is satisfactory.

In order to examine if N excitation function measured in the present work and incorporated into simulation program SIMNRA (Ref. 17) can interpret properly the experimentally obtained N thick target yield we have performed benchmark experiment. For that, thick BN target was selected. Target was covered with 8 nm Au layer for the normalization purposes. In order to separate ¹⁴N(p,p)¹⁴N spectrum from the background coming from ¹⁰B(p,) ¹⁰B, ¹¹B(p,) ¹¹B as well as possible pile-up contribution, E-E telescope was used. 15.9 m thick E Si detector was placed 15 mm in front of thick energy Si particle detector (300 m). Telescope solid angle was 1.4 msr. E detector was thick enough to stop completely low energy alphas coming from ¹⁰B(p,_)¹⁰B (Q = 1447 keV) and ¹⁰B(p,_)¹⁰B (Q=717 keV) reactions. For more energetic alphas coming from ¹¹B(p,_)¹¹B reaction (Q = 8591 keV), energy loss in E detector was significantly larger than energy loss for protons and signals can be well separated. Fig. 3(a) shows two dimensional maps of E-E coincident events. Small contribution coming from pile-up can be also seen. Projection of all events to the x (energy) axis is done on Fig. 3(b), events belonging only to (p,p) scattering are shown on Fig. 3(c) and those coming from (p,) on Fig. 3(d). Simulations of backscattering spectra of BN target at 150° and two energies 3.24 and 4.50 MeV are shown on Figs. 4(a) and 4(b). In the simulation program (Ref. 17), the step width of incident ions was chosen that for each point in the cross section file there was at least one sub-layer in the simulation program. Stopping power data incorporated in SRIM 2003 were used.¹⁶ For simulation at 3.24 MeV, cross sections from Chiari et al. for ¹⁰B(p,p)¹⁰B and ¹¹B(p,p)¹¹B were applied.¹⁸ For energies higher than 3.8 MeV there are no experimental data for proton elastic scattering from boron and therefore simulation at 4.5 MeV is shown without part belonging to boron. Result of our benchmark experiment shows that difference between fit with our measured cross sections and the best fit to experimental data is in all cases less than 5%.


VI. CONCLUSIONS
We have measured the elastic ¹⁴N(p,p)¹⁴N scattering non-Rutherford cross sections in the energy region from 2.4 to 5 MeV for three different scattering angles, 118°, 150° and 165°. Experimentally obtained excitation functions are in good agreement with theoretical predictions especially for 150° and 165° in the entire energy region where calculations were preformed. The parameterization of the experimental data within physical approach provides a possibility to interpolate/extrapolate the ¹⁴N(p,p)¹⁴N cross section for any scattering angle. Calculations can be made using the on-line SigmaCalc calculator at http://www-nds.iaea.org/sigmacalc/.
ACKNOWLEDGMENTS
The authors are grateful to Dr. C. Jeynes for useful comments and for his help in preparing the manuscript. This work was supported by IAEA contracts #13274 and # CRO 13269/R0. A.G. acknowledges the support of Sandia National Laboratories (USA) through the ISTC project #3748p.

TABLE CAPTIONS:
Table 1. Resonance parameters for ¹⁴N(p,p₀)¹⁴N

Table 2. Energies for definition of boundary conditions (MeV).

Table 3. Optimal parameters of the optical potential obtained for ¹⁴N(p,p₀)¹⁴N

Table 1. Resonance parameters for ^14N(p,p0)^14N
Ep, MeV	lab, keV	J^		1/2, deg	3/2, deg
3.405	27	3/2ˉ	0.0	0	-
3.387	97	3/2^+	1.0	-	0
3.872	60	7/2^+	1.0	-	5
3.903	100	1/2^+	1.0	-	90
4.205	25	3/2^+	1.0	-	90
4.565	10	3/2^+	0.8	0	70
4.581	25	3/2ˉ	1	-	90
4.631	25	1/2ˉ	0.0	0.0	-
4.740	10	3/2ˉ	1.0	-	90
4.780	80	3/2^+	1.0	-	90
4.880	80	5/2ˉ	1.0	-	0
5.046	39	5/2ˉ	0.26	90	0
5.220	160	5/2^+	1.0	-	0

Table 2. Energies for definition of boundary conditions (MeV).
J^/L,S	0,1/2	0,3/2	1,1/2	1,3/2>	2,1/2	2,3/2>	3,1/2	3,3/2	4,1/2	4,3/2 >	5,1/2	5,3/2
1/2^+	1.000	-	-	-	-	3.903	-	-	-	-	-	-
1/2^-	-	-	3.996	3.996	-	-	-	-	-	-	-	-
3/2^+	-	3.900	-	-	3.900	3.440	-	-	-	-	-	-
3/2^-	-	-	3.410	4.580	-	-	-	4.580	-	-	-	-
5/2^+	-	-	-	-	3.880	3.880	-	-	-	4.575	-	-
5/2^-	-	-	-	4.580	-	-	4.580	4.580	-	-	-	-
7/2^+	-	-	-	-	-	3.880	-	-	3.880	3.880	-	-
7/2^-	-	-	-	-	-	-	4.780	4.780	-	-	-	4.780
9/2^+	-	-	-	-	-	-	-	-	3.880	3.880	-	-
9/2^-	-	-	-	-	-	-	-	4.780	-	-	4.780	4.780

Table 3. Optimal parameters of the optical potential obtained for ^14N(p,p0)^14N
VR0 MeV	VR1 MeV	VR2 MeV	VR3 MeV	VR>3 MeV	rR fm	aR fm	WD MeV	rD fm	aD fm	VSO MeV	rSO fm	aSO fm	rC fm
77.64	69.87	70.89	92.16	72.60	1.21	0.66	E-4	1.22	0.40	6.00	1.22	0.40	1.22

FIGURE CAPTIONS:
Fig 1. Evaluated elastic scattering cross sections for N(p,p)N as a function of scattering angle and proton energy.
Fig 2. Differential cross sections for elastic backscattering of protons from nitrogen for: (a) 118°, (b) 150° and (c) 165°.
Fig 3. a) Backscattering of 3.24 MeV protons from thick BN target at 150°; (a) two dimensional E-E coincidence map showing well separated (p,p) from (p,) and pile-up events; (b) all events projected to x (energy) axis; (c) events belonging to (p,p) scattering; (d) contribution coming from (p,).
Fig 4. Comparison between experimental and simulated spectra of BN target at 150° and two proton energies: (a) 3.24 and (b) 4.50 MeV. Solid line is SIMNRA simulation, and circles represent experimental data.
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