Introduction

Experimental data of neutron-induced reactions in the energy range around 13 to 15 MeV are needed to verify the accuracy of nuclear models used in the calculation of cross sections. Furthermore, the data are of considerable importance for practical applications, such as for integral calculations on the first wall, blanket and shield of a conceptual fusion power reactor. The data for gas production via neutron induced reactions are of great importance in the domain of fusion reactor technology, particularly of nuclear transmutation rates, nuclear heating and radiation damage due to gas formation. A lot of experimental data on neutron induced cross sections for fusion reactor technology applications have been reported and great efforts have been devoted to compilations and evaluations [1, 2]. In general, cross sections for the formation of isomeric states are more difficult to be predicted than those for the total reaction channels, since more details on the structure of the residual nucleus have to be taken into account (cf. Ref. [3]). The relative probability of forming isomeric states in a nucleus is mainly governed by the spin state values of the levels involved, and the spin distribution of the excited states of the compound nucleus. The high spin value 16+ and 25/2− of the second isomeric state (m2) of 178Hf and 179Hf (Fig. 1) relative to the corresponding values 0+ and 9/2+ of the ground state (g), offers great sensitivity for the study of the spin distribution of the residual nucleus. The cross sections of (n, 2n), (n, γ), (n, p), (n, d), (n, t)··· reactions for the isotopes of hafnium have been measured by many authors. Here, we adduced several authors [4–33], but the cross sections of natHf(n, x)178m2Hf and natHf(n, x)179m2Hf reactions have been not reported. Most of the authors neglected the effect of (n, n′), (n, γ) reactions or deducted the effect indirectly when they calculated the cross section of (n, 2n) reaction. Thus, we investigate them in order to give the cross section of the element producing certain radioactive product nuclide.

Fig. 1
figure 1

Simplified representation of formation and decay of the ground state and isomeric states of 178Hf [34] and 179Hf [35]. All energies are in keV

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In the present work, the cross-sections of the above mentioned reaction were measured in at neutron energy 14.8 ± 0.2 MeV and a gamma-ray counting technique was applied using gamma-ray spectrometer and data acquisition system. Pure HfO2 was used as the target material. The reaction yields were obtained by absolute measurement of the gamma activities of the product nuclei using a coaxial high-purity germanium detector. The neutron energies in this measurement was determined by cross section ratios for the 90Zr(n, 2n)89m+gZr and 93Nb(n, 2n)92mNb reactions [36].

Experimental

Cross sections were measured by activation and identification of the radioactive products. This technique is very suitable for investigating low-yield reaction products and closely space low-lying isomeric states, provided their lifetimes are not too short. The details have been described over the years in many publications [37–41]. Here we give some salient features relevant to the present measurements.

Samples and irradiations

About 3 g of HfO2 powder of natural isotopic composition (>99.95% pure) was pressed at 10 ton/cm2, and a pellet, 0.3 cm thick and 2.0 cm in diameter was obtained. One such pellet was prepared. Monitor foils of Nb (99.999% pure, 0.20 mm thick) of the same diameter as the pellet were then attached in front and at the back of sample.

Irradiation of the samples was carried out at the ZF-300-II Intense Neutron Generator at Lanzhou University and lasted about 22 h with a yield ~ 0.8 to 2.4 × 1011 s−1. Neutrons were produced by the T(d, n)4He reaction with an effective deuteron beam energy of 125 keV and beam current of 20 mA. The tritium-titanium (T-Ti) target used in the generator was 0.9 mg/cm2 thick. The neutron fluence rate was monitored by the accompanying alpha-particle so that corrections could be made for small variations in the yield. The accompanying α particle monitor is shown in Fig. 2. The Au–Si surface barrier detector used in 135° accompanying α-particle tube was at a distance of 119.7 cm from the target. The groups of samples were placed at 0° angle relative to the beam direction and centered about the T-Ti target at distances of 5 cm. Cross section for 93Nb(n, 2n)92mNb reaction [42] was selected as monitors to measure the reaction cross section on several Hf isotopes.

Fig. 2
figure 2

ZF-300-II Intense Neutron Generator neutron flux monitor

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Determination of the incident neutron energy

In the D-T reaction (Q value of 17.6 MeV), induced by deuterons of energy E d , the kinetic energy E n of the neutrons emitted at angle θ can be estimated [43] from the following expression (1);

$$ \left( {E_{n} } \right)^{{{\tfrac{1}{2}}}} = {\frac{{\left( {M_{d} M_{n} E_{d} } \right)^{{{\tfrac{1}{2}}}} \cos \theta + \left( {M_{d} M_{n} E_{d} \cos^{2} \theta + \left\{ {M_{\alpha } + M_{n} } \right\}\left[ {M_{\alpha } Q + E_{d} \left( {M_{\alpha } - M_{n} } \right)} \right]} \right)^{{{\tfrac{1}{2}}}} }}{{M_{\alpha } + M_{n} }}} $$
(1)

where M d , M n and M α are the masses of deuteron, neutron and alpha particle, respectively. The effective D-T neutron energy at irradiation position was determined by the Nb/Zr method [36, 43–45]. The measured neutron energy was shown in Fig. 3 together with the calculation using Eq. (1). The uncertainty in the neutron energy at ~5 cm was estimated to be 200 keV from a consideration of the sample sizes, d+ beam diameter of about 3–4 mm, and the uncertainty in the Nb/Zr method [36].

Fig. 3
figure 3

Angular dependence of d-T neutron energy. The neutron energies were calculated by choosing the incident deuteron energy Ed = 125 keV. The open circle show experimental data determined by the Nb/Zr method (Lewis and Zieba, [36])

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The neutron energies were obtained both by the Eq. (1) and the Nb/Zr method (Lewis and Zieba [36]). The results are shown in Fig. 3. It can be seen, measured result is somewhat lower than the calculated value at neutron energy of 14.8 MeV. This discrepancy is probably due to the samples corresponding to the solid angle geometry, the neutron transport and scattering processes in the target and integrates over the sample volume. But measured result is consistent with the calculated value of Eq. (1) within the experimental uncertainty.

Measurement of radioactivity

The gamma ray activity of 178m2Hf, 179m2Hf, and 92mNb was determined by a CH8403 coaxial high-purity germanium detector made in the People’s Republic of China with a relative efficiency of 20% and an energy resolution of 3 keV at 1332 keV. The efficiency of the detector was calibrated using the standard gamma source, Standard Reference Material 4275 from the National Institute of Standards and Technology, Washington, D.C., USA. An absolute efficiency calibration curve was obtained at 20 cm from the surface of the germanium crystal. At this distance the coincidence losses can be considered to be negligible. In our case, however, we needed to calibrate the efficiency at 2 cm, the actual counting position used because of the weak activity of the sample. Therefore, we selected a set of single γ-line sources (137Cs, 54Mn) and placed them at two positions (20 and 2 cm) successively to measure their efficiency ratios to be able to evaluate the efficiency ratio curve as a function of energy. The absolute efficiency calibration curve at 2 cm was obtained from the calibration curve at 20 cm and the efficiency ratio curve. The uncertainty in the absolute efficiency curve at 2 cm was estimated to be ~2–3%, while the uncertainty of the activity of the standard source is ~1.0%.

The decay characteristics of the product radioisotopes and the natural abundances of the target isotopes under investigation are summarized in Table 1, [34, 35, 46].

Table 1 Reactions and associated decay data of activation products
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Calculation of cross sections and their uncertainties

The measured cross sections can be calculated by the following formula [cf. 47]:

$$ \sigma_{x} = {\frac{{\left[ {S\varepsilon I_{\gamma } K\eta MD} \right]_{m} }}{{\left[ {S\varepsilon I_{\gamma } K\eta MD} \right]_{x} }}}{\frac{{\left[ {\lambda AFC} \right]_{x} }}{{\left[ {\lambda AFC} \right]_{m} }}}\sigma_{m} $$
(2)

where the subscript m represents the term corresponding to the monitor reaction and subscript x corresponds to the measured reaction, ε full-energy peak efficiency of the measured characteristic gamma-ray, Iγ gamma-ray intensity, η abundance of the target nuclide, M mass of sample, \( D = e^{{ - \lambda t_{1} }} - e^{{ - \lambda t_{2} }} \) counting collection factor, t 1 ,t 2 time intervals from the end of the irradiation to the start and end of counting, respectively, A atomic weight, C measured full energy peak area, λ decay constant, F total correction factor of the activity:

$$ F = f_{s} \times f_{c} \times f_{g} $$
(3)

where f s , f c and f g are correction factors for the self-absorption of the sample at a given gamma-energy, the coincidence sum effect of cascade gamma-rays in the investigated nuclide and in the counting geometry, respectively. Coincidence summing correction factor f c was calculated by the method [48]. In turn, the gamma ray attenuation correction factors f s in the HfO2 foil and the geometry correction f g were calculated by the following Eqs. (4) and (5), respectively.

$$ f_{s} = {\frac{\mu h}{{1 - \exp \left( { - \mu h} \right)}}} $$
(4)
$$ f_{g} = {\frac{{\left( {D + {\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}} \right)^{2} }}{{D^{2} }}} $$
(5)

Here μ (in cm−1) is the linear attenuation coefficient in HfO2 for gamma rays at each of the photon energies E, h (in cm) the thickness of the sample and D is the distance from the measured sample to the surface of the germanium crystal.

K neutron fluence fluctuation factor:

$$ K = \left[ {\sum\limits_{i}^{L} {\Upphi_{i} \left( {1 - e^{{ - \lambda \Updelta t_{i} }} } \right)e^{{ - \lambda T_{i} }} } } \right]/\Upphi S $$
(6)

where L number of time intervals into which the irradiation time is divided, ∆t i duration of the ith time interval, T i time interval from the end of the ith interval to the end of irradiation, Φ i neutron flux averaged over the sample during ∆t i , \( S = 1 - e^{ - \lambda T} \) growth factor of the product nuclide, T total irradiation time, Φ neutron flux averaged over the sample during the total irradiation time T.

One can easily derive the expression of the Eq. (1) \( \eta_{x} \sigma_{x} \) as

$$ \eta_{x} \sigma_{x} = {\frac{{\left[ {S\varepsilon I_{\gamma } \eta KMD} \right]_{m} }}{{\left[ {S\varepsilon I_{\gamma } KMD} \right]_{x} }}}{\frac{{\left[ {\lambda AFC} \right]_{x} }}{{\left[ {\lambda AFC} \right]_{m} }}}\sigma_{m} $$
(7)

Since the samples we selected are natural abundance, furthermore, the hafnium has many isotopes (174Hf, 0.162%; 176Hf, 5.206%; 177Hf, 18.606%; 178Hf, 27.297%; 179Hf, 13.629%; 180Hf, 35.100%) [46], it is obvious that a radioactive nuclide can be produced by several reactions, such as the reaction natHf(n, x)178m2Hf is made up of 179Hf(n, 2n)178m2Hf, 178Hf(n, n′)178m2Hf, 177Hf(n, γ)178m2Hf; natHf(n, x)179m2Hf is made up of 180Hf(n, 2n)179m2Hf, 179Hf(n, n′)179m2Hf, 178Hf(n, γ)179m2Hf and so on. Surely, each of the reactions producing certain radioactive product has different contribution. So in the calculation we just gave the cross section of the element and didn’t import the abundance of the target nuclide into the formula. For the reaction natHf(n, x)178m2Hf, there are three routes to produce 178m2Hf, the first via the 179Hf(n, 2n)178m2Hf reaction, the second through the 178Hf(n, n′)178m2Hf reaction, and the third via 177Hf(n, γ)178m2Hf reaction. So the cross section of natHf(n, x)178m2Hf reaction can be calculated by the following formula:

$$\sigma \left( {^{\text{nat}} {\text{Hf(n,x)}}^{{ 1 7 8 {\text{m2}}}} {\text{Hf}}} \right) = 0.13629\sigma \left( {^{ 1 7 9} {\text{Hf(n,2n)}}^{{ 1 7 8 {\text{m2}}}} {\text{Hf}}} \right) + 0.27297\sigma \left( {^{ 1 7 8} {\text{Hf(n,n}}^{'} )^{{ 1 7 8 {\text{m2}}}} {\text{Hf}}} \right) + 0.18606\sigma \left( {^{ 1 7 7} {\text{Hf(n, }}\gamma )^{{ 1 7 8 {\text{m2}}}} {\text{Hf}}} \right) = \,{\frac{{\left[ {S\varepsilon I_{\gamma } \eta KMD} \right]_{m} }}{{\left[ {S\varepsilon I_{\gamma } KMD} \right]_{x} }}}{\frac{{\left[ {\lambda AFC} \right]_{x} }}{{\left[ {\lambda AFC} \right]_{m} }}}\sigma_{m} $$
(8)

For the reaction natHf(n, x)179m2Hf, there are three routes to produce 179m2Hf, the first via the 180Hf(n, 2n)179m2Hf reaction, the second through the 179Hf(n, n′)179m2Hf reaction, and the third via 178Hf(n, γ)179m2Hf reaction. So the cross section of natHf(n, x)179m2Hf reaction can be calculated by the following formula:

$$ \sigma \left( {^{\text{nat}} {\text{Hf(n,x)}}^{{ 1 7 9 {\text{m2}}}} {\text{Hf}}} \right) = 0.35100\sigma \left( {^{ 1 8 0} {\text{Hf(n,2n)}}^{{ 1 7 9 {\text{m2}}}} {\text{Hf}}} \right) + 0.13629\sigma \left( {^{ 1 7 9} {\text{Hf(n,}}n^{'} )^{{ 1 7 9 {\text{m2}}}} {\text{Hf}}} \right) + 0.27297\sigma \left( {^{ 1 7 8} {\text{Hf(n,}}\gamma )^{{ 1 7 9 {\text{m2}}}} {\text{Hf}}} \right) = {\frac{{\left[ {S\varepsilon I_{\gamma } \eta KMD} \right]_{m} }}{{\left[ {S\varepsilon I_{\gamma } KMD} \right]_{x} }}}{\frac{{[\lambda AFC]_{x} }}{{\left[ {\lambda AFC} \right]_{m} }}}\sigma_{m} $$
(9)

The main contributions to the uncertainty of measurement in our work result from counting statistics (0.1–15%), standard cross sections uncertainties (1%), detector efficiency (2–3%), weight of samples (0.1%), self-absorption of gamma-ray (0.5%) and the coincidence sum effect of cascade gamma-rays (0–5%), the uncertainties of irradiation, cooling and measuring times (0.1–1%), etc. And some other uncertainty contribution form the parameters of the measured nuclei and standard nuclei, such as, uncertainties of the branching ratio of the characteristic gamma rays, uncertainties of the half life of the radioactive product nuclei and so on all are considered.

Results and conclusions

Results

Using the relative activation technique of neutron, the sample was sandwiched between two Nb foils which were used as monitor. Because the cross section of 93Nb(n, 2n)92mNb reaction has been measured by many authors, we can select it as a standard. With the irradiation taking place, the neutron flux for the sample was the same as the flux for the monitors. So we can find an equation of neutron flux between them, and avoid importing the neutron flux into the calculation.

After having been irradiated, the samples were cooled for 5–24 days, and one sample was measured with a 20% relative efficiency HPGe detector. One of the gamma-ray spectra is shown in Fig. 4. Cross-section values for natHf(n, x)178m2Hf and natHf(n, x)179m2Hf reactions were obtained relative to those of the 93Nb(n, 2n)92mNb reaction. The cross sections for the 93Nb(n, 2n)92mNb reaction, which is 459.7 ± 5.0 mb at 14.8 ± 0.2 MeV incident neutron energies, was obtained by interpolating the evaluated values of the literature [42]. The cross-sections measured in the present work are summarized in Table 2.

Fig. 4
figure 4

The γ-ray spectra of about 24 days after the end of irradiation

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Table 2 Summary of cross-section measurements
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Conclusions

We have measured the activation cross sections for natHf(n, x)178m2Hf and natHf(n, x)179m2Hf reactions on hafnium isotopes induced by 14.8 ± 0.2 MeV neutrons. It should be mentioned that this work presents the first cross-sections for formation of 178m2Hf and 179m2Hf through reactions on natural hafnium. The high precision of the cross-sections obtained in the earlier measurement was limited, because there are 174Hf, 176Hf, 177Hf, 178Hf, 179Hf, and 180Hf isotopes in the natural hafnium and overlapping of the gamma-peaks from different isotopes could not be neglected in the gamma-spectrum measurements with the use of NaI(Tl) detectors. The decay data such as half-life and branching ratio of the radioactive product nuclei were imprecise at that early time. In recent years, the reliabilities of the results have improved with the use of the Ge semiconductor detectors and the more accurate above-mentioned parameters. A good plan of irradiation and cooling time is also important and in our experiment we selected different irradiation time according to the half-life of the product nuclei. In addition, the present measurements were performed in the Low Background Laboratory of Lanzhou University and disturbance from environmental radiation was reduced to a very low level. In conclusion, our data would improve the quality of the neutron cross-section database.