**2. Theoretical evaluation**

Let us consider the neutron-absorbing material in which the following processes occur under the influence of neutrons: nucleus X + neutron ! nucleus Y in the excited state ! nucleus Y in the isomeric (metastable) state ! nucleus Z in the ground state. For example: Gd<sup>155</sup> + n ! Gd156\* ! Gd156m ! Gd156. Nuclei Y and Z also undergo radiation capture that is they are "shot." Isomer Gd156m has the half-life period of 1.3 μs and decays emitting gamma-quantum with the energy of 2.1376 MeV. In **Figure 1**, the scheme of the process is shown.

In **Table 1**, the parameters of two Gd isotopes in ground and isomeric states are presented.

Before appearing in a metastable state, Gd<sup>156</sup> nucleus is in an excited state. The typical nucleus lifetime in the excited state is �10–14 s, which is nine orders of

### **Figure 1.**

*Pumping of the active medium formed by gadolinium isotopes nuclei [3].*


### **Table 1.**

*Parameters of gadolinium isotopes nuclei.*

*Application of the Gadolinium Isotopes Nuclei Neutron-Induced Excitation Process DOI: http://dx.doi.org/10.5772/intechopen.85596*

magnitude longer than the nuclear interaction time. Therefore, the nucleus in the excited state can gain and conserve energy ΔE transferred to it as a result of neutron scattering on it. Energy supplied to the nucleus when the neutron dissipates on it depends on the nucleus mass—the less the nucleus mass is, the bigger energy the neutron gives to it during dissipation. In the ideal case, energy Δ*E* transferred at dissipation is the value equal to the difference between the energy of metastable state and energy of excitation. For the given value of the neutron energy *E*n, the value of the energy Δ*E* transferred at inelastic scattering on the nucleus can be determined from the transcendent equation:

$$\frac{\Delta E^{i}}{E\_{\text{n}}} = \frac{2A}{\left(A+1\right)^{2}} \left(1 + \frac{A+1}{2} \cdot \frac{\Delta E^{i}}{E\_{\text{n}}} - 0.07A^{2/3}E\_{\text{n}}\sqrt{1 - \frac{A+1}{A} \cdot \frac{\Delta E^{i}}{E\_{\text{n}}}}\right),\tag{1}$$

where *A* is the nucleus mass number. The equation is obtained on the assumption that the transferred energy Δ*E* is equal to the energy of nucleus excitation from the ground state. If the nucleus has several excitation levels (i = 1, 2, 3…), the equation allows determining the value of the neutron energy *En*, providing transfer of the nucleus to the corresponding excitation level.

The number of collisions (dissipations) of the neutrons with the nuclei of active medium occurring in the medium volume unit per time unit can be determined by the following relation:

$$
\Phi \sigma n\_{mc} \tag{2}
$$

where Ф is the neutron flux density, σ is the microscopic cross section of inelastic neutron scattering on the nuclei, and nnuc is the number of nuclei per medium volume unit. In this case, the scattering frequency experienced by neutrons in the active medium is determined by the following relation:

$$
\nu \sigma n\_{nuc} \tag{3}
$$

where <sup>υ</sup> is the neutron velocity (*<sup>υ</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffi 2*E=m* p ).

For example, to transfer nuclei of 54Xe<sup>130</sup> isotope from the ground state to the excited state, taking into account that they have three excitation levels (0.54, 1.21, and 1.95 MeV), the presence of neutrons with the energies of 0.709, 1.285, and 2.005 MeV, correspondingly, is required in the flux. To transfer nuclei of 10Ne<sup>22</sup> isotope from the ground state to the excited state, which also has three excitation levels, the presence of neutrons with the energies of 2.075, 3.747, and 4.859 MeV is required in the flux. The average neutron energy of the fission spectrum is 2 MeV. The average neutron spectrum energy of the nuclear reactor (even fast neutron reactors) is significantly lower. Besides, to transfer isotope nuclei to the excited state by direct scattering of neutrons on nuclei, it is necessary to "choose" isotopes not only with bigger specific binding energy of nucleons in the nucleus but also with small value of the neutron absorption cross section. Therefore, to accumulate nuclei in the excited state, it is reasonable to obtain them as a product of the reaction of neutron radiative capture by nuclei with a mass number smaller by one unity. The daughter nucleus is formed in the excited state and, if required, gains an additional energy due to neutron scattering on it. As a result, the daughter nucleus appears in the metastable state.

particular their nuclear safety. The possibility of accumulation and uncontrolled release of excess energy in neutron-absorbing materials because of potential accumulation of excess energy in isomeric states of atomic nuclei (for example, hafnium or gadolinium) comprising some of them was paid attention to [2].

Let us consider the neutron-absorbing material in which the following processes

In **Table 1**, the parameters of two Gd isotopes in ground and isomeric states are

Before appearing in a metastable state, Gd<sup>156</sup> nucleus is in an excited state. The

**Nucleus Half-life Isotopic content in natural mixture Spin and nucleus parity**

Gd<sup>155</sup> Stable 14.80% 3/2- Gd155m 31.97 ms — 11/2- Gd<sup>156</sup> Stable 20.47% 0+ Gd156m 1.3 μs — 7-

typical nucleus lifetime in the excited state is �10–14 s, which is nine orders of

occur under the influence of neutrons: nucleus X + neutron ! nucleus Y in the excited state ! nucleus Y in the isomeric (metastable) state ! nucleus Z in the ground state. For example: Gd<sup>155</sup> + n ! Gd156\* ! Gd156m ! Gd156. Nuclei Y and Z also undergo radiation capture that is they are "shot." Isomer Gd156m has the half-life period of 1.3 μs and decays emitting gamma-quantum with the energy of

2.1376 MeV. In **Figure 1**, the scheme of the process is shown.

*Pumping of the active medium formed by gadolinium isotopes nuclei [3].*

**2. Theoretical evaluation**

*Rare Earth Elements and Their Minerals*

presented.

**Figure 1.**

**Table 1.**

**68**

*Parameters of gadolinium isotopes nuclei.*

To evaluate the possibility of energy accumulation in isomeric nuclei states due to radiation neutron capture in such material, it is required to solve the differential equations system:

$$\begin{cases} \frac{d\mathbf{x}}{dt} = -\sigma\_1 \mathbf{x} \Phi \\ \frac{d\mathbf{y}}{dt} = \sigma\_1 \mathbf{x} \Phi - \sigma\_2 \mathbf{x} \Phi - \lambda \mathbf{y} \\ \frac{d\mathbf{z}}{dt} = -\sigma\_3 \mathbf{z} \Phi + \lambda \mathbf{y} \end{cases} \tag{4}$$

"shooting" by neutrons and the rate of their ground-state transition even at flux

The possibility of pumping the medium formed by hafnium nuclei with gamma-quanta was studied in [5]. External gamma-quantum flow cannot provide conditions for population inversion of metastable nuclei energy levels. To compare

the ability to accumulate energy in isomeric conditions at nuclei excitation, according to the scheme presented in **Figure 1**, the stable isotope 72Hf<sup>178</sup> was considered. Metastable nuclei of hafnium-178m2 form from the nuclei of hafnium-178 (stable isotope with 27.28% content in natural mixture). According to contemporary data [6–8], the energy of emitted gamma-quantum is 2.446 MeV at transition to the ground state and the half-life of 31.0 years correspond to metastable nuclei of hafnium-178m2. These parameters are different for metastable nuclei of hafnium-178m3 (higher energy level) and are equal to 2.534 MeV and 68 μs, and

1.147 MeV and 468 μs for metastable nuclei of hafnium-178m1.

scattering, even if the neutron flux density of <sup>Ф</sup> � <sup>10</sup><sup>14</sup> cm�<sup>2</sup> <sup>s</sup>

further increase in the neutron flux density leads to reducing the time interval after which the excess energy begins to accumulate. As a result, rapid accumulation of excess energy in the metastable state of gadolinium-156 isotope nuclei should be

The metastable nuclei of hafnium-178m2 form not only at inelastic scattering of fast neutrons on nuclei of hafnium-178, but also at radiation neutron capture by the nuclei of hafnium-177 (stable isotope with 18.6% content in natural mixture). As a result of neutron capture, the main nucleus of hafnium-178\* forms in a very excited state. The excitation energy is equal to the sum of neutron-binding energy in the nucleus and neutron kinetic energy. Lifetime of the compound nucleus in a state of excitement is not more than 10�<sup>13</sup> s, excitation is removed by emission of highenergy gamma-quantum, and the nucleus transfers either to the ground or one of

Inelastic scattering cross section on nuclei of hafnium-178 does not exceed 2.5 barns in a wide range of neutron energies, which leads to impossibility to accumulate considerable amount of energy in isomeric states only by means of inelastic

*z t*ð Þ <sup>≥</sup>1 due to only inelastic scattering will be achieved in a very big period of time. The balance of hafnium-178 nuclei in the isomeric state **m2** improves, if it is taken into account that they form as a result of radiation neutron capture by nuclei of hafnium-177. Cross section of this process is hundreds of barn for thermal neutrons and is more than 1 barn for neutrons with the energy to 100 eV. The condition

*z t*ð Þ <sup>≥</sup>1 can be achieved in a significantly shorter period of time with account of radiation neutron capture, but if it is taken into account that as a result of neutron capture by nuclei of hafnium-178 and its isomers (the cross section of the process for thermal neutrons is tens of barn) all these nuclei disappear, the condition can be

To perform the research of excess energy accumulation in Gd, the following

Gd2O3 is placed in a cylindrical volume made of pure tungsten. Further, the cylinder is placed in the active core of the reactor unit. Uranium-graphite reactor is chosen as a reactor unit for the purpose of investigating the sample in thermal

Several versions of the tungsten bulb with graphite reflector and without reflector are considered in the work (see **Table 2**). Graphite block serves as a reflector.

The neutronic calculation was performed using a WIMSD-5B.12 specialized program (OECD Nuclear Energy Agency). The program WIMS is applied for

system to place Gd2O3 in the reactor core was used (see **Figure 2**).

Isotopic composition of Gd consists of 50% Gd<sup>155</sup> and 50% Gd156.

�1

. The condition

�1 . The

densities of resonant and thermal neutrons of the order of �10<sup>13</sup> cm�<sup>2</sup> <sup>s</sup>

*Application of the Gadolinium Isotopes Nuclei Neutron-Induced Excitation Process*

expected at moderate neutron flux densities.

*DOI: http://dx.doi.org/10.5772/intechopen.85596*

the metastable states.

not achieved in principle.

neutrons spectrum [9–13].

*y t*ð Þ

*y t*ð Þ

**71**

Here *x*(*t*), *y*(*t*), *z*(*t*) – Gd155, Gd156m, Gd156 nuclei concentration, respectively; Ф is the neutron flux density; *σ* is the micro-cross section of radiation neutron capture (*σ*1—for Gd<sup>155</sup> nuclei, *σ*2—for Gd156m nuclei, *σ*3—for Gd<sup>156</sup> nuclei), and *λ* is the decay constant of isomers nuclei Gd156m.

Solution of the system of equations gives the formulae to determine the possibility to achieve the condition at which the nuclei concentration in isomeric state *y* (*t*) becomes bigger or equal to the concentration of nuclei in the ground state *z*(*t*) influenced by neutrons with the flux density Ф to 10<sup>16</sup> cm�<sup>2</sup> s �1 :

$$\frac{y(t)}{z(t)} \approx \frac{\lambda t - (\sigma\_1 - \sigma\_2)\Phi t}{S\lambda},\tag{5}$$

where

$$S = \frac{\mathbf{1} - (\boldsymbol{\lambda} + 2\sigma\_3 \Phi)t}{\boldsymbol{\lambda} + \sigma\_3 \Phi} - \frac{\mathbf{1} - (\sigma\_1 - \sigma\_2 + 2\sigma\_3)\Phi t}{(\sigma\_1 - \sigma\_2 + \sigma\_3)\Phi} + \frac{(\boldsymbol{\lambda} - (\sigma\_1 - \sigma\_2)\Phi)(\mathbf{1} - \sigma\_3 \Phi t)}{(\sigma\_1 - \sigma\_2 + \sigma\_3)\Phi(\boldsymbol{\lambda} + \sigma\_3 \Phi)}. \tag{6}$$

The ratio *y t*ð Þ *z t*ð Þ is the ratio of the concentration of Gd156m nuclei to the concentration of Gd<sup>156</sup> nuclei. When this ratio becomes greater than 1, that means that, starting from a certain point in time, the concentration of Gd156m nuclei becomes greater than the concentration of Gd<sup>156</sup> nuclei. It is taken into account that the Gd156m nuclei transfer to the ground state with the emission of gamma-quants that act on the Gd<sup>156</sup> nuclei, transferring them to the excited metastable state.

When neutrons with the flux density Ф = 1013 cm�<sup>2</sup> s �<sup>1</sup> influence the neutrons absorber formed by gadolinium nuclei, the condition *y t*ð Þ *z t*ð Þ <sup>≈</sup>1 is achieved within several tens of seconds. It is explained by almost unique combination of absorbing properties of two isotopes of gadolinium (Gd155 and Gd156) in both thermal and resonant energy regions of neutrons.
