Low Thermal Neutron Resonances in Piezo Crystalline Structures

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http://dx.doi.org/10.3938/NPSM.67.940

Low Thermal Neutron Resonances in Piezo Crystalline Structures

Nurgali Takibayev

^∗

· Baurzhan Abdykadyrov

Institute of Experimental and Theoretical Physics, al-Farabi Kazakh National University, Almaty, Kazakhstan (Received 15 February 2017 : revised 10 May 2017 : accepted 11 May 2017)

We investigate the new type neutron resonances that appear in the scattering of neutrons on systems of few-nuclei at certain energies of neutrons and certain distances between the nuclei.

These new type resonances are related to the presence of resonances in the two-body neutron- nucleus subsystems and to their co-ordination and co-operation in the system of a neutron and heavy nuclei. We demonstrate that the new neutron resonances can be detected in the scattering of neutrons on special piezocrystals created with¹¹³Cd isotopes. The calculations are carried out for resonances in a wide region of distances between nuclei fixed in crystalline nodes and in the region of thermal neutrons. For other Cd isotopes, for example, ¹¹⁴Cd, the calculations do not show any neutron resonances in the low thermal range. Remarkably, the crystals created on a base of Cd have piezoelectric features that allow changes in the crystalline parameters. This property can allow the distance between the nuclei to be varied in order to estimate the presence or the lack of the three-body-type resonances.

PACS numbers: 21.45.+v, 28.20.Cz, 25.70.Ef, 61.12.Ex, 77.65.–j

Keywords: Neutron resonances in few-body systems, Piezo crystalline targets, Low thermal neutrons

I. INTRODUCTION

Recently, neutron-nucleus resonances at low thermal energies were found for some isotopes, such as

113Cd¹⁴⁹Sm,¹⁵⁵Gd [1]. In this study, we investigate the model where the neutron scattering on a target (a crystal including Cd isotopes with resonances in the low thermal range) gives us new resonances of the few-body type. It is noticeable that Cd crystals demonstrate piezoelectric features which allow a change of the spatial parameters in the crystal.

We have chosen the ¹¹³Cd isotope because its lowest neutron resonance lies in the low thermal range on the energy scale, while other cadmium isotopes have no neutron resonances in this energy region.

The parameters of the ¹¹³Cd lowest energy neutron resonance are: the energy 2.77 meV, the wavelength is 5.44 Å [1]. The abundance of the ¹¹³Cd isotope is

∗E-mail: takibayev@gmail.com

12.22%. The next resonance level has the energy 7 eV and the wavelength 0.178 Å [2].

Here, we consider a new type of three-body resonances formed by rescattering of a neutron on two heavy centers, i.e., rescattering of a neutron on a system of two nuclides fixed in neighboring nodes of a crystal. This phenomenon represents a three-body scattering problem: a neutron interacts with two neighbor nuclei successively [3]. In this case, the resonant intensification of the scattering amplitudes of neutrons arises at the specific values of the neutron energy and, of course, at certain distances between the nuclides.

So, the new neutron resonances appear at the neutron rescattering in the subsystem of neighboring heavy¹¹³Cd nuclei. Such neutron resonances become stronger at specific distances between the nuclei and the resonances van- ish if the distances are larger or less than these resonance distances [4–6].

The Cd isotopes are considered because the corresponding electric piezo effect allows us to vary the dis-

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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tances between the nuclei in CdS and CdSe crystals [7–

10].

Our theoretical results demonstrate the facts of appearance and disappearance of the three-body resonances in the CdS and CdSe crystals at alteration of the crystalline parameters or energy of neutrons.

II. THEORETICAL DESCRIPTION

The quantum scattering theory for three-body systems is based on Faddeev equations [11]:

Tij= tiδij+∑

k̸=i

tiG0Tkj , i, j, k = 1, 2, 3 , (1)

where total T -matrix T = ∑

i,jTij, and i, j correspond to the initial and final interacting pairs of the system.

The conditions of existence and uniqueness of the solutions are satisfied in the Eq. (1). As it is usually accepted in the Faddeev equations, here i, j, k are the numbers of pairs, which are denoted by the numbers of the third particles [11]. Then ti is the pair t-matrix associated with the pair interaction potential Vi and determined in the system of three particles: ti = Vi+ ViG0ti, where G0 is the free Green’s function for the three-particle system.

The conditions of existence and uniqueness of solutions are satisfied in the Eq. (1).

Hereinafter, we consider the problem of neutron resonance scattering on a subsystem of two nuclei with heavy masses M2 ≈ M3 ≫ mn, where m1 = m_n = m is the mass of neutron. Note that the nuclei fixed at the crystal nodes take the mass of the whole crystal (or the domain of the crystal), and reflect the incoming neutron as the nucleus with the effective mass many times larger than its own mass like in Mőssbauer effect. For simplicity, we take into account the units with ℏ = 1 and c = 1.

In simple case of neutron scattering on the nucleus, we can write the two-body t-matrix in the resonance energy region in the Breit-Wigner form:

ti(k0; ⃗k, ⃗k^′) = ¯νi(⃗k)· ηi(k0)· νi(⃗k^′) ,

η_i(k₀) = 1/(E₀− ER,i)+ iΓ_i/2) , (2) where ER,i is the energy of neutron-nucleus resonance in the two-body system i, Γi is the resonance width, E0 = k²₀/2mn is the kinetic energy of the neutron,

k0 is the initial wave-number of the neutron. In the case of S-wave, the form-factor νi is taken in the form νi=√

πΓi/(k· mn). The law of the energy conservation leads to k = |⃗k| = |⃗k^′| and k = k0 in the physical area that gives the well known behavior of neutron-nucleus cross-section∼ 1/k0 (see, for example [2]).

At the next step we consider the three-body system in the resonance energy region, where a free neutron is scattered by the subsystem of two heavy nuclei. The compacted part T_ij^c of the T -matrix can be written as T_ij^c = T_ij− tiδ_ij. Hereinafter, we assume that the two heavy nuclei are fixed in the neighboring nodes of the crystalline lattice, and their effective mass becomes many times more than the neutron mass. Then, in the frame of Faddeev equations [11,14], using Eq. (1) and expressions (2), we can write the solutions for the T_ij^c-matrices in the forms:

T_ij^c = 1 m· k0

√Γi· ηi(k0)· Mij· ηj(k0)·√

Γj , (3) where i, j = 2, 3 are the numbers of related nuclei, and

Mij = Mij(⃗r, ⃗r^′) = M_ij⁺(⃗r)δ(⃗r + ⃗r^′) , (4) for j̸= i ,

M_ii = M_ii(⃗r, ⃗r^′) = M_ii⁻(⃗r)δ(−⃗r + ⃗r^′) , (5) in the case j = i.

The cardinal distinction between the two-body t- matrices and the three-body T^c-matrix is due to the Mi,j-matrix that appears as a result of successive re- scatterings of the neutron on two neighboring heavy nu- clei (i.e., on two heavy centers). The fact is that three- body resonances of the Mi,j(⃗r)-matrix depend on initial values of the wave-number k0of the neutron and also on the correspondent distances d = 2|⃗r| between the nuclei.

In Eqs. (4) and (5) ⃗r corresponds to the positions of the first nucleus that interacts with an incoming neutron, and ⃗r^′corresponds to the position of the last nucleus that interacts with the out-coming neutron. The reference point corresponds to the half of the distance between the two nuclei.

We have to complete the construction of three-body amplitudes taking Eqs. (4) and (5) between “bra” and

“ket” wave-functions in order to obtain the neutron scattering amplitudes on the subsystem of two heavy nuclei fixed in the neighboring nodes of the crystal.

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For simplicity we assume that heavy nuclei are strictly fixed at the points ⃗R₂ and ⃗R₃ = ⃗R₂+ ⃗d. We also intro- duce the wave-functions for these scattering centers in the form [4]

Ψi(⃗r, ⃗Ri) = C exp [

−(⃗r− ⃗Ri)² 2∆²

]

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for the heavy nuclei localized at the points ⃗R_i, i = 2, 3. The obvious normalization of the wave-functions

< Ψi|Ψi >= 1 gives C² = (π∆)⁻³ and the condition is valid even at ∆→ 0.

To obtain the scattering amplitude of neutron on the subsystem of two heavy nuclei, it is necessary to deter- mine the expression: < Ψin|T |Ψf >, where < Ψ_in| =<

χ₁Ψ₂Ψ₃| and |Ψf >=|χ1Ψ₂Ψ₃>,|χ1> is the free wave function for the neutron. The positions of heavy nuclei are specified in the c.m. of the three-body system R⃗2= ⃗d/2 and ⃗R3=−⃗d/2.

The related part of the neutron scattering amplitude on the subsystem of two heavy nucleus is determined by the form

f^c( ⃗d/2, p₀) = 1 m k0

∑

i,j=2,3

η_i(k₀)√ Γ_iΓ_j

× < Ψi(⃗r, ⃗Ri)|Mij(⃗r, ⃗r^′)|Ψj(⃗r^′, ⃗Rj) > ηj(k0).(7) For convenience, we retain the variables ⃗r and ⃗r ^′ in Mij(⃗r, ⃗r^′) that must be substituted by d/2 and−d/2 at the final stage following to Eqs. (4) and (5). In this case, the neutron does not change its own wave number, and the three-body T_ij^c-matrices keep this value k = k^′ = k₀ same as in the two-body problem [4].

Accordingly, one can obtain the non-diagonal and di- agonal Mi,j-matrices in the following forms:

M_ij⁺(⃗r, k₀) = 1

Dii(⃗r, k0)J_ij(⃗r, k₀), (8) M_ii⁻(⃗r, k0) = 1

D_ii(⃗r, k₀)Jij(⃗r, k0)· ηj(k0)· Jji(−⃗r, k0).

(9) Here Jij(⃗r, k₀) and D_ii(⃗r, k₀) can be written in terms of functions and variables of the two-body subsystems:

J_ij(⃗r, k₀) = 2m

∫

d⃗k· exp(i⃗k⃗r)ν_i(⃗k)· ¯νj(⃗k)

k²₀− k²+ iγ , (10) and

Dii(⃗r, k0) = δij− Jij(⃗r)ηj(k0)Jji(−⃗r)ηi(k0) . (11)

Fig. 1. (Color online) Enhancement factor ξ =|F (k0; ⃗r)|² for neutron resonance scattering on the subsystem¹¹³Cd +¹¹³Cd. Designations I, II, III correspond to the curves with different neutron wave numbers (in⁻¹): k0= 1.278 for I; k0= 1.15 for II; k0= 0.988 for III. The values d = 2r correspond to the distances (in Å) between the nuclei at the maximum of the resonance curves. Then d≈ 0.9 for I; d≈ 2.5 for II; d ≈ 5.0 for III.

Note that the diagonal elements Jii(⃗r, k0)≡ 0, while the diagonal elements Dii(⃗r, k₀) are not equal to zero, but D_ij(⃗r, k₀) = 0 if j̸= i.

It is remarkable that this dependence leads to an additional gain of resonance effects in the three-body system, for example, to appearance of new resonances at definite values of the wave-numbers k0and the definite values of distances between the nuclei d. New resonance peaks are increasing or decreasing with changing distances between the nuclei.

Note that the original two-body resonance peak does not display like three-body resonances, it depends only on energy of neutron but not on distances between the nuclei.

So, the result of division for the three-body T^c-matrix on the two-body t-matrix can be written in the resonance area in the form

F (k₀; ⃗r) =∑

i,j

¯

ν_iη_i· Mi,j· ηjν_j/t_i . (12)

Then, we can determine the additional enhancement fac- tor as ξ =|F (k0; ⃗r)|²which is the ratio between the cross section of neutron resonances of three body type and the two-body (neutron-nucleus) one.

The calculations are illustrated at Fig. 1. It is important that if the distance between the nuclei increases and

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Fig. 2. (Color online) Enhancement factor ξ for neutron resonance scattering on the subsystem ¹¹³Cd + ¹¹⁴Cd system. Designations I, II, III correspond to the curves with different neutron wave numbers (in⁻¹): k0= 1.452 for I; k0= 0.9872 for II; k0= 0.6388 for III. The values d = 2r correspond to the distances (in Å) between the nuclei at the maximum of the resonance curves. Then d≈ 0.08 for I; d ≈ 0.084 for II; d ≈ 0.3 for III.

becomes far from the resonance distances, the three-body resonance effect disappears. The same effect happens when the neutron wave-number goes out from the resonance area. If the distances between these nuclei are decreasing, the other two-body neutron-nucleus resonances enter in force and create new three-body resonances at higher energies.

Our goal is to study the three-body resonance effects that arise in the scattering of neutrons at a subsystem of two nuclear isotopes. The resonances appear only at certain distances between these two heavy nuclei and only at corresponding energies of the neutrons. To assure that the distance between the nuclei in the target remains unchanged during the neutron irradiation, we propose to use the crystalline target. It is important that one can change the distance between the nuclei in the target using the piezoelectric effect.

In this case the summarizing in the above equations is to be taken over i, j = 2, 3, because i, j = 1 are re- sponsible for the direct nucleus-nucleus scattering which is impossible inside of the crystal at low energies and at low temperatures.

It is also very important to create the pure enough crystalline targets with necessary isotopes of heavy nuclei Cd fixed in the neighboring nodes of the crystal.

Fig. 3. (Color online) Enhancement factor ξ for neutron resonance scattering on the subsystem ¹¹⁴Cd + ¹¹⁴Cd.

Designations I, II, III correspond to the curves with different neutron wave numbers (in ⁻¹): k0 = 1.278 for I;

k0 = 1.15 for II; k0 = 0.14 for III. The values d = 2r correspond to the distances (in Å) between the nuclei at the maximum of the resonance curves. Then d≈ 0.006 for I; d≈ 0.02 for II; d ≈ 0.04 for III.

III. CALCULATED NEW NEUTRON RESONANCES

Remarkable that the calculations of the enhancement factor in three-body systems (and in few-body ones, too) show the appearance of new additional neutron resonances which depend on distances between the nuclei (see Figs. 1, 2, and 3). In order to determine the suitable isotopes and crystals on their base, one should select corresponding conditions and nuclei to obtain clearer effects. Of course, it is important to create special targets and run the experiments at very low temperatures.

Moreover, to assume high resolution, the initial beam of low energy neutrons should be highly collimated in terms of energy.

The combinations based on Cd were found to be more suitable among the elements with low thermal neutron resonances. For example, the only ¹¹³Cd has the low thermal resonances, while others like¹⁰⁶Cd -¹¹²Cd, and

114Cd,¹¹⁶Cd do not have any resonances in the low thermal energies.

Here we consider the parameters of CdS and CdSe crystals with the hexagonal (h) and the face-centered cubic (c) structures [6]. In the case of cubic structures the distances (d) between two neighboring Cd nuclei are equal to d = 3.748 Å for CdS and d = 3.917 Å for CdSe.

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Accordingly, in the case of the hexagonal structures one can obtain d = 4.116 Å for CdS and d = 4.292 Å for CdSe.

At first, we consider the crystal that contained isotopes of ¹¹³Cd and determine the neutron resonances of the three-body type which can appear in scattering of neutrons on the subsystems of two¹¹³Cd isotopes. Fig. 1 shows the resonance curves where the peaks correspond to the new neutron resonances.

After that we consider the crystal that contains two types of isotopes: ¹¹³Cd and ¹¹⁴Cd. Then, we determine the neutron resonances of the three-body type in the case. Note, the lowest neutron resonance in the two- body system n+¹¹⁴Cd has the energy 56.4 eV, i.e., situ- ated at higher energy.

The calculations of the three-body system n+¹¹³Cd+¹¹⁴Cd that contains two different isotopes give us a new situation: the resonances of three-body types may appear in this case at much smaller distances between these isotopes (see Fig. 2). It is clear that these three-body resonances cannot be registered in ordinary conditions. Such resonances appear and become very important at very high density, when the crystal could be squeezed with the super pressure.

In the case of the three-body system n+¹¹⁴Cd+¹¹⁴Cd, neutron resonances appear in the region of even smaller distances between these isotopes (see Fig. 3).

IV. DISCUSSION

Theoretical results demonstrate the existence of the three-body resonances in the scattering of a neutron on crystals contained the ¹¹³Cd isotopes. It means that the neutron resonances in the three-body system can appear at relatively large distances between the ¹¹³Cd isotopes, up to sizes comparable with crystalline parameters. There are several crystals that have the piezoelectric properties and neutron-nucleus resonances at very low energies. CdS and CdSe crystals are among them. There is a remarkable property of these crystals:

the distance between the nuclei can be changed using the piezo-electric effect.

At the same time, other Cd-isotopes have neutron resonances at few eV and more [1,2]. It means that the resonance distances between the nuclei in three-body system must be for a hundred times less than the electron orbits.

Indeed, our calculations give for the resonances in the system n + ¹¹³Cd + ¹¹³Cd the following values: k0 = 0.988 Å⁻¹ and dres ≈ 5.0 (Fig. 1). Other isotope pairs, for example,¹¹³Cd +¹¹⁴Cd or¹¹⁴Cd +¹¹⁴Cd, give the three-body resonances at much shorter distances between the nuclei (Fig. 2 and Fig. 3).

Note that only few type of nucleus have the neutron resonances at very low energies, for instance, ¹⁵⁵Gd,

157Gd, ¹⁴⁹Sm, or ¹⁵²Sm and some others. The parameters of the two-body resonances in low energy region were taken from [1].

New experiments are needed to demonstrate the existence of the neutron resonances of the three-body type.

They can lead to finding of new properties and processes that take place in crystalline structures. It could be a good chance to verify three-body effects using the piezocrystals with unique isotopes which have the low energy neutron resonances [5].

In the case of positive results, these experiments can give the first step for investigation of the neutron resonances that appear in crystals at changing their parameters. Further, very interesting effects could be obtained for the crystalline structure properties under very high pressures. The neutron resonances that appear in the neutron star envelopes at specific depths should depend on a sort of nuclei.

It is important to note that under huge pressure the matter of the neutron star envelopes becomes more simple because the atoms are destroyed and create a crystalline cubic lattice with nuclei fixed in the nodes and this whole construction is plunged into degenerated electron Fermi liquid [12,13].

The density of matter in neutron star envelopes, where free neutrons appear and the crystalline structures exist, corresponds to ρ ≈ 10⁷ - 10¹⁴ g/cm³ [15]. The lattice parameters are reducing from 10³fm to 40 fm, after that the crystalline lattice is destroyed.

Note that the resonance sources can enhance each other, as can be seen from the theoretical analysis of the appearance of new resonances in the three-body problem. Amplification would be associated not only with the presence of resonance sources, but also with certain characteristics of the environment itself. At the same time, organized structures, such as crystals, can stimulate the corresponding resonant phenomena.

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The parameters of the three-body resonances are related to the characteristics of the neutron resonances of nuclear isotopes and the lattice constants. The latter can change under the influence of pressure or piezo-effect.

This is in the case of laboratory experiments.

In neutron stars, the lattice parameters change with the depth of the corresponding crystalline layer in the envelopes of neutron stars. Accordingly, the number of neutron resonant levels, which take part in the resonance interactions, increase with the deep of layer. The new resonances can stimulate nuclear reactions and unusual processes, which appear only under certain conditions, for example, only in the certain deep layers of the crystalline envelopes of neutron stars.

Of course, we try to consider these extreme conditions where the neutron resonances of three- and few- body type stimulate the accompanied reactions and processes in the neutron star envelopes, and lead to the corresponding resonance forces [4,5].

Therefore, the experiments with the neutron resonances of three-body type in the terrestrial conditions are very important to confirm the strengthening of neutron resonances of few-body type in the neutron star envelopes.

V. CONCLUSIONS

In this study, we conducted a theoretical analysis and calculations of the new phenomena associated with the appearance of neutron resonances of the three-body type in CdS and CdSe crystals. There were also investigated same crystalline structures with different contents of cadmium isotopes ¹¹³Cd and ¹¹⁴Cd: the physico-chemical characteristics of the crystals were identical, but the dif- ferences appeared owing to their nuclear physical properties.

We calculated new neutron resonances for the systems consisted of a neutron and two nuclei in the energy range close to conventional neutron-nucleus resonances. The study of the three-body resonance was focused at scattering of neutrons on a subsystem of two isotopes when distances between these nuclei are taken into account. It was assumed that the distance between the nuclei can be changed using the piezoelectric properties of the selected crystals CdS and CdSe.

All of this opens up opportunities to use specialized piezo-crystals and to assure control over the target parameters in the experiments with thermal resonance neutrons. For example, it is possible to set up the thermal neutron scattering experiments with different targets such as CdS and CdSe crystals.

Note, the neutron three-body resonances have a specific origin - they happen due to the neutron resonance rescattering on two heavy nuclei. As in the M¨ossbauer effect, a crystal structure accepts the recoil momentum, which strengthens the neutron resonances. It means that the new neutron resonances could be detected more ef- fectively at very low temperatures.

ACKNOWLEDGEMENTS

The authors would like to thank Dr. A. Yermilov, Prof. Kiyoshi Kato and Prof. V. S. Vasilevsky for helpful discussions. The work was carried out in the frame of the Project IPS-3106 of MES RK.

REFERENCES

[1] Thermal neutron resonances - NIST Center for Neu- tron Research, https://www.ncnr.nist.gov/resour ces/activation/resonance.html (accessed May 11, 2017).

[2] S. F. Mughabghab, Atlas of Neutron Resonances (Elsevier BV, 2006).

[3] N. Zh. Takibayev, Phys. Atom. Nuclei 71, 460 (2008).

[4] N. Zh. Takibayev, Few-Body Systems 50, 311 (2011).

[5] N. Takibayev, Few-Body Systems 54, 447 (2013).

[6] N. Takibayev and B. Abdykadyrov, Adv. Stud.

Theor. Phys. 9, 617 (2015).

[7] W. J. Minkus, Phys. Rev. 138, A1277 (1965).

[8] D. A. Grynko, O. M. Fedoryak, M. Kratzer, O. B.

Smirnov and P. S. Smertenko et al., Proc. NAP 3, 02NEA07 (2014).

[9] K.-H. Schulte-Schrepping and M. Piscator, Cad- mium and Cadmium Compounds in Ullmann’s En- cyclopedia of Industrial Chemistry (Willey-VCH, Weinheim, 2007), p. 499.

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[10] B. Moon, J. Lee and H. Jung, Thin Solid Films 511, 299 (2006).

[11] L. D. Faddeev, Mathematical Aspects of the Thre- Body Problem in Quantum Scattering Theory (Davey, New York, 1965), p. 114.

[12] D. A. Kirzhnits, J. Exp. Theor. Phys. 38, 503 (1960).

[13] D. A. Kirzhnits, Physics-Uspekhi (Advances in Physical Sciences) 3, 489 (1971).

[14] L. D Faddeev and S. P. Merkuriev, Quantum Scat- tering Theory for Several Particle Systems (Kluwer Academic Publishers: Boston, Dordrech, 1993), p.

405.

[15] S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars (New York: John Wiley and Sons, 1983), p. 400.