ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 11, с. 1436-1442
ЯДРА
PHOTO-NEUTRON CROSS-SECTION CALCULATIONS
OF 142,143,144,145,146,i50Nd RARE-EARTH ISOTOPES
FOR (Y,n) REACTION
©2014 A. Kaplan1)*, H. Ozdogan1), A. Aydin2), E. Tel3)
Received November 28, 2013; in final form, May 5, 2014
The theoretical photo-neutron cross sections for (7, n) reaction have been calculated on 142,143,144,145,146,i50Nd rare-earth isotopes at photon energies of 8-23 MeV using the PCROSS, TALYS 1.2, and EMPIRE 3.1 computer codes. TALYS 1.2 two-component exciton model and EMPIRE 3.1 exciton model has been used to calculate the pre-equilibrium photo-neutron cross sections. PCROSS Weisskopf-Ewing model has been used for the reaction equilibrium cross-section calculations. The obtained cross sections have been compared with each other and against the experimental values existing in the EXFOR database. Generally, pre-equilibrium model cross-section calculations are in good agreement with the experimental data for all reactions along the incident photon energy in this study.
DOI: 10.7868/S0044002714100092
1. INTRODUCTION
Photonuclear reaction data are very important for understanding of the structure and dynamics of the atomic nucleus. Moreover, photo-neutron cross sections are also important for some applications as: analysis of radiation transport and shielding, absorbed dose calculations in the human body throughout photon-radiotherapy, fission and fusion reactor technology, activation analysis including protections and material analysis studies for photon-induced nuclear reactions and transmutation of nuclear waste [1,2].
The nuclear reaction models are generally required to get the prediction of the reaction cross sections, especially if no experimental data are obtained or in cases where it is difficult to carry out the experimental measurements [3-16]. Photon-induced nuclear reaction cross-section evaluation for materials attaches special importance to use of reaction systematics.
The equilibrium reactions occur on a very much longer time scale (=10_16 to 10_1S s). Equilibrium processes play an important role in nuclear reactions induced by light projectiles with incident energies up to about 10 MeV. While the particles at low energies
1)Siileyman Demirel University, Arts and Sciences Faculty,
Physics Department, Isparta, Turkey.
2)Kirikkale University, Arts and Sciences Faculty, Physics
Department, Kirikkale, Turkey.
3)Osmaniye Korkut Ata University, Arts and Sciences Faculty,
Physics Department, Osmaniye, Turkey.
E-mail: abdullahkaplan@sdu.edu.tr
from a nuclear reaction mechanism are emitted by a statistical process that can be described adequately by compound nucleus theories, at higher emission energies, the pre-equilibrium processes are important mechanisms in nuclear reactions induced by light projectiles with incident energies above 10 MeV.
In this study, the theoretical (7, n)-reaction cross sections of 142>143>144>145>146>150Nd rare-earth isotopes in photon-induced reactions have been studied. The cross-section calculations of 142Nd(Y, n)141 Nd, 143 Nd(Y,n)142 Nd, 144 Nd(7,n)143 Nd, 145 Nd(7, n)144Nd, 146Nd(7,n)145Nd, and 150Nd(7, n)149Nd reactions have been carried out in the incident photon energies between 8 and 23 MeV. Reaction cross sections as a function of photon energy have been calculated using PCROSS [17], TALYS 1.2 [18, 19], and EMPIRE 3.1 [20, 21] computer codes. TALYS 1.2 and EMPIRE 3.1 exciton models have been used to calculate the pre-equilibrium photo-neutron cross sections. PCROSS Weisskopf— Ewing (WE) [22] model have been used for the reaction equilibrium cross-section calculations. The calculated results have been compared with each other and with the experimental values existing in the EXFOR database [23].
2. CALCULATION METHODS
Photo-neutron cross sections as a function of photon energy have been calculated using PCROSS code for the WE model, TALYS 1.2 code for the two-component exciton model, and EMPIRE 3.1 code for the exciton model.
The equilibrium particle emission is given by the WE Model in which angular momentum conservation is neglected. In the process, the basic parameters are inverse reaction cross section, binding energies, the pairing and the level-density parameters. The reaction cross section for incident channel a and exit channel b can be written as
aabE = Vab(Emc)
г
b
ЕГь''
V
where Einc is the incident energy. In Eq. (1), rb can be also expressed as
)
_ 2sb + 1
ui(E) =
1 exp
2y/a{E - D)
^48
E-D
a = -Ô 9-n2
The pre-equilibrium model of TALYS is the two-component exciton model of Kalbach [32]. In the two-component model, the neutron and proton type of the created particles and holes is explicitly followed throughout the reaction. The exciton model cross section is given as
daEM
(1)
dEk
aCFx
(4)
Peq
peq
pv
x ^k (pn ,h ,Pv ,hv, Ek ) x
(2)
where U, ¡ib, sb are the excitation energy of the residual nucleus, the reduced mass, and the spin, respectively. The total single-particle level density is taken as
(3)
Here, a™, E, D, and g are the inverse reaction cross section, the excitation energy of the compound nucleus, the pairing energy, and the single-particle level density, respectively.
TALYS [18, 19] is a nuclear reaction simulation code for the estimation and analysis of nuclear reactions that include protons, neutrons, photons, tritons, deuterons, 3He, and alpha particles in the energy range of 1 keV-200 MeV. For this, TALYS integrates the optical model, direct, pre-equilibrium, fission and statistical nuclear reaction models in one calculation scheme and thereby gives a prediction for all open reaction channels. In TALYS, several options are included for the choice of different parameters such as 7-strength functions, nuclear level densities and nuclear model parameters [24]. The 7-ray strength function is obtained from the compilation by Kopecky and Uhl [25] and the nuclear level density is also based on an approach using the Fermi-gas model [26]. The pre-equilibrium reactions were considered by the two-component exciton model [27]. The effects of direct-like interactions due to complex particles (clusters) were calculated by the phenomenological model of Kalbach [28]. This model performs the pre-equilibrium calculations by taking into account the contributions from the break-up, transfer, and knockout reactions [29]. TALYS contains the ECIS-06 code [30] for optical model and direct reaction calculations. The default optical model potentials were from the compilation of Koning and Delaroche [31].
X ^pre (pn i hn, pv, hu ) j
where pn (pv) is the proton (neutron) particle number and hn (hv) the proton (neutron) hole number. The initial proton and neutron particle numbers are pPn = = Zp and p0 = Np, with Zp(Np) the proton (neutron) number of the projectile. Generally, hn = pn — pi and hv = pv — p0 so that the initial hole numbers are zero, i.e., hi = h0 = 0 for primary pre-equilibrium emission [33].
The emission rate Wk has been derived by Cline and Blann [34] from the principle of microreversibility, and can easily be generalized to a two-component version [35]. The emission rate for an ejectile k with relative mass ¡k and spin sk is
Wk p,h,pv,hv,Ek)= (5)
2sfc + 1 T? ( TP \
"Atfc-C'fcCfc.inv l-C'fcJ X
Ш
p - Zk,Pv - Nk,hv, Etot - Ek)
u(pn,hn,pv, hv,Etot)
where ak>.inv (Ek) is the inverse reaction cross section, again calculated with the optical model, Zk (Nk)isthe charge (neutron) number of the ejectile and Etot is the total energy of the composite system [27]. The details of the other code model parameters and options of TALYS 1.2 can be found in [18, 19].
The exciton model assumes that after the initial interaction between the incident particle and the target nucleus the excited system can pass through a series of stages of increasing complexity before equilibrium is reached, and emission may occur from these stages giving the pre-equilibrium particles [36, 37].
The Empire 3.1 includes the pre-equilibrium mechanism as defined in the exciton model [36], as based on the solution of the master equation [34] in the form proposed by Cline [38] and Ribansky [39]
-qt=o(n) = \+ (E,n + 2) t (n + 2)+ (6) + A_ (E,n - 2) t (n - 2) -- [A+ (E, n) + A+ (E, n)+L (E, n)] t (n),
Pn = РП Pv = PV
6
Cross section, mb
8 12 16 20 24
Photon energy, MeV
Fig. 1. The comparison of calculated photo-neutron cross sections of142Nd(7, n)141 Nd reaction with the values reported in [23,47, 48].
where qt (n) is the initial occupation probability of the composite nucleus in the state with the exciton number n, (E, n) and A_ (E, n) are the transition rates for decay to neighboring states, and L(E,n) is the total emission rate integrated over emission energy for particles (protons n, neutrons v and clusters) and Y rays.
The pre-equilibrium spectra can be calculated as
-"T"1" i£b) — &a,b №nc) h (E\nc) X (7) d£b
Wb (E,n,£b) T (n),
n
where arab (Einc) is the cross section of the reaction (a,b), Wb (E,n,£b) is the probability of emission of a particle of type b (or gamma ray) with energy £b from a state with n excitons and excitation energy E of the compound nucleus, and Da,b (Einc) is the depletion factor, which takes into account the flux loss as a result of the direct reaction processes. Kalbach's method [34, 38, 40] was implemented for the calculation of the nucleon emission rate. The expression for the nucleon emission probability is derived applying the principle of detailed balance in a way similar to what is done in the evaporation model [20,21].
Nuclear reaction cross sections are important for a variety of applications in the areas of astrophysics, nuclear energy, and national security. When
these cross sections cannot be measured directly or predicted reliably, it becomes necessary to develop indirect methods for determining the relevant reaction rates.
The prediction of nuclear reaction cross sections above the resonance region depends on seve
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