ISSN (0970-2083)

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National Research Tomsk Polytechnic University (TPU), 634050, Tomsk, Russian Federation

- Corresponding Author:
- A. Naymushin

National Research Tomsk Polytechnic University (TPU)

634050, Tomsk, Russian Federation

**Email:**agn@tpu.ru

**Received date:** 06 July, 2016; **Accepted date:** 09 August, 2016

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Journal of Industrial Pollution Control

Analytical scheme of defect formation process was designed, describing temporal dynamics of number of atoms in lattice nodes as well as point and complex defects. According to this scheme, a system of differential equations was made. Analysis of solution of the equation system and experimental data on stored energy (Wigner energy) for industrial graphite reactors allowed to determine the dependence of constant recombination of point defects on irradiation temperature. Calculated and experimental asymptotes of Wigner energy dependence on irradiation time of graphite were compared.

Graphite reactor, Wigner energy, Irradiation temperature, Irradiation time, Energy determination.

Extension of lifetime of operating graphite reactors and reactor decommissioning are of great importance nowadays. Solving these problems requires correct estimations of reactor graphite life-time as well as energy stored in it (Wigner energy).

Neutron irradiation of graphite alters its properties because of damaging its lattice structure. During moderation process neutron energy is transferred to atoms of carbon which can be dislocated in the lattice relative to the initial location. Many of these shifted (primary knocked out) atoms having high kinetic energy can induce displacement of other atoms, slow down, etc. For example, neutron with 1 MeV energy can induce up to 900 atom displacements losing its energy till the energy of thermal neutron. To displace an atom in graphite lattice it is required about 25 eV of energy. Many displaced atoms immediately return on vacant places. A large number of atoms occupy intermediate position. This can be individual atoms as well as groups of atoms. They have significant impact on numerous properties of the material and this significance depends on dose and temperature of irradiation (Virgilev, 2001).

Temperature – one of the main factors affecting the degree of radiation damage in materials’ structure
(Glushkov *et al*., 1985; Baybakov *et al*., 2015;
Nesterov *et al*., 2013; Avdohin *et al*., 2013). Neutron
bombardment leads to formation of point defects,
the fate of which is determined by temperature
conditions. Migration of defects to sinks, annihilation
of Frankel pairs, formation of complexes and other
diffusion processes are related with temperature.
The number of initially knocked out atoms at the
moment of interaction of radiation with the material
at low and high temperatures is practically the same,
however, mobility of these atoms at high temperature
is higher, i.e. they annihilate faster. This leads to
decrease of defects’ concentration and therefore to
lesser change of properties during irradiation.

Decrease of irradiation temperature and attendant
gamma-radiation flux density in the range 100–
300°С due to decrease of thermal and radiation
gamma-annealing leads to increase in concentration
of defects and consequently to decrease of critical
neutron fluence (Avdohin *et al*., 2013; Karpuhin et
al., 1997). Critical fluence – is the fast neutron fluence
that enables polycrystalline graphite to compensate
shrinkage. The value of this fluence defines the
lifetime of reactor graphite and limit concentration of
defects because during further growth of fast neutron fluence thermal-physical and strength properties of
graphite deteriorate abruptly.

In higher temperature range (higher 300°С) prevail
much more complex defects than point defects.
These complex defects have little effect on lattice
parameters but take part in formation of additional
basal planes in the graphite irreversibly changing
form of graphite crystallites (Karpuhin *et al*.,
1997; Golovatsky *et al*., 2011; Baybakov *et al*., 2015;
Golovatsky *et al*., 2011; Nesterov, 2013).

Nowadays research in the field of radiation damage of reactor graphite is mainly experimental.

This paper sets the goal of analytical description of damaging and recovery process of the crystal structure of reactor graphite to determine the dependence of Wigner energy on the fluence of damaging neutrons.

Formulation of the problem requires selecting 3 types of atoms in graphite structure:

• Atoms present in the lattice. Its concentration is denoted N.

• Atoms relating to point defects – N_{t}.

• Atoms creating complex defects – N_{c}.

The sum of these atoms represents the concentration of all atom types and is defined by the relation:

(1)

where М – molar mass of carbon; ρ – density of reactor graphite.

The problem is formulated using the following approximations:

• Process of damaging of the graphite does not lead to significant change of specific volume.

• Number of point defects formed during the
interaction of neutron with nuclei of carbon is
equal for atoms present in the lattice and for
atoms forming complex defects. The number
of defects formed per neutron with energy of
1 MeV is value of 10^{3} order.

• Recombination constant of point defects λ
is a sum of constant fraction of point defects
returning to the initial state (recombining)
λ_{a}and constant fraction of defects that are
transformed into complex defects λ_{c}. The
recombination constant of point defects is
defined by the relation:

(2)

where all the values depend on equivalent irradiation temperature (Т) and on attendant gamma-radiation flux density .

The scheme of defect formation process for graphite
is shown in **Fig. 1** and is described as follows:

Change of the number of atoms in the lattice ( dN / dT ) is affected by two competing processes:
the first one – increase due to recombination process
of point defects (transition of atoms of point defects
to the lattice nodes – λ_{a} N_{t} ); the second one –
decrease due to interaction of neutron flux with the
lattice (knocking out of atoms from the lattice and
formation of point defects – Ôσ_{s} nN ).

Change of the number of atoms forming complex
defects ( dN_{c} / dT ) is affected by two competing
processes: the first one – increase due to transition
of atoms of point defects into complex defect ( Nt λ_{c} ); the second one – decrease due to effect of neutron
flux on complex defects (transition of atoms of
complex defects into point defects –Ôσ_{s}nN_{c} ).

Change of the number of atoms of point defects (
dN_{t} / dT ) is affected by two competing processes:
the first one – decrease due to recombination process
of point defects and its transition into complex
defects ( λ N_{t} ); the second one – increase due to effect
of neutron flux on the lattice and complex defects
(transition of atoms in the lattice and complex defects
into point defects –Ôσ_{s} n(N + N_{c} ) .

Thus, the system of differential equations describing the change of the number of atoms in the lattice as well as change of complex and point defects is as follows:

(3)

where n – the number of formed point defect per
one act of neutron scattering from nucleus of carbon;
t – time; Ф – damaging neutrons flux density; – σ_{s} scattering microscopic cross-section for the
damaging neutrons.

The solution of this system are the following expressions:

(4)

where

N_{oc} , N_{o} – concentrations of atoms of point defects,
complex defects and atoms in the lattice at initial
time, respectively.

The amount of energy stored in graphite is in direct proportion to the amount of point defects. This allows determining the dependence of recombination constant of point defects on irradiation temperature. Experimental data on Wigner energy for the reactors of Siberian Chemical Combine allowed determining this dependence.

To displace one atom in the lattice it is required
around 25 eV of energy (E_{d} ). It can be assumed that
the same amount of energy is released when atom
returns to vacant place in the lattice. Solution of the
system of differential equations (3) defines the ratio
for the amount of point defects:

(5)

In this ratio the first term describes increase of the
number of point defects during irradiation process;
the second term describes decrease of the number of
point defects present in graphite at the beginning of
irradiation. Thus, for sufficiently long time (about one
year) this ratio tends asymptotically. The dependence
of Wigner energy per weight unit of graphite
on irradiation temperature for Ö =10^{13} cm^{−2}s^{−1} , is defined by
the relation:

(6)

Then the problem is reduced to determining the
function, which approximates the dependence of
recombination constant on temperature λ (T ) .
Where in values E and T are known (experimental
data), the end result is shown in **Fig. 2**.

Exponential dependence λ (T ) can be excluded from
consideration. The dynamics of concentration of point
defects vs. irradiation time for different irradiation
temperatures and, accordingly, Wigner energy
vs. fluence at constant flux density of damaging
neutrons can be described by using polynomial and
power dependencies of recombination constant on irradiation temperature. **Fig. 3** shows calculated
dependence of Wigner energy on the fluence. The
dependence is obtained assuming that all neutrons
are damaging and irradiation conditions are
constant. Experimental dependence of store energy
on neutron fluence is shown in **Fig. 4**.

As seen from results, maximum deviation of the calculated value from the experimental one corresponds to temperature of 260°С which is quite close to the transition region (around 300°С). In this region, one would expect a significant deviation due to uncertainty of the experimental data on critical fluence vs. irradiation temperature dependence.

Thus, the dependence of recombination constant on graphite irradiation temperature at the conditions present in industrial graphite reactors is determined in this paper. To specify the type of functional dependence of critical fluence on irradiation temperature and attendant gamma-radiation flux density it is required to carry out further analysis to select specific function type describing recombination of point defects and transition of point defects into complex defects .

of lattice nodes’ concentration on irradiation temperature is in satisfactory agreement with the experimental data obtained by defining the value of stored energy in graphite blocks and reactor core bushings of graphite reactors as a function of irradiation temperature.

This work was performed on the unique scientific IRT-T equipment and financially supported by Government represented by the Ministry of Education and Science of the Russian Federation (RFMEFI59114X0001).

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