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Point defects created by the collisions of high-energy particles and their subsequent evolutions form the foundation for all observed irradiation effects. Qualitative analysis is performed for the local and global behaviors of the planar system of nonlinear ordinary differential equations for the point defects balance model. The results indicated that the evolution behavior for the in-pile process is qualitatively very similar to the more simple annealing process, but very different from the degenerated systems that possess analytical solutions. However, quantitatively, irradiations in the in-pile process will shift the stable node away from the defect free state and change the local behaviors. A too strong irradiation may result in a nonphysical stable node and produce amorphous states, thus making the model inadequate.

In irradiate environments, solid atoms can be displaced from their lattice sites and become interstitial atoms by the collisions of high-energy particles [

Experimental and theoretical studies have shown that the point defects evolution is mainly governed by three different mechanisms, the production rate, the recombination rate, and the annihilation rate [

By using the defect reaction rate theory, a planar system of nonlinear ordinary differential equations (ODEs), named the point defect balance equations (PDBEs), has been proposed by Lomer [

In this paper, the qualitative (geometrical) method of ODEs [

The Frenkel defects are generated by collisions cascade and can be lost through either recombination or reaction with defect sinks. So the defect concentration of vacancy and interstitial is the balance between (1) local production rate, (2) reaction with other species, and (3) diffusion into or out of the local volume. The balance equations of the irradiation-induced point defects are [

The parameters

Let

For simplicity, the superscript of

We will discuss the following two different processes separately.

(1) Annealing:

(2) In-pile irradiation:

First, we will discuss the properties of the singular points and the local behaviors near them during annealing process. In system (

If

The local behaviors near the two singular points are easily obtained as follows.

If

At

Both eigenvalues of

The phase portraits are shown in Figure

(a) Phase portraits near stable node

If

At

The eigenvalues of

The phase portraits are shown in Figure

(a) Phase portraits near saddle point

After introducing the source term, the singular points become as follows.

If

Compared with the annealing process, after introducing the source term, we know that the amount of the singular point has not been changed. But the position of the stable node

If

At

The trace and the determinant of

So we have

By definition, we have

If

Similarly, at

It is easy to find that

As shown in Figure

Phase portraits near (a) stable node

Next, we analyze the effects of defects production rate on the eigenvalues and eigenvectors. As shown in Figure

(a) Eigenvalue

(a) Eigenvalue

Under some extreme conditions, one or more effects in system (

(1) Sink-Free Limit (SFL):

(2) Low Temperature Limit (LTL):

(3) High Temperature Limit (HTL):

Compared to UD, the singular points degenerate to singular lines in SFL.

(i) If

If

If

(i) if

(ii) if

(iii) if

For SFL system, the balance equations (

Phase portraits near singular lines of system (

If only vacancy-sink reaction is negligible, the singular line is quite different compared to the previous system.

If

If

For LTL system, for vacancy-sink reactions are too small and negligible, we have

Phase portraits near singular lines or points of system (

At high temperatures, the recombination does not contribute much and could be neglected. System (

If

(i)

(ii)

(iii)

Before we discuss the global behaviors of system (

There is no closed trajectory of system (

At

Assuming there exists a closed trajectory crossing

The closed trajectory of system (

If

If

If

If

So there is no closed trajectory in

If

Phase plane portraits of system (

The phase plane portraits of SFL system are shown in Figure

Phase plane portraits of system (

The phase plane portraits of LTL system are shown in Figure

Phase plane portraits of system (

Phase plane portraits of HTL system are shown in Figure

Phase plane portraits of system (

The global phase portraits for annealing and in-pile processes cannot be easily constructed from the phase plane behaviors of degenerated systems.

Next, we analyze the behaviors at infinity of system (

If

At

The trace of (

For nonhyperbolic singular point

Letting

Inserting (

So, if

(1)

(2)

Transform

System (

If

Similarly, we can prove

(1)

(2)

So, if

For degenerate systems, we could get the following conclusions by the similar process.

If

If

The behaviors of global phase portraits on Poincaré disk are similar to the phase plane portraits (Figure

Global phase portraits on Poincaré disk of system (

The global phase portraits of degenerate system on Poincaré disk are shown in Figure

Global phase portraits on Poincaré disk of system (

From the definition of vacancy and interstitial, we know that

A singular point is admissible if it is inside

A trajectory starting from a point

A trajectory starting from a point

During the annealing process, the stable node

(i) The stable node

(ii) The trajectories are partly admissible if and only if

(iii) The trajectories are always admissible if and only if

Letting

we get admissible condition (

Assuming

And (

Then, the partly admissible condition (

As shown in Figure

(a) Admissible and (b) partly admissible phase plane portraits.

During annealing process, or if the point production rate is small during in-pile process, the stable node and trajectories are always admissible. If the defect production rate is large, the trajectories are partially admissible or inadmissible. The balance equation is no longer applicable.

For better illustration, the evolutions of point defect concentrations and irradiation enhanced diffusion coefficients of UD, SFL, and HTL system during in-pile process will be presented in the following.

The temperature and defects generation rate are taken as

Calculating parameters for ND, SFL, and HTL system.

| | | | | | |
---|---|---|---|---|---|---|

ND | | 0.0047 | 14.7 | | | |

SFL | | 0.0047 | 14.7 | 0 | | |

HTL | | 0.0047 | 14.7 | | | |

In the experiments, the defect concentrations are usually hard to be traced. But it is possible to use the Zener relaxation time [

As shown in Figure

The evolutions of (a) point defects concentration and (b) irradiation enhanced diffusion coefficient of UD, SFL, and HTL system.

Initially, the concentrations of point defects are too small and result in the small effect of recombination or reaction with sinks. So they increase linearly. With the increase of concentrations, the generation of the point defects is compensated by the recombination, corresponding to the quasi-steady state. After that, the interstitials start to find the sinks and annihilate, and the concentration decreases. At the same time, vacancy concentration goes on increasing for less recombination.

As shown in Figure

In this paper, the point defect balance equations are analyzed by the qualitative method of ordinary differential equations for the first time. The behaviors of the defects evolution during the annealing and the in-pile processes have been studied. We get the following conclusions:

(1) Two singular points exist for both annealing and in-pile processes. One is a stable node and the other is a saddle point. The stable node is at the origin for annealing such that the steady state is defect free. It is shifted to a defective state when the material is in-pile. Under very large irradiations, the shift can be too much such that the stable node goes outside the physically admissible region. Then, the material becomes amorphous and the PDBE model is no longer physically relevant.

(2) Local behavior near the stable node is affected by the irradiation as well. While the trajectories approach the origin with the two axes as the stable subspaces for annealing, the stable subspaces are rotated to be nonorthogonal under irradiations. Moreover, even when the stable node and the initial state are both inside the admissible region, it can happen that the trajectory goes outside for a while before entering again.

(3) No closed trajectory can exist and the global behavior is determined by the two singular points and the other six at infinity. The global phase portraits for annealing and in-pile processes are qualitatively very similar. But they are very different and cannot be easily constructed from the phase plane behaviors of degenerated systems.

The calculating data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (nos. 11272092, 11772094, and 11461161008) and the National Science and Technology Major Project of China (nos. 2016YFB0700103, 2017ZX06002006).