# Qualitative Analysis of the Evolution Behaviors of Irradiation-Induced Point Defects

Authors: Hao Huang; Shurong Ding; Yongzhong Huo
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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.


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## Body

## 1. Introduction

In irradiate environments, solid atoms can be displaced from their lattice sites and become interstitial atoms by the collisions of high-energy particles [1, 2]. The generated point defects (vacancies and interstitials, known as the Frenkel pairs) and their subsequent evolutions form the foundation for all observed irradiation effects on the physical and mechanical properties of materials. Notably irradiation-induced mechanical degradations include the volumetric swelling, creep instabilities, increase in hardness, loss of ductility, and also cracking [1, 3]. Thus, a proper understanding of their evolution behavior is at the central of the radiation damage.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 [4–12]. The rate of point defects production depends on the irradiation conditions and the material types. The irradiation generated pairs of vacancies and interstitials can get lost due to their recombination and the recombination rate is mainly determined by their diffusions. They can also be lost to sinks such as voids, dislocation loops, and grain boundaries. The annihilation rates depend on their diffusions and the sink concentrations and strengths [1, 13].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 [9] for the evolution of the spatially averaged point defects concentrations (cv(t), ci(t)) for materials under irradiations. The system is quadratic in cv and ci with a constant production rate and four additional material constants representing the recombination and the annihilation mechanisms, respectively. The system has been used widely as a standard model for the evolution behavior of the irradiation-induced point defects. Analytical solutions were obtained for several degenerated systems in which some mechanisms are neglected and the corresponding material constants are assumed null. Although such analytical solutions are very helpful, they cannot be directly extrapolated to the original nondegenerated (UD) system. Many numerical solutions have been obtained and can provide useful information [5, 14, 15]. However, a proper understanding of the solution behaviors of PDBEs and the effect of the three different mechanisms are still lacking.In this paper, the qualitative (geometrical) method of ODEs [16, 17] is utilized to study the solution behavior of PDBEs. After introducing the planar system, the singular points are obtained and their local behaviors are discussed in the next section. In Section 3, the global behavior is analyzed in the phase plane. Considering the behavior at infinity, the global phase portraits are also obtained on the Poincaré disk. Possible physical admissible conditions are discussed in Section 4. A numerical example is also presented before the conclusion to illustrate the difference between degenerated and nondegenerated systems.


## 2. Balance Equations and Local Behaviors Near Singular Points

### 2.1. Point Defects Balance Equations (PDBEs)

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 [1, 5](1)dcvdt=K0-Kivcvci-Kvscscvdcidt=K0-Kivcvci-Kiscsciwhere(1)
cv,i is the atom fraction of vacancy/interstitial, or vacancy/interstitial concentration;(2)
K0 is the defect production rate by radiation;(3)
Kiv is the recombination rate coefficient of vacancy and interstitial;(4)
Kv,is is the vacancy/interstitial-sink reaction rate coefficients;(5)
cs is the sink density.The parametersK0, Kiv, Kv,is, and cs in (1) are nonnegative constants. And we have Kis>Kvs for sinks absorb interstitials at faster rates than vacancies.Letx=cv and y=ci; if Kiv>0, system (1) becomes(2)dxdt=-xy-ax+c=P~x,ydydt=-xy-by+c=Q~x,ywhere(3)a=KvscsKiv,b=KiscsKiv,c=K0Kiv,t~=KivtFor simplicity, the superscript oft~, P~, and Q~ in the following contents will be neglected.We will discuss the following two different processes separately.(1) Annealing:K0=0, i.e., c=0, no point defect source. Thus, the defects could only annihilate by recombination or in sinks.(2) In-pile irradiation:K0>0, i.e., c>0. Thus, the point defects are generated as the material is exposed in irradiate environments.


### 2.2. Singular Points and Local Behaviors

#### 2.2.1. Annealing:c=0

First, we will discuss the properties of the singular points and the local behaviors near them during annealing process. In system (2), by letting P(x,y)=Q(x,y)=0, we can get the singular points.Theorem 1.
Ifa,b>0 and c=0, system (2) has two singular points(4)x1,y1=0,0,x2,y2=-b,-aThe local behaviors near the two singular points are easily obtained as follows.Theorem 2.
Ifa,b>0 and c=0, the singular point (x1,y1)=(0,0) (4) is a stable node. And the local behaviors of system (2) near the stable node are(5)xtyt=exp⁡-at00exp⁡-btx0y0and the trajectories satisfy(6)x-bya=Const.Proof.
Atx1,y1=0,0, the Jacobi matrix of system (2) is(7)Jx1,y1=-a00b
Both eigenvalues ofJx1,y1 are negative real number and (x1,y1)=(0,0) is a stable node. The local behaviors could be easily obtained.The phase portraits are shown in Figure1(a); the decrease of y is much faster than x, because sinks absorb interstitials at faster rates than vacancies, i.e., b>a>0. Besides, as shown in Figure 1(b), x/x0 versus y/y0 under the different value of δ=a/b, it is easy to find that the smaller δ leads to the steeper curve for the faster decrease of y.Figure 1
(a) Phase portraits near stable node(0,0) and (b) y/y0 versus x/x0 under different δ=a/b if c=0.
(a) (b)Theorem 3.
Ifa,b>0 and c=0, the singular point (x2,y2)=(-b,-a) (4) is a saddle point. And the local behaviors of system (2) near the saddle point are(8)xt-x2yt-y2=P2exp⁡abt00exp⁡-abtP2-1x0-x2y0-y2where(9)P2=1-1ababand the trajectories satisfy(10)ax+b2-by+a2=Const.Proof.
At(x2,y2)=(-b,-a), the Jacobi matrix of system (2) is(11)Jx1,y1=0ba0
The eigenvalues ofJ(x2,y2) are λ1,2=±ab: one is a positive and the other is a negative real number, so (x2,y2) is a saddle point. And the local behaviors could be easily obtained.The phase portraits are shown in Figure2(a). For the same constant, the trajectories of different δ=a/b in the first quadrant are shown in Figure 2(b). It could be found that the larger δ leads to the larger slope of the asymptotic line and the larger y for the same x.Figure 2
(a) Phase portraits near saddle point(x2,y2) and (b) local trajectories under different δ=a/b if c=0.
(a) (b)


#### 2.2.2. In-Pile Irradiation:c>0

After introducing the source term, the singular points become as follows.Theorem 4.
Ifa,b>0 and c>0, system (2) has two singular points(12)x1,2=-b21∓1+4cab,y1,2=-a21∓1+4cabCompared 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(x1,y1) is shifted to the first quadrant. And (x2,y2) is still in the third quadrant. The local behaviors near the singular points are as follows.Theorem 5.
Ifa,b>0 and c>0, the singular point (x1,y1) (12) of system (2) is a stable node in the first quadrant, and the local behaviors near the stable node are(13)xt-x1yt-y1=P1exp⁡λ1+t00exp⁡λ1-tP1-1x0-x1y0-y1where P1=v1+,v1- and the eigenvalues λ1± and v1± eigenvectors of the Jacobi matrix are(14)λ1±=-a+b41+1+4cab±Δ12(15)v1±==11+φ1±21φ1±where(16)Δ1=14a+b21+1+4cab2-16ab1+4cab(17)φ1±=b-a1+4c/ab+1∓2Δ12b1+4c/ab-1Proof.
At(x1,y1) (12), the Jacobi matrix of system (2) is(18)Jx1,y1=-a21+1+4cabb21-1+4caba21-1+4cab-b21+1+4cab
The trace and the determinant ofJ(x1,y1) are(19)T1=-a+b21+1+4cab(20)J1=ab1+4cab
So we haveT1<0, J1>0. And it is easy to find that Δ1≥0 for a+b2≥4ab and 1+1+4c/ab2≥41+4c/ab. The eigenvalues of Jacobi matrix are(21)λ1±=T1±Δ12
By definition, we haveT1±Δ1<0 and λ1±<0. So the singular point (x1,y1) is a stable node.Theorem 6.
Ifa,b>0 and c>0, the singular point (x2,y2) (12) of system (2) is a saddle point in the third quadrant. And the local behaviors near the saddle point are(22)xt-x2yt-y2=P2′exp⁡λ2+t00exp⁡λ2-tP2′-1x0-x2y0-y2where P2′=v2+,v2- and the eigenvalues λ2± and v2± eigenvectors of the Jacobi matrix are(23)λ2±=-a+b41-1+4cab±Δ22(24)v2±==11+φ2±21φ2±where(25)Δ2=14a+b21-1+4cab2+16ab1+4cab(26)φ2±=b-a1+4c/ab-1±2Δ22b1+4c/ab+1Proof.
Similarly, atx1,y1 (12), the eigenvalues of the Jacobi matrix are(27)λ2±=T2±Δ22where(28)T2=-a+b21-1+4cab
It is easy to find thatT2 and Δ2>0. Besides, T2+Δ2>0, T2-Δ2<0, and λ2+>0>λ2-. So the singular point (x2,y2) is a saddle point.As shown in Figure3, in the in-pile process, x or y will no longer approach or deviate the singular points along the coordinate axis. Because the eigenvectors are different from the annealing process, also, y decreases faster than x for sinks absorb interstitials at faster rates than vacancies. The trajectories tend to the eigenvector v1+ near the stable node (x1,y1). The trajectories tend to v2+ near the saddle point (x2,y2).Figure 3
Phase portraits near (a) stable node(x1,y1) and (b) saddle point (x2,y2) if c>0.
(a) (b)Next, we analyze the effects of defects production rate on the eigenvalues and eigenvectors. As shown in Figure4, with the increase of the defects production rate c, λ1+ will always decrease. But λ1- will decrease first and then remain almost constant for c large enough. Define θ1±=arctan(φ1±) as the angle between v1± and x axis. We found that, with the increase of c, θ1± will decrease first and then remain almost constant for c large enough. The effects of c on λ2± and φ2± are similar, as shown in Figure 5.Figure 4
(a) Eigenvalueλ1± and (b) angle between eigenvector v1± and x axis θ1± versus c.
(a) (b)Figure 5
(a) Eigenvalueλ2± and (b) angle between eigenvector v2± and x axis θ2± versus c.
(a) (b)


### 2.3. Some Degenerated Systems

Under some extreme conditions, one or more effects in system (2) are small enough and can be neglected. We get some degenerated systems.(1) Sink-Free Limit (SFL):cs=0, i.e., a=b=0, negligible defects-sink reactions. Thus, the point defects can only annihilate due to recombination and system (2) becomes symmetric.(2) Low Temperature Limit (LTL):Kvs≈0, i.e., a=0, negligible vacancy-sink reaction. Thus, the vacancies can only annihilate due to recombination.(3) High Temperature Limit (HTL):Kiv≈0. The defect annihilation rate at the sinks keeps the concentration of interstitial low, such that recombination does not contribute much.Compared to UD, the singular points degenerate to singular lines in SFL.Theorem 7.
(i) Ifa=b=c=0, the singular lines of system (2) are x=0 and y=0; ii) if a=b=0 and c>0, the singular lines of system (2) are xy=c.Theorem 8.
Ifa=b=0, the singular lines of system (2) are asymptotically stable only if x,y>0 and unstable if x,y<0. (x,y)=(0,0) is a saddle-node.Corollary 9.
Ifa=b=0, the trajectories of system (2) satisfy y=x-Δc0, where Δc0=x0-y0. And the solutions are as follows:
(i) ifc=0 and Δc0=x0-y0,(29)yt=xt=x01+x0t,
(ii) ifc=0 and Δc0≠x0-y0,(30)xt=12x0-y0-Δs+Δs1-x0+y0-Δsx0+y0+Δsexp⁡-Δst-1yt=xt+Δc0,where Δs=Δc02,
(iii) ifc>0, the solutions have the same form as (30), but Δs=Δc02+4c.For SFL system, the balance equations (2) are symmetric. Lines P=0 and Q=0 are the same, which is xy=c. The stable node (x1,y1) degrades to asymptotic stable part of the singular line xy=c(x>0), and the saddle-node (x2,y2) degrades to unstable part of xy=c(x<0). The local behaviors near the singular lines are shown in Figure 6.Figure 6
Phase portraits near singular lines of system (2) if (a) a=b=c=0 and (b) a=b=0,c>0.
(a) (b)If only vacancy-sink reaction is negligible, the singular line is quite different compared to the previous system.Theorem 10.
Ifa=0, b>0, the singular line of system (2) is y=0 if c=0. System (2) does not have any singular point if c>0.Corollary 11.
Ifa=0, b>0, and c=0, the singular line y=0 is asymptotically stable only if x>-b and unstable if x<-b. And the trajectories satisfy(31)y-x+bln⁡x=Const.For LTL system, for vacancy-sink reactions are too small and negligible, we havea=0 and b>0. During annealing process, the stable node (x1,y1) degrades to asymptotic stable part of the singular line y=0(x>-b), and the saddle-node (x2,y2) degrades to unstable part y=0(x<-b). In in-pile process, the stable node and the saddle-node could not be found. The local behaviors near the singular line are shown in Figure 7.Figure 7
Phase portraits near singular lines or points of system (2) if (a) a=0,b>0,c=0 and (b) a=0,b>0,c>0.
(a) (b)At high temperatures, the recombination does not contribute much and could be neglected. System (1) becomes a linear system, and we can easily get the following analytical solutions.Theorem 12.
IfKiv=0, and
(i)Kv,iscs>0, the only stable node is x∞,y∞=K0/Kvscs,K0/Kiscs and the solutions are(32)xt=x∞+x0-x∞exp⁡-Kvscstyt=y∞+y0-y∞exp⁡-Kiscst;the trajectories satisfy(33)x-x∞-by-y∞a=Const;
(ii)cs=0, there is no singular point; the solutions and the trajectories are(34)x=x0+K0t,y=y0+K0t,x-y=Const;
(iii)Kvs=0, Kis>0, and cs=0, the singular line is y=0 if K0=0 and there is no singular point if K0>0. The solutions are(35)xt=x0+K0tyt=y∞+y0-y∞exp⁡-Kiscst


## 2.1. Point Defects Balance Equations (PDBEs)

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 [1, 5](1)dcvdt=K0-Kivcvci-Kvscscvdcidt=K0-Kivcvci-Kiscsciwhere(1)
cv,i is the atom fraction of vacancy/interstitial, or vacancy/interstitial concentration;(2)
K0 is the defect production rate by radiation;(3)
Kiv is the recombination rate coefficient of vacancy and interstitial;(4)
Kv,is is the vacancy/interstitial-sink reaction rate coefficients;(5)
cs is the sink density.The parametersK0, Kiv, Kv,is, and cs in (1) are nonnegative constants. And we have Kis>Kvs for sinks absorb interstitials at faster rates than vacancies.Letx=cv and y=ci; if Kiv>0, system (1) becomes(2)dxdt=-xy-ax+c=P~x,ydydt=-xy-by+c=Q~x,ywhere(3)a=KvscsKiv,b=KiscsKiv,c=K0Kiv,t~=KivtFor simplicity, the superscript oft~, P~, and Q~ in the following contents will be neglected.We will discuss the following two different processes separately.(1) Annealing:K0=0, i.e., c=0, no point defect source. Thus, the defects could only annihilate by recombination or in sinks.(2) In-pile irradiation:K0>0, i.e., c>0. Thus, the point defects are generated as the material is exposed in irradiate environments.


## 2.2. Singular Points and Local Behaviors

### 2.2.1. Annealing:c=0

First, we will discuss the properties of the singular points and the local behaviors near them during annealing process. In system (2), by letting P(x,y)=Q(x,y)=0, we can get the singular points.Theorem 1.
Ifa,b>0 and c=0, system (2) has two singular points(4)x1,y1=0,0,x2,y2=-b,-aThe local behaviors near the two singular points are easily obtained as follows.Theorem 2.
Ifa,b>0 and c=0, the singular point (x1,y1)=(0,0) (4) is a stable node. And the local behaviors of system (2) near the stable node are(5)xtyt=exp⁡-at00exp⁡-btx0y0and the trajectories satisfy(6)x-bya=Const.Proof.
Atx1,y1=0,0, the Jacobi matrix of system (2) is(7)Jx1,y1=-a00b
Both eigenvalues ofJx1,y1 are negative real number and (x1,y1)=(0,0) is a stable node. The local behaviors could be easily obtained.The phase portraits are shown in Figure1(a); the decrease of y is much faster than x, because sinks absorb interstitials at faster rates than vacancies, i.e., b>a>0. Besides, as shown in Figure 1(b), x/x0 versus y/y0 under the different value of δ=a/b, it is easy to find that the smaller δ leads to the steeper curve for the faster decrease of y.Figure 1
(a) Phase portraits near stable node(0,0) and (b) y/y0 versus x/x0 under different δ=a/b if c=0.
(a) (b)Theorem 3.
Ifa,b>0 and c=0, the singular point (x2,y2)=(-b,-a) (4) is a saddle point. And the local behaviors of system (2) near the saddle point are(8)xt-x2yt-y2=P2exp⁡abt00exp⁡-abtP2-1x0-x2y0-y2where(9)P2=1-1ababand the trajectories satisfy(10)ax+b2-by+a2=Const.Proof.
At(x2,y2)=(-b,-a), the Jacobi matrix of system (2) is(11)Jx1,y1=0ba0
The eigenvalues ofJ(x2,y2) are λ1,2=±ab: one is a positive and the other is a negative real number, so (x2,y2) is a saddle point. And the local behaviors could be easily obtained.The phase portraits are shown in Figure2(a). For the same constant, the trajectories of different δ=a/b in the first quadrant are shown in Figure 2(b). It could be found that the larger δ leads to the larger slope of the asymptotic line and the larger y for the same x.Figure 2
(a) Phase portraits near saddle point(x2,y2) and (b) local trajectories under different δ=a/b if c=0.
(a) (b)


### 2.2.2. In-Pile Irradiation:c>0

After introducing the source term, the singular points become as follows.Theorem 4.
Ifa,b>0 and c>0, system (2) has two singular points(12)x1,2=-b21∓1+4cab,y1,2=-a21∓1+4cabCompared 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(x1,y1) is shifted to the first quadrant. And (x2,y2) is still in the third quadrant. The local behaviors near the singular points are as follows.Theorem 5.
Ifa,b>0 and c>0, the singular point (x1,y1) (12) of system (2) is a stable node in the first quadrant, and the local behaviors near the stable node are(13)xt-x1yt-y1=P1exp⁡λ1+t00exp⁡λ1-tP1-1x0-x1y0-y1where P1=v1+,v1- and the eigenvalues λ1± and v1± eigenvectors of the Jacobi matrix are(14)λ1±=-a+b41+1+4cab±Δ12(15)v1±==11+φ1±21φ1±where(16)Δ1=14a+b21+1+4cab2-16ab1+4cab(17)φ1±=b-a1+4c/ab+1∓2Δ12b1+4c/ab-1Proof.
At(x1,y1) (12), the Jacobi matrix of system (2) is(18)Jx1,y1=-a21+1+4cabb21-1+4caba21-1+4cab-b21+1+4cab
The trace and the determinant ofJ(x1,y1) are(19)T1=-a+b21+1+4cab(20)J1=ab1+4cab
So we haveT1<0, J1>0. And it is easy to find that Δ1≥0 for a+b2≥4ab and 1+1+4c/ab2≥41+4c/ab. The eigenvalues of Jacobi matrix are(21)λ1±=T1±Δ12
By definition, we haveT1±Δ1<0 and λ1±<0. So the singular point (x1,y1) is a stable node.Theorem 6.
Ifa,b>0 and c>0, the singular point (x2,y2) (12) of system (2) is a saddle point in the third quadrant. And the local behaviors near the saddle point are(22)xt-x2yt-y2=P2′exp⁡λ2+t00exp⁡λ2-tP2′-1x0-x2y0-y2where P2′=v2+,v2- and the eigenvalues λ2± and v2± eigenvectors of the Jacobi matrix are(23)λ2±=-a+b41-1+4cab±Δ22(24)v2±==11+φ2±21φ2±where(25)Δ2=14a+b21-1+4cab2+16ab1+4cab(26)φ2±=b-a1+4c/ab-1±2Δ22b1+4c/ab+1Proof.
Similarly, atx1,y1 (12), the eigenvalues of the Jacobi matrix are(27)λ2±=T2±Δ22where(28)T2=-a+b21-1+4cab
It is easy to find thatT2 and Δ2>0. Besides, T2+Δ2>0, T2-Δ2<0, and λ2+>0>λ2-. So the singular point (x2,y2) is a saddle point.As shown in Figure3, in the in-pile process, x or y will no longer approach or deviate the singular points along the coordinate axis. Because the eigenvectors are different from the annealing process, also, y decreases faster than x for sinks absorb interstitials at faster rates than vacancies. The trajectories tend to the eigenvector v1+ near the stable node (x1,y1). The trajectories tend to v2+ near the saddle point (x2,y2).Figure 3
Phase portraits near (a) stable node(x1,y1) and (b) saddle point (x2,y2) if c>0.
(a) (b)Next, we analyze the effects of defects production rate on the eigenvalues and eigenvectors. As shown in Figure4, with the increase of the defects production rate c, λ1+ will always decrease. But λ1- will decrease first and then remain almost constant for c large enough. Define θ1±=arctan(φ1±) as the angle between v1± and x axis. We found that, with the increase of c, θ1± will decrease first and then remain almost constant for c large enough. The effects of c on λ2± and φ2± are similar, as shown in Figure 5.Figure 4
(a) Eigenvalueλ1± and (b) angle between eigenvector v1± and x axis θ1± versus c.
(a) (b)Figure 5
(a) Eigenvalueλ2± and (b) angle between eigenvector v2± and x axis θ2± versus c.
(a) (b)


## 2.2.1. Annealing:c=0

First, we will discuss the properties of the singular points and the local behaviors near them during annealing process. In system (2), by letting P(x,y)=Q(x,y)=0, we can get the singular points.Theorem 1.
Ifa,b>0 and c=0, system (2) has two singular points(4)x1,y1=0,0,x2,y2=-b,-aThe local behaviors near the two singular points are easily obtained as follows.Theorem 2.
Ifa,b>0 and c=0, the singular point (x1,y1)=(0,0) (4) is a stable node. And the local behaviors of system (2) near the stable node are(5)xtyt=exp⁡-at00exp⁡-btx0y0and the trajectories satisfy(6)x-bya=Const.Proof.
Atx1,y1=0,0, the Jacobi matrix of system (2) is(7)Jx1,y1=-a00b
Both eigenvalues ofJx1,y1 are negative real number and (x1,y1)=(0,0) is a stable node. The local behaviors could be easily obtained.The phase portraits are shown in Figure1(a); the decrease of y is much faster than x, because sinks absorb interstitials at faster rates than vacancies, i.e., b>a>0. Besides, as shown in Figure 1(b), x/x0 versus y/y0 under the different value of δ=a/b, it is easy to find that the smaller δ leads to the steeper curve for the faster decrease of y.Figure 1
(a) Phase portraits near stable node(0,0) and (b) y/y0 versus x/x0 under different δ=a/b if c=0.
(a) (b)Theorem 3.
Ifa,b>0 and c=0, the singular point (x2,y2)=(-b,-a) (4) is a saddle point. And the local behaviors of system (2) near the saddle point are(8)xt-x2yt-y2=P2exp⁡abt00exp⁡-abtP2-1x0-x2y0-y2where(9)P2=1-1ababand the trajectories satisfy(10)ax+b2-by+a2=Const.Proof.
At(x2,y2)=(-b,-a), the Jacobi matrix of system (2) is(11)Jx1,y1=0ba0
The eigenvalues ofJ(x2,y2) are λ1,2=±ab: one is a positive and the other is a negative real number, so (x2,y2) is a saddle point. And the local behaviors could be easily obtained.The phase portraits are shown in Figure2(a). For the same constant, the trajectories of different δ=a/b in the first quadrant are shown in Figure 2(b). It could be found that the larger δ leads to the larger slope of the asymptotic line and the larger y for the same x.Figure 2
(a) Phase portraits near saddle point(x2,y2) and (b) local trajectories under different δ=a/b if c=0.
(a) (b)


## 2.2.2. In-Pile Irradiation:c>0

After introducing the source term, the singular points become as follows.Theorem 4.
Ifa,b>0 and c>0, system (2) has two singular points(12)x1,2=-b21∓1+4cab,y1,2=-a21∓1+4cabCompared 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(x1,y1) is shifted to the first quadrant. And (x2,y2) is still in the third quadrant. The local behaviors near the singular points are as follows.Theorem 5.
Ifa,b>0 and c>0, the singular point (x1,y1) (12) of system (2) is a stable node in the first quadrant, and the local behaviors near the stable node are(13)xt-x1yt-y1=P1exp⁡λ1+t00exp⁡λ1-tP1-1x0-x1y0-y1where P1=v1+,v1- and the eigenvalues λ1± and v1± eigenvectors of the Jacobi matrix are(14)λ1±=-a+b41+1+4cab±Δ12(15)v1±==11+φ1±21φ1±where(16)Δ1=14a+b21+1+4cab2-16ab1+4cab(17)φ1±=b-a1+4c/ab+1∓2Δ12b1+4c/ab-1Proof.
At(x1,y1) (12), the Jacobi matrix of system (2) is(18)Jx1,y1=-a21+1+4cabb21-1+4caba21-1+4cab-b21+1+4cab
The trace and the determinant ofJ(x1,y1) are(19)T1=-a+b21+1+4cab(20)J1=ab1+4cab
So we haveT1<0, J1>0. And it is easy to find that Δ1≥0 for a+b2≥4ab and 1+1+4c/ab2≥41+4c/ab. The eigenvalues of Jacobi matrix are(21)λ1±=T1±Δ12
By definition, we haveT1±Δ1<0 and λ1±<0. So the singular point (x1,y1) is a stable node.Theorem 6.
Ifa,b>0 and c>0, the singular point (x2,y2) (12) of system (2) is a saddle point in the third quadrant. And the local behaviors near the saddle point are(22)xt-x2yt-y2=P2′exp⁡λ2+t00exp⁡λ2-tP2′-1x0-x2y0-y2where P2′=v2+,v2- and the eigenvalues λ2± and v2± eigenvectors of the Jacobi matrix are(23)λ2±=-a+b41-1+4cab±Δ22(24)v2±==11+φ2±21φ2±where(25)Δ2=14a+b21-1+4cab2+16ab1+4cab(26)φ2±=b-a1+4c/ab-1±2Δ22b1+4c/ab+1Proof.
Similarly, atx1,y1 (12), the eigenvalues of the Jacobi matrix are(27)λ2±=T2±Δ22where(28)T2=-a+b21-1+4cab
It is easy to find thatT2 and Δ2>0. Besides, T2+Δ2>0, T2-Δ2<0, and λ2+>0>λ2-. So the singular point (x2,y2) is a saddle point.As shown in Figure3, in the in-pile process, x or y will no longer approach or deviate the singular points along the coordinate axis. Because the eigenvectors are different from the annealing process, also, y decreases faster than x for sinks absorb interstitials at faster rates than vacancies. The trajectories tend to the eigenvector v1+ near the stable node (x1,y1). The trajectories tend to v2+ near the saddle point (x2,y2).Figure 3
Phase portraits near (a) stable node(x1,y1) and (b) saddle point (x2,y2) if c>0.
(a) (b)Next, we analyze the effects of defects production rate on the eigenvalues and eigenvectors. As shown in Figure4, with the increase of the defects production rate c, λ1+ will always decrease. But λ1- will decrease first and then remain almost constant for c large enough. Define θ1±=arctan(φ1±) as the angle between v1± and x axis. We found that, with the increase of c, θ1± will decrease first and then remain almost constant for c large enough. The effects of c on λ2± and φ2± are similar, as shown in Figure 5.Figure 4
(a) Eigenvalueλ1± and (b) angle between eigenvector v1± and x axis θ1± versus c.
(a) (b)Figure 5
(a) Eigenvalueλ2± and (b) angle between eigenvector v2± and x axis θ2± versus c.
(a) (b)


## 2.3. Some Degenerated Systems

Under some extreme conditions, one or more effects in system (2) are small enough and can be neglected. We get some degenerated systems.(1) Sink-Free Limit (SFL):cs=0, i.e., a=b=0, negligible defects-sink reactions. Thus, the point defects can only annihilate due to recombination and system (2) becomes symmetric.(2) Low Temperature Limit (LTL):Kvs≈0, i.e., a=0, negligible vacancy-sink reaction. Thus, the vacancies can only annihilate due to recombination.(3) High Temperature Limit (HTL):Kiv≈0. The defect annihilation rate at the sinks keeps the concentration of interstitial low, such that recombination does not contribute much.Compared to UD, the singular points degenerate to singular lines in SFL.Theorem 7.
(i) Ifa=b=c=0, the singular lines of system (2) are x=0 and y=0; ii) if a=b=0 and c>0, the singular lines of system (2) are xy=c.Theorem 8.
Ifa=b=0, the singular lines of system (2) are asymptotically stable only if x,y>0 and unstable if x,y<0. (x,y)=(0,0) is a saddle-node.Corollary 9.
Ifa=b=0, the trajectories of system (2) satisfy y=x-Δc0, where Δc0=x0-y0. And the solutions are as follows:
(i) ifc=0 and Δc0=x0-y0,(29)yt=xt=x01+x0t,
(ii) ifc=0 and Δc0≠x0-y0,(30)xt=12x0-y0-Δs+Δs1-x0+y0-Δsx0+y0+Δsexp⁡-Δst-1yt=xt+Δc0,where Δs=Δc02,
(iii) ifc>0, the solutions have the same form as (30), but Δs=Δc02+4c.For SFL system, the balance equations (2) are symmetric. Lines P=0 and Q=0 are the same, which is xy=c. The stable node (x1,y1) degrades to asymptotic stable part of the singular line xy=c(x>0), and the saddle-node (x2,y2) degrades to unstable part of xy=c(x<0). The local behaviors near the singular lines are shown in Figure 6.Figure 6
Phase portraits near singular lines of system (2) if (a) a=b=c=0 and (b) a=b=0,c>0.
(a) (b)If only vacancy-sink reaction is negligible, the singular line is quite different compared to the previous system.Theorem 10.
Ifa=0, b>0, the singular line of system (2) is y=0 if c=0. System (2) does not have any singular point if c>0.Corollary 11.
Ifa=0, b>0, and c=0, the singular line y=0 is asymptotically stable only if x>-b and unstable if x<-b. And the trajectories satisfy(31)y-x+bln⁡x=Const.For LTL system, for vacancy-sink reactions are too small and negligible, we havea=0 and b>0. During annealing process, the stable node (x1,y1) degrades to asymptotic stable part of the singular line y=0(x>-b), and the saddle-node (x2,y2) degrades to unstable part y=0(x<-b). In in-pile process, the stable node and the saddle-node could not be found. The local behaviors near the singular line are shown in Figure 7.Figure 7
Phase portraits near singular lines or points of system (2) if (a) a=0,b>0,c=0 and (b) a=0,b>0,c>0.
(a) (b)At high temperatures, the recombination does not contribute much and could be neglected. System (1) becomes a linear system, and we can easily get the following analytical solutions.Theorem 12.
IfKiv=0, and
(i)Kv,iscs>0, the only stable node is x∞,y∞=K0/Kvscs,K0/Kiscs and the solutions are(32)xt=x∞+x0-x∞exp⁡-Kvscstyt=y∞+y0-y∞exp⁡-Kiscst;the trajectories satisfy(33)x-x∞-by-y∞a=Const;
(ii)cs=0, there is no singular point; the solutions and the trajectories are(34)x=x0+K0t,y=y0+K0t,x-y=Const;
(iii)Kvs=0, Kis>0, and cs=0, the singular line is y=0 if K0=0 and there is no singular point if K0>0. The solutions are(35)xt=x0+K0tyt=y∞+y0-y∞exp⁡-Kiscst


## 3. Global Behaviors and Phase Plane Portraits

### 3.1. Nonexistence of Closed Trajectories

Before we discuss the global behaviors of system (2), we should prove the nonexistence of closed trajectory first.Lemma 13.
There is no closed trajectory of system (2) that can cross the x or y axis.Proof.
Atx axis, y≡0, we have y˙=c≥0 in system (2). And y always increases if c>0 and unchanged if c=0.
Assuming there exists a closed trajectory crossingx axis, y must decrease at one point and increase at another point, and y˙=c≥0 is no longer satisfied, such that there is no closed trajectory crossing x axis. Similarly, there is no closed trajectory crossing y axis.Corollary 14.
The closed trajectory of system (2) must be completely inside one of the quadrants of x-y plane; i.e.,(36)Ω1=x>0,y>0,Ω2=x<0,y>0Ω3=x<0,y<0,Ω4=x>0,y<0Theorem 15.
Ifc=0 and a,b>0, system (2) has no closed trajectory.Proof.
Ifc=0, the stable node is x1,y1=0,0 and the saddle point is x2,y2=-b,-a∈Ω3. By Lemma 13, the closed trajectory could only exist inside the quadrant which contains singular point. Besides, the singular point in Ω3 is a saddle point, so system (2) has no closed trajectory.Theorem 16.
Ifc>0 and a,b>0, system (2) has no closed trajectory.Proof.
Ifc>0, the stable node (x1,y1)∈Ω1 and the saddle point (x2,y2)∈Ω3. There is no closed trajectory in Ω2, Ω3, and Ω4. Inside Ω1, we have(37)∂P∂x+∂Q∂y=-a+b+x+y<0
So there is no closed trajectory inΩ1.


### 3.2. Phase Plane Portraits

Ifa,b>0, the phase plane portraits of system (2) are shown in Figure 8. For annealing (c=0) and in-pile (c>0) process, the portraits are qualitative similar. Only if x0 and y0 are small enough (lower than the asymptotic line L1), the trajectories will approach to negative infinity. Others will approach to the stable node (x1,y1) (12).Figure 8
Phase plane portraits of system (2) with a,b>0 and (a) c=0 and (b) c>0.
(a) (b)The phase plane portraits of SFL system are shown in Figure9. During annealing process, as shown in Figure 9(a), in the first quadrant, if x0>y0, the trajectories evolve to x axis and only vacancies are left at last. Otherwise if x0<y0, only interstitials are left at last. Only if x0=y0, the system becomes defect free in the end. During the in-pile process, for the introducing of source term, both defects always exist at last.Figure 9
Phase plane portraits of system (2) with a=b=0 and (a) c=0 and (b) c>0.
(a) (b)The phase plane portraits of LTL system are shown in Figure10. During annealing process, in the first quadrant, if x0>0, the trajectories evolve to axis and only vacancies are left at last. And if x0=0, the trajectories will evolve to origin, as shown in Figure 10(a). But during in-pile process, for there is no singular point, we could not find any stable phenomenon in the finite condition, and more information will be presented in the next section.Figure 10
Phase plane portraits of system (2) with a=0,b>0 and (a) c=0 and (b) c>0.
(a) (b)Phase plane portraits of HTL system are shown in Figure11. If Kv,iscs>0, the stable node is (x∞,y∞), so any trajectory will approach (x∞,y∞) when t→∞.Figure 11
Phase plane portraits of system (1) if Kiv=0 and (a) Kv,iscs>0, (b) Kv,iscs=0,K0>0, (c) Kvscs=0,Kiscs>0,K0=0, and (d) Kvscs=0,Kiscs>0,K0>0.
(a) (b) (c) (d)The global phase portraits for annealing and in-pile processes cannot be easily constructed from the phase plane behaviors of degenerated systems.


### 3.3. Global Phase Trajectory on Poincaré Disk

Next, we analyze the behaviors at infinity of system (2). First, transform x and y in Poincaré and time(38)u=yx,z=1x,dt=zdτSystem (2) becomes(39)dudτ=z2Q1z,uz-uz2P1z,uz=uu-1+au-buz+c1-uz2dzdτ=-z3P1z,uz=zu+az-cz2Let z=0 and the singular points of system (39) are(40)C0,0andB1,0Theorem 17.
Ifa,b>0, the singular point of system (39) C0,0 is a saddle-node. And B1,0 is an unstable node.Proof.
AtC(0,0), the Jacobi matrix of (39) is(41)J0,0=-1000
The trace of (41) is T=-1<0 and the determinant J=0, so C(0,0) is nonhyperbolic singular point. Similarly, at B(1.0), the trace is T=2>0 and the determinant J=1>0, so B(1,0) is an unstable node.
For nonhyperbolic singular pointC(0,0), system (39) can be written as(42a)u˙=uu-1+au-buz+c1-uz2=-u+Ψu,z(42b)z˙=zu+az-cz2=φu,z
Lettingu˙=0, we have(43)u=c+b-a28+oz2
Inserting (43) into (42b), we have the expansion(44)φu,z=az2+b-a28z3+oz4
So, if
(1)a≠0, the minimal order of x in φ is m=2 and C(0,0) is a saddle-node;
(2)a=0 and b≠0, it has m=3 and C(0,0) is an unstable node.Transformx and y in Poincaré and time:(45)v=xy,z=1y,dt=zdτSystem (2) becomes(46)dvdτ=vv-1+a-buuz+c1-vz2dzdτ=zv+bz-cz2Theorem 18.
Ifa,b>0, the singular point of system (46) D(0,0) is a saddle-node.Proof.
Similarly, we can proveD(0,0) is nonhyperbolic singular point and when
(1)b≠0, singular point D(0,0) is a saddle-node;
(2)b=0 and a≠0, singular point D(0,0) is an unstable node.
So, ifa,b>0, D(0,0) is a saddle-node.For degenerate systems, we could get the following conclusions by the similar process.Theorem 19.
Ifa=b=0, the singular point C(0,0) is a nonisolated point and B(1,0) is an unstable node and D(0,0) is a nonisolated point.Theorem 20.
Ifa=0,b>0, the singular point B1,0 is an unstable node and D0,0 is a saddle-node. C0,0 is a nonisolated point if c=0 and is a saddle-node if c>0.The behaviors of global phase portraits on Poincaré disk are similar to the phase plane portraits (Figure8). As for the infinity, we found that B1, C2, and D2 are unstable nodes. B2 and (x1,y1) are stable node, and C1, D1, and (x2,y2) are saddle point, as shown in Figure 12.Figure 12
Global phase portraits on Poincaré disk of system (2) if (a) a,b>0,c=0 and (b) a,b>0,c>0.
(a) (b)The global phase portraits of degenerate system on Poincaré disk are shown in Figure13. For SFL system, the singular points degrade to singular line xy=c, such that C(0,0) and D(0,0) change to the nonisolated points. But the behaviors at B(1,0) are unchanged. For LTL system, the behaviors at C(0,0) and B(1,0) are unchanged. But C(0,0) becomes a nonisolated point if c=0, for singular points become singular line y=0. And C(0,0) becomes a saddle-node if c>0 for (x1,y1) and (x1,y1) shift to (+∞,0) and (-∞,0), respectively.Figure 13
Global phase portraits on Poincaré disk of system (2) if (a) a=b=c=0, (b) a=b=0, c>0, (c) a=0,b>0,c=0, and (d) a=0,b>0,c>0.
(a) (b) (c) (d)


## 3.1. Nonexistence of Closed Trajectories

Before we discuss the global behaviors of system (2), we should prove the nonexistence of closed trajectory first.Lemma 13.
There is no closed trajectory of system (2) that can cross the x or y axis.Proof.
Atx axis, y≡0, we have y˙=c≥0 in system (2). And y always increases if c>0 and unchanged if c=0.
Assuming there exists a closed trajectory crossingx axis, y must decrease at one point and increase at another point, and y˙=c≥0 is no longer satisfied, such that there is no closed trajectory crossing x axis. Similarly, there is no closed trajectory crossing y axis.Corollary 14.
The closed trajectory of system (2) must be completely inside one of the quadrants of x-y plane; i.e.,(36)Ω1=x>0,y>0,Ω2=x<0,y>0Ω3=x<0,y<0,Ω4=x>0,y<0Theorem 15.
Ifc=0 and a,b>0, system (2) has no closed trajectory.Proof.
Ifc=0, the stable node is x1,y1=0,0 and the saddle point is x2,y2=-b,-a∈Ω3. By Lemma 13, the closed trajectory could only exist inside the quadrant which contains singular point. Besides, the singular point in Ω3 is a saddle point, so system (2) has no closed trajectory.Theorem 16.
Ifc>0 and a,b>0, system (2) has no closed trajectory.Proof.
Ifc>0, the stable node (x1,y1)∈Ω1 and the saddle point (x2,y2)∈Ω3. There is no closed trajectory in Ω2, Ω3, and Ω4. Inside Ω1, we have(37)∂P∂x+∂Q∂y=-a+b+x+y<0
So there is no closed trajectory inΩ1.


## 3.2. Phase Plane Portraits

Ifa,b>0, the phase plane portraits of system (2) are shown in Figure 8. For annealing (c=0) and in-pile (c>0) process, the portraits are qualitative similar. Only if x0 and y0 are small enough (lower than the asymptotic line L1), the trajectories will approach to negative infinity. Others will approach to the stable node (x1,y1) (12).Figure 8
Phase plane portraits of system (2) with a,b>0 and (a) c=0 and (b) c>0.
(a) (b)The phase plane portraits of SFL system are shown in Figure9. During annealing process, as shown in Figure 9(a), in the first quadrant, if x0>y0, the trajectories evolve to x axis and only vacancies are left at last. Otherwise if x0<y0, only interstitials are left at last. Only if x0=y0, the system becomes defect free in the end. During the in-pile process, for the introducing of source term, both defects always exist at last.Figure 9
Phase plane portraits of system (2) with a=b=0 and (a) c=0 and (b) c>0.
(a) (b)The phase plane portraits of LTL system are shown in Figure10. During annealing process, in the first quadrant, if x0>0, the trajectories evolve to axis and only vacancies are left at last. And if x0=0, the trajectories will evolve to origin, as shown in Figure 10(a). But during in-pile process, for there is no singular point, we could not find any stable phenomenon in the finite condition, and more information will be presented in the next section.Figure 10
Phase plane portraits of system (2) with a=0,b>0 and (a) c=0 and (b) c>0.
(a) (b)Phase plane portraits of HTL system are shown in Figure11. If Kv,iscs>0, the stable node is (x∞,y∞), so any trajectory will approach (x∞,y∞) when t→∞.Figure 11
Phase plane portraits of system (1) if Kiv=0 and (a) Kv,iscs>0, (b) Kv,iscs=0,K0>0, (c) Kvscs=0,Kiscs>0,K0=0, and (d) Kvscs=0,Kiscs>0,K0>0.
(a) (b) (c) (d)The global phase portraits for annealing and in-pile processes cannot be easily constructed from the phase plane behaviors of degenerated systems.


## 3.3. Global Phase Trajectory on Poincaré Disk

Next, we analyze the behaviors at infinity of system (2). First, transform x and y in Poincaré and time(38)u=yx,z=1x,dt=zdτSystem (2) becomes(39)dudτ=z2Q1z,uz-uz2P1z,uz=uu-1+au-buz+c1-uz2dzdτ=-z3P1z,uz=zu+az-cz2Let z=0 and the singular points of system (39) are(40)C0,0andB1,0Theorem 17.
Ifa,b>0, the singular point of system (39) C0,0 is a saddle-node. And B1,0 is an unstable node.Proof.
AtC(0,0), the Jacobi matrix of (39) is(41)J0,0=-1000
The trace of (41) is T=-1<0 and the determinant J=0, so C(0,0) is nonhyperbolic singular point. Similarly, at B(1.0), the trace is T=2>0 and the determinant J=1>0, so B(1,0) is an unstable node.
For nonhyperbolic singular pointC(0,0), system (39) can be written as(42a)u˙=uu-1+au-buz+c1-uz2=-u+Ψu,z(42b)z˙=zu+az-cz2=φu,z
Lettingu˙=0, we have(43)u=c+b-a28+oz2
Inserting (43) into (42b), we have the expansion(44)φu,z=az2+b-a28z3+oz4
So, if
(1)a≠0, the minimal order of x in φ is m=2 and C(0,0) is a saddle-node;
(2)a=0 and b≠0, it has m=3 and C(0,0) is an unstable node.Transformx and y in Poincaré and time:(45)v=xy,z=1y,dt=zdτSystem (2) becomes(46)dvdτ=vv-1+a-buuz+c1-vz2dzdτ=zv+bz-cz2Theorem 18.
Ifa,b>0, the singular point of system (46) D(0,0) is a saddle-node.Proof.
Similarly, we can proveD(0,0) is nonhyperbolic singular point and when
(1)b≠0, singular point D(0,0) is a saddle-node;
(2)b=0 and a≠0, singular point D(0,0) is an unstable node.
So, ifa,b>0, D(0,0) is a saddle-node.For degenerate systems, we could get the following conclusions by the similar process.Theorem 19.
Ifa=b=0, the singular point C(0,0) is a nonisolated point and B(1,0) is an unstable node and D(0,0) is a nonisolated point.Theorem 20.
Ifa=0,b>0, the singular point B1,0 is an unstable node and D0,0 is a saddle-node. C0,0 is a nonisolated point if c=0 and is a saddle-node if c>0.The behaviors of global phase portraits on Poincaré disk are similar to the phase plane portraits (Figure8). As for the infinity, we found that B1, C2, and D2 are unstable nodes. B2 and (x1,y1) are stable node, and C1, D1, and (x2,y2) are saddle point, as shown in Figure 12.Figure 12
Global phase portraits on Poincaré disk of system (2) if (a) a,b>0,c=0 and (b) a,b>0,c>0.
(a) (b)The global phase portraits of degenerate system on Poincaré disk are shown in Figure13. For SFL system, the singular points degrade to singular line xy=c, such that C(0,0) and D(0,0) change to the nonisolated points. But the behaviors at B(1,0) are unchanged. For LTL system, the behaviors at C(0,0) and B(1,0) are unchanged. But C(0,0) becomes a nonisolated point if c=0, for singular points become singular line y=0. And C(0,0) becomes a saddle-node if c>0 for (x1,y1) and (x1,y1) shift to (+∞,0) and (-∞,0), respectively.Figure 13
Global phase portraits on Poincaré disk of system (2) if (a) a=b=c=0, (b) a=b=0, c>0, (c) a=0,b>0,c=0, and (d) a=0,b>0,c>0.
(a) (b) (c) (d)


## 4. Admissibility and Numerical Examples

From the definition of vacancy and interstitial, we know thatcv,i=0 represents the perfect atom lattice and cv,i=1 means that the lattice is occupied by vacancy or interstitial completely. Thus, for system (2), only if 0≤x(t),y(t)≤1, the concentration could be thought as admissible. So we define the admissible zone(47)Ωad=x,y∣0≤x≤1,0≤y≤1=0,1×0,1Definition 21.
A singular point is admissible if it is insideΩad.Definition 22.
A trajectory starting from a pointx0,y0∈Ωad is admissible if xt,yt∈Ωad for any t>0.Definition 23.
A trajectory starting from a pointx0,y0∈Ωad is partly admissible if xt,yt∉Ωad for sufficiently large t.


### 4.1. Admissibility of Singular Points and Trajectories

During the annealing process, the stable nodex1,y1=0,0 and any trajectory starting from x0,y0∈Ωad are always inside Ωad. Thus, they are always admissible. However, during the in-pile process, the stable singular point can be shifted outside the admissible zone, and we have the following.Theorem 24.
(i) The stable node(x1,y1) (12) is admissible if and only if(48)c≤min⁡b+ba,a+ab
(ii) The trajectories are partly admissible if and only if(49)min⁡a,b<c≤min⁡b+ba,a+ab
(iii) The trajectories are always admissible if and only if(50)c≤min⁡a,bProof.
Letting(51)0≤x1=-b21-1+4cab≤10≤y1=-a21-1+4cab≤1
we get admissible condition (48) for the stable node.
Assuminga<b, only if line P=0 is negative at x=1, the trajectories are admissible; that is,(52)yx=1=-ax+cx=-a+c≤0
And (52) can be expand to (50) after adding case a>b.
Then, the partly admissible condition (49) is the difference set of (48) and (50).As shown in Figure14(a), if the parameters adopted satisfy the admissible condition (50), any trajectory starting from (x0,y0)∈Ωad is always inside Ωad for any t>0. But if the parameters adopted satisfy the partly admissible condition (49), as shown in Figure 14(b), even though every trajectory reaches (x1,y1) at last, we could clearly find that some trajectories starting inside Ωad can go outside during the process.Figure 14
(a) Admissible and (b) partly admissible phase plane portraits.
(a) (b)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.


### 4.2. Numerical Examples for Defects Evolutions

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 asT=550K, K0=1×10-3s-1. According to the literature [4], the parameters for calculation are shown in Table 1.Table 1
Calculating parameters for ND, SFL, and HTL system.
K v i s - 1 K v s s - 1 K i s s - 1 c s c v s c i s ND 1.47 × 1 0 11 0.0047 14.7 1 × 1 0 5 4.6268 × 1 0 - 6 1.4701 × 1 0 - 9 SFL 1.47 × 1 0 11 0.0047 14.7 0 0.25 × 1 0 - 8 0.25 × 1 0 - 8 HTL 0 0.0047 14.7 1 × 1 0 5 0.2141 6.8 × 1 0 - 5In the experiments, the defect concentrations are usually hard to be traced. But it is possible to use the Zener relaxation time [5, 7, 18] to study the diffusion processes during irradiation. Zener relaxation could be also noted as the enhanced diffusion coefficient about [5, 19](53)Drad=Dvcv+DiciAs shown in Figure15(a), the evolutions of vacancy and interstitial of UD system are quite different from the degenerate systems. Initially, both concentrations increase linearly following K0t. Then, the curve has a platform known as quasi-steady state [1], which is exactly the stable state of the SFL system. After that, the vacancy concentration goes on increasing but interstitial decreases. The value of cv is much larger than ci at the stable node. The stable concentrations of HTL system are larger than others.Figure 15
The evolutions of (a) point defects concentration and (b) irradiation enhanced diffusion coefficient of UD, SFL, and HTL system.
(a) (b)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 Figure15(b), it is easy to find that the evolutions of Drad are similar to the evolutions of interstitial concentration. Dici determines the Drad most of time, but the contributions of vacancy and interstitial are the same at last. The results qualitatively agree to the experimental observations [5].


## 4.1. Admissibility of Singular Points and Trajectories

During the annealing process, the stable nodex1,y1=0,0 and any trajectory starting from x0,y0∈Ωad are always inside Ωad. Thus, they are always admissible. However, during the in-pile process, the stable singular point can be shifted outside the admissible zone, and we have the following.Theorem 24.
(i) The stable node(x1,y1) (12) is admissible if and only if(48)c≤min⁡b+ba,a+ab
(ii) The trajectories are partly admissible if and only if(49)min⁡a,b<c≤min⁡b+ba,a+ab
(iii) The trajectories are always admissible if and only if(50)c≤min⁡a,bProof.
Letting(51)0≤x1=-b21-1+4cab≤10≤y1=-a21-1+4cab≤1
we get admissible condition (48) for the stable node.
Assuminga<b, only if line P=0 is negative at x=1, the trajectories are admissible; that is,(52)yx=1=-ax+cx=-a+c≤0
And (52) can be expand to (50) after adding case a>b.
Then, the partly admissible condition (49) is the difference set of (48) and (50).As shown in Figure14(a), if the parameters adopted satisfy the admissible condition (50), any trajectory starting from (x0,y0)∈Ωad is always inside Ωad for any t>0. But if the parameters adopted satisfy the partly admissible condition (49), as shown in Figure 14(b), even though every trajectory reaches (x1,y1) at last, we could clearly find that some trajectories starting inside Ωad can go outside during the process.Figure 14
(a) Admissible and (b) partly admissible phase plane portraits.
(a) (b)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.


## 4.2. Numerical Examples for Defects Evolutions

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 asT=550K, K0=1×10-3s-1. According to the literature [4], the parameters for calculation are shown in Table 1.Table 1
Calculating parameters for ND, SFL, and HTL system.
K v i s - 1 K v s s - 1 K i s s - 1 c s c v s c i s ND 1.47 × 1 0 11 0.0047 14.7 1 × 1 0 5 4.6268 × 1 0 - 6 1.4701 × 1 0 - 9 SFL 1.47 × 1 0 11 0.0047 14.7 0 0.25 × 1 0 - 8 0.25 × 1 0 - 8 HTL 0 0.0047 14.7 1 × 1 0 5 0.2141 6.8 × 1 0 - 5In the experiments, the defect concentrations are usually hard to be traced. But it is possible to use the Zener relaxation time [5, 7, 18] to study the diffusion processes during irradiation. Zener relaxation could be also noted as the enhanced diffusion coefficient about [5, 19](53)Drad=Dvcv+DiciAs shown in Figure15(a), the evolutions of vacancy and interstitial of UD system are quite different from the degenerate systems. Initially, both concentrations increase linearly following K0t. Then, the curve has a platform known as quasi-steady state [1], which is exactly the stable state of the SFL system. After that, the vacancy concentration goes on increasing but interstitial decreases. The value of cv is much larger than ci at the stable node. The stable concentrations of HTL system are larger than others.Figure 15
The evolutions of (a) point defects concentration and (b) irradiation enhanced diffusion coefficient of UD, SFL, and HTL system.
(a) (b)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 Figure15(b), it is easy to find that the evolutions of Drad are similar to the evolutions of interstitial concentration. Dici determines the Drad most of time, but the contributions of vacancy and interstitial are the same at last. The results qualitatively agree to the experimental observations [5].


## 5. Conclusions

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.



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