2. The calculation method

#### 2.1 Linear method

This method was reported by Kiritani et al. [2] and Tabata et al. [14] in which several factors affecting the migration energy were discussed in detail. According to their analysis, the variations of the concentrations of interstitial atoms (Ci) and lattice vacancies (Cv) with the point defect mobility (M) at high temperature (at which vacancies are highly mobile) are given as follows:

$$\frac{d\mathbf{C}\_i}{dt} = \mathbf{P} - Z\_{i\nu}(\mathbf{M}\_i + \mathbf{M}\_v)\mathbf{C}\_i\mathbf{C}\_v - Z\_{i\mathbf{L}}\mathbf{M}\_i\mathbf{C}\_{s\mathbf{L}}\mathbf{C}\_i - \mathbf{M}\_i\mathbf{C}\_{si}\mathbf{C}\_i \tag{1}$$

and

Measurement of Vacancy Migration Energy by Using HVEM DOI: http://dx.doi.org/10.5772/intechopen.87131

$$\frac{d\mathbf{C}\_v}{dt} = P - Z\_{iv}(\mathbf{M}\_i + \mathbf{M}\_v)\mathbf{C}\_i\mathbf{C}\_v - Z\_{vL}\mathbf{M}\_v\mathbf{C}\_{vL}\mathbf{C}\_v - \mathbf{M}\_v\mathbf{C}\_{sv}\mathbf{C}\_v \tag{2}$$

where Ziv, ZL, and Cs denote the recombination coefficient, the number of reaction sites of dislocation loops and point defects, and concentration of sinks, respectively. And, t denotes the time. First term P is the production rate of free Frenkel pairs induced by HVEM. The second term is the mutual annihilation rate between interstitials and vacancies with the mutual annihilation cross-section of Ziv. The third terms are the annihilation rate of each kind of point defect to dislocation loops whose concentration of atomic sites as sinks is CsL and with absorption cross-sections ZiL and ZvL at each site for each point defect. The last terms are the annihilation at the surface sink. When a steady state of point defect concentrations is established, there is:

$$\frac{dC\_i}{dt} = \frac{dC\_v}{dt} = \mathbf{0} \tag{3}$$

Then, both Eqs. (1) and (2) are equal to zero, and one can obtain the following equation:

$$(\mathbf{Z}\_{i\mathbf{L}}\mathbf{C}\_{s\mathbf{L}} + \mathbf{C}\_{s\mathbf{i}})\mathbf{M}\_{i}\mathbf{C}\_{i} = (\mathbf{Z}\_{v\mathbf{L}}\mathbf{C}\_{s\mathbf{L}} + \mathbf{C}\_{sv})\mathbf{M}\_{v}\mathbf{C}\_{v} \tag{4}$$

The most dominant terms of the point defect annihilation are the second terms in Eqs. (1) and (2), at least for the "thick-foil" case in which the last terms in the equations never become dominant. Then one can obtain the following, under the condition of Mi ≫ Mv:

$$P = Z\_{iv} \mathcal{M}\_i \mathcal{C}\_i \mathcal{C}\_v \tag{5}$$

Using Eqs. (4) and (5), the following equation could be obtained:

$$\mathbf{M}\_v \mathbf{C}\_v = \beta \mathbf{M}\_i \mathbf{C}\_i = \frac{\beta \mathbf{P}}{Z\_{iv} \mathbf{C}\_v} = \sqrt{\frac{\beta \mathbf{P} \mathbf{M}\_v}{Z\_{iv}}} \tag{6}$$

where β = (ZiLCsL + Csi)/(ZvLCsL + Csv). Since Csi and Csv are considered to be equal to each other and stay constant as long as the loop remains at the same fixed position in a specimen foil, the value β is thought to be slightly larger than unity because of the slightly larger capture cross-section ZiL for interstitial than that for vacancy, ZvL, at a dislocation loop and can be taken as constant (at least at a fixed temperature) though it varies slowly with the increase of CsL by the growth of the dislocation loops [2].

The growth speed of interstitial loop of size R is expressed by the arrival rate to the loop of interstitial atoms in excess of vacancies and is:

$$\frac{d\mathbf{R}}{dt} = a(Z\_{iL}M\_iC\_i - Z\_{vL}M\_vC\_v) = a\left(\frac{\frac{Z\_{iL}}{\sqrt{\beta}} - \sqrt{\beta}Z\_{vL}}{\sqrt{Z\_{iv}}}\right)\sqrt{P}\sqrt{M\_v} \tag{7}$$

where a is the increase of loop size by the absorption of one point defect per one site on the dislocation [2]. ZiLMiCi and ZvLMvCv are the ensemble of point defect fluxes captured by the loop. In order to reduce the effect of loop's size, the loops with same/similar initial size should be selected during experimental operation, which can be done relatively easily. Then, the equation can be written as:

#### Fusion Energy

$$\frac{dR}{dt} = \mathbf{C} \cdot \exp\left(\frac{-E\_m^v}{2kT}\right) \tag{8}$$

where k is the Boltzmann constant and C is an experimental constant. The above equation can be rewritten as:

$$\ln\left(\frac{dR}{dt}\right) = \ln C - \frac{E\_m^v}{2kT} \tag{9}$$

Using this equation, the vacancy migration energy can therefore be obtained from the slope of a linear relationship between the logarithm of the growth rate of dislocation loops and the inverse of temperature. Here, we called this method "linear method."

#### 2.2 Numerical method

Using the "linear method" with Eq. (9), the vacancy migration energy can be calculated simply. However, the position of the dislocation loop must be deep in the thick foil to ensure the surface effect could be avoided. This makes the experiment more complex and difficult. Considering the surface effect of the sample, another brilliant derivation was developed by not avoiding the third and fourth terms of Eqs. (1) and (2) by Wan et al. [11].

When the point defect concentration is in a steady state, there is dCi/dt = dCv/ dt = 0. And one can obtain an equation under the condition of Mi ≫ Mv:

$$P = Z\_{\rm in} \mathbf{M}\_i \mathbf{C}\_i \mathbf{C}\_v + Z\_{vL} \mathbf{M}\_v \mathbf{C}\_{vL} \mathbf{C}\_v + \mathbf{M}\_v \mathbf{C}\_{\rm w} \mathbf{C}\_v = \frac{Z\_{\rm in}}{\beta \mathbf{M}\_v} \left( \mathbf{M}\_v \mathbf{C}\_v \right)^2 + \left( Z\_{vL} \mathbf{C}\_{zL} + \mathbf{C}\_{\rm w} \right) \mathbf{M}\_v \mathbf{C}\_v \tag{10}$$

The former equation can be written as:

$$\frac{A\_1}{M\_v}(M\_v \mathbf{C}\_v)^2 + B\_1 M\_v \mathbf{C}\_v - P = \mathbf{0} \tag{11}$$

where A1 and B1 are constant. The root of this equation is:

$$\mathbf{M}\_{v}\mathbf{C}\_{v} = \left(-\mathbf{B}\_{1} \pm \sqrt{\mathbf{B}\_{1}^{2} + \frac{4PA\_{1}}{\mathbf{M}\_{v}}}\right)\frac{\mathbf{M}\_{v}}{2A\_{1}}\tag{12}$$

Substituting Eq. (12) into Eq. (7), one can obtain:

$$\frac{d\mathbf{R}}{dt} = a(Z\_{i\mathcal{L}}M\_i\mathbf{C}\_i - Z\_{v\mathcal{L}}M\_v\mathbf{C}\_v) = a\left(\frac{Z\_{i\mathcal{L}}}{\beta} - Z\_{v\mathcal{L}}\right)M\_v\mathbf{C}\_v = \left(B\_2 \pm \sqrt{B\_2^2 + \frac{A\_2}{M\_v}}\right)M\_v\tag{13}$$

where A2 and B2 are constant. This equation can be changed to be:

$$\exp\left(\frac{E\_m}{kT}\right)\left(\frac{dR}{dt}\right)^2 - B\_3\left(\frac{dR}{dt}\right) - A\_3 = 0\tag{14}$$

Letting dR/dt = V, the above equation can be expressed as:

Measurement of Vacancy Migration Energy by Using HVEM DOI: http://dx.doi.org/10.5772/intechopen.87131

$$\exp\left(\frac{E\_m}{kT}\right)V^2 - B\_3V - A\_3 = 0\tag{15}$$

By HVEM observation, V1, V2, and V3 at temperatures T1, T2, and T3 can be obtained, respectively. Using these data in Eq. (15) and getting rid of the constant, there is:

$$\frac{V\_1 - V\_2}{V\_2 - V\_3} = \frac{V\_1^2 \exp\left(\frac{E\_m}{kT\_1}\right) - V\_2^2 \exp\left(\frac{E\_m}{kT\_2}\right)}{V\_2^2 \exp\left(\frac{E\_m}{kT\_2}\right) - V\_3^2 \exp\left(\frac{E\_m}{kT\_3}\right)}\tag{16}$$

Letting (V1 � V2)/(V2 � V3)=V0 > 0 and substituting it in Eq. (16), the vacancy migration energy is given as follows:

$$a\_1 \cdot \exp\left(E\_m \cdot b\_1\right) + a\_2 \cdot \exp\left(E\_m \cdot b\_2\right) - \mathbf{1} = \mathbf{0} \tag{17}$$

where

$$a\_1 = \frac{V\_1^2}{(1+V\_0)V\_2^2}, a\_2 = \frac{V\_0 V\_3^2}{(1+V\_0)V\_2^2}, b\_1 = \frac{1}{k} \left(\frac{1}{T\_1} - \frac{1}{T\_2}\right), \ b\_2 = \frac{1}{k} \left(\frac{1}{T\_3} - \frac{1}{T\_2}\right)$$

and where k is the Boltzmann constant. Eq. (17) can be solved by using a computer. A Python program example for solving Eq. (17) is given in the Appendix. According to Eq. (17), two mathematical results would be obtained. In this case it is necessary to select one of these two results by the physical meaning. Sometimes, one will have a little difficulty in choosing the correct result. Probably for this reason, the application of this method had been limited.

The derivation was discussed here again to help select the result of Eq. (17). Two roots were induced by the Eq. (11) firstly. We can avoid one root directly in Eq. (12) by the physical meaning:

2A1Cv . 0

where A1 = Ziv/β > 0, Cv is the concentration of the vacancy. So, Eq. (12) could be rewritten as:

$$M\_v C\_v = \left(-B\_1 + \sqrt{B\_1^2 + \frac{4PA\_1}{M\_v}}\right) \frac{M\_v}{2A\_1} \tag{18}$$

The Eq. (13) can also be changed to:

$$\frac{\text{dR}}{\text{dt}} = a(Z\_{iL}M\_iC\_i - Z\_{vL}M\_vC\_v) = \left(B\_2 + \sqrt{B\_2^2 + \frac{A\_2}{M\_v}}\right)M\_v \tag{19}$$

There should be only one root for Em in the Eq. (15) in which A3 and B3 are constant. Thus, one knows to choose the big one of the two results from Eq. (17). Only one result will be obtained. This may encourage a broader range of applications of this method as it considers the surface effect intrinsically. We called this method "numerical method" here because of numerical calculating of the vacancy migration energy by iterated operation [15].

### 2.3 Nonlinear method

Using linear method, the vacancy migration energy could be obtained simply with no consideration of the surface effect. In order to ensure the surface effect could be avoided, however, the position of the dislocation loop must be deep which is difficult sometimes and will make the HVEM experiment more complex. Considering other sink effects such as surface, small void, precipitates, and grain boundary, some terms were added in Eqs. (1) and (2):

$$\frac{dC\_i}{dt} = P - Z\_{i\nu}(M\_i + M\_\nu)C\_i\mathbf{C}\_v - Z\_{i\mathcal{L}}M\_i\mathbf{C}\_{i\mathcal{L}}\mathbf{C}\_i - Z\_{i\nu}M\_i\mathbf{C}\_{i\nu}\mathbf{C}\_i - Z\_{i\mathcal{R}}M\_i\mathbf{C}\_{i\mathcal{P}}\mathbf{C}\_i - Z\_{i\mathcal{B}}M\_i\mathbf{C}\_{i\mathcal{B}}\mathbf{C}\_i - M\_i\mathbf{C}\_{i\mathcal{C}}\mathbf{C}\_{i\mathcal{R}}\tag{20}$$

$$\frac{d\mathbf{C}\_{\nu}}{dt} = \mathbf{P} - \mathbf{Z}\_{\nu}(\mathbf{M}\_{i} + \mathbf{M}\_{\nu})\mathbf{C}\_{i}\mathbf{C}\_{\nu} - \mathbf{Z}\_{\nu\text{L}}\mathbf{M}\_{\nu}\mathbf{C}\_{\text{L}}\mathbf{C}\_{\nu} - \mathbf{Z}\_{\nu\text{S}}\mathbf{M}\_{\nu}\mathbf{C}\_{\nu\text{S}}\mathbf{C}\_{\nu} - \mathbf{Z}\_{\nu\text{R}}\mathbf{M}\_{\nu}\mathbf{C}\_{\text{p}}\mathbf{C}\_{\nu} - \mathbf{Z}\_{\nu\text{B}}\mathbf{M}\_{\nu}\mathbf{C}\_{\text{L}}\mathbf{C}\_{\nu} - \mathbf{M}\_{\nu}\mathbf{C}\_{\nu}\mathbf{C}\_{\nu}\tag{21}$$

where Ziv, ZL, and Cs denote the recombination coefficient, the number of reaction sites of dislocation loops, and point defects and density of sinks, respectively. The first term P is the production rate of free Frenkel pairs induced by HVEM. The second term is the mutual annihilation between interstitials and vacancies. The third terms are the annihilation rate of each kind of point defect to dislocation loops whose concentration of atomic sites as sinks is CsL and with absorption cross-sections ZiL and ZvL at each site for each point defect. The fourth, fifth, and sixth terms are the ensemble of annihilation of each kind of point defect to void, precipitates, and grain boundary, respectively. It is worth noting that these three kinds of sinks mean a very small one or inconspicuous one which could not be observed by HVTEM at the same magnification times as observing the whole loop. One can avoid it directly during the TEM observation if the void or precipitates are big enough, which will not be discussed here. The seventh terms express the escape of point defects to the surface sink. Under the condition of Mi >> Mv and the steady state of point defect concentrations (dCi dt <sup>¼</sup> dCv dt ¼ 0), Eqs. (20) and (21) can be written as:

$$\mathbf{P} = Z\_{\text{iv}}\mathbf{M}\_{\text{i}}\mathbf{C}\_{\text{i}}\mathbf{C}\_{\text{v}} + Z\_{\text{v}\mathbf{L}}\mathbf{M}\_{\text{v}}\mathbf{C}\_{\text{i}\mathbf{L}}\mathbf{C}\_{\text{v}} + Z\_{\text{v}\mathbf{o}}\mathbf{M}\_{\text{v}}\mathbf{C}\_{\text{v}}\mathbf{C}\_{\text{v}} + Z\_{\text{v}\mathbf{P}}\mathbf{M}\_{\text{v}}\mathbf{C}\_{\text{i}}\mathbf{C}\_{\text{v}} + Z\_{\text{v}\mathbf{B}}\mathbf{M}\_{\text{v}}\mathbf{C}\_{\text{i}\mathbf{S}}\mathbf{C}\_{\text{v}} + M\_{\text{v}}\mathbf{C}\_{\text{v}}\mathbf{C}\_{\text{v}}\tag{22}$$

Then, there is:

$$\frac{\mathbf{Z\_{iv}}}{\rho \mathbf{M}\_{v}} \left(\mathbf{M}\_{v} \mathbf{C}\_{v}\right)^{2} + \left(\mathbf{Z\_{vL}} \mathbf{C\_{iL}} + \mathbf{Z\_{vo}} \mathbf{C\_{w}} + \mathbf{Z\_{vp}} \mathbf{C\_{iP}} + \mathbf{Z\_{vB}} \mathbf{C\_{iB}} + \mathbf{C\_{sv}}\right) \mathbf{M}\_{v} \mathbf{C\_{v}} - P = \mathbf{0} \tag{23}$$

where β = (ZiLCsL + ZioCso + ZiPCsP + ZiBCsB + Csi) / (ZvLCsL + ZvoCso + ZvPCsP + ZvBCsB + Csv).

Set a1= Ziv <sup>β</sup> and b1= ZvLCsL þ ZvoCso þ ZvPCsP þ ZvBCsB þ Csv. According to the physical meaning, a1 and b1 are constant larger than zero. The above equation can be written as:

$$\frac{a\_1}{M\_v}(M\_v \mathbf{C}\_v)^2 + b\_1 M\_v \mathbf{C}\_v - P = \mathbf{0} \tag{24}$$

The root of Eq. (24) is:

$$M\_v \mathcal{C}\_v = \left( -b\_1 \pm \sqrt{b\_1^2 + \frac{4Pa\_1}{M\_v}} \right) \frac{M\_v}{2a\_1} \tag{25}$$

Measurement of Vacancy Migration Energy by Using HVEM DOI: http://dx.doi.org/10.5772/intechopen.87131

According to the physical meaning, MvCv should be larger than zero, while �b<sup>1</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 <sup>1</sup> <sup>þ</sup> <sup>4</sup>Pa<sup>1</sup> Mv � � q Mv <sup>2</sup>a<sup>1</sup> < 0. Anyway, substituting Eq. (25) into Eq. (7), the following equation can be obtained:

$$\frac{\text{dr}}{\text{dt}} = a \left( \frac{\text{Z}\_{\text{iL}}}{\beta} - Z\_{vL} \right) \text{M}\_v \text{C}\_v = \left( b\_2 \pm \sqrt{b\_2^2 + \frac{a\_2}{M\_v}} \right) \text{M}\_v \tag{26}$$

where a is the change of loop radius by the absorption of one point defect per one site on the dislocation [2].

 $\left(\frac{\mathbf{Z}\_{i\text{L}}}{\beta} - \mathbf{Z}\_{v\text{L}}\right) \approx \left(\mathbf{Z}\_{i\text{L}} - \mathbf{Z}\_{v\text{L}}\right) = L\_{bias}$ , thus,  $\mathbf{a}\_2 = \frac{a^2 \text{PL}\_{bias}^2}{a\_1}$  and  $\mathbf{b}\_2 = \frac{-\mathbf{b}\_1 a \text{L}\_{bias}}{2a\_1}$  Eq. (26) can be changed into:

$$\exp\left(\frac{E\_m}{kT}\right)\left(\frac{dr}{dt}\right)^2 - b\_3\left(\frac{dr}{dt}\right) - a\_3 = 0\tag{27}$$

where a3 = 1/4a2, b3 = b2. The above equation can be written as:

$$\exp\left(\frac{E\_m}{kT}\right) = a\_3 \left(\frac{dt}{dr}\right)^2 + b\_3 \left(\frac{dt}{dr}\right) \tag{28}$$

As a3, b3 and <sup>b</sup><sup>2</sup> 3 <sup>4</sup>a<sup>3</sup> <sup>¼</sup> <sup>b</sup><sup>2</sup> 2 <sup>a</sup><sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>2</sup> 1 <sup>4</sup>Pa<sup>1</sup> are constant. There is a constant d to hold the following equation:

$$d \ast \exp\left(\frac{E\_m}{kT}\right) = a\_3 \left(\frac{dt}{dr} + \frac{b\_3}{2a\_3}\right)^2\tag{29}$$

From the above equation, one can obtain:

$$\frac{1}{T} = \frac{2k}{E\_m} \ast \ln\left(\frac{dt}{dr} + B\_1\right) + C\_1 \tag{30}$$

where B1 = b3/2a3 and C1 is constant. By HVEM observation and getting dt/dr [1/(dr/dt)] data at different temperatures, the Em can be obtained by this equation. In order to plot easily, the equation can be rewritten as:

$$\frac{1000}{T} = \mathbf{A} \ast \ln\left(\frac{1}{(dr/dt)} + B\right) + \mathbf{C} \tag{31}$$

Note: A = 2 k\*1000/Em,B= �b1/(a\*P) is a negative constant relating to total effects of all other sinks. Thus, from the function of 1/(dr/dt) and 1/T, the Em can be obtained by plotting the curve according to Eq. (31).
