Gamma radiation induced resistivity changes in Iron

Abstract

Monte Carlo Code JA-IPU is used for estimation of Frenkel pairs and their effect on change of resistivity of Iron on irradiation by gamma spectrum of Co⁶⁰. The Code includes three cascade processes of incident gamma, produced electrons and recoiled atoms and simulation of the lattice structure of the target material. Change in experimentally measured resistivity of Iron is found to vary with number of Frenkel pairs as ~(x − 1) ln N _d .

Appendix

Based on the linear relationship between N _d and T _dam given in Fig. 2 a phenomenological approach based on the contribution of Frenkel pairs towards resistivity is developed as follows,

$$dN_{d} \propto dT_{dam}$$

(2)

$$dN_{d} = k dT_{dam}$$

Constant, k depends on the spectrum of incident particle. On integration we can obtain total number of Frenkel pairs, N _d

$$\smallint dN_{d} = k \int_{{E_{d} }}^{{T_{dam} }} {dT_{dam } }$$

(3)

$$N_{d} = k\left( {T_{dam} - E_{d} } \right)$$

(4)

$\frac{{N_{d} }}{n} = k\left( {\frac{{T_{dam} }}{n} - \frac{{E_{d} }}{n}} \right)$ can be used for calculations per incident particle.The density of displaced atoms may be written as,

$$n_{d} = \frac{{N_{d} }}{V} = k\frac{{\left( {T_{dam} - E_{d} } \right)}}{V}$$

(5)

$$V = \frac{{N_{d} }}{{n_{d} }}$$

(6)

Assuming n_d is roughly constant, disordered volume may be written as,

$$dV = \left( {\frac{1}{{n_{d} }}} \right)dN_{d}$$

From the Eq. (6),

$$\frac{dV}{V} = \frac{{dN_{d} }}{{N_{d} }}$$

(7)

Therefore, using Eq. (2) one can write

$$\frac{dV}{V} = \frac{k}{{N_{d} }}dT_{dam}$$

(8)

In an explanation of resistivity, Brinkman [26] has related it to the disordered volume. Following this, we can write that,

$$\frac{{\rho - \rho_{0} }}{{\rho_{D} - \rho_{0} }} = \frac{dV}{V}$$

(9)

where $\rho_{0}$ and $\rho_{D}$ correspond to the resistivity of pristine and highly disordered state of a sample respectively.

In the situation of non-availability of data of resistivity, $\rho_{D}$ we have assigned $\rho_{D} = x\rho_{0}$. Here, parameter x ≥ 0 corresponds to a disordered state of irradiated material and it is also spectrum dependent. For example dV = 0 corresponds to $\rho = \rho_{0}$ and dV = V corresponds to $\rho_{D} = \rho$. In that sense higher values of x may lead to more interesting results. Thus, relation (9) can be written as,

$$\frac{d\rho }{{\left( {x\rho_{0} - \rho_{0} } \right)}} = \frac{dV}{V}$$

(10)

Putting expression for $\frac{dV}{V}$ from the Eq. (8), change in resistivity can be approximated by the following relation,

$$d\rho = \left( {x\rho_{0} - \rho_{0} } \right)\left( {\frac{k}{{N_{d} }}} \right)dT_{dam}$$

$$d\rho = \rho_{0} \left( {x - 1} \right)\left( {\frac{k}{{ N_{d} }}} \right)dT_{dam}$$

(11)

where both ‘k’ and ‘x’ are spectrum dependent. Thus, there is a composite independent variable, $\frac{{dT_{dam} }}{{N_{d} }}$ on which the resistivity depends linearly.

In case of one irradiation spectrum like Co⁶⁰ gamma spectrum, $k = \frac{{d N_{d} }}{{ d T_{dam} }}$, Therefore, the Eq. (11) can be written as,

$$\left( {\rho - \rho_{0} } \right) = \rho_{0} \left( {x - 1} \right)\ln N_{d}$$

(12)

or there is a linear relation between $\rho$ and $\ln N_{d}$,

$$\rho = \rho_{0} + {\text{Z}}\ln N_{d}$$

(13)

where $Z = \left( {{\text{x}} - 1} \right)\rho_{0}$. Taking data from Table 2 in the following Fig. 4 normalized resistivity, $\rho /\rho_{0}$ has been plotted as a function of (x − 1) ln N _d and it can be seen that the normalized resistivity increases linearly with (x − 1) ln N _d .

Fig. 4

Measured normalized resistivity plotted as a function of (x − 1) ln N _d