**2. Experimental and software-supported investigations**

Since science of metals is generally experimental science, the depth, objectivity and reliability of our understanding of investigated phenomena related to materials electronic and defect structure are determined by capacity of the technical means and methods of investigation used in order to solve the problem.

The method of positron spectroscopy is the most important instrument of investigation in this case. Not only did the relativistic quantum mechanical theory developed by Dirac (1928) explain the main properties of electron and obtain the right values of its spin and magnetic moment, but also it determined the positron existence probability. Positron is the antiparticle of electron with the mass *m*e+ = *m*e-= 9.1 10-28 g same as electron's, rest energy *m*o*с*<sup>2</sup> = 0.511 MeV and elemental, but opposite in sign, electron charge *е* = 1.6 10-19 K and spin *S*= 1/2*h*.

Natural positron sources normally do not exist. Therefore positrons are usually obtained from nuclear reactions in different nuclear power plants. The principal criteria for choosing positron sources are the cost and half-life period. The most widespread is the sodium isotope 22Na obtained from 25Mg by reaction (*p*,*α*). It is convenient in all respects and easy-to-use in the positron spectroscopy experiments, as well as for angular distributions measurements, Doppler broadening of annihilation line, positrons life time and counting rate of 3γ – coinci‐ dence. 22Na nuclear decay occurs by the following scheme: 11 *<sup>N</sup>*<sup>22</sup> *<sup>a</sup>*→<sup>10</sup> *<sup>N</sup>*<sup>22</sup> *<sup>e</sup>* <sup>+</sup> *<sup>e</sup>* <sup>+</sup> <sup>+</sup> *γЯ*. In this nuclear decay reaction the 22Nа nuclear is produced in an excited state with the time of life less than 10-12 s. At the return to the ground state it emits the nuclear quantum with energy *Е*=1.28 MeV, which effectively testifies positron production.

In spite of considerable amount of realized investigations, the analysis of the obtained results justifies the following findings: by the time of preparation of this work, the lack of the systematic information was experienced about the character of the radiation damageability of some perspective constructional refractory metals and their alloys, which firstly undergo polymorphous or phase transformations; influence of the type and concentration of alloying elements on the character of the structural disturbances at plastic deformation and radiation exposure in conditions of vacancy and vacancy-impurity complexes formation, packing defects, dislocation loops subject to material history, fluence, energy, flux, nature of ionizing

There was no a thorough research of the influence of preliminary thermochemical treatment, includinghydrogenandotheratomicgasessaturationandcyclicthermalshockswithanaccount of reconstruction of electron structure and density of pulse distribution of electrons in the field of defect production on the materials' final properties. Availability of such data would com‐ plete a full picture of purposeful properties changes and make possible working the materials with predetermined properties. This problem definition caused by demands of the state-of-theart science and technology appears to be strategically important area of research in the fields of

Therefore, the main goal of the present work, which is based on the authors' own research, is investigation and establishment of regularities of the electron structure alteration and its correlation with different titanium alloys crystal lattice defects created as a result of deforma‐

Since science of metals is generally experimental science, the depth, objectivity and reliability of our understanding of investigated phenomena related to materials electronic and defect structure are determined by capacity of the technical means and methods of investigation used

The method of positron spectroscopy is the most important instrument of investigation in this case. Not only did the relativistic quantum mechanical theory developed by Dirac (1928) explain the main properties of electron and obtain the right values of its spin and magnetic moment, but also it determined the positron existence probability. Positron is the antiparticle

Natural positron sources normally do not exist. Therefore positrons are usually obtained from nuclear reactions in different nuclear power plants. The principal criteria for choosing positron sources are the cost and half-life period. The most widespread is the sodium isotope 22Na obtained from 25Mg by reaction (*p*,*α*). It is convenient in all respects and easy-to-use in the positron spectroscopy experiments, as well as for angular distributions measurements, Doppler broadening of annihilation line, positrons life time and counting rate of 3γ – coinci‐

= 0.511 MeV

of electron with the mass *m*e+ = *m*e-= 9.1 10-28 g same as electron's, rest energy *m*o*с*<sup>2</sup>

and elemental, but opposite in sign, electron charge *е* = 1.6 10-19 K and spin *S*= 1/2*h*.

radiation, temperature of irradiation and postradiational annealing.

tion radiation and complex thermochemical treatment.

in order to solve the problem.

116 Titanium Alloys - Advances in Properties Control

**2. Experimental and software-supported investigations**

physics of metals, physics of radiation damage and radiative study of materials.

The essence of using positrons for solid structure probing is explained in the following. A positron emitted by a source while penetrating in solid to a certain depth subject to energy, experiences numerous collisions with the atoms of the solid, and consequently this positron gradually loses its velocity and at the end gains energy that corresponds to environment's absolute temperature: *E*0 = *kT*=0.025 eV, where *k* is the Boltzmann constant. This process is referred to as the positron thermolysis. The fundamental result of this phenomenon is the positron thermalization time, during which the positron dissipates its initial energy. Its calculated value is 3 10-12 s. [1].

Positron thermolysis process occurs during the time, which is considerably shorter than its life time before annihilation. This circumstance serves as grounds for using positrons in order to study the properties of condensed matters, because the conduction electrons, with which positron interacts, occupy the energetic band of the range of several electron-volt and more importantly positron does not contribute to the total pulse and energy of the pair and hence can be neglected. Therefore, the information, which is carried by the positron annihilation photons, corresponds to solid electrons state, in which positron's thermolysis and interaction and annihilation processes have occurred.

Annihilation is the act of mutual destruction of a particle and its appropriate antiparticle. While no absolute destruction of either matter or energy occurs, instead there is a mutual transfor‐ mation of particles and energy transitions from one form to another.

Due to the law of charge parity a positron in singlet state (1S0) decays with emission of even number (usually two) gamma-ray quanta. A positron in triplet state annihilates with emission of odd number (usually three) photons. The probability of the 3γ - process is lower by more than two orders than the probability of the 2γ - process. Therefore, all basic research that is oriented towards studying properties of condensed state properties is performed around this phenomenon.

If an annihilation pair is found in the state of rest in center-of-mass system (v=0), then in laboratory system of coordinates two photons would be emitted strictly in opposite directions at sin*θ* =0 (Fig.1a). As a result of interaction with the medium's electrons and phonons, positron completely thermalizes and in essence is in state of rest. However, we cannot state the same about electron, the other immediate participant of the annihilation process. At the same time pulse transverse component leads to deflection of γ1 and γ2 photons from collinearity:

$$P\_Z = m\_0 c \theta\_z;\tag{1}$$

This circumstance is initiating development of the method of measuring angular distribution of annihilatedphotons(ADAP).Thepurposeofthemethodistoobtaininformationaboutelectrons

distribution function in momentum space. To this effect, it is supposed that length of the detector (ℓ), which registers the annihilated photons, must be much longer than its width (*δ*), and substantially smaller than distance (*L*) from positron source to detector. These facts correspond to so-called long-slot geometry of experiment, which is schematically represented in Fig. 1a.

2 2 2 <sup>0</sup> ( ) (0)( ) ( ) (0)exp( / ) *NN fN*

q q

<sup>&</sup>gt; = (5)

Physicochemical and Radiation Modification of Titanium Alloys Structure

*F F*

q q

£

q q

The constant factors *N*p(0), *N*g(0) and *N*0 in equation (4) define the intensity of the Gaussian parabola at *θ=0* and the background level of random coincidence, respectively. Besides the angular distribution there also exist other characteristics, which describe the regularities of electron-positron annihilation phenomena (EPA). The most important among them are the positron lifetime and Doppler broadening of annihilation line (DBAL). The logical interrelation

The positrons annihilation process in solids can be described by a set of parameters. But, the most informative for material properties characteristics are those, which successfully fit into different physical regularities, i.e. those that carry in themselves one or another physical meaning. One of these parameters can be the values of probability of positrons annihilation with free and bound electrons. These parameters are derived from processing of experimental angular distributions spectra of annihilation emission (4). The area under each component (*SP*, *Sg*) is usually defined by integration. Knowing the total area under the whole curve

*N* (*θ*)*dθ*, we can calculate the positron annihilation probability with free electrons and

as well as the redistribution of positron annihilation probability between the free electrons and

*WP SP S WG Sg S* = = / 0, / 0, (6)

*<sup>g</sup> N* (4)

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119

 q= - + -+ *P F <sup>g</sup>*

> 1 ( ) <sup>0</sup>

of these three methods of positron annihilation is schematically depicted in Fig.2

*<sup>f</sup> ï ðè*

q

**Figure 2.** Schematic diagram of different methods of electron-positron annihilation (EPA)

*S*<sup>0</sup> = *∫* −*∞*

+*∞*

electrons of ion core, respectively:

the electrons of ion core:

*ï ðè*

The normalizing factor *f*(*θ*) in this equation takes only the following values:

q

 qq

**Figure 1.** Line and slot geometry circuit with pulse decomposition е- -е+ - pair on components (a) and Fermi surface cross-section for gas of free electrons (b)

At rest positron the impulse of annihilated photons is defined by electron impulse. The latter is uniformly distributed on whole Fermi sphere for ideal gas of electrons [2,3]. Therefore, ADAP measuring boils down to choosing thin sphere layer on distance *P*z from its center located perpendicularly of this component of impulse (Fig. 1b). In this case angular correlation spectrum *N*(*θ*) must be of shape of reverse parabola, which mathematically can be described by the following equation:

$$N\_P\left(\theta\right) = N(0)(\theta\_{\mathbb{F}}^2 - \theta^2) \text{ for } \theta \le \theta\_{\mathbb{F}}\tag{2}$$

This distribution vanishes outside *θ*F and this area corresponds to the boundary Fermi momentum*P*F= *m*о*c θ*F. Besides the parabolic component the spectrum also contains a wide angular component caused by positrons annihilation with inner electrons of ion core with the impulse that considerably exceeds the Fermi momentum. The regularities of positron annihi‐ lation in this case have sufficiently reasonable description by Gaussian function:

$$N\_{\mathcal{S}}\left(0\right) = N\_{\mathcal{S}}\left(0\right) \exp\left(-\theta \bigvee\_{\mathcal{S}} \theta\_{\mathcal{S}}\right) \tag{3}$$

where *θ*g is a Gaussian parameter and determines the penetration depth of positron's wave functions into the ion core. Hence, the ADAP general curve for any materials can be presented as the following:

$$N(\theta) = N\_P(0)(\theta\_{\text{F}}^2 - \theta^2)f(\theta) + N\_{\text{g}}(0)\exp(-\theta^2/\theta\_{\text{g}}) + N\_0 \tag{4}$$

The normalizing factor *f*(*θ*) in this equation takes only the following values:

distribution function in momentum space. To this effect, it is supposed that length of the detector (ℓ), which registers the annihilated photons, must be much longer than its width (*δ*), and substantially smaller than distance (*L*) from positron source to detector. These facts correspond to so-called long-slot geometry of experiment, which is schematically represented in Fig. 1a.

(a) (b)

At rest positron the impulse of annihilated photons is defined by electron impulse. The latter is uniformly distributed on whole Fermi sphere for ideal gas of electrons [2,3]. Therefore, ADAP measuring boils down to choosing thin sphere layer on distance *P*z from its center located perpendicularly of this component of impulse (Fig. 1b). In this case angular correlation spectrum *N*(*θ*) must be of shape of reverse parabola, which mathematically can be described

( ) 2 2 (0)( ) for *N N PF F*

This distribution vanishes outside *θ*F and this area corresponds to the boundary Fermi momentum*P*F= *m*о*c θ*F. Besides the parabolic component the spectrum also contains a wide angular component caused by positrons annihilation with inner electrons of ion core with the impulse that considerably exceeds the Fermi momentum. The regularities of positron annihi‐

 qq

*g*

q

q

è ø

=- £ (2)

 q q

lation in this case have sufficiently reasonable description by Gaussian function:

*N N*

() () <sup>2</sup> 0 0 exp , *g g*

æ ö <sup>=</sup> ç ÷ -

where *θ*g is a Gaussian parameter and determines the penetration depth of positron's wave functions into the ion core. Hence, the ADAP general curve for any materials can be presented


(3)

**Figure 1.** Line and slot geometry circuit with pulse decomposition е-

q

cross-section for gas of free electrons (b)

118 Titanium Alloys - Advances in Properties Control

by the following equation:

as the following:

$$f(\theta) = \begin{array}{ccccc} 1 & \text{ $\vec{r} \, \delta \vec{r}$ } & \|\theta\| \, \|\theta\_{\vec{r}}\| \\ 0 & \text{ $\vec{r} \, \delta \vec{r}$ } & \|\theta\| \, \|\theta\_{\vec{r}}\| \end{array} \tag{5}$$

The constant factors *N*p(0), *N*g(0) and *N*0 in equation (4) define the intensity of the Gaussian parabola at *θ=0* and the background level of random coincidence, respectively. Besides the angular distribution there also exist other characteristics, which describe the regularities of electron-positron annihilation phenomena (EPA). The most important among them are the positron lifetime and Doppler broadening of annihilation line (DBAL). The logical interrelation of these three methods of positron annihilation is schematically depicted in Fig.2

**Figure 2.** Schematic diagram of different methods of electron-positron annihilation (EPA)

The positrons annihilation process in solids can be described by a set of parameters. But, the most informative for material properties characteristics are those, which successfully fit into different physical regularities, i.e. those that carry in themselves one or another physical meaning. One of these parameters can be the values of probability of positrons annihilation with free and bound electrons. These parameters are derived from processing of experimental angular distributions spectra of annihilation emission (4). The area under each component (*SP*, *Sg*) is usually defined by integration. Knowing the total area under the whole curve +*∞*

*S*<sup>0</sup> = *∫* −*∞ N* (*θ*)*dθ*, we can calculate the positron annihilation probability with free electrons and

electrons of ion core, respectively:

$$\text{WPP} = \text{SP } / \text{SO} / \text{WG} = \text{Sg } / \text{SO} \text{ } \tag{6}$$

as well as the redistribution of positron annihilation probability between the free electrons and the electrons of ion core:

$$F = \mathsf{WP} \; / \, \mathsf{WG} = \mathsf{S} \mathsf{P} / \, \mathsf{S} \mathsf{g}. \tag{7}$$

While comparing results of one set of measurements, which are related to thermal, deformative or radiative influences on the investigated materials, one can use a special configuration

> *t f V V Ñ t f c c*

where *N* is a counting rate value in the structurally sensitive region of the annihilated photons angular distribution spectrum; subscripts v and c of *N* refer to a positron annihilation with free and core electrons, respectively; superscripts t and f correspond to positron annihilation from the trapped and free states. It is assumed that this parameter does not depend on defects

The basic specifications of the experimental spectrometer of annihilated photons angular

**•** The time resolution on the fast channel equals to 100 ns and on the slow channel ranges

**•** The movable detector step width is set stepwise by 0.25, 0.5 and 1.0 mrad; in all the setup permits to measure up to 50 values of coinciding gamma photons intensity in one direction

**•** The counting rate instability in the course of three days of continuous work does not exceed

**•** The allowed maximum intensity of incoming information on the slow channel is no worse

**•** The maximal vacuum in the measuring chamber is no worse than 10−4 Pa at the temperature

The reliability of positron investigations results depends on a number of reasons which are as

The preparation of investigated objects of different composition was realized in the high temperature electroarc furnace on a copper bottom with nonexpendable electrodes. After batch charging and before alloying, a vacuum of ~10−2 Pa was created in the furnace. After that a high purity argon was introduced into the furnace, in which all the processes of melting were conducted in this atmosphere. For homogeneity, ingots were repeatedly melted (up to 5–6

from 1mm strips, which repeatedly underwent plastic deformation (*ε* = 50%) by rolling at room

far as possible taken into account during the process of experimental investigations.

Bk (10 mKi).

C up to 1–2mm strips and then

C during 2 hours. The annealed samples were prepared

,


Physicochemical and Radiation Modification of Titanium Alloys Structure

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121

parameter sensible to the presence of only one kind of defect in the crystal [4]:

concentration in the material and is determined by its structure only.

**•** Angular resolution of the setup changes within the range of 0.5–1.5 mrad.

distributions with line and slot geometry are the following:

within the interval of 0.3–1.0µs.

two standard deviations.

s−1.

annealed at 10−5 Pa vacuum at 900 0

than 3 × 105

of 300 – 1100 К.

from the spectrum maximum position.

**•** The positrons source activity of 22Na is 3.7 × 108

times).The finished ingots were rolled at a temperature of 9000

*N N <sup>R</sup> N N*

The example of decomposition of experimental spectra to components is shown in Fig. 3. Due to parabolic component spreading in the *θ* = *θ*F region, the Fermi angle value is usually determined by extrapolation.

**Figure 3.** The decomposition of the angular correlation spectra into components

The changes in the investigated material structure are by all means reflected on the spectra form and lead to redistribution of positron annihilation probabilities. In this case after normalization to a single area, they can be built on one axis for comparison purposes (Fig. 4).

**Figure 4.** The APAD spectra normalized to a single area for annealed (1) and deformed (2) titanium

While comparing results of one set of measurements, which are related to thermal, deformative or radiative influences on the investigated materials, one can use a special configuration parameter sensible to the presence of only one kind of defect in the crystal [4]:

*F WP WG SP Sg* = = / /. (7)

The example of decomposition of experimental spectra to components is shown in Fig. 3. Due to parabolic component spreading in the *θ* = *θ*F region, the Fermi angle value is usually

The changes in the investigated material structure are by all means reflected on the spectra form and lead to redistribution of positron annihilation probabilities. In this case after normalization to a single area, they can be built on one axis for comparison purposes (Fig. 4).

**Figure 4.** The APAD spectra normalized to a single area for annealed (1) and deformed (2) titanium

**Figure 3.** The decomposition of the angular correlation spectra into components

determined by extrapolation.

120 Titanium Alloys - Advances in Properties Control

$$R\_{\tilde{N}} = \left| \frac{N\_V^t - N\_V^f}{N\_c^t - N\_c^f} \right| \tag{8}$$

where *N* is a counting rate value in the structurally sensitive region of the annihilated photons angular distribution spectrum; subscripts v and c of *N* refer to a positron annihilation with free and core electrons, respectively; superscripts t and f correspond to positron annihilation from the trapped and free states. It is assumed that this parameter does not depend on defects concentration in the material and is determined by its structure only.

The basic specifications of the experimental spectrometer of annihilated photons angular distributions with line and slot geometry are the following:


The reliability of positron investigations results depends on a number of reasons which are as far as possible taken into account during the process of experimental investigations.

The preparation of investigated objects of different composition was realized in the high temperature electroarc furnace on a copper bottom with nonexpendable electrodes. After batch charging and before alloying, a vacuum of ~10−2 Pa was created in the furnace. After that a high purity argon was introduced into the furnace, in which all the processes of melting were conducted in this atmosphere. For homogeneity, ingots were repeatedly melted (up to 5–6 times).The finished ingots were rolled at a temperature of 9000 C up to 1–2mm strips and then annealed at 10−5 Pa vacuum at 900 0 C during 2 hours. The annealed samples were prepared from 1mm strips, which repeatedly underwent plastic deformation (*ε* = 50%) by rolling at room temperature. The true content of the components in the check samples of the material was determined by chemical and spectral analysis methods. Then the surfaces of these samples were thoroughly burnished, and the samples were polished electrochemically in the solution of the following composition: HF – one part, HNO3 – three parts and H2O – two parts. The prepared samples were flushed with flowing water and wiped with alcohol. The temperature measurement of samples at irradiation and annealing was performed by thermometry methods. All measurement processes of annihilation characteristics of investigated materials after different kinds of influences (plastic deformation, radiative and other complex physico‐ chemical and thermal treatments) were conducted at room temperature.

**2.1. Structural transformations in plastically deformed alloys of the Ti-Zr system**

relative changes Δ*F* and Δ*θ*F were used.

annihilation occurs.

Influence of plastic deformation on the structural damages formation was investigated on the alloys that contained 0; 2.7; 8.3; 17.0; 22.0 and 39.0 at.% Zr. Plastic deformation *ε* = 50% considerably changes shape of the curves by decreasing width on the half height (FWHM) and increase of counting rate at maximum of *N*(0) relative to the initial (annealed) state, which is the effect of crystal structure defects occurrence in the material (Fig.4). For interpretation of investigation results the above mentioned annihilation parameters *F*=SP/Sg, *θ*F = *P*F/*mc* and their

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123

In order to establish regularities of changes of annihilation parameters depending on the structure damages level, Ti and its alloy Ti – 2.7 at.% Zr underwent plastic deformation by different degrees in the range from *ε* = 5 up to 80 %. It is determined that main changes of the annihilation parameter occur when *ε* takes values up to 30%, whereupon it reaches saturation. Though for Ti–2.7 at.% Zr alloy it reaches saturation considerably quicker than for pure Ti (Fig. 5). Evidently it can be stated that with an increase of deformation degree the defects concentration in metals also increases and at certain level virtually all positrons are captured and annihilated in defects. In such state the annihilation parameters are typical for deformation defects. In other words, in the conditions of ultimate strains the EPA characteristics contain information about structure of the crystal regions, in which the vast majority of positrons

**Figure 5.** The influence of deformation degree and alloying element concentration on titanium alloys structural-sensi‐

In order to determine the phase composition, titanium and Ti–2.7at.% Zr alloy underwent Xray analysis using the DRON-2 diffractometer in the filtered CuK<sup>α</sup> radiation with the help of

tive characteristics: 1-Ti; 2- Ti-2.7 at.% Zr; 3- N(0)/S0 relationship; 4- the β-phase content in alloy
