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Self-organizing processes in metals by low-energy ion beams
INIOR
Nuclear Instn.ments and Methods in Physics Research B80/81 (1993) 115-119 North-Holland
ÇPIf_~r~oa i~jY1a
Bsam Interactlona with Materials&Atoms
nrnresses in metals by low-energy ion beams
IN. Te~eshko a, V.I . KhodyreV a, V .M. Tereshko b , E .A. Lipsky a, AN . Goncharenya a and S . Ofori-Sey " Mechanical Engineering Institute, Mogilev 212005, Belarus Moscow State University, Moscow 117234, Russian Federation
The process of relaxation in atomic lattices subjected to the bombardment with low-energy ions was investigated. The method of molecular dynamics was applied to one-dimensional crystal lattices, and a special apprc:;'mate method, based on equations of classical mechanics, was worked out for two- and three-dimensional models . In linear crystal lattices the atomic oscillations are damped and the lattices relax to the initial non-excised state . In nonlinear crystal lattices, energy transfer takes ~lace between the excited oscillators, and as a result new stable structural shies of crvctal Lattices are formed. I. ,introduction Traditional radiation solid-state physics is based on the study of defects in materials under conditions of intense and high-energy irradiation . The process of charged particle interaction with the surface of materials has been investigated thoroughly and reported in refs . [1,2]. It has been shown in ref. [3] that low-energy ion bombardment on the samples of armco-iron, electrolytic copper, Nj 3Fe alloy and high-speed steel leads to an increase in dislocation density up to the depth of 10 mm from the irradiated surface . The specimens were exposed to irradiation in a specially constructed plasmatron, the average energy of ions of the vacuum residual gases b(:ing within the limits of 1 to 2 keV. When irradiated, the specimens experienced neither mechanical nor thermal stresses, but we observed all kinds of dislocational structures even at a large depth from the irradiated surface. Such structures are usually observed only under strong plastic deformations . It has also been observed that a decrease in energy from 2 to 1 keV leads to an increase of the depth of the modified layer . It has also been observed that disclination loops are formed after irradiation . The formation of fragmented structures was observed on the specimens having a small initial dislocation density . It has been shown in ref. [4] that the process of low-energy ion bombardment leads to an increase in the microhardness of specimens and a decrease of their electroresistance . These modifications in materials could be understood within the concept of active self-organizing processes in crystal lattices. Such selforganizing processes lead to the formation of new structural collective states of atoms [5]. The main ob-
jective of the present paper is to investigate and theoreVc--iiy aodei sucn stai,;~s . 2. Theoretical approach For one-dimensional atomic lattices a model was developed by means of molecular dynamics. The Morse potential was used to describe the potential for atomic oscillator interaction : U(r) = J{exp[ -2a(r-ra)] -2exp [ -a(r-ro) ]), (1) where, J, a are the parameters of the dissociation energy of a pair of atoms and the degree of the potential unharmonicity, respectively: Ar = (r - r o) is the displacement from the equilibrium position . Decomposing the potential (1) into a Taylor series we obtain F= -
dU(r) dr
= -2a 2 0r+3a 3Ar 2 -2 .3a°Or 3 .
(2)
Let K=2a2 , A=3a 3, B=2.3a 4, where K, A, B are the coefficients of elasticity, quadratic and cubic nonlinearity, respectively. These coefficients were defined considering the theory of elasticity. For a chain of n coupled oscillators the system of equations can be written as d2 yt tn -= -K'y + Ay, -Byl +K(y 2 -y,) dt 2
i
d - A(y2 yt)2 +B(y2 - yt)3- P'
0168-583X/93/$06.00 C 1993 - Elsevier Science Publishers B.V. Ali rights reserved
dtl "
lb. BASIC INTERACTIONS (b)
LV. Tereshko et al / Self-organizing processes in metals
m
a -y,
dt2 =
- K(Y,
.05 0
_ Y_ - Y,-1) +A(Yi 1)2 3
d y, -A(yt+1-Y,)2+B(y,+1-y)3-ßdt' 71
d 2yn ât2
=
- K(Y
-
-
B(yn -
Yn-1)
+A(Yn
F 2
W
2
- Yo-1)
N
À
Yn _ 1) 3 - K'yn + Ayn - BYn
d yn (3) i=2 , . . . ,n _ 1 _ß .dt where y,, j = 1,- - -, n, is the displacement of the jth oscillator from the equilibrium position ; K', K are the coefficients of elasticity at the boundary and in internal areas respectively; ß', /3 are damping factors at the boundary and in internal areas; A, B are coefficients of quadratic and cubic nonlinearitl . The equation system (3) was solved by means of the Runge-Kutta method [6]. The initial conditions (displacement and speed) of the atoms and coefficients K, K', A, B, /3', )3 were varied . As a result of the ion bombardment, it was assumed that an arbitrary atom in the chain gains an impulse m(dy1 /dt), which varied over a wide range (fig. l b). On the basis of calculations, graphs for the observation of the relaxation process of each atom after the initial influence were constructed, as well as graphs for the observation of the corresponding location of the atomic chain as a whole for a given period of time . In
T
aoo " 0.
2 4
8
12
16
OSCILLATOR
217
24
28
NUMBER
Fig. 2. Displacement of all atoms of the linear one-dimen sional lattice from their equilibrium position at time (1) t = 65, (2) 1 = 75,(3)
1
= 125.
the one-dimensional model, yt is the longitudinal displacement of atoms in the chain, but in the graphs it is represented as a cross oscillation for visualization . The values of displacement and time we are interested in are represented as dimensionless, i.e. divided by characteristic scales of displacement and time, respectively. A special model for calculating lattices of different dimensions, using the equations of classical dynamics enabling the description of the evolution of atom ensemble has been developed . At the assigned values of A, B, K, K', 0', !? and initial conditions (aggregates of all initial coordinates and speeds), an arbitrary atom receives an impulse from an impinging ion (fig. la), and the subsequent behaviour of each atom is investigated . Graphs for atom displacement along the coordinate axes, as well as phase trajectories of each atom on any coordinate plane are made . In linear approximation A = 0, B = 0. We call such lattices linear ones . If A 0 and B * 0, the lattices are called nonlinear. 3. Results 3.1. Scenarios of self-organization in one-dimensional systems
a Y Fig. 1 . Interaction between an impinging ion and a crystal lattice : (a) three-dimensional lattice, (b) one-dimensional lattice (chain) exposed to a direct impact .
Fig. 2 illustrates the results of a numerical calculation of the relaxation process for a one-dimensional linear atomic chain (n = 25, A - 0, B = 0) after its first atom received an impulse from an external ion impact (fig . lb) . The displacement from the equilibrium position, of each of the atomic oscillators, is assigned to one axis and the oscillator number to the other. Curves 1, 2, 3 show the results after different times have elapsed since the end of the external influence (tl < t,
i. V. Tereshko e1 al. / Sei;-orgultiziflg processee b~ -!als 1,3544 ß540 1,35)6 2
1,3532 1,1529
ä 0
1,J524 0521 f,3516 . 1,3512 OSCILLATOR Q
NUMBER
i
VQ hn ô
OSCILLATOR
b
NUMBER
Fig. 3 . Displacement of all atoms of the nonlinear one-dimensional lattice from their equilibrium position : (a) A = 3, B = 2: (1) 1= 65, (2) 1= 75; (b) t =125: (1) A = 3, B = 2, (2) A = 5, B=2.
< t 3) . The chain scattered by one ion impact forms a damped immobile semiwave and the whole system comes back to the initial equilibrium position some time later. A nonlinear atomic chain behaves quite differently. In the system of coupled oscillators nonlinear oscillations are excited, which result in the formation and development of metastable atomic clusters (fig. 3a). During the relaxation process the system goes through intermediate states, represented as "a devil's staircase" (curve 1) and ends into a new collective state corresponding to the total displacement of the atom group to level yi = 1 .352, j =1, - - -, n (curve 2) . The state is stable and, compared with the initial state, can be determined as a highly excited state of the atom aggregate . The change of nonlinear parameters causes a new type of self-organization where stable clusters are formed in the final state . The periodic spacing of the lattice inside the clusters does not correspond to
the initial one . Some clusters are separated into regions having a negative density fluctuation. Fig . 3b shows different kinds of self-organization ; curve 1 corresponds to the first scenario according to which a displaced chain, without tàtanging the lattice period, is formed (A = 3, B = 2). Cvrve 2 corresponds to cluster formation of the chain (A = 5 . B = 2). Thus, changes in the nonlinearity coefficients predict wC. possible types of structures and ways in which they are formed . There is a wide spectrum of non-equilibriurr : stable collective states of lattice oscillators. From the physical point of view such stable structures of the lattice can be interpreted as long-lived states of atcm groups, the lifetime being far greater than the time of atomic relaxations. To purposefully construct a certain structure one should change the internal parameters of the system (A, B, K, K') or its controlling parameters, such as the amount of external influence. In a real crystal tie local displacements along the atom chains can lead to breaking up of the atom planes, i .e. to dislocation formation . Clustering of atom chains leads to local changes of the lattice spacings, to the formation of dislocation loops, and to internal stresses . When the energy of impinging ions is high enough, there occur strong oscillations of the first atoms of the chain which lead to the breaking up of the atom bonds and to radiation defects, i.e. to vacancies and displaced atoms. Nonlinear oscillations are not excited in this case and, consequently, self-organization effects do not appear . 3 .2. Self-organization in three-dimensional systems In the given numerical experiment a homogeneous three-dimensional cubic lattice was bombarded by external ions where each atom was in equilibrium (the scheme is shown in fig. la). Fig . 4 shows the dependence of atom displacement along the x-axis on time elapsed after stopping the external influence . Curves 1 and 2 are related to the nonlinear three-dimensional lattice (atom nos. 25 and 28, in accordance with the notations in fig . 1a) . It is observed that the relaxation of the lattice also leads to the formation of a new stable structural state in nonlinear oscillations where each atom is practically in the displaced state but without breaking up of atomic bonds. Curve 3 corresponds to atom no. 25 in the linear three-dimensional lattice . We showed that when nonlinearity is absent each atom of the crystal lattice performing a damping oscillation under the influence of primary excitation relaxes towards its initial state . Dependences of each atom displacement on the serial atom number of the lattice (lattice 4 x 4 x 4, fig . 1a) were constructed for the time close to the process of full stabilization of the lattice after an isolated ion Ib. BASIC INTERACTIONS (b)
L V.
Tereshko of
al. / Self-organizing processes in metals impact. Fig. Sa shows that in the linear !attize all atoms (except the chain which was exposed to ion impact) have been stabilized in the initial state . In the nonlinear lattice (fig . 5b) all atoms are displaced from the equilibrium state. The latter figure allows a conclusion to be made about the character of noT line- oscillations in the overall volume of the material . In the finite state. the whole volume of an atomic lattice irradiated by low-energy ions respresents a highly fragmented distorted structure .
1200
Z
W W h4 0
4. Discussion 20
40
60 T :Mf
80
fou
120
Dependence of atom displacement along the x-axis on the relaxation time in the three-dimensionai lattice: (1) nonlinear lattice, atom no. 25, (2) nonlinear lattice, atom no . 28, (3) linear lattice, atom no. 25 . Fig . 4.
F z F û ° , c
0
â %0
20 .i0 OSCILLATOR
a
50 40 NUMBER
60
70
The above mentioned results can be explained from general synergetic assumptions. In a nonlinear medium forced out of stable equilibrium the collective effect of atom interactions leads to the formation of new structural states . The specific kind of these new heterogeneous structures depends on the properties of the potential, characterising the atomic oscillator bend betwcen themselves . At the decomposed potential new highly excited stable states are defined by the relation of the quadratic and cubic nonlinearity coefficients . We have made calculations at the decomposed Morse potential and showed that all the above mentioned results and conclusions are confirmed . In this case new heterogeneous structures are being formed in the initial homogeneous crystal medium too, the specific kind of which are defined by the unharmonicity of the potential . Fig. 6 shows the initial state of the atomic cubic three-dimensional lattice in the plane (x, y), as well as the finite state after the ion influence . The latter state has a stable character and is a highly fragmented structure . Each atom of the lattice has passed over to a 9
19
6
Fig . 5. Displacement of all the atoms of the three-dimensional lattice: (a) linear lattice, (b) nonlinear lattice.
Fig.
6.
x Fragmented structures in the crystal lattice exposed to low-energy ions .
LV. Tereshko et al.
/ 3elf-organiz "ng processes in metals
new non-equiiibrium stable state as a result of excita . fion of nonlinear oscillations within i, " iat?ice. It is . very important to note that this process affects the overall volume of the irradiated crystal . It shouid be noted that in high-energy excitation the radiation-induced defects are actually formed in the boundary surface of the irradiated sample. There exists an upper and lower boundary for the excitation energy of nonlinear oscillations. The lower boundary depends on the potential unharmonicity, the upper one is defined by the energy of the atomic-bond breakdown, i.e . by the energy forming vacancie, and displaced atoms . Powerful oscillations of the atomic oscillators up to the bond breakdown occur at activation energies of point defect formation and the abovementioned effects are no longer relevant. 5. Conclusions
The following conclusions can be made : 1) Low-energy ion bombardment on a crystal lattice leads to nonlinear oscillations of atomic oscillators, which can result in the formation of stable non-equi-
librium structures of the lattice . These structures determine the new physical and mechanical properties of irradiated materials . 2) High-energy, in contrast to low-energy, ions result in the breakdown of atomic bonds, and the abovementioneJ effects are no longer relevant . References
H . Thompson, Defects and Radiation Damage in Metals (Cambridge University Press, 1969) . [2] F.F . Komarov, Ion Beam Modification of Metals (Gordon and Breach, Philadelphia, 1992). [3] E .V. Kozlov, I .V. Tereshko, V .I. Khodyrev, N.A . Popova, L .N. Ignatenko and E.A . Lipsky, 1zv. Vuzov Fiz. 1 (1992) 14, in Russian. [41 I.V . Tereshko, I .I. Silin, V .I. Khodyrev and E .A. Lipsky, Proc . Conf. on Physics of Metals, Kuibyshev, 1989, p. 17. [5] I .V . Tereshko, V .I. Khodyrev, V .M. Tereshko, E .A. Lipsky, A.V. Goncharenya and S . Ofosi-Sey, Computer modelling on modification of solids 5y low-energy ion beams, preprint no. 1, Mech. Eng . Inst ., Mogilev (1991) . [6] J. Stoer and R . Bulir-sch, Introduction to Numerical Analysis (Springer, New York, 1980). [1]
Ib. BASIC INTERACTIONS (b)
Nuclear Instn.ments and Methods in Physics Research B80/81 (1993) 115-119 North-Holland
ÇPIf_~r~oa i~jY1a
Bsam Interactlona with Materials&Atoms
nrnresses in metals by low-energy ion beams
IN. Te~eshko a, V.I . KhodyreV a, V .M. Tereshko b , E .A. Lipsky a, AN . Goncharenya a and S . Ofori-Sey " Mechanical Engineering Institute, Mogilev 212005, Belarus Moscow State University, Moscow 117234, Russian Federation
The process of relaxation in atomic lattices subjected to the bombardment with low-energy ions was investigated. The method of molecular dynamics was applied to one-dimensional crystal lattices, and a special apprc:;'mate method, based on equations of classical mechanics, was worked out for two- and three-dimensional models . In linear crystal lattices the atomic oscillations are damped and the lattices relax to the initial non-excised state . In nonlinear crystal lattices, energy transfer takes ~lace between the excited oscillators, and as a result new stable structural shies of crvctal Lattices are formed. I. ,introduction Traditional radiation solid-state physics is based on the study of defects in materials under conditions of intense and high-energy irradiation . The process of charged particle interaction with the surface of materials has been investigated thoroughly and reported in refs . [1,2]. It has been shown in ref. [3] that low-energy ion bombardment on the samples of armco-iron, electrolytic copper, Nj 3Fe alloy and high-speed steel leads to an increase in dislocation density up to the depth of 10 mm from the irradiated surface . The specimens were exposed to irradiation in a specially constructed plasmatron, the average energy of ions of the vacuum residual gases b(:ing within the limits of 1 to 2 keV. When irradiated, the specimens experienced neither mechanical nor thermal stresses, but we observed all kinds of dislocational structures even at a large depth from the irradiated surface. Such structures are usually observed only under strong plastic deformations . It has also been observed that a decrease in energy from 2 to 1 keV leads to an increase of the depth of the modified layer . It has also been observed that disclination loops are formed after irradiation . The formation of fragmented structures was observed on the specimens having a small initial dislocation density . It has been shown in ref. [4] that the process of low-energy ion bombardment leads to an increase in the microhardness of specimens and a decrease of their electroresistance . These modifications in materials could be understood within the concept of active self-organizing processes in crystal lattices. Such selforganizing processes lead to the formation of new structural collective states of atoms [5]. The main ob-
jective of the present paper is to investigate and theoreVc--iiy aodei sucn stai,;~s . 2. Theoretical approach For one-dimensional atomic lattices a model was developed by means of molecular dynamics. The Morse potential was used to describe the potential for atomic oscillator interaction : U(r) = J{exp[ -2a(r-ra)] -2exp [ -a(r-ro) ]), (1) where, J, a are the parameters of the dissociation energy of a pair of atoms and the degree of the potential unharmonicity, respectively: Ar = (r - r o) is the displacement from the equilibrium position . Decomposing the potential (1) into a Taylor series we obtain F= -
dU(r) dr
= -2a 2 0r+3a 3Ar 2 -2 .3a°Or 3 .
(2)
Let K=2a2 , A=3a 3, B=2.3a 4, where K, A, B are the coefficients of elasticity, quadratic and cubic nonlinearity, respectively. These coefficients were defined considering the theory of elasticity. For a chain of n coupled oscillators the system of equations can be written as d2 yt tn -= -K'y + Ay, -Byl +K(y 2 -y,) dt 2
i
d - A(y2 yt)2 +B(y2 - yt)3- P'
0168-583X/93/$06.00 C 1993 - Elsevier Science Publishers B.V. Ali rights reserved
dtl "
lb. BASIC INTERACTIONS (b)
LV. Tereshko et al / Self-organizing processes in metals
m
a -y,
dt2 =
- K(Y,
.05 0
_ Y_ - Y,-1) +A(Yi 1)2 3
d y, -A(yt+1-Y,)2+B(y,+1-y)3-ßdt' 71
d 2yn ât2
=
- K(Y
-
-
B(yn -
Yn-1)
+A(Yn
F 2
W
2
- Yo-1)
N
À
Yn _ 1) 3 - K'yn + Ayn - BYn
d yn (3) i=2 , . . . ,n _ 1 _ß .dt where y,, j = 1,- - -, n, is the displacement of the jth oscillator from the equilibrium position ; K', K are the coefficients of elasticity at the boundary and in internal areas respectively; ß', /3 are damping factors at the boundary and in internal areas; A, B are coefficients of quadratic and cubic nonlinearitl . The equation system (3) was solved by means of the Runge-Kutta method [6]. The initial conditions (displacement and speed) of the atoms and coefficients K, K', A, B, /3', )3 were varied . As a result of the ion bombardment, it was assumed that an arbitrary atom in the chain gains an impulse m(dy1 /dt), which varied over a wide range (fig. l b). On the basis of calculations, graphs for the observation of the relaxation process of each atom after the initial influence were constructed, as well as graphs for the observation of the corresponding location of the atomic chain as a whole for a given period of time . In
T
aoo " 0.
2 4
8
12
16
OSCILLATOR
217
24
28
NUMBER
Fig. 2. Displacement of all atoms of the linear one-dimen sional lattice from their equilibrium position at time (1) t = 65, (2) 1 = 75,(3)
1
= 125.
the one-dimensional model, yt is the longitudinal displacement of atoms in the chain, but in the graphs it is represented as a cross oscillation for visualization . The values of displacement and time we are interested in are represented as dimensionless, i.e. divided by characteristic scales of displacement and time, respectively. A special model for calculating lattices of different dimensions, using the equations of classical dynamics enabling the description of the evolution of atom ensemble has been developed . At the assigned values of A, B, K, K', 0', !? and initial conditions (aggregates of all initial coordinates and speeds), an arbitrary atom receives an impulse from an impinging ion (fig. la), and the subsequent behaviour of each atom is investigated . Graphs for atom displacement along the coordinate axes, as well as phase trajectories of each atom on any coordinate plane are made . In linear approximation A = 0, B = 0. We call such lattices linear ones . If A 0 and B * 0, the lattices are called nonlinear. 3. Results 3.1. Scenarios of self-organization in one-dimensional systems
a Y Fig. 1 . Interaction between an impinging ion and a crystal lattice : (a) three-dimensional lattice, (b) one-dimensional lattice (chain) exposed to a direct impact .
Fig. 2 illustrates the results of a numerical calculation of the relaxation process for a one-dimensional linear atomic chain (n = 25, A - 0, B = 0) after its first atom received an impulse from an external ion impact (fig . lb) . The displacement from the equilibrium position, of each of the atomic oscillators, is assigned to one axis and the oscillator number to the other. Curves 1, 2, 3 show the results after different times have elapsed since the end of the external influence (tl < t,
i. V. Tereshko e1 al. / Sei;-orgultiziflg processee b~ -!als 1,3544 ß540 1,35)6 2
1,3532 1,1529
ä 0
1,J524 0521 f,3516 . 1,3512 OSCILLATOR Q
NUMBER
i
VQ hn ô
OSCILLATOR
b
NUMBER
Fig. 3 . Displacement of all atoms of the nonlinear one-dimensional lattice from their equilibrium position : (a) A = 3, B = 2: (1) 1= 65, (2) 1= 75; (b) t =125: (1) A = 3, B = 2, (2) A = 5, B=2.
< t 3) . The chain scattered by one ion impact forms a damped immobile semiwave and the whole system comes back to the initial equilibrium position some time later. A nonlinear atomic chain behaves quite differently. In the system of coupled oscillators nonlinear oscillations are excited, which result in the formation and development of metastable atomic clusters (fig. 3a). During the relaxation process the system goes through intermediate states, represented as "a devil's staircase" (curve 1) and ends into a new collective state corresponding to the total displacement of the atom group to level yi = 1 .352, j =1, - - -, n (curve 2) . The state is stable and, compared with the initial state, can be determined as a highly excited state of the atom aggregate . The change of nonlinear parameters causes a new type of self-organization where stable clusters are formed in the final state . The periodic spacing of the lattice inside the clusters does not correspond to
the initial one . Some clusters are separated into regions having a negative density fluctuation. Fig . 3b shows different kinds of self-organization ; curve 1 corresponds to the first scenario according to which a displaced chain, without tàtanging the lattice period, is formed (A = 3, B = 2). Cvrve 2 corresponds to cluster formation of the chain (A = 5 . B = 2). Thus, changes in the nonlinearity coefficients predict wC. possible types of structures and ways in which they are formed . There is a wide spectrum of non-equilibriurr : stable collective states of lattice oscillators. From the physical point of view such stable structures of the lattice can be interpreted as long-lived states of atcm groups, the lifetime being far greater than the time of atomic relaxations. To purposefully construct a certain structure one should change the internal parameters of the system (A, B, K, K') or its controlling parameters, such as the amount of external influence. In a real crystal tie local displacements along the atom chains can lead to breaking up of the atom planes, i .e. to dislocation formation . Clustering of atom chains leads to local changes of the lattice spacings, to the formation of dislocation loops, and to internal stresses . When the energy of impinging ions is high enough, there occur strong oscillations of the first atoms of the chain which lead to the breaking up of the atom bonds and to radiation defects, i.e. to vacancies and displaced atoms. Nonlinear oscillations are not excited in this case and, consequently, self-organization effects do not appear . 3 .2. Self-organization in three-dimensional systems In the given numerical experiment a homogeneous three-dimensional cubic lattice was bombarded by external ions where each atom was in equilibrium (the scheme is shown in fig. la). Fig . 4 shows the dependence of atom displacement along the x-axis on time elapsed after stopping the external influence . Curves 1 and 2 are related to the nonlinear three-dimensional lattice (atom nos. 25 and 28, in accordance with the notations in fig . 1a) . It is observed that the relaxation of the lattice also leads to the formation of a new stable structural state in nonlinear oscillations where each atom is practically in the displaced state but without breaking up of atomic bonds. Curve 3 corresponds to atom no. 25 in the linear three-dimensional lattice . We showed that when nonlinearity is absent each atom of the crystal lattice performing a damping oscillation under the influence of primary excitation relaxes towards its initial state . Dependences of each atom displacement on the serial atom number of the lattice (lattice 4 x 4 x 4, fig . 1a) were constructed for the time close to the process of full stabilization of the lattice after an isolated ion Ib. BASIC INTERACTIONS (b)
L V.
Tereshko of
al. / Self-organizing processes in metals impact. Fig. Sa shows that in the linear !attize all atoms (except the chain which was exposed to ion impact) have been stabilized in the initial state . In the nonlinear lattice (fig . 5b) all atoms are displaced from the equilibrium state. The latter figure allows a conclusion to be made about the character of noT line- oscillations in the overall volume of the material . In the finite state. the whole volume of an atomic lattice irradiated by low-energy ions respresents a highly fragmented distorted structure .
1200
Z
W W h4 0
4. Discussion 20
40
60 T :Mf
80
fou
120
Dependence of atom displacement along the x-axis on the relaxation time in the three-dimensionai lattice: (1) nonlinear lattice, atom no. 25, (2) nonlinear lattice, atom no . 28, (3) linear lattice, atom no. 25 . Fig . 4.
F z F û ° , c
0
â %0
20 .i0 OSCILLATOR
a
50 40 NUMBER
60
70
The above mentioned results can be explained from general synergetic assumptions. In a nonlinear medium forced out of stable equilibrium the collective effect of atom interactions leads to the formation of new structural states . The specific kind of these new heterogeneous structures depends on the properties of the potential, characterising the atomic oscillator bend betwcen themselves . At the decomposed potential new highly excited stable states are defined by the relation of the quadratic and cubic nonlinearity coefficients . We have made calculations at the decomposed Morse potential and showed that all the above mentioned results and conclusions are confirmed . In this case new heterogeneous structures are being formed in the initial homogeneous crystal medium too, the specific kind of which are defined by the unharmonicity of the potential . Fig. 6 shows the initial state of the atomic cubic three-dimensional lattice in the plane (x, y), as well as the finite state after the ion influence . The latter state has a stable character and is a highly fragmented structure . Each atom of the lattice has passed over to a 9
19
6
Fig . 5. Displacement of all the atoms of the three-dimensional lattice: (a) linear lattice, (b) nonlinear lattice.
Fig.
6.
x Fragmented structures in the crystal lattice exposed to low-energy ions .
LV. Tereshko et al.
/ 3elf-organiz "ng processes in metals
new non-equiiibrium stable state as a result of excita . fion of nonlinear oscillations within i, " iat?ice. It is . very important to note that this process affects the overall volume of the irradiated crystal . It shouid be noted that in high-energy excitation the radiation-induced defects are actually formed in the boundary surface of the irradiated sample. There exists an upper and lower boundary for the excitation energy of nonlinear oscillations. The lower boundary depends on the potential unharmonicity, the upper one is defined by the energy of the atomic-bond breakdown, i.e . by the energy forming vacancie, and displaced atoms . Powerful oscillations of the atomic oscillators up to the bond breakdown occur at activation energies of point defect formation and the abovementioned effects are no longer relevant. 5. Conclusions
The following conclusions can be made : 1) Low-energy ion bombardment on a crystal lattice leads to nonlinear oscillations of atomic oscillators, which can result in the formation of stable non-equi-
librium structures of the lattice . These structures determine the new physical and mechanical properties of irradiated materials . 2) High-energy, in contrast to low-energy, ions result in the breakdown of atomic bonds, and the abovementioneJ effects are no longer relevant . References
H . Thompson, Defects and Radiation Damage in Metals (Cambridge University Press, 1969) . [2] F.F . Komarov, Ion Beam Modification of Metals (Gordon and Breach, Philadelphia, 1992). [3] E .V. Kozlov, I .V. Tereshko, V .I. Khodyrev, N.A . Popova, L .N. Ignatenko and E.A . Lipsky, 1zv. Vuzov Fiz. 1 (1992) 14, in Russian. [41 I.V . Tereshko, I .I. Silin, V .I. Khodyrev and E .A. Lipsky, Proc . Conf. on Physics of Metals, Kuibyshev, 1989, p. 17. [5] I .V . Tereshko, V .I. Khodyrev, V .M. Tereshko, E .A. Lipsky, A.V. Goncharenya and S . Ofosi-Sey, Computer modelling on modification of solids 5y low-energy ion beams, preprint no. 1, Mech. Eng . Inst ., Mogilev (1991) . [6] J. Stoer and R . Bulir-sch, Introduction to Numerical Analysis (Springer, New York, 1980). [1]
Ib. BASIC INTERACTIONS (b)