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Driven mobility of self-interstitial defects under electron irradiation
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 256 (2007) 253–259 www.elsevier.com/locate/nimb
Driven mobility of self-interstitial defects under electron irradiation S.L. Dudarev
a,*
, P.M. Derlet b, C.H. Woo
c
a
c
EURATOM/UKAEA Fusion Association, Culham Science Centre, Oxfordshire OX14 3DB, UK b Paul Scherrer Institute, CH 5232 Villigen PSI, Switzerland Department of Electronics and Information Engineering, Polytechnic University of Hong-Kong, Hung Hom, Kowloon, Hong Kong Available online 23 January 2007
Abstract In situ electron microscope observations of defects show that the incident high-energy electrons influence the evolution of microstructure of an irradiated material, reducing the number of defects seen in the field of view of the microscope. We investigate the origin of this phenomenon, using tungsten as a case study. We find that displacements of atoms due to electron impacts give rise to the stochastic jumps of a defect over several interatomic distances, but the frequency of these events under normal conditions is too low to influence the high thermal mobility of defects in a pure material. At the same time our analysis shows that migration of defects driven by electron impacts provides the dominant mechanism of diffusion in a material where the defects are pinned by solute atoms or impurities. 2006 Elsevier B.V. All rights reserved. PACS: 61.80.Az; 66.10.Cb; 67.80.Mg Keywords: Tungsten; Radiation defects; Modelling; Diffusion of defects; Electron irradiation; Electron microscopy; Molecular dynamics
1. Introduction Direct in situ electron microscope observation of evolution of microstructure under ion irradiation is one of the very few experimental methods capable of providing information about changes in the microstructure of an irradiated material occurring in real time [1–4]. Historically the very first direct electron microscope observation of migration of dislocations in a metal [5] was performed using an in situ technique, where the intense beam of high-energy electrons used for diffraction imaging caused the rapid motion of dislocation lines, attracting the attention of Whelan and Horne operating the microscope [6]. Very recently, in situ electron microscope observation of migration of prismatic dislocation loops [7] revealed new mechanisms of interaction between radiation and induced defects. These observations showed that migration of prismatic loops is significantly impeded by solute atoms or impurities
*
Corresponding author. Tel.: +44 1235 463513; fax: +44 1235 463435. E-mail address: [email protected] (S.L. Dudarev).
0168-583X/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.12.013
[8], resulting in the relatively high characteristic activation energy for migration Ea 1.3 eV. In situ observations of radiation damage effects occurring in FeCr alloys under heavy-ion irradiation also showed that the incident highenergy electrons induce stochastic motion of dislocation loops [9] and reduce the volume density of defects in the regions illuminated by the electron beam [10]. Since the known mechanisms of migration of radiation defects in materials are almost exclusively related to thermal activation, it is tempting to attribute the above observations to the local heating of the specimen by the beam of the incident high-energy electrons. However, numerical estimates show that the rapid dissipation of heat results in the temperature of a metallic foil illuminated by the high-energy electrons in a microscope increasing by only a few degrees centigrade [11], and local heating cannot account for the observed significant changes in the dynamics of microstructural evolution in the regions of the specimen illuminated by the beam of electrons. In this paper, we describe a mechanism for migration of defects that is likely to be responsible for these somewhat puzzling changes in the dynamics of microstructural
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S.L. Dudarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 253–259
evolution of an irradiated material during in situ electron microscope imaging. We find that the random displacements of atoms resulting from direct impacts of high-energy electrons stimulate migration of defects in a material even in the case where the ‘‘normal’’ thermally activated migration is suppressed by solute atoms or impurities. An unexpected and interesting aspect of this stimulated, or driven, migration of defects under electron irradiation is that only impacts of electrons on atoms forming the core of an interstitial defect contribute significantly to the fluctuating random force giving rise to the electron-irradiation-driven diffusion. The paper is organized as follows. We start from describing the basic formalism and molecular dynamics simulations of thermally activated diffusion of defects. Then we consider effects of electron impacts on atoms. We rationalize and interpret results of simulations using the multistring Frenkel–Kontorova model of an interstitial defect. Using this model and molecular-dynamics simulations, we investigate the part played by direct impacts of high-energy electrons on atoms in the core of the defect, and compare the diffusion coefficient characterizing this mechanism of migration with the observed values of diffusion constants describing migration of defects in an ultrapure and in a real material. 2. Thermally activated migration of crowdion defects A recent density functional study of self-interstitial atom defects in body-centred cubic (bcc) metals [12] shows that in all the non-magnetic bcc metals, including tungsten, the linear 1 1 1 crowdion configuration of a single self-interstitial atom defect has the lowest formation energy. Hence the long-range migration of interstitial defects in tungsten observed experimentally at very low temperatures [14] is associated with the translational one-dimensional mode of diffusion of the defects along one of the 1 1 1 crystallographic directions. The atomic structure of a 1 1 1 crowdion defect is shown in Fig. 1. The 1 1 1 string of atoms shown in darker colour contains an extra self-interstitial atom. The number of atoms in all the other 1 1 1 strings remains the same as in a perfect lattice. All the strings in the structure shown in Fig. 1 are deformed due to the presence of an extra self-interstitial atom in the simulation cell. The central string is in the state of compression while some parts of the neighbouring strings are under tension. In molecular-dynamics simulations Brownian motion of the crowdion was followed over a relatively long interval of time of the order of 1 ns. The Finnis–Sinclair-type manybody interatomic potential describing interaction between atoms in tungsten was constructed by adjusting the form of two functions q(r) and V(r) entering the expression for the total energy of a configuration ffi X sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X X E ¼ A qðrab Þ þ V ðrab Þ; rab ¼ jra rb j: ð1Þ a
b6¼a
a;b;b6¼a
Fig. 1. The fully relaxed structure of a 1 1 1 crowdion defect in tungsten. Atoms shown in darker colour form a collective object (a crowdion) that is able to glide in the 1 1 1 crystallographic direction through the lattice [15]. Atomic bonds linking the nearest neighbour atoms are shown to illustrate the symmetry of the defect.
Functions q(r) and V(r) were fitted to match the expression for the energy (1), as well as its first, second and the third derivatives, to the observed values of elastic constants, the equilibrium lattice parameter, the cohesive energy, and the low-temperature thermal expansion coefficient. Since fitting a many-body potential only to the equilibrium lattice structure does not guarantee the desired accuracy of the calculated defect formation energies, we included the formation energies of a vacancy and of several self-interstitial defect configurations [12] into the objective function for the fitting algorithm. The resulting values of the defect formation energies calculated using the new many-body potential are given in Table 1. Functions q(r) and V(r) are approximated by cubic splines qðrÞ ¼
7 X
3
qi ðri rÞ Hðri rÞ;
i¼1
V ðrÞ ¼
9 X
ð2Þ 3
V i ðri rÞ Hðri rÞ;
i¼1
Table 1 Comparison between the formation energies of point defects in tungsten evaluated using the many-body potential described in this paper, and the values found using density functional theory (DFT) [12] Formation energy (eV)
Vacancy
111
110
Tetra
100
Octa
MD DFT Difference
3.557 3.56 0.003
9.492 9.551 0.06
9.776 9.844 0.07
10.919 11.05 0.13
11.342 11.49 0.15
11.573 11.68 0.11
Many-body potential calculations were performed using a 2001 atom cell. DFT values refer to a 129 atom cell.
S.L. Dudarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 253–259
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Table 2 Parameters of the many-body potential of interaction between atoms in tungsten used in molecular dynamics simulations ˚ 3) ˚ ˚ qi (A rVi ðAÞ i rqi ðAÞ
˚ 3) Vi (eV*A
1 2 3 4 5 6 7 8 9
0.1036435865158945 0.2912948318493851 2.096765499656263 19.16045452701010 41.01619862085917 46.05205617244703 26.42203930654883 15.35211507804088 14.12806259323987
3.321600 3.165333333 3.009066666 2.852800 2.741100 2.604045 2.466990
43.30492851387751 43.19656227920225 122.2997872906446 203.2056800288892 9.7911640643983748 Æ 103 159.8326343460623 24.69647315288123
4.268900 3.985680 3.702460 3.419240 3.136020 2.852800 2.741100 2.604045 2.466990
These parameters form a part of a more extensive set of potentials describing interatomic interactions in vanadium, niobium, tantalum, molybdenum and tungsten [16]. The value of parameter A in Eq. (1) is A = 2.133860034294818 eV.
where H(x) is the Heaviside function, H(x) = 1 for x > 0 and H(x) = 0 for x < 0. The positions of the knot points and values of coefficients qi and Vi entering Eq. (2) are given in Table 2. The potential has not been corrected in the core region to match the universal potential [13] that approximates the energy of interaction between high-energy ions at short distances. The derivation of the correct form of the potential in the region of intermediate energies warrants a further study addressing the asymptotic behaviour of the band term and the effect of electron energy losses. At a finite temperature the collective defect object shown in Fig. 1 performs random Brownian motion in the lattice. To investigate the statistics of this motion we carried out molecular dynamics simulations under thermal equilibrium conditions at several different temperatures. The simulations were performed using a relatively large cell containing 60,000 atoms. In each simulation, Brownian motion of the crowdion was followed over a relatively long interval of time of the order of 1 ns. A typical trajectory of Brown-
40
ian motion of a crowdion defect is shown in Fig. 2. In what follows we show that Brownian motion of the defect is driven by displacements of atoms from their equilibrium positions in the lattice. But first we investigate the statistics of atomic displacements associated with impacts of highenergy electrons on atoms. 3. Collisions of high-energy electrons with atoms The double differential cross-section of scattering of a high-energy electron by an atom in a lattice has the form [17] d2 r dr p p0 E ¼ ; S ; ð3Þ dodE do h h where do denotes an element of the solid angle and E = hx is the energy lost by the electron. The structure factor S(j,x) is given by Z 1 X 1 1 Sðj; xÞ ¼ eEn =kB T dteixt 2ph 1 Z n X hnj expðij ^uð0Þjlihlj expðij ^ uðtÞjni; ð4Þ
Coordinates of the defect (Angstrom)
l
X coordinate Y coordinate Z coordinate
20
0
-20
-40
-60
-80
-100 0
200
400
600
800
Time (picoseconds) Fig. 2. A trajectory of the centre of a crowdion defect performing Brownian motion in crystalline tungsten at T = 800 K. The diffusion coefficient corresponding to this trajectory approximately equals D = 0.52 Æ 108 m2/s.
P where j = (p p 0 )/h, Z ¼ n expðEn =k B T Þ is the partition function, and ^uð0Þ and ^uðtÞ are the Heisenberg operators of atomic displacements. In the case of a Bravais lattice the operator of atomic displacement expressed in terms of phonon creation and annihilation operators has the form [18] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h X h ^uðtÞ ¼ eðf; aÞ^af;a eifRa ixðf;aÞt 2MN xðf; aÞ 0 f;a i þ e ðf; aÞ^ayf;a eifRa þixðf;aÞt ; ð5Þ where summation over f and a is performed over the wave vectors and branches of the phonon spectrum of the material. e(f, a) is the polarization vector of a phonon excitation and x(f, a) is the frequency of this excitation. To find the cross-section of generation of phonons on electron impact we expand the exponential functions of displacements in (4) as a product of the individual Taylor
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series, and note that only terms linear and quadratic in atomic displacements contribute to the probability of creation and annihilation of phonon excitations. The cubic 1=2 terms are of the order of N 0 and can be safely neglected. The resulting expression for the structure factor is Z 1 1 dt exp½ixt þ /ðtÞ /ð0Þ; ð6Þ Sðj; xÞ ¼ 2p h 1 where /ðtÞ ¼
h X ½j eðf; aÞ2 2MN 0 f;a xðf; aÞ ½ nðf; aÞ þ 1eixðf;aÞt þ nðf; aÞeixðf;aÞt ;
ð7Þ
R = 2E0(m/M ) transferred on impact is R = 0.6 eV. For the current density of the incident beam j = 1024 m2 s1 [8] the average time between electron impacts is s (jrtr)1 0.08 s. The atom that receives an impact is displaced from its equilibrium position by the vector u, the magnitude of which can be estimated from the relation 2 u2 =2. Depending on the choice of the estimate for R ¼ Mx we find that the the effective frequency of vibrations x ˚ and 1 A ˚ . These value of u lies in the range between 0.2 A values are many times the amplitude of thermal vibrations of atoms at room temperature. Hence the collisions of high-energy electrons with atoms in a crystal give rise to the relatively infrequent but fairly large abrupt displacements of atoms from the lattice sites.
and nðf; aÞ ¼
1 expð hxðf; aÞ=k B T Þ 1
4. Driven diffusion of crowdions
is the average equilibrium occupation number of a phonon state. The Fourier-transform of the exponent of /(t) in (6) describes processes of multiple generation and absorption of phonons occurring in a collision between the incident high-energy electron and the atom. The average energy transferred to phonons in the collision is Z 1 h d exp½/ðtÞ /ð0Þ dðhxÞ hxSðj; xÞ ¼ i dt 1 t¼0 ¼
2 j2 h : 2M
ð8Þ
To derive this equation, we used the condition of completeness of a set of polarization vectors XZ d3 f X0 e ðf; aÞej ðf; aÞ ¼ dij : 3 i ð2pÞ Brillouin zone a Formula (8) has a simple meaning, namely, it shows that the average energy transferred to an atom in a lattice on impact with a high-energy electron equals the classical recoil energy (p p 0 )2/2M, where p p 0 is the momentum transferred to an atom by a high-energy electron and M is the mass of the atom. Eq. (8) can be recast in a more convenient form by noting that in an elastic collision p Æ p 0 = p2cosh, where h is the angle of scattering, and 2 2 Z Z 2 dr hj dr ðp p0 Þ do ¼ do do 2M do 2M Z p2 dr do ð1 cos hÞ ¼ do M m ð9Þ ¼ 2E0 rtr ; M where E0 = p2/2m is the energy of the incident high-energy electrons and rtr is the transport cross-section of scattering. The transport cross-section of scattering of a 100 keV incident electron by an atom of tungsten is rtr = 1.28 · 1019 cm2 [19], and the average recoil energy
What is the origin of the force that gives rise to the stochastic motion of the defect illustrated in Fig. 2? To address this question we need to develop a model describing the crowdion as an individual object characterized by the position of its centre of mass, and by parameters describing its interaction with the surrounding crystal lattice. The field of atomic displacements in the crowdion is well approximated by the solution of the sine-Gordon equation [15]. This suggests that the interaction of a crowdion with random displacements of atoms in the lattice can be described by the multistring Frenkel–Kontorova model. This model takes into account displacements of atoms in all the strings forming the crystal lattice. The Lagrangian of the multistring Frenkel–Kontorova model has the form [15] " # 1 X X M z_ 2n;j a 2 L¼ ðznþ1;j zn;j aÞ 2 2 j n¼1
1 Mx2 a2 X X 2 pðzn;j zn;jþh Þ sin : ð10Þ a 2p2 j;h n¼1 In this equation, summation over vector index j = (jx,jy) is performed over atomic strings in the plane normal to the axes of the strings, and summation over h is performed over the nearest neighbour strings (for example, in the bcc lattice each h1 1 1i atomic string has six nearest neighbour strings). n is the index of an atom in a string, M is the mass of an atom, and variable zn,j is the generalized coordinate of atom n in string j. a is the equilibrium distance between atoms in a h1 1 1i string, a is a factor characterizing the strength of elastic interaction between two neighbouring atoms in the same string, and x2 is a parameter describing the strength of interaction between atoms in the neighbouring strings. In the limiting case where the crystal contains no interstitial defects, atoms are situated ð0Þ at the equilibrium positions zn;j ¼ an. To investigate the migration of defects driven by electron impacts it is convenient to introduce displacements of atoms from their equilibrium positions un,j = zn,j an
S.L. Dudarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 253–259
as a set of variables describing an atomic configuration. The equations of motion now acquire the form d2 un;j ¼ aðunþ1;j þ un1;j 2un;j Þ dt2
Mx2 a X 2pðun;j un;jþh Þ sin : p a h
-0.05
ð11Þ
To derive the equation of motion for a crowdion defect we write the field of atomic displacements in the form of a sum ZðtÞ un;j ðtÞ ¼ Uj n ð12Þ þ Un;j ðtÞ; a where Uj ðnÞ is the strain field of the defect (this field is assumed to be centred around the instantaneous position of the defect ZðtÞ) and Un,j(t) are the stochastic displacements associated with thermal phonon excitations and with random electron impacts. Substituting (12) into (11) and performing the summation over n, after some algebra we find that the coordinate ZðtÞ of the crowdion defect located in the central string j = 0, satisfies the equation
X d2 ZðtÞ ZðtÞ 2 p 2 U0 n ¼ 4x ½Un0 ðtÞ Un;h ðtÞ sin : dt2 a a n;h ð13Þ Using the analytical solution for the displacement field of the defect [15]
ZðtÞ 2a ZðtÞ 1 U0 n arctan exp N n ¼ ; a p a ð14Þ where N is a parameter characterizing the width of the core region of the crowdion, we arrive at the equation of motion of the defect in the form X d2 ZðtÞ ¼ 4x2 ½Un0 ðtÞ Un;h ðtÞ 2 dt n;h
ZðtÞ 2 1 n : cosh N a
0
ð15Þ
This equation shows that the force acting on a crowdion in a lattice is proportional to the difference between the stochastic (i.e. not associated with the strain field of the crowdion itself) displacements of atoms in the central string containing the interstitial atom, and in the strings surrounding it. The weighting factor cosh2 ½ðn ZðtÞ=aÞ=N, which determines the relative contribution of displacements of atoms to the force acting on a crowdion, is proportional to the square of the strain field of the crowdion
1 o ZðtÞ 1 ZðtÞ U0 n cosh1 N1 n ¼ : a on a pN a ð16Þ Fig. 3 shows that the analytical Frenkel–Kontorova solution (14) agrees very well with results of atomistic simula-
Strain (Un+1-Un)/a
M
257
the crowdion solution empirical potential
-0.1
-0.15
-0.2
0
5 10 15 20 Index of an atom in the string
Fig. 3. Comparison of the strain field of a 1 1 1 crowdion in tungsten given by the solution of the Frenkel–Kontorova model (14) and plotted for N ¼ 1:8 and Z=a ¼ 11:5, with the discrete values of strain calculated from the coordinates of atoms forming the central string of the structure shown in Fig 1.
tions performed using the potential described in Table 2. The comparison between the analytical formula (14) and atomistic simulations confirms the accuracy of approximations underlying the derivation of the equation of motion of the crowdion (15). The examination of Eq. (15) also shows that only displacements of atoms in the core of the defect are significant in terms of inducing its migration through the lattice. To verify this conclusion we performed molecular dynamics simulations where various atoms in the cell containing a crowdion defect were given random initial impacts representing recoils due to collisions of electrons with atoms. A series of snapshots describing one of the simulations is shown in Fig. 4. This figure illustrates the fact that only the electron impacts on atoms situated in the core of the defect give rise to a significant displacement of the defect from its original position. This is confirmed by a simulation where the shock wave of a fairly large collision cascade occurring a small distance away from the core of the defect did not result in any appreciable movement of the defect. The latter fully agrees with the equation of motion (15), according to which a coherent displacement of atoms due to a wave propagating through the core of a defect does not result in a significant force acting on the defect. What is the value of the diffusion coefficient associated with the electron impacts? Since the impacts occur with equal probability in the central string and in the neighbouring strings surrounding the crowdion, the force acting on the defect randomly changes sign, giving rise to the stochastic Brownian motion of the defect in the 1 1 1 direction. The resulting coefficient of diffusion can be estimated as D jrtr l2 ;
ð17Þ
where l is the average displacement of a defect due to a single impact of a high-energy electron on an atom situated in
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0 ps
0.162 ps
(a) 0.516 ps
(b)
5. Summary
66.28 ps
(c)
the presence of solute atoms or impurities. Furthermore, electron irradiation catalyzes the thermally activated diffusion by unpinning the defects from impurities. Hence we arrive at the conclusion that while driven diffusion of defects is too slow if compared with the rate of migration in an ideal pure material found in molecular dynamics simulations, it probably provides the dominant mechanism of migration of defects in a real material where thermally activated migration is impeded by impurities or solute atoms.
(d)
Fig. 4. A sequence of snapshots illustrating the mechanism of driven migration of a defect under electron irradiation. A small displacement of an atom situated in the core of the crowdion in the (1 2 3) direction due to an electron impact seen in (a) gives rise to a small local cascade (b), (c), ˚ in which in the limit t ! 1 displaces the defect by approximately 16.5 A the 1 1 1 direction. In this simulation the defect moved to its final position after approximately 1 ps, and remained stationary until the end of the simulation.
˚ and the core region of the defect. Assuming that l = 10 A 1 17 2 jrtr = 10 s , we find that D 10 m /s. Hence the rate of driven diffusion is approximately nine orders of magnitude lower than the rate of diffusion of defects in a pure material estimated from the simulated trajectory of Brownian motion shown in Fig. 2. However, given that diffusion driven by electron impacts is independent of the temperature of the material, we find that it becomes the dominant mechanism of migration in the limit of relatively low temperatures in the case where the defects are pinned by solute atoms or impurities. For example, Arakawa [8] finds that the coefficient of diffusion of small dislocation loops in relatively pure iron is well approximated by the expression 1:3 eV 6 D 9:5 10 exp m2 =s; kBT which for room temperature T = 300 K gives D = 2.47 Æ 1028 m2/s. This value is 11 orders of magnitude lower than the above estimate for the coefficient of driven diffusion. While pinning of defects by impurities or solute atoms impedes the thermally activated mobility of defects, it does not affect the mechanism of driven migration associated with electron impacts. Indeed, the recoil energy is comparable with the characteristic energy of pinning, and hence the long-range displacements of defects resulting from impacts of high-energy electrons on atoms are almost unaffected by
We performed a comprehensive theoretical investigation of a new mechanism of diffusion of defects in a material under in situ high-energy electron irradiation in an electron microscope. We found that the migration of defects may result from impacts of high-energy electrons on atoms in the core region of a defect. Atomistic simulations and numerical estimates show that driven diffusion likely provides the dominant mechanism of migration of defects in real materials irradiated by a beam of high-energy electrons. Acknowledgements The authors thank Dr. M. L. Jenkins and Dr. K. Arakawa for stimulating discussions. Work at UKAEA was funded by the UK Engineering and Physical Sciences Research Council (EPSRC), by EURATOM and by the EXTREMAT integrated project. Two of the authors, C.H.W. and S.L.D., gratefully acknowledge support of PolyU Grants 5322/04E and PolyU HZF20. References [1] M.L. Jenkins, M.A. Kirk, Characterisation of Radiation Damage by Transmission Electron Microscopy, Institute of Physics, Bristol, 2001. [2] T.L. Daulton, M.A. Kirk, L.E. Rehn, Philos. Mag. A 80 (2000) 809. [3] T.L. Daulton, M.A. Kirk, L.E. Rehn, J. Nucl. Mater. 276 (2000) 258. [4] M.L. Jenkins, G.J. Hardy, M.A. Kirk, Mater. Sci. Forum 15–18 (1987) 901. [5] P.B. Hirsch, R.W. Horne, M.J. Whelan, Philos. Mag. 1 (1956) 677. [6] M.J. Whelan, J. Electron Microsc. Tech. 3 (1986) 109. [7] K. Arakawa, M. Hatanaka, E. Kuramoto, K. Ono, H. Mori, Phys. Rev. Lett. 96 (2006) 125506. [8] K. Arakawa, Dynamics of Nanometer-Sized Interstitial-Type Dislocation Loops in Iron by In-situ Transmission Electron Microscopy, a presentation given at UKAEA Culham in February 2006, unpublished. [9] Z. Yao, M.L. Jenkins, M.A. Kirk, Program and Abstracts of the 23rd Symposium on Effects of Radiation on Materials, San Jose, California, 13–15 June, 2006 (ASTM International, West Conshohocken, PA) p. 58. [10] M.L. Jenkins, private communication. [11] L. Reimer, Transmission Electron Microscopy, Springer, Berlin, 1984, p. 427. [12] D. Nguyen-Manh, A.P. Horsfield, S.L. Dudarev, Phys. Rev. B 73 (2006) 020101. [13] V.Y. Young, N. Welcome, G.B. Hoflund, Phys. Rev. B 48 (1993) 2981. [14] F. Dausinger, H. Schultz, Phys. Rev. Lett. 35 (1975) 1773. [15] S.L. Dudarev, Philos. Mag. 83 (2003) 3577.
S.L. Dudarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 253–259 [16] P.M. Derlet, D. Nguyen-Manh, S.L. Dudarev, Phys. Rev. B, submitted for publication. [17] L.-M. Peng, S.L. Dudarev, M.J. Whelan, High-Energy Electron Diffraction and Microscopy, Oxford University Press, 2004.
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[18] W. Marshall, S.W. Lovesey, Theory of Thermal Neutron Scattering, Clarendon Press, Oxford, 1971, p. 73. [19] R. Mayol, F. Salvat, Atomic Data Nucl. Data Tables 65 (1997) 55.
Nuclear Instruments and Methods in Physics Research B 256 (2007) 253–259 www.elsevier.com/locate/nimb
Driven mobility of self-interstitial defects under electron irradiation S.L. Dudarev
a,*
, P.M. Derlet b, C.H. Woo
c
a
c
EURATOM/UKAEA Fusion Association, Culham Science Centre, Oxfordshire OX14 3DB, UK b Paul Scherrer Institute, CH 5232 Villigen PSI, Switzerland Department of Electronics and Information Engineering, Polytechnic University of Hong-Kong, Hung Hom, Kowloon, Hong Kong Available online 23 January 2007
Abstract In situ electron microscope observations of defects show that the incident high-energy electrons influence the evolution of microstructure of an irradiated material, reducing the number of defects seen in the field of view of the microscope. We investigate the origin of this phenomenon, using tungsten as a case study. We find that displacements of atoms due to electron impacts give rise to the stochastic jumps of a defect over several interatomic distances, but the frequency of these events under normal conditions is too low to influence the high thermal mobility of defects in a pure material. At the same time our analysis shows that migration of defects driven by electron impacts provides the dominant mechanism of diffusion in a material where the defects are pinned by solute atoms or impurities. 2006 Elsevier B.V. All rights reserved. PACS: 61.80.Az; 66.10.Cb; 67.80.Mg Keywords: Tungsten; Radiation defects; Modelling; Diffusion of defects; Electron irradiation; Electron microscopy; Molecular dynamics
1. Introduction Direct in situ electron microscope observation of evolution of microstructure under ion irradiation is one of the very few experimental methods capable of providing information about changes in the microstructure of an irradiated material occurring in real time [1–4]. Historically the very first direct electron microscope observation of migration of dislocations in a metal [5] was performed using an in situ technique, where the intense beam of high-energy electrons used for diffraction imaging caused the rapid motion of dislocation lines, attracting the attention of Whelan and Horne operating the microscope [6]. Very recently, in situ electron microscope observation of migration of prismatic dislocation loops [7] revealed new mechanisms of interaction between radiation and induced defects. These observations showed that migration of prismatic loops is significantly impeded by solute atoms or impurities
*
Corresponding author. Tel.: +44 1235 463513; fax: +44 1235 463435. E-mail address: [email protected] (S.L. Dudarev).
0168-583X/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.12.013
[8], resulting in the relatively high characteristic activation energy for migration Ea 1.3 eV. In situ observations of radiation damage effects occurring in FeCr alloys under heavy-ion irradiation also showed that the incident highenergy electrons induce stochastic motion of dislocation loops [9] and reduce the volume density of defects in the regions illuminated by the electron beam [10]. Since the known mechanisms of migration of radiation defects in materials are almost exclusively related to thermal activation, it is tempting to attribute the above observations to the local heating of the specimen by the beam of the incident high-energy electrons. However, numerical estimates show that the rapid dissipation of heat results in the temperature of a metallic foil illuminated by the high-energy electrons in a microscope increasing by only a few degrees centigrade [11], and local heating cannot account for the observed significant changes in the dynamics of microstructural evolution in the regions of the specimen illuminated by the beam of electrons. In this paper, we describe a mechanism for migration of defects that is likely to be responsible for these somewhat puzzling changes in the dynamics of microstructural
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evolution of an irradiated material during in situ electron microscope imaging. We find that the random displacements of atoms resulting from direct impacts of high-energy electrons stimulate migration of defects in a material even in the case where the ‘‘normal’’ thermally activated migration is suppressed by solute atoms or impurities. An unexpected and interesting aspect of this stimulated, or driven, migration of defects under electron irradiation is that only impacts of electrons on atoms forming the core of an interstitial defect contribute significantly to the fluctuating random force giving rise to the electron-irradiation-driven diffusion. The paper is organized as follows. We start from describing the basic formalism and molecular dynamics simulations of thermally activated diffusion of defects. Then we consider effects of electron impacts on atoms. We rationalize and interpret results of simulations using the multistring Frenkel–Kontorova model of an interstitial defect. Using this model and molecular-dynamics simulations, we investigate the part played by direct impacts of high-energy electrons on atoms in the core of the defect, and compare the diffusion coefficient characterizing this mechanism of migration with the observed values of diffusion constants describing migration of defects in an ultrapure and in a real material. 2. Thermally activated migration of crowdion defects A recent density functional study of self-interstitial atom defects in body-centred cubic (bcc) metals [12] shows that in all the non-magnetic bcc metals, including tungsten, the linear 1 1 1 crowdion configuration of a single self-interstitial atom defect has the lowest formation energy. Hence the long-range migration of interstitial defects in tungsten observed experimentally at very low temperatures [14] is associated with the translational one-dimensional mode of diffusion of the defects along one of the 1 1 1 crystallographic directions. The atomic structure of a 1 1 1 crowdion defect is shown in Fig. 1. The 1 1 1 string of atoms shown in darker colour contains an extra self-interstitial atom. The number of atoms in all the other 1 1 1 strings remains the same as in a perfect lattice. All the strings in the structure shown in Fig. 1 are deformed due to the presence of an extra self-interstitial atom in the simulation cell. The central string is in the state of compression while some parts of the neighbouring strings are under tension. In molecular-dynamics simulations Brownian motion of the crowdion was followed over a relatively long interval of time of the order of 1 ns. The Finnis–Sinclair-type manybody interatomic potential describing interaction between atoms in tungsten was constructed by adjusting the form of two functions q(r) and V(r) entering the expression for the total energy of a configuration ffi X sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X X E ¼ A qðrab Þ þ V ðrab Þ; rab ¼ jra rb j: ð1Þ a
b6¼a
a;b;b6¼a
Fig. 1. The fully relaxed structure of a 1 1 1 crowdion defect in tungsten. Atoms shown in darker colour form a collective object (a crowdion) that is able to glide in the 1 1 1 crystallographic direction through the lattice [15]. Atomic bonds linking the nearest neighbour atoms are shown to illustrate the symmetry of the defect.
Functions q(r) and V(r) were fitted to match the expression for the energy (1), as well as its first, second and the third derivatives, to the observed values of elastic constants, the equilibrium lattice parameter, the cohesive energy, and the low-temperature thermal expansion coefficient. Since fitting a many-body potential only to the equilibrium lattice structure does not guarantee the desired accuracy of the calculated defect formation energies, we included the formation energies of a vacancy and of several self-interstitial defect configurations [12] into the objective function for the fitting algorithm. The resulting values of the defect formation energies calculated using the new many-body potential are given in Table 1. Functions q(r) and V(r) are approximated by cubic splines qðrÞ ¼
7 X
3
qi ðri rÞ Hðri rÞ;
i¼1
V ðrÞ ¼
9 X
ð2Þ 3
V i ðri rÞ Hðri rÞ;
i¼1
Table 1 Comparison between the formation energies of point defects in tungsten evaluated using the many-body potential described in this paper, and the values found using density functional theory (DFT) [12] Formation energy (eV)
Vacancy
111
110
Tetra
100
Octa
MD DFT Difference
3.557 3.56 0.003
9.492 9.551 0.06
9.776 9.844 0.07
10.919 11.05 0.13
11.342 11.49 0.15
11.573 11.68 0.11
Many-body potential calculations were performed using a 2001 atom cell. DFT values refer to a 129 atom cell.
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Table 2 Parameters of the many-body potential of interaction between atoms in tungsten used in molecular dynamics simulations ˚ 3) ˚ ˚ qi (A rVi ðAÞ i rqi ðAÞ
˚ 3) Vi (eV*A
1 2 3 4 5 6 7 8 9
0.1036435865158945 0.2912948318493851 2.096765499656263 19.16045452701010 41.01619862085917 46.05205617244703 26.42203930654883 15.35211507804088 14.12806259323987
3.321600 3.165333333 3.009066666 2.852800 2.741100 2.604045 2.466990
43.30492851387751 43.19656227920225 122.2997872906446 203.2056800288892 9.7911640643983748 Æ 103 159.8326343460623 24.69647315288123
4.268900 3.985680 3.702460 3.419240 3.136020 2.852800 2.741100 2.604045 2.466990
These parameters form a part of a more extensive set of potentials describing interatomic interactions in vanadium, niobium, tantalum, molybdenum and tungsten [16]. The value of parameter A in Eq. (1) is A = 2.133860034294818 eV.
where H(x) is the Heaviside function, H(x) = 1 for x > 0 and H(x) = 0 for x < 0. The positions of the knot points and values of coefficients qi and Vi entering Eq. (2) are given in Table 2. The potential has not been corrected in the core region to match the universal potential [13] that approximates the energy of interaction between high-energy ions at short distances. The derivation of the correct form of the potential in the region of intermediate energies warrants a further study addressing the asymptotic behaviour of the band term and the effect of electron energy losses. At a finite temperature the collective defect object shown in Fig. 1 performs random Brownian motion in the lattice. To investigate the statistics of this motion we carried out molecular dynamics simulations under thermal equilibrium conditions at several different temperatures. The simulations were performed using a relatively large cell containing 60,000 atoms. In each simulation, Brownian motion of the crowdion was followed over a relatively long interval of time of the order of 1 ns. A typical trajectory of Brown-
40
ian motion of a crowdion defect is shown in Fig. 2. In what follows we show that Brownian motion of the defect is driven by displacements of atoms from their equilibrium positions in the lattice. But first we investigate the statistics of atomic displacements associated with impacts of highenergy electrons on atoms. 3. Collisions of high-energy electrons with atoms The double differential cross-section of scattering of a high-energy electron by an atom in a lattice has the form [17] d2 r dr p p0 E ¼ ; S ; ð3Þ dodE do h h where do denotes an element of the solid angle and E = hx is the energy lost by the electron. The structure factor S(j,x) is given by Z 1 X 1 1 Sðj; xÞ ¼ eEn =kB T dteixt 2ph 1 Z n X hnj expðij ^uð0Þjlihlj expðij ^ uðtÞjni; ð4Þ
Coordinates of the defect (Angstrom)
l
X coordinate Y coordinate Z coordinate
20
0
-20
-40
-60
-80
-100 0
200
400
600
800
Time (picoseconds) Fig. 2. A trajectory of the centre of a crowdion defect performing Brownian motion in crystalline tungsten at T = 800 K. The diffusion coefficient corresponding to this trajectory approximately equals D = 0.52 Æ 108 m2/s.
P where j = (p p 0 )/h, Z ¼ n expðEn =k B T Þ is the partition function, and ^uð0Þ and ^uðtÞ are the Heisenberg operators of atomic displacements. In the case of a Bravais lattice the operator of atomic displacement expressed in terms of phonon creation and annihilation operators has the form [18] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h X h ^uðtÞ ¼ eðf; aÞ^af;a eifRa ixðf;aÞt 2MN xðf; aÞ 0 f;a i þ e ðf; aÞ^ayf;a eifRa þixðf;aÞt ; ð5Þ where summation over f and a is performed over the wave vectors and branches of the phonon spectrum of the material. e(f, a) is the polarization vector of a phonon excitation and x(f, a) is the frequency of this excitation. To find the cross-section of generation of phonons on electron impact we expand the exponential functions of displacements in (4) as a product of the individual Taylor
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series, and note that only terms linear and quadratic in atomic displacements contribute to the probability of creation and annihilation of phonon excitations. The cubic 1=2 terms are of the order of N 0 and can be safely neglected. The resulting expression for the structure factor is Z 1 1 dt exp½ixt þ /ðtÞ /ð0Þ; ð6Þ Sðj; xÞ ¼ 2p h 1 where /ðtÞ ¼
h X ½j eðf; aÞ2 2MN 0 f;a xðf; aÞ ½ nðf; aÞ þ 1eixðf;aÞt þ nðf; aÞeixðf;aÞt ;
ð7Þ
R = 2E0(m/M ) transferred on impact is R = 0.6 eV. For the current density of the incident beam j = 1024 m2 s1 [8] the average time between electron impacts is s (jrtr)1 0.08 s. The atom that receives an impact is displaced from its equilibrium position by the vector u, the magnitude of which can be estimated from the relation 2 u2 =2. Depending on the choice of the estimate for R ¼ Mx we find that the the effective frequency of vibrations x ˚ and 1 A ˚ . These value of u lies in the range between 0.2 A values are many times the amplitude of thermal vibrations of atoms at room temperature. Hence the collisions of high-energy electrons with atoms in a crystal give rise to the relatively infrequent but fairly large abrupt displacements of atoms from the lattice sites.
and nðf; aÞ ¼
1 expð hxðf; aÞ=k B T Þ 1
4. Driven diffusion of crowdions
is the average equilibrium occupation number of a phonon state. The Fourier-transform of the exponent of /(t) in (6) describes processes of multiple generation and absorption of phonons occurring in a collision between the incident high-energy electron and the atom. The average energy transferred to phonons in the collision is Z 1 h d exp½/ðtÞ /ð0Þ dðhxÞ hxSðj; xÞ ¼ i dt 1 t¼0 ¼
2 j2 h : 2M
ð8Þ
To derive this equation, we used the condition of completeness of a set of polarization vectors XZ d3 f X0 e ðf; aÞej ðf; aÞ ¼ dij : 3 i ð2pÞ Brillouin zone a Formula (8) has a simple meaning, namely, it shows that the average energy transferred to an atom in a lattice on impact with a high-energy electron equals the classical recoil energy (p p 0 )2/2M, where p p 0 is the momentum transferred to an atom by a high-energy electron and M is the mass of the atom. Eq. (8) can be recast in a more convenient form by noting that in an elastic collision p Æ p 0 = p2cosh, where h is the angle of scattering, and 2 2 Z Z 2 dr hj dr ðp p0 Þ do ¼ do do 2M do 2M Z p2 dr do ð1 cos hÞ ¼ do M m ð9Þ ¼ 2E0 rtr ; M where E0 = p2/2m is the energy of the incident high-energy electrons and rtr is the transport cross-section of scattering. The transport cross-section of scattering of a 100 keV incident electron by an atom of tungsten is rtr = 1.28 · 1019 cm2 [19], and the average recoil energy
What is the origin of the force that gives rise to the stochastic motion of the defect illustrated in Fig. 2? To address this question we need to develop a model describing the crowdion as an individual object characterized by the position of its centre of mass, and by parameters describing its interaction with the surrounding crystal lattice. The field of atomic displacements in the crowdion is well approximated by the solution of the sine-Gordon equation [15]. This suggests that the interaction of a crowdion with random displacements of atoms in the lattice can be described by the multistring Frenkel–Kontorova model. This model takes into account displacements of atoms in all the strings forming the crystal lattice. The Lagrangian of the multistring Frenkel–Kontorova model has the form [15] " # 1 X X M z_ 2n;j a 2 L¼ ðznþ1;j zn;j aÞ 2 2 j n¼1
1 Mx2 a2 X X 2 pðzn;j zn;jþh Þ sin : ð10Þ a 2p2 j;h n¼1 In this equation, summation over vector index j = (jx,jy) is performed over atomic strings in the plane normal to the axes of the strings, and summation over h is performed over the nearest neighbour strings (for example, in the bcc lattice each h1 1 1i atomic string has six nearest neighbour strings). n is the index of an atom in a string, M is the mass of an atom, and variable zn,j is the generalized coordinate of atom n in string j. a is the equilibrium distance between atoms in a h1 1 1i string, a is a factor characterizing the strength of elastic interaction between two neighbouring atoms in the same string, and x2 is a parameter describing the strength of interaction between atoms in the neighbouring strings. In the limiting case where the crystal contains no interstitial defects, atoms are situated ð0Þ at the equilibrium positions zn;j ¼ an. To investigate the migration of defects driven by electron impacts it is convenient to introduce displacements of atoms from their equilibrium positions un,j = zn,j an
S.L. Dudarev et al. / Nucl. Instr. and Meth. in Phys. Res. B 256 (2007) 253–259
as a set of variables describing an atomic configuration. The equations of motion now acquire the form d2 un;j ¼ aðunþ1;j þ un1;j 2un;j Þ dt2
Mx2 a X 2pðun;j un;jþh Þ sin : p a h
-0.05
ð11Þ
To derive the equation of motion for a crowdion defect we write the field of atomic displacements in the form of a sum ZðtÞ un;j ðtÞ ¼ Uj n ð12Þ þ Un;j ðtÞ; a where Uj ðnÞ is the strain field of the defect (this field is assumed to be centred around the instantaneous position of the defect ZðtÞ) and Un,j(t) are the stochastic displacements associated with thermal phonon excitations and with random electron impacts. Substituting (12) into (11) and performing the summation over n, after some algebra we find that the coordinate ZðtÞ of the crowdion defect located in the central string j = 0, satisfies the equation
X d2 ZðtÞ ZðtÞ 2 p 2 U0 n ¼ 4x ½Un0 ðtÞ Un;h ðtÞ sin : dt2 a a n;h ð13Þ Using the analytical solution for the displacement field of the defect [15]
ZðtÞ 2a ZðtÞ 1 U0 n arctan exp N n ¼ ; a p a ð14Þ where N is a parameter characterizing the width of the core region of the crowdion, we arrive at the equation of motion of the defect in the form X d2 ZðtÞ ¼ 4x2 ½Un0 ðtÞ Un;h ðtÞ 2 dt n;h
ZðtÞ 2 1 n : cosh N a
0
ð15Þ
This equation shows that the force acting on a crowdion in a lattice is proportional to the difference between the stochastic (i.e. not associated with the strain field of the crowdion itself) displacements of atoms in the central string containing the interstitial atom, and in the strings surrounding it. The weighting factor cosh2 ½ðn ZðtÞ=aÞ=N, which determines the relative contribution of displacements of atoms to the force acting on a crowdion, is proportional to the square of the strain field of the crowdion
1 o ZðtÞ 1 ZðtÞ U0 n cosh1 N1 n ¼ : a on a pN a ð16Þ Fig. 3 shows that the analytical Frenkel–Kontorova solution (14) agrees very well with results of atomistic simula-
Strain (Un+1-Un)/a
M
257
the crowdion solution empirical potential
-0.1
-0.15
-0.2
0
5 10 15 20 Index of an atom in the string
Fig. 3. Comparison of the strain field of a 1 1 1 crowdion in tungsten given by the solution of the Frenkel–Kontorova model (14) and plotted for N ¼ 1:8 and Z=a ¼ 11:5, with the discrete values of strain calculated from the coordinates of atoms forming the central string of the structure shown in Fig 1.
tions performed using the potential described in Table 2. The comparison between the analytical formula (14) and atomistic simulations confirms the accuracy of approximations underlying the derivation of the equation of motion of the crowdion (15). The examination of Eq. (15) also shows that only displacements of atoms in the core of the defect are significant in terms of inducing its migration through the lattice. To verify this conclusion we performed molecular dynamics simulations where various atoms in the cell containing a crowdion defect were given random initial impacts representing recoils due to collisions of electrons with atoms. A series of snapshots describing one of the simulations is shown in Fig. 4. This figure illustrates the fact that only the electron impacts on atoms situated in the core of the defect give rise to a significant displacement of the defect from its original position. This is confirmed by a simulation where the shock wave of a fairly large collision cascade occurring a small distance away from the core of the defect did not result in any appreciable movement of the defect. The latter fully agrees with the equation of motion (15), according to which a coherent displacement of atoms due to a wave propagating through the core of a defect does not result in a significant force acting on the defect. What is the value of the diffusion coefficient associated with the electron impacts? Since the impacts occur with equal probability in the central string and in the neighbouring strings surrounding the crowdion, the force acting on the defect randomly changes sign, giving rise to the stochastic Brownian motion of the defect in the 1 1 1 direction. The resulting coefficient of diffusion can be estimated as D jrtr l2 ;
ð17Þ
where l is the average displacement of a defect due to a single impact of a high-energy electron on an atom situated in
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0 ps
0.162 ps
(a) 0.516 ps
(b)
5. Summary
66.28 ps
(c)
the presence of solute atoms or impurities. Furthermore, electron irradiation catalyzes the thermally activated diffusion by unpinning the defects from impurities. Hence we arrive at the conclusion that while driven diffusion of defects is too slow if compared with the rate of migration in an ideal pure material found in molecular dynamics simulations, it probably provides the dominant mechanism of migration of defects in a real material where thermally activated migration is impeded by impurities or solute atoms.
(d)
Fig. 4. A sequence of snapshots illustrating the mechanism of driven migration of a defect under electron irradiation. A small displacement of an atom situated in the core of the crowdion in the (1 2 3) direction due to an electron impact seen in (a) gives rise to a small local cascade (b), (c), ˚ in which in the limit t ! 1 displaces the defect by approximately 16.5 A the 1 1 1 direction. In this simulation the defect moved to its final position after approximately 1 ps, and remained stationary until the end of the simulation.
˚ and the core region of the defect. Assuming that l = 10 A 1 17 2 jrtr = 10 s , we find that D 10 m /s. Hence the rate of driven diffusion is approximately nine orders of magnitude lower than the rate of diffusion of defects in a pure material estimated from the simulated trajectory of Brownian motion shown in Fig. 2. However, given that diffusion driven by electron impacts is independent of the temperature of the material, we find that it becomes the dominant mechanism of migration in the limit of relatively low temperatures in the case where the defects are pinned by solute atoms or impurities. For example, Arakawa [8] finds that the coefficient of diffusion of small dislocation loops in relatively pure iron is well approximated by the expression 1:3 eV 6 D 9:5 10 exp m2 =s; kBT which for room temperature T = 300 K gives D = 2.47 Æ 1028 m2/s. This value is 11 orders of magnitude lower than the above estimate for the coefficient of driven diffusion. While pinning of defects by impurities or solute atoms impedes the thermally activated mobility of defects, it does not affect the mechanism of driven migration associated with electron impacts. Indeed, the recoil energy is comparable with the characteristic energy of pinning, and hence the long-range displacements of defects resulting from impacts of high-energy electrons on atoms are almost unaffected by
We performed a comprehensive theoretical investigation of a new mechanism of diffusion of defects in a material under in situ high-energy electron irradiation in an electron microscope. We found that the migration of defects may result from impacts of high-energy electrons on atoms in the core region of a defect. Atomistic simulations and numerical estimates show that driven diffusion likely provides the dominant mechanism of migration of defects in real materials irradiated by a beam of high-energy electrons. Acknowledgements The authors thank Dr. M. L. Jenkins and Dr. K. Arakawa for stimulating discussions. Work at UKAEA was funded by the UK Engineering and Physical Sciences Research Council (EPSRC), by EURATOM and by the EXTREMAT integrated project. Two of the authors, C.H.W. and S.L.D., gratefully acknowledge support of PolyU Grants 5322/04E and PolyU HZF20. References [1] M.L. Jenkins, M.A. Kirk, Characterisation of Radiation Damage by Transmission Electron Microscopy, Institute of Physics, Bristol, 2001. [2] T.L. Daulton, M.A. Kirk, L.E. Rehn, Philos. Mag. A 80 (2000) 809. [3] T.L. Daulton, M.A. Kirk, L.E. Rehn, J. Nucl. Mater. 276 (2000) 258. [4] M.L. Jenkins, G.J. Hardy, M.A. Kirk, Mater. Sci. Forum 15–18 (1987) 901. [5] P.B. Hirsch, R.W. Horne, M.J. Whelan, Philos. Mag. 1 (1956) 677. [6] M.J. Whelan, J. Electron Microsc. Tech. 3 (1986) 109. [7] K. Arakawa, M. Hatanaka, E. Kuramoto, K. Ono, H. Mori, Phys. Rev. Lett. 96 (2006) 125506. [8] K. Arakawa, Dynamics of Nanometer-Sized Interstitial-Type Dislocation Loops in Iron by In-situ Transmission Electron Microscopy, a presentation given at UKAEA Culham in February 2006, unpublished. [9] Z. Yao, M.L. Jenkins, M.A. Kirk, Program and Abstracts of the 23rd Symposium on Effects of Radiation on Materials, San Jose, California, 13–15 June, 2006 (ASTM International, West Conshohocken, PA) p. 58. [10] M.L. Jenkins, private communication. [11] L. Reimer, Transmission Electron Microscopy, Springer, Berlin, 1984, p. 427. [12] D. Nguyen-Manh, A.P. Horsfield, S.L. Dudarev, Phys. Rev. B 73 (2006) 020101. [13] V.Y. Young, N. Welcome, G.B. Hoflund, Phys. Rev. B 48 (1993) 2981. [14] F. Dausinger, H. Schultz, Phys. Rev. Lett. 35 (1975) 1773. [15] S.L. Dudarev, Philos. Mag. 83 (2003) 3577.
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[18] W. Marshall, S.W. Lovesey, Theory of Thermal Neutron Scattering, Clarendon Press, Oxford, 1971, p. 73. [19] R. Mayol, F. Salvat, Atomic Data Nucl. Data Tables 65 (1997) 55.