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The vacancy formation energy for zirconium
JOURNAL
OF NUCLEAR
MATERIALS
54 (1974)
THE VACANCY
259-264.0
NORTH-HOLLAND
FORMATION
ENERGY
PUBLISHING
COMPANY
FOR ZIRCONIUM
M. ANTONINI and V. BORTOLANI* Comitato Nazionale Energia Nucleare, 21020 Ispra. Varese, Italy
and C. BERTONI Istituto di Fisica dell’hiversita,
Received
41100 Modena, Italy
11 June
1974
The vacancy formation energy of zirconium has been calculated by means of a microscopic model which takes into account the electron-electron and electron-ion interaction. The result obtained is in reasonable agreement with the presently available experimental data. The connections between the vacancy formation energy and the basic mechanisms for irradiation creep in this material are discussed. L’encrgie de formation d’une lacune dans le zirconium a @t6 calcul6e j I’aidc d’un modtle microscopique qui tient compte des interactions Clectroli-Electron et Electron-ion. Le resultat obtenu est en accord raisonnablc avec les don&es expkrimentales actuellcment disponiblcs. Les relations entre l’dncrgie dc formation d’une lacune et les mkcanismes fondamentaux du fluage sous irradiation dans le zirconium sont discut6s. Die Energie zur Bildung von Zirkon-Leerstellcn wurde mit Hilfe eines mikroskopischen Modells berechnet, das die Elektron-Elektron- und Elektron-Ion-Wechselwirkung beriicksichtigt. Die Ergebnisse stimmen mit den derzeit verfiigbaren experimcntellcn Werten recht gut iibcrcin. Der Zusammenhang zwischen der Energie zur Leerstellenbildung und dem grundlegenden Mcchanismus des bestrahlungsinduzierten Kriechcns in diescm Material wird diskutiert.
1. Introduction Most of the proposed theories concerning the variation of strength and creep of zirconium base alloys during neutron irradiation are founded on the formation and kinetics of point defects, produced during irradiation at an excess concentration with respect to the thermal equilibrium value [ 11. Among these theories, particularly relevant are the yielding creep theory originally proposed by Roberts and Cottrell [2] and subsequently applied to zirconium alloys by Hesketh [3], various mechanisms of radiation-enhanced thermal creep which have been reviewed by Gilbert [4] and a multiple mechanism suggested by Nichols [5], where the formation and interactions of various kinds of atomic defects operate * Permanent Modena,
address: Italy.
Istituto
di Fisica dell’universiti,
41100
either independently or in sequence to increase the creep rate in different ranges of applied stresses. It can easily be shown, as has already been clarified by Piercy [6], that one of the main quantities which needs to be known in all the mentioned theories is the chemical potential for vacancies in the damage region. This quantity is in turn controlled by the formation energy of a vacancy at thermal equilibrium. Therefore the evaluation of the formation energy of a vacancy in zirconium is important to assess the variation of mechanical properties of this material during neutron irradiation [7] . The measurement of the vacancy formation energy for zirconium is a rather complicated task, due to the structural transition which takes place in zirconium at about 860°C. The relatively low value of 0.7-0.9 eV found by Swanson [8] in the quenched metal is likely to be a lower limit, since his measurements were heavily affected by impurities, which were present at this
260
M. Antonini et al., Vacancy formation energy for zirconium
temperature with a concentration similar to that of vacancies. A rough estimate can be made by using the relation between the Debye temperature and vacancy formation energy suggested by Mukhejee [9] in closepacked metals. A value of 1.2 eV is obtained in this way. The empirical character of the Mukhejee relation however prevents this result from being considered a very reliable one. We shall describe in ihe following a calculation of the vacancy formation energy in zirconium. It is based on an atomistic approach described in sect. 2. The method requires us to set up an effective electronion potential whose expression for a transition metal is indicated and discussed in sect. 3. The calculation and the obtained value will finally be reported and discussed in section 4. Similar calculations have been performed in Aluminum and Alkali Metals by Chang and Falikov [lo], Chang [ 1 l] and by Ho [ 121. These materials, however, do not present the same difficulties as zirconium does because of its nature of transition metal with hexagonal structure.
2. Formulation of the problem The calculation of the formation energy of a vacancy in a metal can be divided into three contributions, relative to the various phases of the forming processes [ 121. The first energy contribution is the amount of work necessary to take the vacant atom to the surface. The energy to be evaluated is the total energy of N atoms sitting in (N+ 1) lattice sites added to the variation of surface energy related to the volume increase. In the second phase, by keeping constant the volume reached at the end of the first stage, atoms are allowed to redistribute around each lattice site as a consequence of the interaction with the vacant site. Finally, in the third stage, the total energy of the system is minimized by allowing a uniform compression to the equilibrium volume. In our calculation, we shall assume that the uniform compression at the end of stage III compensates the volume expansion which takes place during stage 1, so that the vacancy formation energy can be caIculated at constant volume. Moreover, we shall neglect
the atomic relaxation. This approximation, which s seems reasonable [ 121, permits a considerable simplification. The formation energy is therefore the difference between the total energy of N atoms in a perfect lattice of volume n and the energy of N atoms in a lattice having N+ 1 lattice sites and the same volume s1. The evaluation of the total energy of a metal, which has to take into account the many-body interactions among the conduction electrons and between conduction electrons and ions, may conveniently be performed within the frame of the theory of pseudopotentials [13]. Moreover, this theory allows one to divide the total energy into one contribufion, En, which is volume-dependent and a second contribution, E,, which depends on the ion configuration [ 141. This last term has been evaluated both in the perfect lattice and in the lattice with a vacant site mentioned above. The final expression obtained for the formation energy 17, is the following one:
s
*
(1)
4=40
The derivation of eq. (1), together with the detailed meaning of the symbols, is described in the appendix. We shall notice here that the “energy-wave number characteristic” Fb(q) (eq. (A.8)), which appears in eq. (I), contains both the effects of the conduction electrons and the effective electron-ion interaction potential. The evaluation of EF by means of eq. (1) requires the construction of an accurate pseudopotential for zirconium. The choice of such a potential is discussed in sect. 3.
3. Choice of the effective potential The adopted form of the pseudopotential is of the type proposed by Heine-Abarenkov (151, in the modified version given by AnimaIu [ 161, in order to take into account the resonant effects of the energy levels associated to the d-electrons which are present in a transition metal. It has the following form:
261
M. Antonini et al., Vacancy formation energy for zirconium
w, 9
= qQqE)
r CR,
= z/r
r>R,.
(2)
The quantities PIare projection operators which project the potential to the 1 component of angular momentum. R, is the width of the potential hole and A@‘) represents the depth of the hole (see fig. 1). Z is the valence. The values of A&5’) for a transition metal are given by Animalu [ 161 and have been chosen in order to reproduce the experimental atomic energy levels. We wish to point out that the matrix elements (klWlk’), where (kl and lk’) represent the,initial and final states of the electron gas, respectively, when constructed by utilizing eq. (2) depend separately on the wavenumbers k and k’. These matrix elements correspond therefore to a non-local interaction. In order to obtain the local interaction required by the theory developed in the appendix, one can use the following average process: I
k~ (klWlk+q)
d3k
0
Ek-E w7) = + 1 k+q > I0 * E~_E d9k k+q
(3)
Table 1 Values of quantitites which intervene in the computation of the pseudopotential (eqs. (2) and (3)) and of the total energy (eqs. (A.l) to (A.1 1)).
=
= 3.2312 5.1477 = 1.593
A A
=
1.75 = 23.1
A radius of atomic sphere A3 atomic volume
RM Ao (eF) Al (eF) AZ (EF)
=4 = 1.05 = 31.28 =46.24 = 40.80
valence A eV pseudopotential parameters at eV the Fermi level eV
kF 01
= 1.72 = 1.79
A-’ Fermi’s wavenumber Madelung constant
A B
= 0.9 = 0.335
a C
cla TS
a0 z
coefficients of eq. (A.ll).
a) Ref. [ 181.
non local character and the energy dependence of the pseudopotential. Moreover the proposed form of the pseudopotential allows us to calculate the formation energy EF without introducing arbitrary parameters.
4. Results and discussion
where
q=k’-k is the wave vector transfer and k, is the Fermi wavenumber. The average process indicated in eq. (3) allows one, as has been shown elsewhere [ 171, to take into account most of the effects which arise from the
The values of the quantities which have to be known in order to compute the effective potential W(4) expressed by eq. (3) are reported in table 1, together with those which appear in eqs. (A. I)-(A. 11) in the
-10
Fig. 1. Schematic representation tential (see eq. (2) of the text).
lattice parameters a)
of the Heine-Abarenkov po-
I!,, .o
.L
.8
(q& 1 ’
c
I,,,,,,,,
1.2
Fig. 2. Wavenumber dependence W(q) in zirconium.
1.6
2.0
2.L
2.8
of the effective potential
262
M. Antonini et al., Vacancy formation energy for zirconium
Fig. 3. Wavenumber dependence of the “energy wavenumber characteristic” FN(~). FN(q) has been normalized to a unit maximum value by means of eq. (4). In order to show up the presence of oscillations, we have amplified the ordinate starting from q = 2 KF.
Appendix. The effective potential W(q) obtained for zirconium has been drawn in fig. 2. W(q) has been computed by numerical evaluation of the principal part of eq. (3). The energy wave number characteristic F&) has been normalized by means of the relationship
q2q)
FJ4) =__ F&d. 2ne2
(4)
This has been calculated using eq. (AS) and is plotted in fig. 3. In consideration of the difficulties involved in making a correct choice of the pseudopotential for a transition metal, already mentioned in the previous section, we have verified the reliability of the obtained FN(q), by using it in the computation of the phonon dispersion relations in zirconium. These relations have been accurately measured by Bezdek et al. [ 191. The calculated curves are in good qualitative agreement with those reported by these authors, both in the basal plane and in the [OOOl] direction. A more detailed discussion concerning this comparison will be reported in a separate paper. By introducing the form of Fb(q) obtained in eq. (1) the formation energy has been calculated. The numerical computation has been executed by truncating F(q) at the value qmax = 2.4 (qa)/2n. This value Of qmax allows to obtain a good convergence in eq. (1). The final value obtained for the vacancy formation energy in zirconium is EF = 1.8 eV. This has to be
considered as an upper value, since the local atomic relaxation and the volume change have been neglected. When taking into account these two addition contributions, a smaller value of EF should result, in anaiogy to what has been found for other metals [lo- 121. As has already been mentioned in the introduction, we do not dispose of reliable experimental results. However it can be concluded that our upper estimate is in reasonable agreement with an experimental lower limit of about 1 eV [8]. The reasonable value which has been obtained by the proposed microscopic method is encouraging to perform a more refined calculation, which would take into account the energy contributions deriving from atomic relaxation and volume change. This extension also allows one to evaluate the vacancy formation volume, which may be partially connected with the observed elongation under irradiation.
Appendix In this appendix we shall derive eq. (1) which provides the formation energy of a vacancy, within the frame of the theory of pseudopotentials. In this theory, the crystal potential, which is strong in proximity to the atomic nuclei, is replaced by an effective weaker potential which reproduces the energy eigenvalues of the system [20]. The main advantage of this substitution is to allow one to use a perturbation approach when computing the physical properties of the material. In the theory of pseudopotentials (14), the total energy per ion Etot is given by: Etot =Ea +E,
(‘4.1)
with Es=Ee
‘Eb.
(A.2)
The term E, in eq. (A.l) depends on the volume R and will not be examined here. The term E, depends on the ionic structure. It is the sum of the direct Coulomb interaction between ions, E,, and of the indirect interaction via the electron gas. The term E, has the following expression [ 131:
E, = C;, GP*(q)F,(4)+ Cl@(q) _z2
P2 n II2
Y-Z2
i&
)
(A.3)
263
AK Antonini et al., Vacancy formation energy for zirconium
where: Fe(q) = 2n$
+ oq
exp(-q2/4[)
(A.4)
eF, k, are the Fermi energy and Fermi wavenumber respectively. The function G(q) represents the exchange and correlation effects of the electron gas. We shall use the approximate expression suggested by Singwi et al. [21] :
and
4(q)=
SI
-$y W,)
exp(-x2)&.
[“‘rl
In eqs. (A.3) (A.4) and A.5), Z is the valence, a0 the atomic volume. 6 is a numerical parameter which is introduced to obtain a faster convergence when performing the numerical computation. S(q) is the structure factor, which is given by:
c4.6) where N is the total number of atoms, rl is the atomic position and q the wavevector. In a perfect lattice, S(q) = 0 except when 4 equals a reciprocal lattice vector ~7~.The prime which appears in eq. (A.3) indicates that the term q. = 0 has to be excluded from the summation. The term Fb has the following expression: E, = ~;,S(q)S*(q)F&).
(A.7)
where Fb(q) is the energy wavenumber characteristic. The detailed expression of F,,(q) depends on the adopted pseudopotential. For a local pseudopotential, Fb(q) has the form: F&)
= IW(q)12 x(q)/+).
1- -
871
x(4) 11-a?)1 >
(A.1 1)
The values of the constants A and B are reported for zirconium in table 1. The formation energy of a vacancy, according to the mechanism of formation described in sect. 2, turns out to be given by: EF = E&
- E&,
(A.12)
where EF,,, the total energy of the perfect lattice, can be computed by equations (A.2) to (A.1 l), by making use of the perfect lattice structure factor. Eit is computed by means of the same equations, by making use of the structure factor which is appropriate to the metal which contains a vacancy. By introducing the vacancy at the origin, the structure factor of the defect lattice has the form: N+l
scq)=k C
,-4-r/
-$
.
(A.13)
1-I
By applying the necessary substitutions, ally obtains: E,
one fin-
52O
=J Fb(q)d3q (27Q3
- 2$ s
.
iA.8)
(A.14)
4=40
W(q) is the matrix element (k IW(r)lk), where k and k’ are the initial and final electron wavenumbers respectively. For a local pseudopotential, such a matrix element depends only on the difference q = k-k’. In eq. (A.8) e(q) is the dielectric function. It has the following form:
e(q) =
B)),
GO=A(l-exp(-(-$2
(A.51
In eq. (A. 14), (Yis the Madelung constant and rs is the radius of the atomic sphere. Eq. (A.14) allows one to calculate the formation energy of a vacancy. It coincides with eq. (1) reported in sect. 2 of the text.
(A-9)
floq2
References
where
[l] C.D. Williams, R. W. Gilbert, Radiation damage in reac(A.lO)
tor materials, Vol. 1 (International Atomic Energy Agency, Vienna, 1969) p. 235. [2] A.C. Roberts and A.H. Cottrell, Phil. Mag. 1 (1956) 711.
264
M. Antonini et al., Vacancy formation energy for zirconium
[ 31 [4] [5] [6] [ 71
R.V. Hesketh, J. Nucl. Mater. 26 (1968)-77. F.R. Gilbert, Reactor Technology 14 (1971) 258. F.A. Nichols, J. Nucl. Mater. 37 (1970) 59. G.R. Piercy, J. Nucl. Mater. 26 (1968) 18. D.L. Douglass, The metallurgy of zirconium, Atomic Energy Review, suppl. 1971 (IAEA, Vienna, 1971) p. 313. [8] M.L. Swanson, G.R. Piercy, G.V. Kidson and A.F. Quenneville, J. Nucl. Mater. 34 (1970) 340. [9] K. Mukherjee, Phil. Mag. 12 (1965) 915. [lo] R. Chang and L.M. FaIikov, J. Phys. Chem. Solids 32 (1971j 465. [ll] R. Chang, J. Phys. Chem. Solids 32 (1971) [ 121 P.S. Ho, Phys. Rev. B3 (1971) 4035.
1409.
[ 131 W.A. Harrison, Pseudopotential in the theory of metals (W.A. Benjamin, New York, 1966) ch. V. [ 141 Ibid, Chs. II and III. 1151 V. Heine and I. Abarenkov, Phil. Mag. 9 (1964) 45 1. [ 161 A.O.E. Animalu, Phys. Rev. B8 (1973) 35. [ 171 C. Bertoni, V. Bortolani, C. Calandra and I:. Nizzoli, J. Phys. F (Metal Physics), 4 (1974) 19. [IS] W.B. Pearson, A handbook of lattice spacings and structures of metals and alloys (Pergamon Press, 1959). [ 191 H.F. Bezdek, R.E. Schmunk and L. Finegold, Phys. Stat. Sol. 42 (1970) 275. [ 201 F. Basiani and V. Celli, J. Phys. Chem. Solids 20 (1961) 64. [ 211 KS. Singwi, A. Sjolander, M.P. Tosi and R.M. Land, Phys. Rev. Bl (1970) 1.044.
OF NUCLEAR
MATERIALS
54 (1974)
THE VACANCY
259-264.0
NORTH-HOLLAND
FORMATION
ENERGY
PUBLISHING
COMPANY
FOR ZIRCONIUM
M. ANTONINI and V. BORTOLANI* Comitato Nazionale Energia Nucleare, 21020 Ispra. Varese, Italy
and C. BERTONI Istituto di Fisica dell’hiversita,
Received
41100 Modena, Italy
11 June
1974
The vacancy formation energy of zirconium has been calculated by means of a microscopic model which takes into account the electron-electron and electron-ion interaction. The result obtained is in reasonable agreement with the presently available experimental data. The connections between the vacancy formation energy and the basic mechanisms for irradiation creep in this material are discussed. L’encrgie de formation d’une lacune dans le zirconium a @t6 calcul6e j I’aidc d’un modtle microscopique qui tient compte des interactions Clectroli-Electron et Electron-ion. Le resultat obtenu est en accord raisonnablc avec les don&es expkrimentales actuellcment disponiblcs. Les relations entre l’dncrgie dc formation d’une lacune et les mkcanismes fondamentaux du fluage sous irradiation dans le zirconium sont discut6s. Die Energie zur Bildung von Zirkon-Leerstellcn wurde mit Hilfe eines mikroskopischen Modells berechnet, das die Elektron-Elektron- und Elektron-Ion-Wechselwirkung beriicksichtigt. Die Ergebnisse stimmen mit den derzeit verfiigbaren experimcntellcn Werten recht gut iibcrcin. Der Zusammenhang zwischen der Energie zur Leerstellenbildung und dem grundlegenden Mcchanismus des bestrahlungsinduzierten Kriechcns in diescm Material wird diskutiert.
1. Introduction Most of the proposed theories concerning the variation of strength and creep of zirconium base alloys during neutron irradiation are founded on the formation and kinetics of point defects, produced during irradiation at an excess concentration with respect to the thermal equilibrium value [ 11. Among these theories, particularly relevant are the yielding creep theory originally proposed by Roberts and Cottrell [2] and subsequently applied to zirconium alloys by Hesketh [3], various mechanisms of radiation-enhanced thermal creep which have been reviewed by Gilbert [4] and a multiple mechanism suggested by Nichols [5], where the formation and interactions of various kinds of atomic defects operate * Permanent Modena,
address: Italy.
Istituto
di Fisica dell’universiti,
41100
either independently or in sequence to increase the creep rate in different ranges of applied stresses. It can easily be shown, as has already been clarified by Piercy [6], that one of the main quantities which needs to be known in all the mentioned theories is the chemical potential for vacancies in the damage region. This quantity is in turn controlled by the formation energy of a vacancy at thermal equilibrium. Therefore the evaluation of the formation energy of a vacancy in zirconium is important to assess the variation of mechanical properties of this material during neutron irradiation [7] . The measurement of the vacancy formation energy for zirconium is a rather complicated task, due to the structural transition which takes place in zirconium at about 860°C. The relatively low value of 0.7-0.9 eV found by Swanson [8] in the quenched metal is likely to be a lower limit, since his measurements were heavily affected by impurities, which were present at this
260
M. Antonini et al., Vacancy formation energy for zirconium
temperature with a concentration similar to that of vacancies. A rough estimate can be made by using the relation between the Debye temperature and vacancy formation energy suggested by Mukhejee [9] in closepacked metals. A value of 1.2 eV is obtained in this way. The empirical character of the Mukhejee relation however prevents this result from being considered a very reliable one. We shall describe in ihe following a calculation of the vacancy formation energy in zirconium. It is based on an atomistic approach described in sect. 2. The method requires us to set up an effective electronion potential whose expression for a transition metal is indicated and discussed in sect. 3. The calculation and the obtained value will finally be reported and discussed in section 4. Similar calculations have been performed in Aluminum and Alkali Metals by Chang and Falikov [lo], Chang [ 1 l] and by Ho [ 121. These materials, however, do not present the same difficulties as zirconium does because of its nature of transition metal with hexagonal structure.
2. Formulation of the problem The calculation of the formation energy of a vacancy in a metal can be divided into three contributions, relative to the various phases of the forming processes [ 121. The first energy contribution is the amount of work necessary to take the vacant atom to the surface. The energy to be evaluated is the total energy of N atoms sitting in (N+ 1) lattice sites added to the variation of surface energy related to the volume increase. In the second phase, by keeping constant the volume reached at the end of the first stage, atoms are allowed to redistribute around each lattice site as a consequence of the interaction with the vacant site. Finally, in the third stage, the total energy of the system is minimized by allowing a uniform compression to the equilibrium volume. In our calculation, we shall assume that the uniform compression at the end of stage III compensates the volume expansion which takes place during stage 1, so that the vacancy formation energy can be caIculated at constant volume. Moreover, we shall neglect
the atomic relaxation. This approximation, which s seems reasonable [ 121, permits a considerable simplification. The formation energy is therefore the difference between the total energy of N atoms in a perfect lattice of volume n and the energy of N atoms in a lattice having N+ 1 lattice sites and the same volume s1. The evaluation of the total energy of a metal, which has to take into account the many-body interactions among the conduction electrons and between conduction electrons and ions, may conveniently be performed within the frame of the theory of pseudopotentials [13]. Moreover, this theory allows one to divide the total energy into one contribufion, En, which is volume-dependent and a second contribution, E,, which depends on the ion configuration [ 141. This last term has been evaluated both in the perfect lattice and in the lattice with a vacant site mentioned above. The final expression obtained for the formation energy 17, is the following one:
s
*
(1)
4=40
The derivation of eq. (1), together with the detailed meaning of the symbols, is described in the appendix. We shall notice here that the “energy-wave number characteristic” Fb(q) (eq. (A.8)), which appears in eq. (I), contains both the effects of the conduction electrons and the effective electron-ion interaction potential. The evaluation of EF by means of eq. (1) requires the construction of an accurate pseudopotential for zirconium. The choice of such a potential is discussed in sect. 3.
3. Choice of the effective potential The adopted form of the pseudopotential is of the type proposed by Heine-Abarenkov (151, in the modified version given by AnimaIu [ 161, in order to take into account the resonant effects of the energy levels associated to the d-electrons which are present in a transition metal. It has the following form:
261
M. Antonini et al., Vacancy formation energy for zirconium
w, 9
= qQqE)
r CR,
= z/r
r>R,.
(2)
The quantities PIare projection operators which project the potential to the 1 component of angular momentum. R, is the width of the potential hole and A@‘) represents the depth of the hole (see fig. 1). Z is the valence. The values of A&5’) for a transition metal are given by Animalu [ 161 and have been chosen in order to reproduce the experimental atomic energy levels. We wish to point out that the matrix elements (klWlk’), where (kl and lk’) represent the,initial and final states of the electron gas, respectively, when constructed by utilizing eq. (2) depend separately on the wavenumbers k and k’. These matrix elements correspond therefore to a non-local interaction. In order to obtain the local interaction required by the theory developed in the appendix, one can use the following average process: I
k~ (klWlk+q)
d3k
0
Ek-E w7) = + 1 k+q > I0 * E~_E d9k k+q
(3)
Table 1 Values of quantitites which intervene in the computation of the pseudopotential (eqs. (2) and (3)) and of the total energy (eqs. (A.l) to (A.1 1)).
=
= 3.2312 5.1477 = 1.593
A A
=
1.75 = 23.1
A radius of atomic sphere A3 atomic volume
RM Ao (eF) Al (eF) AZ (EF)
=4 = 1.05 = 31.28 =46.24 = 40.80
valence A eV pseudopotential parameters at eV the Fermi level eV
kF 01
= 1.72 = 1.79
A-’ Fermi’s wavenumber Madelung constant
A B
= 0.9 = 0.335
a C
cla TS
a0 z
coefficients of eq. (A.ll).
a) Ref. [ 181.
non local character and the energy dependence of the pseudopotential. Moreover the proposed form of the pseudopotential allows us to calculate the formation energy EF without introducing arbitrary parameters.
4. Results and discussion
where
q=k’-k is the wave vector transfer and k, is the Fermi wavenumber. The average process indicated in eq. (3) allows one, as has been shown elsewhere [ 171, to take into account most of the effects which arise from the
The values of the quantities which have to be known in order to compute the effective potential W(4) expressed by eq. (3) are reported in table 1, together with those which appear in eqs. (A. I)-(A. 11) in the
-10
Fig. 1. Schematic representation tential (see eq. (2) of the text).
lattice parameters a)
of the Heine-Abarenkov po-
I!,, .o
.L
.8
(q& 1 ’
c
I,,,,,,,,
1.2
Fig. 2. Wavenumber dependence W(q) in zirconium.
1.6
2.0
2.L
2.8
of the effective potential
262
M. Antonini et al., Vacancy formation energy for zirconium
Fig. 3. Wavenumber dependence of the “energy wavenumber characteristic” FN(~). FN(q) has been normalized to a unit maximum value by means of eq. (4). In order to show up the presence of oscillations, we have amplified the ordinate starting from q = 2 KF.
Appendix. The effective potential W(q) obtained for zirconium has been drawn in fig. 2. W(q) has been computed by numerical evaluation of the principal part of eq. (3). The energy wave number characteristic F&) has been normalized by means of the relationship
q2q)
FJ4) =__ F&d. 2ne2
(4)
This has been calculated using eq. (AS) and is plotted in fig. 3. In consideration of the difficulties involved in making a correct choice of the pseudopotential for a transition metal, already mentioned in the previous section, we have verified the reliability of the obtained FN(q), by using it in the computation of the phonon dispersion relations in zirconium. These relations have been accurately measured by Bezdek et al. [ 191. The calculated curves are in good qualitative agreement with those reported by these authors, both in the basal plane and in the [OOOl] direction. A more detailed discussion concerning this comparison will be reported in a separate paper. By introducing the form of Fb(q) obtained in eq. (1) the formation energy has been calculated. The numerical computation has been executed by truncating F(q) at the value qmax = 2.4 (qa)/2n. This value Of qmax allows to obtain a good convergence in eq. (1). The final value obtained for the vacancy formation energy in zirconium is EF = 1.8 eV. This has to be
considered as an upper value, since the local atomic relaxation and the volume change have been neglected. When taking into account these two addition contributions, a smaller value of EF should result, in anaiogy to what has been found for other metals [lo- 121. As has already been mentioned in the introduction, we do not dispose of reliable experimental results. However it can be concluded that our upper estimate is in reasonable agreement with an experimental lower limit of about 1 eV [8]. The reasonable value which has been obtained by the proposed microscopic method is encouraging to perform a more refined calculation, which would take into account the energy contributions deriving from atomic relaxation and volume change. This extension also allows one to evaluate the vacancy formation volume, which may be partially connected with the observed elongation under irradiation.
Appendix In this appendix we shall derive eq. (1) which provides the formation energy of a vacancy, within the frame of the theory of pseudopotentials. In this theory, the crystal potential, which is strong in proximity to the atomic nuclei, is replaced by an effective weaker potential which reproduces the energy eigenvalues of the system [20]. The main advantage of this substitution is to allow one to use a perturbation approach when computing the physical properties of the material. In the theory of pseudopotentials (14), the total energy per ion Etot is given by: Etot =Ea +E,
(‘4.1)
with Es=Ee
‘Eb.
(A.2)
The term E, in eq. (A.l) depends on the volume R and will not be examined here. The term E, depends on the ionic structure. It is the sum of the direct Coulomb interaction between ions, E,, and of the indirect interaction via the electron gas. The term E, has the following expression [ 131:
E, = C;, GP*(q)F,(4)+ Cl@(q) _z2
P2 n II2
Y-Z2
i&
)
(A.3)
263
AK Antonini et al., Vacancy formation energy for zirconium
where: Fe(q) = 2n$
+ oq
exp(-q2/4[)
(A.4)
eF, k, are the Fermi energy and Fermi wavenumber respectively. The function G(q) represents the exchange and correlation effects of the electron gas. We shall use the approximate expression suggested by Singwi et al. [21] :
and
4(q)=
SI
-$y W,)
exp(-x2)&.
[“‘rl
In eqs. (A.3) (A.4) and A.5), Z is the valence, a0 the atomic volume. 6 is a numerical parameter which is introduced to obtain a faster convergence when performing the numerical computation. S(q) is the structure factor, which is given by:
c4.6) where N is the total number of atoms, rl is the atomic position and q the wavevector. In a perfect lattice, S(q) = 0 except when 4 equals a reciprocal lattice vector ~7~.The prime which appears in eq. (A.3) indicates that the term q. = 0 has to be excluded from the summation. The term Fb has the following expression: E, = ~;,S(q)S*(q)F&).
(A.7)
where Fb(q) is the energy wavenumber characteristic. The detailed expression of F,,(q) depends on the adopted pseudopotential. For a local pseudopotential, Fb(q) has the form: F&)
= IW(q)12 x(q)/+).
1- -
871
x(4) 11-a?)1 >
(A.1 1)
The values of the constants A and B are reported for zirconium in table 1. The formation energy of a vacancy, according to the mechanism of formation described in sect. 2, turns out to be given by: EF = E&
- E&,
(A.12)
where EF,,, the total energy of the perfect lattice, can be computed by equations (A.2) to (A.1 l), by making use of the perfect lattice structure factor. Eit is computed by means of the same equations, by making use of the structure factor which is appropriate to the metal which contains a vacancy. By introducing the vacancy at the origin, the structure factor of the defect lattice has the form: N+l
scq)=k C
,-4-r/
-$
.
(A.13)
1-I
By applying the necessary substitutions, ally obtains: E,
one fin-
52O
=J Fb(q)d3q (27Q3
- 2$ s
.
iA.8)
(A.14)
4=40
W(q) is the matrix element (k IW(r)lk), where k and k’ are the initial and final electron wavenumbers respectively. For a local pseudopotential, such a matrix element depends only on the difference q = k-k’. In eq. (A.8) e(q) is the dielectric function. It has the following form:
e(q) =
B)),
GO=A(l-exp(-(-$2
(A.51
In eq. (A. 14), (Yis the Madelung constant and rs is the radius of the atomic sphere. Eq. (A.14) allows one to calculate the formation energy of a vacancy. It coincides with eq. (1) reported in sect. 2 of the text.
(A-9)
floq2
References
where
[l] C.D. Williams, R. W. Gilbert, Radiation damage in reac(A.lO)
tor materials, Vol. 1 (International Atomic Energy Agency, Vienna, 1969) p. 235. [2] A.C. Roberts and A.H. Cottrell, Phil. Mag. 1 (1956) 711.
264
M. Antonini et al., Vacancy formation energy for zirconium
[ 31 [4] [5] [6] [ 71
R.V. Hesketh, J. Nucl. Mater. 26 (1968)-77. F.R. Gilbert, Reactor Technology 14 (1971) 258. F.A. Nichols, J. Nucl. Mater. 37 (1970) 59. G.R. Piercy, J. Nucl. Mater. 26 (1968) 18. D.L. Douglass, The metallurgy of zirconium, Atomic Energy Review, suppl. 1971 (IAEA, Vienna, 1971) p. 313. [8] M.L. Swanson, G.R. Piercy, G.V. Kidson and A.F. Quenneville, J. Nucl. Mater. 34 (1970) 340. [9] K. Mukherjee, Phil. Mag. 12 (1965) 915. [lo] R. Chang and L.M. FaIikov, J. Phys. Chem. Solids 32 (1971j 465. [ll] R. Chang, J. Phys. Chem. Solids 32 (1971) [ 121 P.S. Ho, Phys. Rev. B3 (1971) 4035.
1409.
[ 131 W.A. Harrison, Pseudopotential in the theory of metals (W.A. Benjamin, New York, 1966) ch. V. [ 141 Ibid, Chs. II and III. 1151 V. Heine and I. Abarenkov, Phil. Mag. 9 (1964) 45 1. [ 161 A.O.E. Animalu, Phys. Rev. B8 (1973) 35. [ 171 C. Bertoni, V. Bortolani, C. Calandra and I:. Nizzoli, J. Phys. F (Metal Physics), 4 (1974) 19. [IS] W.B. Pearson, A handbook of lattice spacings and structures of metals and alloys (Pergamon Press, 1959). [ 191 H.F. Bezdek, R.E. Schmunk and L. Finegold, Phys. Stat. Sol. 42 (1970) 275. [ 201 F. Basiani and V. Celli, J. Phys. Chem. Solids 20 (1961) 64. [ 211 KS. Singwi, A. Sjolander, M.P. Tosi and R.M. Land, Phys. Rev. Bl (1970) 1.044.