- Email: [email protected]
Domain walls in ω-phase transformations
PHYSICAD Physica D 123 (1998) 368-379
ELSEVIER
Domain walls in co-phase transformations Mahdi Sanati a.b.,, Avadh Saxena
b
a Department of Physics, University of Cincinnati, Cincinnati, 0H45221, USA b Theoretical Division, Los Alamos National Laboratory, Los Alarnos, NM 87545, USA
Abstract
The/%phase (body-centered cubic: b.c.c.) to w-phase transformation in certain elements (e.g. Zr) and alloys (e.g. Zr-Nb) is induced either by quenching or application of pressure. The w-phase is a metastable state and usually coexists with the /~-matrix in the form of small particles. To study the formation of domain walls in these materials we have extended the Landau model of Cook for the w-phase transition by including a spatial gradient (Ginzburg) term of the scalar order parameter. In general, the Landau free energy is an asymmetric double-well potential. From the variational derivative of the total free energy we obtain a static equilibrium condition. By solving this equation for different physical parameters and boundary conditions, we obtained different quasi-one-dimensional soliton-like solutions. These solutions correspond to three different types of domain walls between the ~o-phase and the/%matrix. In addition, we obtained soliton lattice (domain wall array) solutions, calculated their formation energy and the asymptotic interaction between the solitons. Copyright © 1998 Published by Elsevier Science B.V. Keywords: Ginzburg-Landau model; Asymmetric double-well potential; Domain wall arrays and interaction; w-phase transformation
1. Introduction
The formation of w-phase in elements (e.g. Ti, Zr, Hf) and alloys (e.g. Zr-Nb, Ti-Nb, Ti-Mo, Ti-V) has been the subject of considerable interest and activity since this phase was first reported by Frost et at. [1]. Presence of w-phase in materials changes their mechanical and physical properties [2]. Following the rationale of de Fontaine et al. [3,4] (that there is no physical basis for differentiating between athermal and thermal w-phase) we will consider the accepted mechanism for obtaining w-phase from the/3-phase (b.c.c.): Collapse of the (1 1 1)/3 plane due to a 2/3(1 1 1)~ longitudinal displacement wave. Thus, the w-phase is defined by and involves a collapse of a pair of (1 1 1)~ planes, leaving the neighboring plane unaltered, as depicted in Fig. 1. This type of transformation is called displacive because it involves cooperative movement over small distances which are fractions of lattice translation vectors. In contrast, a diffusion-controlled nucleation and growth transformation is termed replacive. Internal friction studies [5] (on fl-Ti alloys) support this mechanism whereby transformation occurs via atomic shuffles rather than by diffusion. (The shuffle transformation at the unit cell level is a transformation in which atomic displacements are intracellular with little or no pure strain of the lattice). The atomic movements required for the transformation * Corresponding author. Tel.: +1 505 665 0354; fax: +1 505 665 4063; e-mail: [email protected]. 0167-2789/98/$19.00 Copyright © 1998 Published by Elsevier Science B.V. All rights reserved PII S0167-2789(98)00135-3
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
W
W
/,
W
W
369
W
W
Y
. lal
JJjj ('0)
(el
Fig. 1. w-Phase transformation in a binary alloy. (a) Atomic (1 1 1~ planes in a b.c.c, lattice (fl-phase) before the transition, (b) partial collapse and (c) complete collapse of neighboring planes. are -q-an v/3/12 (or ±c~,/6) where a/~ is the lattice parameter of the fl-phase and c,o is the c-axis lattice parameter of the co-phase. The co-collapse may be "complete" or "incomplete" (meaning that the double layers are planar, or rumpled, respectively). Complete collapse produces the ideal hexagonal structure (also known as the ct-phase) with ao, = v/2a# and c~o = x/3/2al~, while partial collapse (< a# x/-3/12) produces the trigonal form of co-phase. As observed in experiments on quenched Zr-Nb alloys [6], the co-phase starts to form at a certain temperature, 7",o. On cooling below To, the process is found to be highly reversible and electron microscopy studies have shown that the new phase is finely distributed in a fl (b.c.c.) matrix [7]. In addition, diffraction experiments [8-10] and electron microscopic images [11] clearly indicate the rod-like inclusions of co-phase embedded in/4-phase matrix along [1 1 1]t~ direction. Based on these observations Horovitz et al. [12,13] attempted to explain the domain walls in cophase by a 1D soliton-like stacking fault model. This model is based on a discrete sine-Gordon type potential within a lattice. The theory assumes that the co-cluster tends to have a smaller lattice constant than that of the fl-matrix. If the difference is too small, the co-cluster is completely locked into the E-matrix and there are no solitons. However, if the difference is sufficiently large then (sine-Gordon type) solitons appear in the ground state, partially unlocking the co-cluster from the/3-matrix. The critical value 6/a# (where 6 is the difference between the lattice constant of the ,o-phase and that of the fl-matrix, a#) for which this transition occurs is estimated to be 5%. The assumption that the ~o-phase has a smaller lattice constant has been proven by high pressure studies on Ti-V alloys [2]. Nevertheless, no critical behavior in 6/a[~ has been observed which seems contrary to the predictions of the model by Horovitz et al. A possible alternative (and competing) model to explain the spacing and width of the domain walls for this ~tructural phase transition would be to add the Ginzburg term to the Landau free energy of the system and solve the equations which are derived from the variation of the total free energy of the system. This has been done in the context of first-order structural transitions using a triple-well (~b6) potential [ 14]. This approach is justified for the co-phase transformation especially when the mismatch of lattice parameters is sufficiently small, so as to be irrelevant tbr the mechanism considered by Horovitz et al. To model the atomic displacements, Cook [ 15,16] has developed a Landau
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
370
free energy for first-order structural phase transitions and used it to model the o-phase transitions in zirconium alloys. Cook's concepts are in substantial agreement with many recent experiments. However, to our knowledge the formation of domain walls between the o-phase and the/3 (parent) phase has not been studied for this type of Landau free energy. Note that the Ginzburg term represents the nonlocal interatomic forces (as a gradient of the order parameter) that give rise to a "phonon anomaly" such as the minimum of the longitudinal phonon branch at wave vector (27r/3) [1 1 1] for the o-phase transformation. The present work investigates the formation of domain walls, resulting from the presence of o-phase particles in a fl (b.c.c.) matrix, by including a spatial gradient term (Ginzburg term) to the Landau model free energy of Cook. From the equilibrium condition, i.e. by solving the equation derived from the variation of the total free energy for different physical parameters and boundary conditions, we find three different solutions (domain walls). For each type of solution we calculate the associated domain wall energy.
2. G e n e r i c m o d e l for o - p h a s e t r a n s f o r m a t i o n s
The Landau free energy for o-phase suggested by Cook has a single component (scalar) order parameter 7, 1 F L = ~1 A ~ ~/ - + ~I B r/3 + ~ C r/ 4 ,
where A is proportional to the quasiharmonic force constant, B and C are proportional to the third- and fourth-order force constants, respectively [16]. If B is negative, A and C are positive, for certain values of the parameters a first-order transition is possible from FL = 0 at 0 ~ 0 (parent phase) to some finite value of r/where FL is also zero. One can obtain the total free energy by adding the Ginzburg term FG = /G(~Tr/)2 to the Landau free energy, 1 4, F T = ½G(Vr/) 2 + ~1 A 0 "~ - + . ~I B r/3 + ~C~/
(1)
where coefficient G in the Ginzburg term is proportional to the appropriate curvature of the phonon dispersion curve of the material. Eq. (1) contains four parameters; by scaling the energy and order parameter it can be reduced to a standard form with two control parameters a and g, 1 4 F T / F ( , = ½g(V~b) 2 + ½a~b2 - /q~3 + UP ,
(2)
where C 4~ = -~] ~,
B4 Fo = ~ ,
g--
GC B2 ,
a=
AC B2 "
The scaled Landau free energy FL is plotted in Fig. 2. The plot shows the variation of Landau free energy with respect to a. The condition q~ = 0 corresponds to the parent phase (b.c.c.) and the other minimum (if present) corresponds to the product phase (o-phase). There are several regimes depending on the parameter a; the condition, "0FL / Oc~=O yields 4~ = 0, then (i) (ii) (iii)
~b = ~(1 + v ~ -
4a),
the following cases are possible: a > 1/4, FL has a real minimum at 4~ -----0, only (Fig. 2(1)). a = 1/4, FL has a real minimum at ~b = 0, and an inflection point at ~b = 1/4. 2/9 <: a < 1/4, FL has a stable minimum at q~ -----0, a metastable minimum at ~ = (1 + ~/1 - 4a)/2, and a relative maximum at q~ = (1 - ~/1 - 4 a ) / 2 (Fig. 2(2)).
371
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
H
I
f
1/
J Fig. 2. Variation of the Landau contribution to the total free energy (Eq. (2)) with q~for various values of the coefficient of the quadratic term, a. The upper (a >_ 1/4) and lower (a _<0) insets show the limit at which the potential becomes a single well. (iv) a = 2/9, FI. has two minima at values 05 = 0 and 05 = 2/3 with FL = 0 and a m a x i m u m at 05 = 1/3 (Fig. 2(3)). This is the condition for first order phase transition. (v) 0 < a < 2/9, FL has a metastable minimum at 05 = 0, and a stable minimum at 4) = (1 + ,/1 - 4 a ) / 2 for 05 > 2/3, with a m a x i m u m at 05 = (1 - ~/1 - 4 a ) / 2 (Fig. 2(4)). (vi) a < 0 is not relevant to the first order transitions (Fig. 2(5)). Domain walls exist in cases (iii)-(v), since only in these regimes parent and product phases coexist. The parameter a is determined from the structural data of the material.
3. Equilibrium conditions and domain wall energies From the variational derivative of the total energy, one obtains the following static equilibrium condition: 0F-r
OFT
Ox 005,
005
--0.
(3)
By substituting Eq. (2) in (3), one obtains g05" - a05 + 052 _ 4,3 = O.
(4)
We will show that the solutions of this differential equation for different physical parameters and boundary conditions (for cases (iii)-(v)) are quasi-one-dimensional solitary-wave solutions. Traveling kink (domain wall) solutions 05(x, t) are obtained from the static solution 05(x) by boosting to velocity v via x --+ (1 - v2) - 1/2 (x - vt). Finally by substituting the solutions of Eq. (4) in
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
372 oo
ftotal =
f
(V~b) 2 -q- 2~b 2 - -~- ~-
(5)
dx
mOG
we will calculate the domain wall energy in each case.
4. Results
After two integrations Eq. (4) leads to
x(~>- x0 =
(¼~_
(6)
~_~3+ ~a~2- :0
with the boundary conditions f0 = l i m x ~ + ~ FL, limx--,+~ q7 = 0 and the choice of origin ~b(x0) = 0. Depending on different boundary conditions one can obtain three types of domain array (lattice) solutions: 1. For a = 2-, 9 f0 = l i m x ~ _ ~ FL(4~2) and f0 = l i m x ~ + ~ FL(~b3) (Fig. 3) a lattice solution between the two product variants is given by (Fig. 4)
4~(x)=
li~l--~sn
,k
;
(=3 g(l~/+kZ),
(7)
where sn(x/(, k) is a Jacobi elliptic function with modulus k and the total free energy per kink-antikink pair for this solution is given by
--
f0
Fig. 3. Free energy for a = 2 / 9 is a double-well potential (the 4~4 model); f0 corresponds to the integration constant in Eq. (6).
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
373
0.8
0.4
0.0 -82.0
0.0 X
82.0
0.8
~) 0,4
J
0.0 -82.0
82.0
0.0 X
Fig. 4. Kink lattice (upper panel) and single kink (lower panel) solutions for the double-well potential in Fig. 3. The kink corresponds to a domain wall between the fl- and w-phase. The kink lattice is an alternating array of fl- and ~o-phase domains.
Ft°tal -- Ek 3v/2 217 (1 ÷ k2) 3/2
k ' 4 K ( k ) - 1-k'2K(k) + ~ E ( k ) 3 3
E(k)
(8) "
Here k '2 = 1 - k 2, n is the number of kinks (or anti-kinks) in the system, Ek is the energy of the single kink 2~ 81 and K (k) and E (k) are complete elliptic integrals of first and second kind, respectively. For k = 1 Eq. (7) gives a kink-type solution (Fig. 4)
l(
q~(x) = ~
1 4- tanh \ 3 ~ / ~ J / '
(9)
which corresponds to the gb4-model solution [ 17]. By letting k ~ 1 or k' --~ 0 and expanding Eq. (8) up to order k '4 w e obtained the asymptotic interaction energy between kink and anti-kink Ft"tal - - Ek(1 - 12e-d/12v/~),
2n which is also in agreement with the q~4-model result [17].
(10)
374
M. Sanati, A, Saxena/Physica D 123 (1998) 368-379
I
I
,
'°
Fig. 5. Free energy for 2/9 < a < 1/4 is an asymmetric double-well potential with r-phase stable and w-phase metastable. 2. For } < a < ¼ and f0 = limx-+±oo FL(4'3) (Fig. 5) a lattice solution between the two product variants is given by 4 ' ( x ) = 4'j +
4'2 -- 4'1
(11)
1 - ot2sn2((x - xo)/(, k ) '
where
/--4'2)(4'4--4'1) k ~- V(4!-~-43- 4'2)(4'3 4'1)'
~ ~-
V/
8g (4'4 - 4'2)(4'3 - 4'1)'
O~=
~//---4'~7_33 -- 4'2. 4'1
This 4' (x) is a periodic function with the period 2 K (k) ( (Fig. 6). The physical meaning of Eq. (11) is that the order parameter 4' (x) takes on two different values, 4'3 and 4'2, alternatively as the coordinate x is varied. In other words Eq. (11) represents an alternating array of parent and product phases. This solution is definitely not the most stable state, but it may exist as a metastable phase when the boundary conditions are appropriate. Total free energy of the system per soliton pair in this case is equal to
Ftotal 2n
-- CiK(k) + C2E(k) + C3H(k),
(12)
where 17(k) is the complete elliptic integral of the third kind and coefficients C1, C2 and C3 are given in Appendix A. When k = 1 (4'3 = 4'4 = 4'0 = (1 + v/] - - 4a)/2), the lattice solution changes to a pulse-type solution (Fig. 6) 4,2 - 4'1
4'(x) = 4'1 + 1 -- ot2 tanh2((x -- xo)/()
(13)
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
0.80
375
I
0.50
0.20 -66.0
I
0.0 X
66.0
0.80
(h 0.50
0.20 -66.0
I
0.0 X
66.0
Fig. 6. Pulse-type soliton lattice (upper panel) and single soliton (lower panel) solutions for the asymmetric double-well potential in Fig. 5. The soliton corresponds to a "slab" of/3-phase sandwiched between the w-phase matrix.
Eq. (13) physically represents the existence of small amount of parent phase in the product matrix. The corresponding energy of a single pulse-type domain wall is given by E =
(t~2 - ~bl )2
(
~ 3ot4 _ 2ot2 + 3 ) 1 + Or" tanh-1 a q-
-iF-
(14)
Since it is difficult to write Eq. (12) as an expansion in k we could not calculate the interaction energy between the solitons with the method that was used in Part 1. However, one can calculate the interaction energy by using the asymptotic expansion of the pulse-type solution [18]. In this case the interaction energy is given by 5 1 2 g 2°t2
U(s) --
(1-Z~3
-2s/~"
e
" ,
(15)
where s is the distance between the two pulse solitons. For calculating the interaction energy, U(s), we choose the metastable minimum of the Landau-free energy as our reference point. 3. For 0 < a < 2 and f0 = limx-~±oc FL(q~2) (Fig. 7) a lattice solution between the two product variants is given by 4~4 - 4~3
Oh(X) = (l)4 -- 1 -/~2sn2((x -- x o ) / ( , k ) '
(16)
376
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
.
.
.
.
.
'f0
Fig. 7. Free energy for 0 < a < 2/9 is an asymmetric double-well potential with w-phase stable and fl-phase metastable.
where fl = V/(~b3 -- ~ 2 ) / ( 4 4 -- 42)"
Here k and ~ are the same as in Part 2. This 4(x) is also a periodic function with the period 2 K ( k ) ( (Fig. 8). The physical meaning of Eq. (16) is that the order parameter 4 (x) takes on two different values, 4~2 and 43, alternatively as the coordinate x is varied. In other words Eq. (16) represents an alternating array of parent and product phases. This solution is definitely not the most stable state, but it may exist as a metastable phase when the boundary conditions are appropriate. Total free energy of the system per soliton pair in this case is equal to Ftotal
2n
- - D 1 K ( k ) + D 2 E ( k ) + D3/7(k),
(17)
where coefficients Dl, D2 and D3 are given in the Appendix. When k = 1 (~bl = ~b2 = 0), the lattice solution will change to a pulse-type solution (Fig. 7) 44 -- ~b3
~b(X) = ~b4 -- 1 - f12 tanh2((x _ x o ) / ( ) "
(18)
Eq. (18) represents the existence of small amount of product phase in the parent matrix. The corresponding energy of this single pulse-type domain wall is given by E
g(~4
-- ~b3) 2 ( - -
1 q- f12
tarth-1/3 +
3/34 -- 2/32 -F 3
)
(19)
By using the same method as in Part 2 the interaction energy between the solitons in dilute limit (k --~ 1) is given by U ( s ) --
512g2/32 -2s/r (1- _-- /32)2-------3 e " " ,
where the definitions of s and the reference point are the same as in Part 2.
(20)
377
M. Sanati, A. Saxena / Physica D 123 (1998) 368-379
0.50
~) 0.25
I
0"~--49.0
0.0 X
49.0
0.0 X
49.0
0.50
~) 0.25
0.00 ~
49.0
~
Fig. 8. Pulse-type soliton lattice (upper panel) and single soliton (lower panel) solutions for the asymmetric double-well potential in Fig. 7. The soliton corresponds to a "slab" of w-phase sandwiched between the w-phase matrix.
5. Discussion and conclusion By augmenting the Landau model of Cook, which is in substantial agreement with many experiments, we proposed a Ginzburg-Landau continuum model for the description of possible domain configurations created from the formation of w-phase in b.c.c, phase of simple elements and alloys. From the nonlinear equation of equilibrium, the static solutions (domain walls) were calculated. The moving domain wall solutions are obtained by boosting the static solutions to velocity v according to x ~ (1 - v 2 ) - l / 2 ( x - v t ) . From the lattice solutions, the kink and pulse-type soliton solutions were derived in the limit k --+ 1. For each case, total energy for the lattice solution and asymptotic interaction energy between solitons in dilute (widely separated) limit were calculated. The results of our model are very different from that of the Horovitz et al. [ 12,13]. The latter model rests on the assumption that soliton-like solutions arise due to a mismatch between the lattice constants of the/3- and to-phases. According to this model there is a critical value of mismatch which is essential for the existence of the soliton-type solutions. This value is estimated to be the 5% of/3-phase lattice constants. However, there exists a number of elements and alloys for which the mismatch between the lattice constants are of the order of 1% - 2% of the/3-phase [2,19]. Although as a first approximation we neglect the mismatch between the lattice constant (i.e. the volumes of the two different phases), the energy coefficients of our model must be extracted from the experimental data similar to other phenomenological Landau models. In particular, the coefficients g and a are obtained from the curvature of the phonon dispersion curves and structural data (e.g., lattice constant as a function of temperature)
378
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
of the material, respectively. Thus, the effect of the mismatch between the lattice constants is implicitly contained in these coefficients. The validity of our model could be verified by applying this model to specific materials and comparing the predictions with experimental data, such as the spacing and width of domain walls. The other important point is the stability of the solutions. The kink-type solutions are known to be linearly stable. The pulse-type solutions are known to be unstable in field theoretic context [20]. Nevertheless, they could exist in real materials as long-lived metastable states by the way of special sample preparation. In other words there is a possibility that the system is trapped in a metastable minimum rather than the global minimum, l Finally, we note that the Landau free energy of the asymmetric double-well type (~2,3,4) is also used to model other kinds of first-order phase transitions such as b.c.c, to f.c.c, reconstructive phase transition [22], martensitic transformation in shape-memory alloys [23-25], nematic to isotropic transitions in liquid crystals [26] and proton transport in hydrogen-bonded chains [27]. Therefore, the generic results of our model could be applied to many of these diverse phase transformations.
Acknowledgements The authors acknowledge useful discussions with S.D. Cai. This work was supported in part by the US NSF and in part by the US DOE. One of us (MS) acknowledges financial support for this work from NSF Grant DMR9531223.
Appendix The coefficients Cl, C2 and C3 in Eq. (12) are Cl=(- (
_~ - F 0 + - - -a~b, ~--+- [ 6 +~ 1][ - ~ +0104+0203 ]) ,
2g ( - 2 a C2== ~-~
+ ~)
C3 = 2~"(~b2 - 4h )
(6 ')
~-~ ,
where the energy of the system is calculated with respect to the metastable minimum, Fo = FL (4~o). The coefficients Dt, D2 and D3 in Eq. (17) are foo + - u + - a~4 D ~ = ~ -(T - F
[ 6 + ~ 1 ] [ - ~ +0104+4,203 ] ) ,
D 2 = ~2g- ( - 2 a + ~ )
D3=--2~(~4-~3)(6 ~7) where the energy of the system is calculated with respect to the metastable minimum, F0 = FL (0) = 0. I For example, when liquid selenium is cooled sufficientlyfast the material is trapped in a glassy state, which is a metastable state. An activation energy is required for the transition from the glassy to crystalline state and as long as the energy barrier is large enough, the glass (metastable state) is practically stable. See [21].
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
References [1] P.D. Frost, W.M. Parris, L.L. Hirsch, J.R. Doig, C.M. Schwartz, Trans. ASM 46 (1954) 231. [2] S.K. Sikka, Y.K. Vohra, R. Chidambaran, Prog. Mat. Sci. 27 (1982) 245-310. [3] D. de Fontaine, N.E. Paton, J.C. Williams, Acta Metall. 19 (1971) 1153. [4] J.C. Williams, D. de Fontaine, N.E. Paton, Met. Trans. 4 (1973) 2701. [5] A.W. Sommer, S. Motokura, K. Ono, O. Buck, Acta Metall. 21 (1973) 489. [6] D.J. Cometto, G.L. Houze, Jr., R.E Hehemann, Trans. AIME 30 (1965) 233. [71 C.W. Dawson, S.L. Sass, Met. Trans. 1 (1970) 2225. [81 D.T. Keating, S.J. LaPlace, J. Phys. Chem. Solids 35 (1974) 879. [91 T.S. Kuan, S.L. Sass, Phil. Mag. A 36 (1977) 1473. [10] W. Lin, H. Spalt, B.W. Batterman, Phys. Rev. B 13 (1976) 5158. [11] M. Fatemi, Phil. Mag. A 50 (1984) 711. [12] B. Horovitz, J.L. Murray, J.A. Krumhansl, Phys. Rev. B 18 (1978) 3549. [ 13] B. Horovitz, in: A.R. Bishop, T. Schnneider (Eds.), Solitons and Condensed Matter Physics, Springer, Berlin, 1978, p. 254. [141 F. Falk, Z. Physik, B 54 (1984) 159. [15] H.E. Cook, Acta Metall. 23 (1975) 1027. [16] H.E. Cook, Acta Metall. 23 (1975) 1041. [17] R. Rajaraman, Phys. Rev. D 15 (1977) 2866. [18] N.S. Manton, Nucl. Phys. B 150 (1979) 397. [19] S. Banerjee, R.W. Cahn, Acta Metall. 31 (1983) 1721. [20] D. Boyanovsky, C. Aragao de Carvalho, Phys. Rev. D 48 (1993) 5850. [21] A. Modinos, Quantum Theory of Matter, Wiley, New York, 1996, p. 271. [22] P. Toledano, V. Dimitriev, Reconstructive Phase Transitions in Crystals and Quasicrystals, World Scientific, Singapore, 1996. [23] O. Nittono, Y. Koyama, Jap. J. Appl. Phys. 21 (1982) 680. [24] G.R. Barsch, J. de Phys. (Paris) IV (C 8) 5 (1995) 119. [25] A. Saxena, G.R. Barsch, Physica D 66 (1993) 195. [26] J.C. Toledano, P. Toledano, The Landau Theory of Phase Transitions, World Scientific, Singapore, 1987. [27] A. Gordon, Z. Phys. B 96 (1995) 517.
379
ELSEVIER
Domain walls in co-phase transformations Mahdi Sanati a.b.,, Avadh Saxena
b
a Department of Physics, University of Cincinnati, Cincinnati, 0H45221, USA b Theoretical Division, Los Alamos National Laboratory, Los Alarnos, NM 87545, USA
Abstract
The/%phase (body-centered cubic: b.c.c.) to w-phase transformation in certain elements (e.g. Zr) and alloys (e.g. Zr-Nb) is induced either by quenching or application of pressure. The w-phase is a metastable state and usually coexists with the /~-matrix in the form of small particles. To study the formation of domain walls in these materials we have extended the Landau model of Cook for the w-phase transition by including a spatial gradient (Ginzburg) term of the scalar order parameter. In general, the Landau free energy is an asymmetric double-well potential. From the variational derivative of the total free energy we obtain a static equilibrium condition. By solving this equation for different physical parameters and boundary conditions, we obtained different quasi-one-dimensional soliton-like solutions. These solutions correspond to three different types of domain walls between the ~o-phase and the/%matrix. In addition, we obtained soliton lattice (domain wall array) solutions, calculated their formation energy and the asymptotic interaction between the solitons. Copyright © 1998 Published by Elsevier Science B.V. Keywords: Ginzburg-Landau model; Asymmetric double-well potential; Domain wall arrays and interaction; w-phase transformation
1. Introduction
The formation of w-phase in elements (e.g. Ti, Zr, Hf) and alloys (e.g. Zr-Nb, Ti-Nb, Ti-Mo, Ti-V) has been the subject of considerable interest and activity since this phase was first reported by Frost et at. [1]. Presence of w-phase in materials changes their mechanical and physical properties [2]. Following the rationale of de Fontaine et al. [3,4] (that there is no physical basis for differentiating between athermal and thermal w-phase) we will consider the accepted mechanism for obtaining w-phase from the/3-phase (b.c.c.): Collapse of the (1 1 1)/3 plane due to a 2/3(1 1 1)~ longitudinal displacement wave. Thus, the w-phase is defined by and involves a collapse of a pair of (1 1 1)~ planes, leaving the neighboring plane unaltered, as depicted in Fig. 1. This type of transformation is called displacive because it involves cooperative movement over small distances which are fractions of lattice translation vectors. In contrast, a diffusion-controlled nucleation and growth transformation is termed replacive. Internal friction studies [5] (on fl-Ti alloys) support this mechanism whereby transformation occurs via atomic shuffles rather than by diffusion. (The shuffle transformation at the unit cell level is a transformation in which atomic displacements are intracellular with little or no pure strain of the lattice). The atomic movements required for the transformation * Corresponding author. Tel.: +1 505 665 0354; fax: +1 505 665 4063; e-mail: [email protected]. 0167-2789/98/$19.00 Copyright © 1998 Published by Elsevier Science B.V. All rights reserved PII S0167-2789(98)00135-3
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
W
W
/,
W
W
369
W
W
Y
. lal
JJjj ('0)
(el
Fig. 1. w-Phase transformation in a binary alloy. (a) Atomic (1 1 1~ planes in a b.c.c, lattice (fl-phase) before the transition, (b) partial collapse and (c) complete collapse of neighboring planes. are -q-an v/3/12 (or ±c~,/6) where a/~ is the lattice parameter of the fl-phase and c,o is the c-axis lattice parameter of the co-phase. The co-collapse may be "complete" or "incomplete" (meaning that the double layers are planar, or rumpled, respectively). Complete collapse produces the ideal hexagonal structure (also known as the ct-phase) with ao, = v/2a# and c~o = x/3/2al~, while partial collapse (< a# x/-3/12) produces the trigonal form of co-phase. As observed in experiments on quenched Zr-Nb alloys [6], the co-phase starts to form at a certain temperature, 7",o. On cooling below To, the process is found to be highly reversible and electron microscopy studies have shown that the new phase is finely distributed in a fl (b.c.c.) matrix [7]. In addition, diffraction experiments [8-10] and electron microscopic images [11] clearly indicate the rod-like inclusions of co-phase embedded in/4-phase matrix along [1 1 1]t~ direction. Based on these observations Horovitz et al. [12,13] attempted to explain the domain walls in cophase by a 1D soliton-like stacking fault model. This model is based on a discrete sine-Gordon type potential within a lattice. The theory assumes that the co-cluster tends to have a smaller lattice constant than that of the fl-matrix. If the difference is too small, the co-cluster is completely locked into the E-matrix and there are no solitons. However, if the difference is sufficiently large then (sine-Gordon type) solitons appear in the ground state, partially unlocking the co-cluster from the/3-matrix. The critical value 6/a# (where 6 is the difference between the lattice constant of the ,o-phase and that of the fl-matrix, a#) for which this transition occurs is estimated to be 5%. The assumption that the ~o-phase has a smaller lattice constant has been proven by high pressure studies on Ti-V alloys [2]. Nevertheless, no critical behavior in 6/a[~ has been observed which seems contrary to the predictions of the model by Horovitz et al. A possible alternative (and competing) model to explain the spacing and width of the domain walls for this ~tructural phase transition would be to add the Ginzburg term to the Landau free energy of the system and solve the equations which are derived from the variation of the total free energy of the system. This has been done in the context of first-order structural transitions using a triple-well (~b6) potential [ 14]. This approach is justified for the co-phase transformation especially when the mismatch of lattice parameters is sufficiently small, so as to be irrelevant tbr the mechanism considered by Horovitz et al. To model the atomic displacements, Cook [ 15,16] has developed a Landau
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
370
free energy for first-order structural phase transitions and used it to model the o-phase transitions in zirconium alloys. Cook's concepts are in substantial agreement with many recent experiments. However, to our knowledge the formation of domain walls between the o-phase and the/3 (parent) phase has not been studied for this type of Landau free energy. Note that the Ginzburg term represents the nonlocal interatomic forces (as a gradient of the order parameter) that give rise to a "phonon anomaly" such as the minimum of the longitudinal phonon branch at wave vector (27r/3) [1 1 1] for the o-phase transformation. The present work investigates the formation of domain walls, resulting from the presence of o-phase particles in a fl (b.c.c.) matrix, by including a spatial gradient term (Ginzburg term) to the Landau model free energy of Cook. From the equilibrium condition, i.e. by solving the equation derived from the variation of the total free energy for different physical parameters and boundary conditions, we find three different solutions (domain walls). For each type of solution we calculate the associated domain wall energy.
2. G e n e r i c m o d e l for o - p h a s e t r a n s f o r m a t i o n s
The Landau free energy for o-phase suggested by Cook has a single component (scalar) order parameter 7, 1 F L = ~1 A ~ ~/ - + ~I B r/3 + ~ C r/ 4 ,
where A is proportional to the quasiharmonic force constant, B and C are proportional to the third- and fourth-order force constants, respectively [16]. If B is negative, A and C are positive, for certain values of the parameters a first-order transition is possible from FL = 0 at 0 ~ 0 (parent phase) to some finite value of r/where FL is also zero. One can obtain the total free energy by adding the Ginzburg term FG = /G(~Tr/)2 to the Landau free energy, 1 4, F T = ½G(Vr/) 2 + ~1 A 0 "~ - + . ~I B r/3 + ~C~/
(1)
where coefficient G in the Ginzburg term is proportional to the appropriate curvature of the phonon dispersion curve of the material. Eq. (1) contains four parameters; by scaling the energy and order parameter it can be reduced to a standard form with two control parameters a and g, 1 4 F T / F ( , = ½g(V~b) 2 + ½a~b2 - /q~3 + UP ,
(2)
where C 4~ = -~] ~,
B4 Fo = ~ ,
g--
GC B2 ,
a=
AC B2 "
The scaled Landau free energy FL is plotted in Fig. 2. The plot shows the variation of Landau free energy with respect to a. The condition q~ = 0 corresponds to the parent phase (b.c.c.) and the other minimum (if present) corresponds to the product phase (o-phase). There are several regimes depending on the parameter a; the condition, "0FL / Oc~=O yields 4~ = 0, then (i) (ii) (iii)
~b = ~(1 + v ~ -
4a),
the following cases are possible: a > 1/4, FL has a real minimum at 4~ -----0, only (Fig. 2(1)). a = 1/4, FL has a real minimum at ~b = 0, and an inflection point at ~b = 1/4. 2/9 <: a < 1/4, FL has a stable minimum at q~ -----0, a metastable minimum at ~ = (1 + ~/1 - 4a)/2, and a relative maximum at q~ = (1 - ~/1 - 4 a ) / 2 (Fig. 2(2)).
371
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
H
I
f
1/
J Fig. 2. Variation of the Landau contribution to the total free energy (Eq. (2)) with q~for various values of the coefficient of the quadratic term, a. The upper (a >_ 1/4) and lower (a _<0) insets show the limit at which the potential becomes a single well. (iv) a = 2/9, FI. has two minima at values 05 = 0 and 05 = 2/3 with FL = 0 and a m a x i m u m at 05 = 1/3 (Fig. 2(3)). This is the condition for first order phase transition. (v) 0 < a < 2/9, FL has a metastable minimum at 05 = 0, and a stable minimum at 4) = (1 + ,/1 - 4 a ) / 2 for 05 > 2/3, with a m a x i m u m at 05 = (1 - ~/1 - 4 a ) / 2 (Fig. 2(4)). (vi) a < 0 is not relevant to the first order transitions (Fig. 2(5)). Domain walls exist in cases (iii)-(v), since only in these regimes parent and product phases coexist. The parameter a is determined from the structural data of the material.
3. Equilibrium conditions and domain wall energies From the variational derivative of the total energy, one obtains the following static equilibrium condition: 0F-r
OFT
Ox 005,
005
--0.
(3)
By substituting Eq. (2) in (3), one obtains g05" - a05 + 052 _ 4,3 = O.
(4)
We will show that the solutions of this differential equation for different physical parameters and boundary conditions (for cases (iii)-(v)) are quasi-one-dimensional solitary-wave solutions. Traveling kink (domain wall) solutions 05(x, t) are obtained from the static solution 05(x) by boosting to velocity v via x --+ (1 - v2) - 1/2 (x - vt). Finally by substituting the solutions of Eq. (4) in
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
372 oo
ftotal =
f
(V~b) 2 -q- 2~b 2 - -~- ~-
(5)
dx
mOG
we will calculate the domain wall energy in each case.
4. Results
After two integrations Eq. (4) leads to
x(~>- x0 =
(¼~_
(6)
~_~3+ ~a~2- :0
with the boundary conditions f0 = l i m x ~ + ~ FL, limx--,+~ q7 = 0 and the choice of origin ~b(x0) = 0. Depending on different boundary conditions one can obtain three types of domain array (lattice) solutions: 1. For a = 2-, 9 f0 = l i m x ~ _ ~ FL(4~2) and f0 = l i m x ~ + ~ FL(~b3) (Fig. 3) a lattice solution between the two product variants is given by (Fig. 4)
4~(x)=
li~l--~sn
,k
;
(=3 g(l~/+kZ),
(7)
where sn(x/(, k) is a Jacobi elliptic function with modulus k and the total free energy per kink-antikink pair for this solution is given by
--
f0
Fig. 3. Free energy for a = 2 / 9 is a double-well potential (the 4~4 model); f0 corresponds to the integration constant in Eq. (6).
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
373
0.8
0.4
0.0 -82.0
0.0 X
82.0
0.8
~) 0,4
J
0.0 -82.0
82.0
0.0 X
Fig. 4. Kink lattice (upper panel) and single kink (lower panel) solutions for the double-well potential in Fig. 3. The kink corresponds to a domain wall between the fl- and w-phase. The kink lattice is an alternating array of fl- and ~o-phase domains.
Ft°tal -- Ek 3v/2 217 (1 ÷ k2) 3/2
k ' 4 K ( k ) - 1-k'2K(k) + ~ E ( k ) 3 3
E(k)
(8) "
Here k '2 = 1 - k 2, n is the number of kinks (or anti-kinks) in the system, Ek is the energy of the single kink 2~ 81 and K (k) and E (k) are complete elliptic integrals of first and second kind, respectively. For k = 1 Eq. (7) gives a kink-type solution (Fig. 4)
l(
q~(x) = ~
1 4- tanh \ 3 ~ / ~ J / '
(9)
which corresponds to the gb4-model solution [ 17]. By letting k ~ 1 or k' --~ 0 and expanding Eq. (8) up to order k '4 w e obtained the asymptotic interaction energy between kink and anti-kink Ft"tal - - Ek(1 - 12e-d/12v/~),
2n which is also in agreement with the q~4-model result [17].
(10)
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M. Sanati, A, Saxena/Physica D 123 (1998) 368-379
I
I
,
'°
Fig. 5. Free energy for 2/9 < a < 1/4 is an asymmetric double-well potential with r-phase stable and w-phase metastable. 2. For } < a < ¼ and f0 = limx-+±oo FL(4'3) (Fig. 5) a lattice solution between the two product variants is given by 4 ' ( x ) = 4'j +
4'2 -- 4'1
(11)
1 - ot2sn2((x - xo)/(, k ) '
where
/--4'2)(4'4--4'1) k ~- V(4!-~-43- 4'2)(4'3 4'1)'
~ ~-
V/
8g (4'4 - 4'2)(4'3 - 4'1)'
O~=
~//---4'~7_33 -- 4'2. 4'1
This 4' (x) is a periodic function with the period 2 K (k) ( (Fig. 6). The physical meaning of Eq. (11) is that the order parameter 4' (x) takes on two different values, 4'3 and 4'2, alternatively as the coordinate x is varied. In other words Eq. (11) represents an alternating array of parent and product phases. This solution is definitely not the most stable state, but it may exist as a metastable phase when the boundary conditions are appropriate. Total free energy of the system per soliton pair in this case is equal to
Ftotal 2n
-- CiK(k) + C2E(k) + C3H(k),
(12)
where 17(k) is the complete elliptic integral of the third kind and coefficients C1, C2 and C3 are given in Appendix A. When k = 1 (4'3 = 4'4 = 4'0 = (1 + v/] - - 4a)/2), the lattice solution changes to a pulse-type solution (Fig. 6) 4,2 - 4'1
4'(x) = 4'1 + 1 -- ot2 tanh2((x -- xo)/()
(13)
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
0.80
375
I
0.50
0.20 -66.0
I
0.0 X
66.0
0.80
(h 0.50
0.20 -66.0
I
0.0 X
66.0
Fig. 6. Pulse-type soliton lattice (upper panel) and single soliton (lower panel) solutions for the asymmetric double-well potential in Fig. 5. The soliton corresponds to a "slab" of/3-phase sandwiched between the w-phase matrix.
Eq. (13) physically represents the existence of small amount of parent phase in the product matrix. The corresponding energy of a single pulse-type domain wall is given by E =
(t~2 - ~bl )2
(
~ 3ot4 _ 2ot2 + 3 ) 1 + Or" tanh-1 a q-
-iF-
(14)
Since it is difficult to write Eq. (12) as an expansion in k we could not calculate the interaction energy between the solitons with the method that was used in Part 1. However, one can calculate the interaction energy by using the asymptotic expansion of the pulse-type solution [18]. In this case the interaction energy is given by 5 1 2 g 2°t2
U(s) --
(1-Z~3
-2s/~"
e
" ,
(15)
where s is the distance between the two pulse solitons. For calculating the interaction energy, U(s), we choose the metastable minimum of the Landau-free energy as our reference point. 3. For 0 < a < 2 and f0 = limx-~±oc FL(q~2) (Fig. 7) a lattice solution between the two product variants is given by 4~4 - 4~3
Oh(X) = (l)4 -- 1 -/~2sn2((x -- x o ) / ( , k ) '
(16)
376
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
.
.
.
.
.
'f0
Fig. 7. Free energy for 0 < a < 2/9 is an asymmetric double-well potential with w-phase stable and fl-phase metastable.
where fl = V/(~b3 -- ~ 2 ) / ( 4 4 -- 42)"
Here k and ~ are the same as in Part 2. This 4(x) is also a periodic function with the period 2 K ( k ) ( (Fig. 8). The physical meaning of Eq. (16) is that the order parameter 4 (x) takes on two different values, 4~2 and 43, alternatively as the coordinate x is varied. In other words Eq. (16) represents an alternating array of parent and product phases. This solution is definitely not the most stable state, but it may exist as a metastable phase when the boundary conditions are appropriate. Total free energy of the system per soliton pair in this case is equal to Ftotal
2n
- - D 1 K ( k ) + D 2 E ( k ) + D3/7(k),
(17)
where coefficients Dl, D2 and D3 are given in the Appendix. When k = 1 (~bl = ~b2 = 0), the lattice solution will change to a pulse-type solution (Fig. 7) 44 -- ~b3
~b(X) = ~b4 -- 1 - f12 tanh2((x _ x o ) / ( ) "
(18)
Eq. (18) represents the existence of small amount of product phase in the parent matrix. The corresponding energy of this single pulse-type domain wall is given by E
g(~4
-- ~b3) 2 ( - -
1 q- f12
tarth-1/3 +
3/34 -- 2/32 -F 3
)
(19)
By using the same method as in Part 2 the interaction energy between the solitons in dilute limit (k --~ 1) is given by U ( s ) --
512g2/32 -2s/r (1- _-- /32)2-------3 e " " ,
where the definitions of s and the reference point are the same as in Part 2.
(20)
377
M. Sanati, A. Saxena / Physica D 123 (1998) 368-379
0.50
~) 0.25
I
0"~--49.0
0.0 X
49.0
0.0 X
49.0
0.50
~) 0.25
0.00 ~
49.0
~
Fig. 8. Pulse-type soliton lattice (upper panel) and single soliton (lower panel) solutions for the asymmetric double-well potential in Fig. 7. The soliton corresponds to a "slab" of w-phase sandwiched between the w-phase matrix.
5. Discussion and conclusion By augmenting the Landau model of Cook, which is in substantial agreement with many experiments, we proposed a Ginzburg-Landau continuum model for the description of possible domain configurations created from the formation of w-phase in b.c.c, phase of simple elements and alloys. From the nonlinear equation of equilibrium, the static solutions (domain walls) were calculated. The moving domain wall solutions are obtained by boosting the static solutions to velocity v according to x ~ (1 - v 2 ) - l / 2 ( x - v t ) . From the lattice solutions, the kink and pulse-type soliton solutions were derived in the limit k --+ 1. For each case, total energy for the lattice solution and asymptotic interaction energy between solitons in dilute (widely separated) limit were calculated. The results of our model are very different from that of the Horovitz et al. [ 12,13]. The latter model rests on the assumption that soliton-like solutions arise due to a mismatch between the lattice constants of the/3- and to-phases. According to this model there is a critical value of mismatch which is essential for the existence of the soliton-type solutions. This value is estimated to be the 5% of/3-phase lattice constants. However, there exists a number of elements and alloys for which the mismatch between the lattice constants are of the order of 1% - 2% of the/3-phase [2,19]. Although as a first approximation we neglect the mismatch between the lattice constant (i.e. the volumes of the two different phases), the energy coefficients of our model must be extracted from the experimental data similar to other phenomenological Landau models. In particular, the coefficients g and a are obtained from the curvature of the phonon dispersion curves and structural data (e.g., lattice constant as a function of temperature)
378
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
of the material, respectively. Thus, the effect of the mismatch between the lattice constants is implicitly contained in these coefficients. The validity of our model could be verified by applying this model to specific materials and comparing the predictions with experimental data, such as the spacing and width of domain walls. The other important point is the stability of the solutions. The kink-type solutions are known to be linearly stable. The pulse-type solutions are known to be unstable in field theoretic context [20]. Nevertheless, they could exist in real materials as long-lived metastable states by the way of special sample preparation. In other words there is a possibility that the system is trapped in a metastable minimum rather than the global minimum, l Finally, we note that the Landau free energy of the asymmetric double-well type (~2,3,4) is also used to model other kinds of first-order phase transitions such as b.c.c, to f.c.c, reconstructive phase transition [22], martensitic transformation in shape-memory alloys [23-25], nematic to isotropic transitions in liquid crystals [26] and proton transport in hydrogen-bonded chains [27]. Therefore, the generic results of our model could be applied to many of these diverse phase transformations.
Acknowledgements The authors acknowledge useful discussions with S.D. Cai. This work was supported in part by the US NSF and in part by the US DOE. One of us (MS) acknowledges financial support for this work from NSF Grant DMR9531223.
Appendix The coefficients Cl, C2 and C3 in Eq. (12) are Cl=(- (
_~ - F 0 + - - -a~b, ~--+- [ 6 +~ 1][ - ~ +0104+0203 ]) ,
2g ( - 2 a C2== ~-~
+ ~)
C3 = 2~"(~b2 - 4h )
(6 ')
~-~ ,
where the energy of the system is calculated with respect to the metastable minimum, Fo = FL (4~o). The coefficients Dt, D2 and D3 in Eq. (17) are foo + - u + - a~4 D ~ = ~ -(T - F
[ 6 + ~ 1 ] [ - ~ +0104+4,203 ] ) ,
D 2 = ~2g- ( - 2 a + ~ )
D3=--2~(~4-~3)(6 ~7) where the energy of the system is calculated with respect to the metastable minimum, F0 = FL (0) = 0. I For example, when liquid selenium is cooled sufficientlyfast the material is trapped in a glassy state, which is a metastable state. An activation energy is required for the transition from the glassy to crystalline state and as long as the energy barrier is large enough, the glass (metastable state) is practically stable. See [21].
M. Sanati, A. Saxena/Physica D 123 (1998) 368-379
References [1] P.D. Frost, W.M. Parris, L.L. Hirsch, J.R. Doig, C.M. Schwartz, Trans. ASM 46 (1954) 231. [2] S.K. Sikka, Y.K. Vohra, R. Chidambaran, Prog. Mat. Sci. 27 (1982) 245-310. [3] D. de Fontaine, N.E. Paton, J.C. Williams, Acta Metall. 19 (1971) 1153. [4] J.C. Williams, D. de Fontaine, N.E. Paton, Met. Trans. 4 (1973) 2701. [5] A.W. Sommer, S. Motokura, K. Ono, O. Buck, Acta Metall. 21 (1973) 489. [6] D.J. Cometto, G.L. Houze, Jr., R.E Hehemann, Trans. AIME 30 (1965) 233. [71 C.W. Dawson, S.L. Sass, Met. Trans. 1 (1970) 2225. [81 D.T. Keating, S.J. LaPlace, J. Phys. Chem. Solids 35 (1974) 879. [91 T.S. Kuan, S.L. Sass, Phil. Mag. A 36 (1977) 1473. [10] W. Lin, H. Spalt, B.W. Batterman, Phys. Rev. B 13 (1976) 5158. [11] M. Fatemi, Phil. Mag. A 50 (1984) 711. [12] B. Horovitz, J.L. Murray, J.A. Krumhansl, Phys. Rev. B 18 (1978) 3549. [ 13] B. Horovitz, in: A.R. Bishop, T. Schnneider (Eds.), Solitons and Condensed Matter Physics, Springer, Berlin, 1978, p. 254. [141 F. Falk, Z. Physik, B 54 (1984) 159. [15] H.E. Cook, Acta Metall. 23 (1975) 1027. [16] H.E. Cook, Acta Metall. 23 (1975) 1041. [17] R. Rajaraman, Phys. Rev. D 15 (1977) 2866. [18] N.S. Manton, Nucl. Phys. B 150 (1979) 397. [19] S. Banerjee, R.W. Cahn, Acta Metall. 31 (1983) 1721. [20] D. Boyanovsky, C. Aragao de Carvalho, Phys. Rev. D 48 (1993) 5850. [21] A. Modinos, Quantum Theory of Matter, Wiley, New York, 1996, p. 271. [22] P. Toledano, V. Dimitriev, Reconstructive Phase Transitions in Crystals and Quasicrystals, World Scientific, Singapore, 1996. [23] O. Nittono, Y. Koyama, Jap. J. Appl. Phys. 21 (1982) 680. [24] G.R. Barsch, J. de Phys. (Paris) IV (C 8) 5 (1995) 119. [25] A. Saxena, G.R. Barsch, Physica D 66 (1993) 195. [26] J.C. Toledano, P. Toledano, The Landau Theory of Phase Transitions, World Scientific, Singapore, 1987. [27] A. Gordon, Z. Phys. B 96 (1995) 517.
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