edge dislocations in anisotropic inhomogeneous media
TRANSCRIPT
Edge Dislocations in Anisotropic Inhomogeneous MediaC. S. Pande and Y. T. Chou Citation: Journal of Applied Physics 43, 840 (1972); doi: 10.1063/1.1661291 View online: http://dx.doi.org/10.1063/1.1661291 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/43/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Diffusion in inhomogeneous and anisotropic media J. Chem. Phys. 119, 5171 (2003); 10.1063/1.1597476 Edge dislocation inside a lamellar inhomogeneity J. Appl. Phys. 64, 1594 (1988); 10.1063/1.341795 Interfacial edge dislocations and dislocation walls in anisotropic two−phase media J. Appl. Phys. 46, 5 (1975); 10.1063/1.321369 Dislocation Displacement Fields in Anisotropic Media J. Appl. Phys. 40, 2177 (1969); 10.1063/1.1657954 Screw Dislocations in Anisotropic Media J. Appl. Phys. 37, 4051 (1966); 10.1063/1.1707974
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
141.212.109.170 On: Thu, 18 Dec 2014 18:30:07
840 W.A. FATE
65 expected from theory. On the basis of either the Leibfried or Lothe calculations Eq. (4) predicts that dct/dT-1 should be frequency independent above the damping maximum. However, this slope was found to change by more than a factor of 2 between 11. 2 and 29.7 MHz, see Fig. 5.
Thus we see that the GL theory cannot explain the frequency dependence of the dislocation attenuation in lead in the large and small wE limits. Since the frequency dependence given by the approximations of Eqs. (3) and (4) is independent of the distribution of loop lengths, a weighted sum of exponential loop-length distributions, used previously to explain the frequency dependence of ctD in copper, 17 cannot explain the present results.
V. SUMMARY AND CONCLUSION
With the assumption that the GL theory correctly describes the dependence of the dislocation component of the ultrasonic attenuation on dislocation damping factor in the small and large wE limits, it has been shown that the Lothe formula for the thermal damping factor correctly describes the temperature dependence of the attenuation in copper and lead. In the case of lead the main qualitative features of the damping maximum predicted by the GL theory were not observed. Moreover, the observed frequency dependence of the attenuation is in disagreement with the GL prediction for any distribution of dislocation loop lengths.
It is concluded that while the GL theory may correctly describe the dependence of the attenuation on damping factor, it does not always correctly predict the frequency dependence of the attenuation or the qualitative properties of the damping maxium.
ACKNOWLEDGMENTS
The author is grateful to Dr. A.D. Brailsford for helpful discussions and to Dr. J. R. Reitz for reading the manuscript.
lG. Leibfried, Z. Physik 127, 344 (1950). 2W.A. Fate, Appl. Phys. Letters 18, 92 (1971). 3J. Lothe, J. Appl. Phys. 33, 2116 (1962). 4A. D. Brailsford, J. Appl. Phys. 41, 4439 (1970). 5A. Granato and K. Lucke, J. Appl. Phys. 27, 583 (1956). 6For a summary see R. W. Morse, in Progress in Cryogenics (Heywood and Company, London, 1959), Vol. 1, p. 219.
7W.A. Fate, Phys. Rev. 172,402 (1968). 8D. s. Woo, Ph. D. thesis (Rensselaer Polytechnic Institute, 1967) (unpublished).
9C.P. Bean, R.W. DeBlois, and L.B. Nesbitt, J. Appl. Phys. 30, 1976 (1959).
10R.E. Love, R. W. Shaw, and W.A. Fate, Phys. Rev. 138, A1453 (1965).
l1A. C. Rose-Innes, Low Temperatures Techniques (English U.P., London, 1964), pp. 83-92.
12Natl. Bur. std. (US) Monograph No. 10 (U.S. GPO, Wash-ington, D. C. , 1958).
13H.J. Hoge, J. Res. Natl. Bur. Std. 44, 321 (1950). HE.R. Grilly, Cryogenics 2, 226 (1962). 15J. P. Frank and D. M. Martin, Can. J. Phys. 39, 1320 (1961). 16For a review see D.O. Thompson and V. K. Pare, in Physical
Acoustics, edited by W.P. Mason (Academic, New York, 1966), Vol. III, Part A, p. 293.
l1R.M. Stern and A. V. Granato, Acta Met. 10, 358 (1962). 18A.D. Brailsford, Phys. Rev. 186, 959 (1969). 19A. Hikata, R.A. Johnson, and C. Elbaum, Phys. Rev. Let
ters 24, 215 (1970). 2oBrailsford, Ref. 4, has shown there are quantitative difficul
ties with this approach. 21A. Seeger as quoted by D. H. Niblett in Ref. 16, Vol. III,
Part A, p. 90. 22A. H. Cottrell, Dislocations and Plastic Flow in Crystals
(Clarendon, Oxford, England, 1953), p. 52. 23W. P. Mason, Physical Acoustics and the Properties of Solids
(Van Nostrand, New York, 1958), p. 267. 24W. P. Mason, Ref. 23, p. 17.
Edge Dislocations in Anisotropic Inhomogeneous Media
C.S. Pande and Y. T. Chou Department of Metallurgy and Materials Science, Lehigh University, Bethlehem, Pennsylvania 18015
(Received 20 September 1971)
A simple method is given for obtaining the elastic stresses of an edge dislocation near a boundary (grain, interphase, or free boundary) using anisotropic elasticity theory. Exact and explicit solutions are obtained for the cases where the adjacent grains possess certain symmetry elements. The stress functions lead to the image force due to the boundary on the dislocation as Fi = - (Kb2q1/47r) (II a), where K and ql are constants involving the elastic constants only, a is the vertical distance of the dislocation from the boundary, and b is the Burgers vector of the dislocation.
I. INTRODUCTION
The elastic stress fields of an edge dislocation in isotropic inhomogeneous media were given by Headl for three types of idealized grain boundaries, viz., welded, slipping, and free boundary. Recently, several workers have reexamined this problem, on the basis of anisotropic elasticity, employing either the theory of Cauchy integral2 ,3 or the theory of Fourier transformations. 4.5
Pastur et al. 2 employ the former approach, first used by Eshelby et al. 6 and later put in a more convenient
J. Appl. Phys., Vol. 43, No.3, March 1972
form by stroh,7 whereas Gemperlova and Saxl4 employ the second approach, first given by Seeger and Schoeck.s These solutions, however, are cumbersome and to some extent inexplicit. Furthermore, it is not easy to simplify these results for the situation when the material possesses certain symmetry elements. Since in many practical applications the anisotropy of the material is important and also the stress fields are needed in the simplest possible form, the problem of an edge dislocation in an anisotropic inhomogeneous medium possessing certain symmetry elements is treated
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
141.212.109.170 On: Thu, 18 Dec 2014 18:30:07
DISLOCATIONS IN ANISOTROPIC MEDIA 841
afresh by a different and simpler treatment which readily gives the elastic fields in a compact and explicit form. A similar treatment for screw dislocations has been given by Chou. 9
II. FORMULATION OF THE PROBLEM
We consider an anisotropic two-phase medium with elastic constants and other parameters distinguished by superscript (1) for x>O and by superscript (2) for x<O. The elastic medium is under conditions of plane strain in the xy plane due to an infinitely long edge dislocation parallel to the z axis, at x == a, Y == O. Further, we assume that the two adjacent phases possess certain symmetry elements such that for both regions10
C 14 =C15 = C16 = C24 ==C25 = C26 = C46 = C56 =0. (1)
Then if u and v denote the x and y displacement components, the following relations hold at x = O.
(i) welded boundary:
u(U =U(2),
v(U =V(2),
<7~~) = <7~;) ,
<7~;) = <7~~) ,
all at x=O;
(2)
(3)
(4)
(5)
(ii) slipping boundary: only conditions (2) and (4) hold, and in addition
<7~;) =<7~~) ==0 at x==O;
(iii) free boundary:
a~~) = <7~;) == 0 at x = O.
(5')
(6)
Equations (2) and (3) can be written in a slightly different but equivalent form as
dV(ll dV(2)
"dy="dy'
Following Eshelby et al. ,6 we assume that the displacements u and v are given as
u ==Re(C1'11+ - C2~,)'
v=Re[-A(C1'11+ +c2~')1,
(7)
(8)
(9)
(10)
where Re denotes the real part of the complex functions and C 1 and C2 are complex constants. The constant A and the functions '11. are defined as
and
A = _A..:...(C:...;6:::;:6.:..e_I "'_, _+.....,V-=.:12,-e_-I_"')
C12 + C66 (11)
(12)
where k J are unknown constants to be determined,
A=(Cll )1/4=(C12 )1/2, (13) C22 C22
C12 == (C 11C22)1/2, (14)
and
(15)
In the following analysis, the anisotropy parameter a is considered to be positive real. In actual practice, a can be complex. 11,12 When a is complex, the analysis is slightly different, but the final stress functions are the same as obtained below. Further, for convenience we consider in detail only the case in which the Burgers vector of the edge dislocation is perpendicular to the boundary (=b.). In this case C1 =C2=CO (see Ref. 6). When the Burgers vector is parallel to the boundary, the method of solution is similar but C1 = - C2 •
USing Eqs. (9)-(12), the boundary conditions at x=O given by Eqs. (4), (5), (7) and (8) can be written respectively, as
Re(E(U cfJ~U) == Re(E(2) cfJ~2»),
Re(G (1) cfJ~U) == Re(G(2) cfJ~2»),
Re( 1'(1) cfJ~U) == Re( 1'(2) cfJ~2»),
Re(A <1)1'(1) cfJ~ll) =Re(A (2)y(2)<1>~2»),
where
y=~ej""
E==C ll - C12yA,
G==C66(y-A),
<1>. == Co'l1: ± Co'l1!,
and .T,I d'I1 .., == dt:'
III. GENERAL METHOD OF SOLUTION
(16)
(17)
(18)
(19)
(20)
The solution of the problem essentially consists in determining the unknown function '11 consistent with the boundary conditions. Since all the boundary conditions are for x = 0, it is much easier first to determine '11' at x == O. Further, Head's result for the isotropic case indicates that '11' is likely to consist of the "homogeneous" term, the "image" term plus extra term or terms. These considerations suggest a trial function '11' of the form
Ccill'l1;o' = K1 + K2 + ~3 + ~4, (21) Zl Z2 Zl Z2
where Zl' Z2' Zl' Z2 contain all the possible sums of ±a and ±Aexp(±ia). Similarly,
Cci2)'I1(2)'=B...+~+ P:J + ~ + e01 E02 ~ iIT2
(22)
and similar expressions for '11~1)' and '11~2)·. Here E:, K 1,
and ~ (i == 1 ,2,3,4) are unknown complex constants to be determined by the boundary conditions,
Z 1 = - a + y(ll y, the "homogeneous" term
Z2 = - a - y(O y, the "image" term
01 == _g+y(2)y,
02 = - g- y(2)y,
(23)
J. Appl. Phys., Vol. 43, No.3, March 1972
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
141.212.109.170 On: Thu, 18 Dec 2014 18:30:07
842 C.S. PANDE AND Y.T. CHOU
and g is an unknown constant. It is easily seen that relations exist between Z's and n's (such relationships are are expected because of the boundary conditions). The relations are
En1=zl' En2=z2 provided
E = y(I) ly(2) , g= alE.
Therefore (22) can be written
(24)
(25)
Cci2)w,<2)' = P" + P:! + ~3 + ~4 • (26) ZI Z2 Zl Zz
Using (21) and (26), and after some manipulations, the boundary conditions (16)-(19) respectively, yield the . following equations: E(1)(K -K) _E(2)(P' -R) +E(1)(K -K) -E(2)(R -P)-O 12 12 34 34-'
(27)
G(l)(K +K) _G(2)(P. +R) +G(I)(K +K) -G(2)(R +P)-O 12 12 g 4 g 4-'
(28)
y(1) (Kl + K2) - y(2) (PI + Fa) + y<U (Kg + K4
) - y(2) (Fa +~) = 0,
(29)
A (1) y(l)(KI -K2) -A (2),,(2)(P1 - Pz)+ A (l),,<1)(Ks -K4)
_A(Z),,(Z) (Fa -~) =0; (30)
whereas the condition (5') gives
G(I)(K1 +Kz) + G(I)(Kg + K4
) = G(Z)(~ + Fa) + G(Z) (Fa +~) =0.
(31)
IV. SOLUTIONS
A. Free Boundary
Out of Eqs. (27)-(30) only (27) and (28) hold, with the terms involving~, Fa, Fa, and ~ vanishing. In order that the stresses may not have a singularity apart from that at the dislocation site (i. e., at x = a), K4 is also equal to 0, and therefore
(C well') =Kl + K2 + !:3 O. x=O Zl Z2 Zl' or,
Cow.W '(x, y)
Kl K2 x- a+ ~05 exp(ia(1)y x+ a+ i\(1) exp(ia(l)y
+ Ks exp(2i a(1» x - a exp(2i all)~ + i\ 1 exp(ia(l)y , (32)
where the first term is thus seen to be the homogeneous term if Kl is given by
Kl = -21
. (2~(1) i~ 2 <u [cg) - cg) exp(U(l!(1)J). (33) 1Tl lZ sin (l!
Then from (27) and (28), omitting the superscript (1):
(E/E)- (GIG»
Kz=Kl (E/E)+(G/G> ' (34)
K 2K; s = - (ElF) + (GIG) .
Substituting for E, E, G and G, we obtain
K2 = iKl tana ,
J. Appl. Phys., Vol. 43, No.3, March 1972
(35)
(36)
K3 =-Klexp(-3ia)/cosa. (37)
Since w(x, y) is completely determined, the stresses are immediately obtained: for x;;, 0
Un = - Ke2be"-2(1.. [(C +3)(x - a}2 + i\zy2J
rr Xl
_1.. [(C + 3)(x + a)Z + ).,2y2J\ X2 J
- Kebei\2 (1.. [2 ~ + C(x + a)2J _1.. [2 ~ + C(X2 + aZ)J\ , rrC X2 Xs J
(38)
U = Kebe (.l. [(x _ a)Z _ i\zo.2]_ 1:. [(x + a)2 _ i\Zo.2J) yy 2rr Xl y X
2 y
+Kebe (1.. (2~+Ci\21) rrC X2
-1:.{2~ +C[(C +4)a2 +2ax+ i\Zy2n), (39) Xs
Uxy = K2ebe (X - a [(x _ a)2 _ ~y2J _ x + a [(x + a)2 _ i\2y2]) rr Xl X2
_Kebe (x+a (2~+Ci\2y2) rrC X2
-.!.. {2(x + a)~ + C[Ca2x + a(x + a)(3x + a) + Xi\Zy2]}\ , Xs /
where
Xl = [(X - a)Z + i\2y2]2 + C(X _ a)2i\Zy2,
Xz = tf + C(X + a)2).,Zy2,
Xg = (~+ Cax)2 + C(X - a)2).ZI,
~ = (x + a)2 + i\2y2 ,
(- ) ~ C66«(;12 - C12) ) 1/2 Ke= C1Z +C 12 C (C +C +2C) , 22 12 12 66
C «(;12 + CI2)«(;12 - C12 - 2C66 ) 4 C - , - < < 00. C12C66
A more detailed discussion is given by Pande and ChOU lS ,14
B. Slipping Boundary
(40)
(41)
Corresponding to the boundary conditions (2), (4), and (5'), Eqs. (28), (29), and (31) hold. In order that the stresses may not have a singularity apart from that at the dislocation site, K4 =Pz =Ps = O. The constants K2,
Ks are then immediately obtained, on solving the Eqs. (28), (29), and (31) for K2 and Kg and on substituting for E(1) and G(l).
K2 = - [K1 exp(- id<1»/cosam]t(1-M) -MKl' (42)
K - _ exp(-3i(l!(1)K1 1-M (43) g - COS(l!(I) 2'
where
(44)
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
141.212.109.170 On: Thu, 18 Dec 2014 18:30:07
DISLOCATIONS IN ANISOTROPIC MEDIA 843
and Kl is given as before by Eq. (33). The stresses for x >0 are then obtained as15
(] = _ Ke be A(l)2 (1.. [(C(l) +3)(x _ a)2 + A(l)ll y2]
xx 27T Xl
K b (1)2 My 2 ~ I-M e eX +- [(C(l) +3)(x+a)2 +X(ll y] --- (1) ~ 2 ~
(] = Kebe (Y [(x _ a)2 _ X (1)2 y2] +~ (x + a)2 - X(l)2lJ\ yy 27T Xl X2 I)
+ 1 - M Kebe (1.. (2 ~ + C(l) A(u2 2) 2 7TC(ll X2 y
_1.. {2~ +C{l)[(C(l) +4)tf +2ax + X(1)2yl}), X3
(] =-- -- x-a -A y Kebe (x-a [( )Z '(l)2 • .2] xy 27T Xl
+M(x+a) [(x+a)2_'i\(1l1lyllr X
2 I)
(46)
+l-M Kebe (x+a (2~+C(t)XW\,2)-.!..{2(x+aH 2 1fC(I) X2 X3
+ C(1)[C(oa2x + a(x + a)(3x + a) + XA (1)2y21}) • (47)
C. Welded Boundary
Equations (27), (28), (29), and (30) hold, and as before K4 = Fa = Ps == O. The constants Kll and Kg are then obtained in the usual way as
K2 =K1[Ql +itana(l)(Qo - C(l)Qz)J
==K1{Ql +Qo - C(l)Q2
- [exp(- ia(1»)/ cosaP)](Qo - CUlQ2)}' (48)
Kg == - [exp(- 3ia(1)/cosa <1lJKJQo + CW N], (49)
where Qo' Ql' Q2' and N are real quantities and are given as
Qo =H;I[2{Cg l + cg» (Cgl - cgl )
+ (C(2l2 _ C(I!)2)(C(1l + C(l»/C(ll
12 12 66 12 66
(C<u 2 CW2)(C(2) + C(Z»/C(Z)] (50) - 12 - 12 66 12 66'
Q1 = (X (1) A (2) HO)-1[4 sina(l) sina(2lCg>C1~) ('i\ (1)2 _ X (2)2)],
(51)
Q - (C-(Zl 2 _ C(2)2)C(l)/H (C<O + C(l) 2 - 12 12 12 0 12 12' (52)
(53)
N-C<lJ(C(Z) _ C(Z»/H . (54) - 12 12 12 0'
(] = Kebe (1.. [(x_a)2_x(1l 2 • .21_(Q +Q _C(l)Q)
yy 21f Xl Y 1 0 2
X 1.. [(x + a)l! - A(02l1) X2
+ Kgb,. I(Q _ C(l)Q )1.. (2~ +C(1l'i\<1l2y2) ITCh) ,~ 0 2 X
2
- (Qo + C(l) N) ~ {2~ + C(l)[(C(1)+4)a2+2ax + A(1)2yaU), (56)
(] = Keb,. (X - a [(x _ a)2 _ A(1)2y21_ (Q + Qo
_ C(llQa) xy 21T Xl 1
X x~a [(x+a)2 - 'i\(1)Zy2J) - !C~~l (Qo - C(l)Qz)
x x +a (2 ~ + C(llX (1)2 yll) _ Qo + C(l) N {2(x +a)~ X2 X3
+ C(ll[CHltfx+ «(x +a)(3x + a) + XA<l)2l ]}). (57)
V. DISCUSSION
A. Stress Component Along the Slip Plane
In the application of the results to practical problems, such as yielding in a heterogeneous material or an anisotropic bicrystal, the most important quantity is the component of the shear stress along the slip plane due to the dislocation, i.e., (]Xy(y=O). Our treatment gives these components in a particularly simple form as
(J' ( -0)- Kebe (~_...!lL + 2aq2(a-x) , xy y- - 21T \'x-a x+a (x+a)[(x+a)2+ CaxJ
x20 (58)
where the quantities ql and q2 are defined in Table 1. It is interesting to note that apart from an additional term in the denominator of the third term which is small for most materials as Co<O (for isotropic materials C = 0), the algebraic form of (J'xy(y=O) is the same as for the isotropic case. The constants are of course different. This means, that when the effect of the additional term is small, the results of the isotropic elasticity can directly be used by simply choosing the appropriate values of the anisotropic constants to replace the iso-
1. Appl. Phys., Vol. 43, No.3, March 1972
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
141.212.109.170 On: Thu, 18 Dec 2014 18:30:07
844 C. S. PANDE AND Y. T. CHOU
TABLE I. Expressions for ql and q2 for the three kinds of boundaries [see Eq. (58)1.
Boundary ql q,. Remarks
Free 1 1 ql and q2 are the same as in the isotropic case
Slipping -M (1-M) -1 <M < 1
Welded Q1 + Qo - Q2 (C(1) + 4) -4N Qo+ C(1)N - 1" til ., 1
tropic ones. The range of values of C for which such a procedure is valid, without significant error, is being investigated and will be reported later.
B. Image Force Due to the Boundary
Another way of studying the interaction of the boundary, with the dislocation is to obtain the stress (Ji' tending to move the dislocation by glide. It is the value of (J"y at the position of the dislocation, excluding the singular term corresponding to the stress due to the dislocation in a homogeneous medium. It is found from (58) that in all the three cases, (Ji and the image force .Fj are given respectively as
(J = _ (Kebell;J!' j 41T fa
and F. = _ (Keb~ql)!. i 41T a' (59)
where the negative sign denotes the stress tending to move the dislocation towards the boundary. The stress (Jj can be thought of as the shear stress which would be produced by a dislocation of Burgers vector - qlbe at the image point (- a, 0). From (59) it is seen that in the case of the free boundary, the dislocation is always attracted to the free surface; in the case of the slipping boundary, the dislocation will be attracted or repelled from the boundary, accordingly as
sina(2)C~~)
is greater or less than
sina(l)Ca>
J. Appl. Phys., Vol. 43, No.3, March 1972
This behavior is different from that of a screw dislocation which is always attracted to the Slipping boundary. In the case of the welded boundary, no such simple interpretation of the image force is possible.
C. Stress Field of a General Dislocation
In Sec. IV, the stresses due to an edge dislocation with Burgers vector (be' 0, 0) are given. The stresses for the dislocation with Burgers vector (0, b", 0) can be obtained in a Similar fashion. 13 These results combined with that of the screw component given by Chou,9 give the stresses of a general dislocation with an arbitrary Burgers vector b==(be , b", b.). The displacements u, v, and w can also be similarly obtained.
ACKNOWLEDGMENT
The authors wish to thank Dr. G. P. Conard II of this department for helpful discussions. The work was supported in part by the National Science Foundation.
lA. K. Head, Proc. Phys. Soc. (London) 66B, 793 (1953). 2L.A. Pastur, E.P. Fel'dman, A.M. Kosevich, and V.M. Kosevich, Sov. Phys. Solid State 4, 1896 (1963).
3M.a. Tucker, Phil. Mag. 19, 1141 (1969). 4J. Gemperlova and 1. Saxl, Czech J. Phys. B18, 1085 (1968). 51. Saxl, Czech. J. Phys. B19, 836 (1969). 6J.D. Eshelby, W. T. Read, and W. Shockley, Acta Met. 1, 251 (1953).
TA. N. Stroh, Phil. Mag. 3, 625 (1958). BA. Seeger and G. Schoeck, Acta Met. 1, 519 (1953). 9y. T. Chou, Phys. Status Solidi 15, 123 (1966).
IOThroughout this paper, when the superscripts are omitted. the statements are valid for both regions.
I1J. P. Hirth, K. Malen, and J. Lothe, Scripta Met. 5, 231 (1971).
12y. T. Chou and G. T. Sha, Scripta Met. 5, 551 (1971). 13C. S. Pande and Y. T. Chou, Phys. Status Solidi (a), 6, 499
(1971) . 14In Ref. 13, due to a typographical error the sign of the third
term in Eq. (31) is given positive. It should be negative as it appears in Eq. (40) of this paper.
15In Eq. (45) -(47) and (55)-(57), the parameters Ke , Xh X2, X3, and ~ refer to region 1.
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
141.212.109.170 On: Thu, 18 Dec 2014 18:30:07