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Chapter
A BRIEF REVIEW ON THE HIGH-Tc SUPERCONDUCTING COMPOUNDS AND AN INTRODUCTION TO THE THEORY
OF ELASTICITY AND THERMAL EXPANSION
1 .I Introduction
The discovery of superconductivity in La-Ba-CuO by Bednorz and
Muller (1) in 1986 near 30 K triggered a flood of research on high-temperature
superconductivity that has resulted in thousands of papers and innumerable
conferences in this field. A sequence of new materials which show
high-temperature superconductivity has subsequently been discovered. Although
each of these materials has its own individual chanckristics, they share a largely
common phenomenology (2).
The direct consequence of the high transition temperature Tc and low
carrier concentrations is that the Bardeen-Cooper-Schreiffer (BCS) coherence
length (3) 5 0 is very small. A representative value of 50 is 10A, which is of the
order of unit cell dimensions in these compounds. The small coherence length
gives a coherence volume so small as to contain only a few Cooper pairs, which
implies that the fluctuations can play a much larger role in these materials than in
the classical superconductors.
After the discovery of high temperature superconductivity in La-Ba-CuO
system, there have been many elemental substitutions and different processing
conditions on the structure and superconducting properties of this oxide. In 1987,
Wu el al. (4) discovered superconductivity above 90 K in the Y-Ba-CuO system.
This discovery triggered a fluny of crystallographic research unprecedented in the
history of materials science (5). This discovery was quickly reproduced and the
90 K phase was identified as YBa~Cu307 and studies of the effects of substitution
and processing conditions were initiated. The cycle has been repeated twice in
1988 with the independent discoveries of superconductivity near 100 K in the
Bi-Sr-Ca-CuO system by Hiroshi Maeda et al. (6) and in the TI-Ba-Ca-CuO system
by Zhengzhi Sheng and Allen Hermann (7,s).
Since the remarkable discovery of high-Tc superconductors, there has been
an enormous effort to elucidate their fundamental properties. Research on the
atomic structures of high-temperature superconductors has played a prominent role
In characterising these materials.
Early in 1987, researchers (9-1 1) found that substitution of Sr for Ba in the
La-Ba-CuO compound raises the transition temperature to approximately 40 K.
The superconducting Laz-,Sr,CuOd is derived from the stoichiometric compound
La2Cu04 by replacing ~ a ~ ' with srZ+ partially, and it has a tetragonal structure at
room temperature. The high-symmetry form which is tetragonal can be adopted by
La2Cu04 at temperatures above 500 K (12) and it is stabilised at lower
temperatures by the partial substitution of Sr or Ba for La (13). Among the high-Tc
cuprates, La2.,SrxCu04 has the simplest crystal structure with the single CuOz
planes separated by the L a 0 charge reservoirs (14). Ramirez (15) recently
observed that by substituting divalent strontium for trivalent lanthanum in La-based
214 compounds, the antiferrornagnetism is destroyed. La2..SrxCu04 compounds
have a structure of perovskite type, belonging to the I4/mmm space group (1 6). The
unit cell contains two formula units with CuOs octahedra occupying the body
centre and eight corner sites. Each copper atom in this T structure is co-ordinated
to four oxygens in a perfect corner-linked array of square planes. Superconducting
behaviour with a transition temperature Tc near 35 K is achieved in LazCu04
when it is doped with strontium on the lanthanum site to form La1.8,Sr0.15Cu04
(17).
Cuprate superconductors have in common a lower carrier density. Their
characteristic structural elements are chains, planes, pyramids or octahedrons
formed by Cu or Bi with unstable valence and oxygen atoms. These building
blocks are assembled by ions which act as spacers and dopants. For the critical
doping level leading to superconductivity, one charge carrier per formula unit
typically is delocalised in the Cu-0 bond (19).
Another class of high-Tc superconductors which aroused much interest are
the Nd-based 214 compounds. Tokura el al. (20) synthesised Ce doped NdzCu04
with an onset of critical temperature at about 25 K. Ndz.,Ce,CuOa was the first
electron superconductor with excess negative charge per unit cell. Its structure is
tetragonal I4lmrnm with Nd, Cu and 0 in sites identical to the T-phase (21).
Copper becomes strictly square-planar co-ordinated, with four Cu-0 bonds of
about 1.9 A" and there are no apical oxygens. The Nd atoms bond to eight oxygens
compared to nine oxygens for larger lanthanum in the T structure. The most
studied among these superconductors is Ndl,~~Ceu,15Cu04 which has a
superconducting transition temperature at 23 K. Intermediate valence state of
cerium ion plays an important role in its properties (22). The structure of
N ~ I . ~ ~ C ~ , , . ~ ~ C U O ~ is tetragonal with I4/mmm space group and the lattice parameters
a rea=b=0 ,395nmandc= 1.208nm(23).
Since the discovery of superconductivity in the bismuth-strontium-calcium-
copper oxides (6) , there have been a number of studies on their physical properties
due to their higher superconducting transition temperature. The Bi-2:2: 1 :2 phase
of bismuth cuprates has the ideal composition of Bi~Sr~CaCu2Og and has Tc of
90 K (24). The structure is nearly tetragonal with I4Jmmm space, group (21) and
the lattice parameters are a = 0.541 nm, b = 0.544 nm and c = 3.078 nm with CuO
layers in the ab-plane (18). The copper and oxygen are in sheets typical of the high
temperature superconductors and are spaced by cations and interleaved with Bi202
layers. Calclum adopts eight co-ordination, similar to the Y environment in
Y-Ba-CuO. There are no oxygens at this level and copper atoms have only five
nearest-neighbours in the square pyramidal co-ordination.
The elastic properties of these high-Tc superconductors are of interest for
both technology and basic research. Anisotropy in elasticity is a fundamental
property of the superconductlng crystals. We study primarily on La-based and
Nd-based 214 compounds and bismuth cuprates. We review here briefly the status
of the experimental measurement on the elastic properties which give additional
insight into the basic microscopic mechanism underlying the condensed state of
these materials.
For high-7c superconductors, majority of the ultrasonic investigations has
been carried out on ceramic materials with material defects such as voids, twins,
microcracks and texture, which lower the sound velocity, necessitating corrections
as large as a factor of 2 or 3 (25). La2,SrxCu04 show extreme acoustic mode
softening below room temperature (26, 27). Using the shear modulus as a probe
Bhattachaqa et al. (28) found evidence for phase transition at 95 K in
LalgSrt2CuO~~. The acoustic mode softening persists down to very low
temperatures and is accompanied by a large ultrasonic attenuation (26, 29, 30).
The structural phase transition is triggered by the softening of two degenerate
transverse optical phonons at the X-point of the first Brillouin zone boundary of the
body-centered tetragonal Bravais lattice (31). In a Lal.8Sro.zCu04 ceramic system,
elastic anomalies are detected at 30 K (27) and are ascribed to the structural phase
transition from the I41mmm phase to the P42fncm phase. Esquinazi el al. (32)
observed a minimum in the Young's modulus near 20 K and a change of slope in
attenuation at 44 K. The differences between reported observations in the phase
transition temperatures could possibly be due to difference in oxygen content, an
effect which has considerable influence on the structural phase transition (33) as
well as on its elastic properties (34).
A large discontinuous drop of acoistic wave velocity is observed near the
transition temperature Tc in Laz..Sr.&u04 (29,35-37). Attenuation normally shows
no special feature at Tc except a maximum below Tc (28, 37). For single crystals
of Laz,SrxCu04, with Sr concentration of 0.12 and with Li impurity, anomalies of
the sound velocity with frequency 50 MHz in the temperature range 60-1 50 K are
observed (39). These anomalies are attributed to the deformation of the soft
optical mode (39). Fukase el al. (40) observed that under magnetic fields, the shear
mode velocity is enhanced in the lower temperature region below Tc in
Lal.aSro.15Cu04 single crystals. Suzuki ef al. (41) measured the longitudinal and
transverse wave velocities over a wide temperature range from 1.8 to 300 K in
single crystals of La1.86Sr0.14C~04 and La1.81sr0.1~Cu04. They also observed a
structural phase transition from tetragonal to orthorhombic phase at 210 K with a
large softening of C6s mode and a small softening of C11 mode. Nohara et al. (42)
observed an enhancement in the elastic constants CII, CM and C66 below 10 K
which suggests the existence of another state different from the P4z/ncm phase in
La1.~5Sr015Cu04. Their finding is an advancement from the earlier suggestion
6
made by Fukase et al. (38) in Laz,SrxCu04 with 0.10 < x 5 0.12 that this new
phase is P4~1ncm itself Fujita and Suzuki (43) found evidence for the structural
phase transition in single crystals of L~I .sS~O.I~CUO~ by studying the anisotropic
response to various sicoustic modes. Hanaguri el af. (44) observed a jump and a
kink in the CU and C33 mode velocity respectively at the superconducting transition
temperature in La1~5Sr0.15Cu04. Nohara el al. (45) measured the longitudinal
elastic constants in single crystals of Lal.8sSr0.1~Cu04 across the superconducting
transition temperature under magnetic fields upto 14 T applied both parallel and
normal to the CuOz-plane. They (45) also found anomalies in elastic constants at
Tc indicating the anisotropic coupling between the high-Tc superconductivity and
the lattice. Mode softening (46) has been found in the transverse elastic constant
(CII-CIZ)/~ below 50 K in single crystalline Lal.saSr0.14Cu04 by ultrasonic
measurements in magnetic fields along the c-axis. Bums (47) showed that the
lattice motion due to shear is responsible for the structural phase transition in
Laz.Sr,CuO4 systems. Ultrasonic measurements (48) performed on single crystals
of Laz.,SrxCu04 with x = 0.09, 0.14 and 0.19, reveal that some elastic constants
increase below the superconducting transition temperature Tc.
Increase in superconducting transition temperature by applying uniaxial
pressure in L ~ I . U ~ S ~ O . ~ ~ C U O ~ has been accomplished recently (49, 50). Locquet
et a1. (51) achieved a doubling of the critical temperature Tc of the Lal,$r~,lCu04
by compressively straining thin films, though their conclusions (51) regarding this
finding have been criticised later (52).
Earlier works on the non-linear elastic properties of La-based 214 ceramic
high-Tc superconductors have been reviewed by many workers (53-56). The central
work on elasticity in single-crystalline Lal..saSro.l4Cu04 is by Migliori et al. (57, 58)
who determined the elastic constants by the method of resonant eigen frequencies.
Fanggao et al. (59) have measured three out of six non-vanishing elastic constants
of La18Sr~.zCuO4 using the ultrasonic technique. Bezuglyi et al. (60) have
determined the elastic constants of Lal.g.1Sro.l,CuO4 from sound velocity
measurements. Fil et al. (61) measured the changes in elastic moduli during the
tetragonal to orthorhombic phase transition in L a l . s ~ S r ~ . ~ E u O ~ single crystals.
The vast majority of high-Tc superconductors are of hole-doped variety.
In Ndl.~5C&l5CuOc one of the electron-doped high-Tc superconductors, the
doping level of Ce changes the physical properties (62). Murayama et al. (63)
reported that the resistively determined superconducting transition temperature of
Ndl.a5C~.lsCuOG, with Tc = 22 K remains unchanged under the application of
hydrostatic pressure upto 2.5 GPa. Bucher et al. (64, 65) also found Tc to be
insensitive to changes in oxygen content.
Al-Kheffaji et al. (66) showed that sound velocity measurements provide
evidence for mode softening in Lal.8Sro.tCuOs but not in Ndl.s~Ceo.1sCuO4.
Fanggao et al. (67) have observed a pronounced change of gradient of the
longitudinal mode velocity with temperature near 220 K in Ndl.8sCeo.l5Cu04.
Pressure dependence of the longtudinal and shear wave velocities in
Ndl.~~Ceo,l,Cu04 , was studied by Fanggao el al. (59). They found a linear
dependence of the ultrasonic wave velocities on pressure for N d l , a C ~ , l j C u O ~ ~
and that of the bulk modulus. Saint-Paul et al. (68) measured the sound velocity in
single crystals of Nd1s~Ceo.1sCuO4 along the z-axis. Three of the six independent
second-order elastic constants of Ndl.g5Cq.l5Cu04 have been measured by Fanggao
el al. (59) using the ultrasonic wave velocity measurements.
The Bi-2:2: 1:2 phase of bismuth cuprates, which has the ideal composition
of Bi2Sr2CaCuzOs has aroused much interest because of its potential applications
and high transition temperature. Groen and Zandbergen (69) have pointed out that
the lone-pair bond from the Bi p-orbital leads to weak binding between the
adjacent BiO layers. Olsen et al. (70) found that the superconductor
Bi2Sr~CaCuzO8 is the most compressible of all the systems with perovskite-like
structure and the volume compressibility K, obtained is 16 x 10" GP~-'. Liu et al.
(71) estimated the volume dependence of Tc from the measurements of pressure
dependence of the transition temperature of a single crystal of Bi-Sr-Ca-CuO. The
pressure dependence of transition temperature Tc has been extensively studied in
bismuth systems by many workers (72-75).
There exist a number of ultrasonic and other measurements on the elastic
properties of Bi~Sr2CaCu208 (54, 56). Ledbetter el al. (76) observed a nearly
regular behaviour of bulk modulus and shear modulus in the temperature range
from 5 to 295 K. Wang (77) showed that there exists a phase-like transition
characterised by a jump of lattice parameters between 100 and 150 K in
Bi~Sr2CaCuzO8. The absolute values of the elastic moduli of flux-line lattice of
Bi~Sr2CaCu~08 have been probed by Yoon el al. (78). Chen et al. (79) measured
the effect of elastic stress on the resistivity of Bi-2:2:1:2 whiskers in the a-
and c-directions. Lubenets et al. (80) observed that the elastic compliance in
Bi-Sr-Ca-CuO crystals is higher than that of the Y-based superconductors.
The ultrasonic longitudinal velocity measurements as a function of
temperature were performed by the pulse transmission technique in
potassium-doped (81) and sodium-doped (82) single crystals of Bi-2:2:1:2.
Aleksandrov et a/. (83) measured the surface wave velocities in the ab-plane and
found that the velocity along the a-axis is higher than that along b-axis in
Bi2Sr2CaCu208. In textured samples, higher values of velocities in the ab-plane
than along c-axis were also found for Bi-2:2:1:2 system (84). The data from high
pressure experiments yield a value of 61 GPa for the bulk modulus of Bi-2:2: 1:2
system (74).
Ultrasonic wave velocity measurements on single crystal Bi2SrzCaCu~08
over a wide temperature range revealed three softening minima between 100 and
250 K (85). Room temperature sound velocity measurements on the (001) and
(0 10) planes of BizSrzCaCuzO8 single crystals were performed using Brillouin light
scattering experiments by Boekholt et al. (86) to determine the elastic constants.
Fanggao et al. (87) studied the hydrostatic pressure dependence of longitudinal and
shear wave velocity in ceramic specimen of single crystal BizSrzCaCuzOs using
ultrasound technique. Wu et al. (88) determined the temperature dependence of
elastic constants in Bi~SrzCaCuzO8 crystals from sound velocity measurements.
The velocities of longitudinal and shear ultrasonic waves propagated in very dense,
highly-textured, ceramic BizSrzCaCu208 were measured (84, 89) as functions of
temperature and hydrostatic pressure.
For anisotropic media, elastic waves are neither longitudinal nor transverse
except for some special symmetry directions. We give below the theory of
elasticity of crystals, particularly applicable to high-Tc superconductors.
1.5 Theory of Eladklty
Cons~der an elastic medium where the co-ordinates of any point can be
denoted as (a,, az, a3). Choose a set of orthonormal vectors el, 4, 4 as the basis
vectors for the co-ordinate system and denote the k'h component of the stress acting
on the plane ei = 0 by a& where i and k are the component indices. Consider the
equilibrium of a small element centered at the point ai and bounded by the plane
ai + ?'id&. Let ui denote the elastic displacement of the point ai of the body and
p the density of this point. The equation of volume element can be derived by
considering the total force acting on the volume element. If we ignore the body
forces, the equations of motion for an elastic solid can be written as (the convention
that repeated indices indicate summation over the indices will be followed here).
where the stress tensor oik is given by
ok = @/&I
where 41 IS the crystal potential and E* are the components of the strain tensor given
by
o~ and EI are symmetric tensors of second rank. According to Hooke's law
or = C*I, ~ l r n (1.4)
The constants C i h form a fourth rank tensor with 81 components.
From equations ( I .2) and (1.4), we have
Hence the elastic constants C& are multiple strain derivatives of the state
functions and since the strains Q,,, are symmetric, the elastic constants possess
complete Voigt symmetry. Thus,
C*l, = Ckilrn = CM = Clmik (I,6)
These quantities are symmetric with respect to interchange of the subscripts.
It will be convenient to abbreviate the double subscript notation to the single
subscript Voigt notation running from 1 to 6, according to the following scheme:
11+1; 2242; 3 3 4 3 ; 2 3 4 ; 13+5 and 1 2 4 .
Hence the matrix of elastic constants Ciu, would contain a 6 x 6 array of
36 independent quantities in the most general case. This number is, however,
reduced to 2 1 by the requirements that the matrices be symmetric on interchange of
double indices. The number of independent elastic constants will be further
reduced by the symmetry operations of the respective crystal classes. The high-Tc
superconducting crystals La1.8Sr0.zCu0~ Ndl.s5Ceo,l~Cu04 and BizSrzCaCu208
belong to the tetragonal14fmmm class which have six independent elastic constants
(90). The elastic constant matrix for this class of compounds is given by
In the equation of motion for an elastic medium, the forces on an element of
volume, are given by the divergence of the stress field.
Using equations (I .3) and (1.4), the equation (1.1) can be written as
where & are the components of the amplitude of vibration, w is the angular
frequency and k 1s the wave vector corresponding to the wavelength h = 2xk. The
resulting equations of motion from equation (1.8) are
substituting k = kii , where 13 is the unit vector, we get
where Tijh = Ci,dp are the reduced elastic constants and v is the phase velocity
given by v = ok. The components of second rank tensor A are given by
Hence equation ( 1 . l I ) can be written as
This shows that u is the eigen vector of tensor A where eigen value is vz. Hence
v2 is the root of the equation
I A - ~ ~ I = 0 (1.14)
This IS the Christoffet equation. The theory of elastic waves generally
reduces to finding u and v for all plane waves propagating in an arbitrary direction
for crystals possessing different symmetries. In this situation, all terms in equation
(1.11) which involve differentiation with respect to co-ordinates other than that
along the propagation direction drop out.
A more fundamental significance to the elastic constants is implied by their
appearance as the second derivatives of elastic energy with respect to strains.
It should be noted here that the stored elastic energy is only a part of the complete
thermodynamic potential of the crystal, since it depends on many other variables.
Also, one can introduce elastic constants as a constitutive, local relation between
stress and strain for materials in which long-range atomic forces are unimportant.
1.6 Finite Strain Theory of Elasticity
Let the position co-ordinates of a material particle in the unstrained state be
( i = 1, 2 3). Let the w-ordinates of the material particle in the strained state be
xi. Consider two material particles locatd at ai and ai + dai. Let their co-ordinates
in the deformed state be xi and xi + dx,. The elements dx, are related to da, by the
equation
& dx, = L d a ,
8%
The convention that repeated indices indicate summation over the indices
will be followed here. ag is the Kronecker delta and &i, are the deformation
parameters. The Jacobian of the transformation
is taken to be positive for all real transfonirations. IfdV. is a volume element in the
natural state and dV, its volume after deformation
where po and p are the densities in the natural and strained states respectively. Let
the square of the length of arc from ai to ai + dai be dl0 in the unstrained state and
dl in the strained state. Then
= (3 dr. - 6*] dqd9 da, da,
where q,k are the Lagrangian strain components which are symmetric with respect
to the interchange of the indices j and k. In terms of &r,
Q. = %(E* +E. +Z&,,C.) (1.19)
The internal energy function U(S, ~ k ) for the material is a function of the
entropy S and Lagrangan strain components. U can be expanded in powers of the
strain parameters about the unstrained state as
The linear term in strain is absent because the unstrained state is one where
U is minimum. We shall define the elastic constants of different orders referred to
the unstrained state as (91)
and
Here the der~vatlves are to be evaluated at equilibrium configuration and constant
entropy C:,ki and CS,,,,, are the adiabatic elastic constants of second- and
third-orders respectively. They are tensors of fourth and sixth ranks. The number
of independent second-order elastic constants and third-order elastic constants for
different crystal classes are tabulated by Bhagavantam (90).
1.7 Quasi-harmonic Theory of Thermal Expansion
In the harmonic approximation, the atoms in a solid are assumed to oscillate
symmetrically about their equilibrium positions which remain unaltered irrespective
of the temperature. The thermal expansion of a solid, therefore, is a property
arising strictly due to the anharmonicity of %e lattice. In the quasi-harmonic
approximation, the oscill&ions are still assumed to be harmonic in nature but the
frequencies are taken to be functions of the strain components in the lattice.
The strained state of the lattice is specified filly by the six strain components
llrs (r, s = 1, 2, 3; ?In = q s r ) .
A normal mode with frequency o(q, j) makes a contribution F(q, j) to the
total vibrational free energy Fvib given by
F(q, j) = K~T[%X + log(1 - ex)] (1.23)
where X = Fim(q, j) 1 KBT, h = h/2x, h being the Plank's constant, KB is the
Boltzmann's constant, T is the absolute temperature and q is the wave vector of the
j" acoustic mode. The total vibrational free energy is, therefore
The thermal coefficients ah of the crystal are obtained as
Here, .I,,, = (%) d
or,,, are the components of the stress tensor, Sh are the compliance coefficients
relating q, and GI,,, and
Here, y, are the generalised Gnineisen parameters (GPs) of the normal mode
frequencies. In equation (1.25). the subscript a' means that all other ail, are to be
held constant while differentiating with respect to crlm and subscript a means that
2 -x -x 2 . all oh are held constant. a[o(q, j)T] = X e /(l-e ) 1s the Einstein specific heat
function. In the quasi-hannonic approximation, the GPs are assumed to be
constants independent of temperature.
It is more advantageous to choose such strains that do not alter the crystal
symmetry, instead of choosing any arbitrary strain, while defining GPs. For
Lal,~Sro,zCu04, N ~ I . ~ J C ~ ~ . I ~ C U O ~ and. Bi2SrzCaCuzOs, there are two principal
thermal expansion coefficients, namely a1 which is the linear expansion coefficient
parallel to the c-axis and a, which is the linear thermal expansion coefficient
perpendicular to the c-axis. Here, it is convenient to use the following strains for
the determination of the thermal expansion.
(i) A uniform longitudinal strain E" along the c-axis. Then all the q, are zero,
except q33 = E" = d log c, where c is the axial length.
(~ i ) A uniform areal strain E' in the basal plane perpendicular to the c-axis.
Then q11 = q 2 ~ = , where A is the area of the basal plane of A
the crystal. All other q, vanish.
From equation (l.25), we now obtain
Va33 = Val = C [2s13yf(q,j) + S33 Y"(q,j)I K B ~ ( o (q,j)) 4.1
and Val, = Vazz = Va,
= c [(%I +slz)y1(s, j) + su ~"(4 , j)l K B O ( ~ (q, j)) (1.28) 4,)
The effective Grimeisen functions are defined as
= [ ( c S , +csZ)U, +cs3arl V/CP
-
Y ,,(TI = [ 2 ~ s , a , +cs,ail VICP (1.29)
The C: are the adiabatic elastic constants, Cp is the specific heat at constant
pressure and V is the volume of the crystal.
Comparing equations (1.29) with (1.28), .we get
- C v "(a j) o[o(q, r ,(T) =
4.1
z .[a (4. j). T] 9.1
The expression (1.30) give the temperature dependence of effective
Grimeisen function.
In the low temperature limit, only the low frequency acoustic modes make a
contribution to the specific heat. The number of such normal modes in the jm
18
acoustic branch is proportional to vf (8,4), where vj(8, 4) is the velocity of the j"
acoustic mode travelling in the direction (8, 4). The GP y(q, j) depends only on the
branch index j and the direction (8,4). It is independent of the magnitude of the
wave vector q. The effective lattice Griineisen functions ?,(T) and T,(T)
approach the limits defined below, at low temperatures.
where y ; (0, 4) and y x0,4) are the GPs for the acoustic modes propagating in the
direction (8, 9). Calculations of the low temperature limit of ?,(T) and 7 ,(T) are
possible knowing the pressure derivatives of the second-order elastic constants
(SOEC) or the third-order elastic constants (TOEC) of the crystal. The evaluation
of the low temperature limits y,(0) and 7 ,(O) for La1.sSr0.tCuO4, N ~ I . s ~ C Q . I ~ C U O ~
and BitSrzCaCuzO~ is given in Chapter 6 of this thesis
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