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Third order elastic constants and vibrational anharmonicity of a semiconducting iron phosphate glass
Journal of Non-Crystalline Solids 44 (1981) 157-169 North-Holland Publishing Company
157
THIRD ORDER ELASTIC CONSTANTS AND V I B R A T I O N A L ANHARMONICITY OF A SEMICONDUCTING IRON PHOSPHATE GLASS M.P. BRASSINGTON, A.J. MILLER, J, PELZL * and G.A. SAUNDERS
School of Physics, University of Bath, Claverton Down, Bath, BA2 7A Y, UK Received 16 May 1980 Revised manuscript received 8 August 1980
From measurements of changes in transit time of 10 MHz ultrasonic waves as a function of hydrostatic and uniaxial stresses, the third order elastic constant of (Fe203)o.38(P2Os)o.62 glass have been determined. The vibrational behaviour of this glass lies intermediate between those of the amorphous form of arsenic and the anomalies characteristic of the silica glasses. The pressure derivative of the bulk modulus (aBs/aP)p=_o is positive (+4.73) but that of the shear modulus (a~/aP)t,=_o is negative (-0.16): the Griineisen mode gamma Vl for longitudinal modes is +1.1 while "rt for transverse modes is -0.3 - under pressure the longitudinal modes stiffen in the usual way but the shear modes soften. To examine the effects of vibrational anharmonicity on the phonon dynamics of a phosphate glass, the displacement amplitudes for the scattered waves in the long wavelength limit have been computed for the possible types of three-phonon interactions.
1. Introduction The hydrostatic and uniaxial pressure derivatives o f ultrasonic wave velocities o f fused silica [1] and Pyrex glass [2] are negative. The peculiarity o f this behaviour becomes strikingly apparent when it is realised that this means that the pressure derivative o f the bulk modulus B is negative (OB/aP)e=o is - 6 . 3 for fused silica [1 ]) so that the compressibility actually increases with pressure. In contrast (~B/ aP)e=-o is positive for the amorphous forms o f the elements selenium (+8.5) [3] and arsenic (+6.42) [4] and also for the chalcogenide glasses [5]. The anharmonicity o f the acoustic vibrational modes o f amorphous arsenic (a-As) is also entirely different from that o f the silica-based glasses: the third order elastic constants (TOEC) are negative for the amorphous element (except C144) [4] but are anomalously positive for both fused silica (except C4s6) [1] and Pyrex (except C144) [2]. Such wide variations in the effect o f pressure on the dispersion curve slopes near wavevector k = 0 in the anharmonicity make it difficult to decide whether there is elastic * On sabbatical leave from: Institut ftir Experimentalphysik VI, Ruhr Universit/it, 4630 Bochum, FRG. 0 0 2 2 - 3 0 9 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 5 0 © North-Holland
158
M.P. Brassington et a L / Third order elastic constants
behaviour which can be said to be typical of a glass. Sato and Anderson [6] argue that the anomalous decreases with pressure of both the bulk and shear moduli of silica and soda-lime-silica glasses are not characteristic of the glassy state but rather that the low coordination of the structure permits large atomic vibrations. Data on other types of glass are needed to resolve this problem. The (Fe2Oa)0.3s(P2Os)0.62 glass studied here is chosen as an example of the semiconducting transition metal phosphate glasses. Conduction takes place in these glasses by electron transfer from low to high valence states of the transition metal ion (which can exist in more than one valence state). Although these conduction processes are of great interest, and have been extensively studied [7,8], little is known about the vibrational properties of such glasses, save for the second order elastic constants (SOEC) [9]. In the present work the effects of hydrostatic pressure and uniaxial stress on the velocity of ultrasonic waves propagating through (Fe2Oa)o.as(P2Os)o.62 have been measured. The results provide the first sets of pressure derivatives of the SOEC and also the TOEC for a glass of this type. The hydrostatic pressure derivatives of the SOEC given quantitative information of the vibrational mode dependences'upon volume. The TOEC are the cubic coefficients of the Hamiltonian with respect to strain and thus depend upon the anharmonicity of the interatomic forces, that is their nonlinearity. Usual practice is to consider the anharmonic effects in terms of the GrOneisen mode gammas and these are compared and contrasted here with those of other glasses. The study of the vibrational anharmonicity is then extended to a computation of the displacement amplitude of the scattered phonon from phonon-phonon interaction processes, to provide basic details of the phonon dynamics in this type of glass.
2. Experimental procedure and remits The phosphate glass was supplied by M. Sayer; details of its preparation have been given by Sayer and Mansingh [7]. While embedded in cold set resin, an ultrasonic sample was cut with a diamond saw into an orthogonal block having dimensions 1.0 X 0.8 X 0.5 cm. A series of aloxite and diamond pastes was then used to lap each surface to a mirror finish with parallelism between pairs of faces better than 10 -s rad. The density was determined as 3010 kg m -3. The SOEC, obtained from pulse echo overlap measurements of the propagation velocities of longitudinal and transverse 10 MHz ultrasonic waves, are listed in table 1 together with the earlier results obtained by Farley and Saunders [9]. The difference between the two sets of data is almost entirely accounted for by the inclusion in the present work of a transducer correction [10] amounting to a 4.6% increase in the longitudinal and a 3% increase in the shear velocity. Ultrasonically determined elastic constants are adiabatic; since neither the thermal expansion coefficient a nor the specific heat Cp are available for this glass, the isothermal constants
M.P. Brassington et al. / Third order elastic constants
Table 1 A comparison of the elastic properties of (Fe~Oa)o.aa (P~0s)o.e~ phous materials Property
(Fez03)o.38(Pzos)o.62
Present kgm-3 -1 ms
Pyrex
Fused silica
[41
[21
[II
191
3010
3010
4770
5120
2501
3043
2990 78.9
1396 29.8
25.1
11.3
62
GPa
83.0 21.3
c44
GPa
21.9
26.9
9.3
Bs
GPa
45.9
ES
GPa
69.5
43.0 66.8
17.5 23.1
0.24
0.25
ez
with those of other amor-
a-As
5253
crs
159
0.21
_ 69.5
13.7 21.9 41.8 84.1 0.16 _
2203 5965 3768 78.3 15.9 31.3 47.0 93.5 0.17
Cl11
K GPa
-450 f 60 a
-465
400
530
Cl12
GPa
-200 + 25 a
-
-33
30
240
cl23
GPa
-160 f 20 a
-
-162
260
50
_-120
90
156
425
GPa
-18+
2a
-
Cl55
GPa
-62t
8a
-
c456
GPa
-22*
3a
Cl44
afi -
(ap1
+64 -108
90
-86
105
495
70 -10
+4.73 a
+6.42
-4.72
-6.3
-0.16 a
+1.73
-2.39
-4.1
+2.34 +1.45
-1.74 a
-2.8 a
-1.50 a
-2.36 a
P=o
YI
+1.1 a
Yt
-0.3 a
-
a) The calculation of these parameters has involved one or more of the following assumptions: BT 2 Bs, ET 2 Es and cT 1 us.
could not be calculated, however, the difference between isothermal (T) and adiabatic (S) elastic constants is expected to be substantially smaller than 1%. For the shear modulus it is usual practice to consider the Lame shear constant /Afor which /_I’ and pT are identical and equal to C44. The Debye temperature 0g has been estimated from the mean sound velocity
+)/3]d’3)
cm(=[($ +
M.P. Brassington et al. / Third order elastic constants
160
using k\47r ]
Cm
(1)
where N, the volume density of vibrational units, is taken to be equal to the number of atoms per unit volume. The Debye temperature calculated from this model, which represents all the .vibrational modes, is 425 K. The hydrostatic pressure dependences of the ultrasonic wave velocities were measured at constant temperature in a piston-cylinder equipment using castor oil as the pressure medium. Uniaxial stresses were applied with a screw press, the force exerted being measured by a proving ring. The small changes in wave transit times t(p) were measured by an automatic frequency controlled, gated carrier, pulse superpos~tion apparatus [11]. At an applied pressure P the "natural" velocity W(p)
o
8.0
x
~-~ 7.0
i"
<1 6.0
g
c ~
.0
-~ 4 0
~ 3.0
~ 2.0
1.0
0
0
I
04
0!2 Pressure
013
0.4
I k bar
Fig. 1. Relative natural velocity change as a function of hydrostatic pressure for 10 MHz ]ongitudinaJ waves i n (Fe203)o.3s(P2Os)0.62. (1 kbax is 10 s Pa).
M.P. Brassington et al. / Third order elastic constants
161
of ultrasound in the sample is given by
W(p)= lo/t(e)
(2)
where lo is the unstrained path length. The measured relative change in this natural velocity under hydrostatic pressure [i.e. AW(p)/Wo=(W(p)- Wo)/Wo; where I¢o = (I¢(e))p:_o] is plotted for longitudinal waves in fig. 1 and for shear waves in fig. 2. When a pressure of 0.3 kbar was exceeded, non-reproducible, nonlinear effects were observed, especially for longitudinal waves, so the results reported here are restricted to the reproducible and reversible region up to 0.3 kBar. For longitudinal modes d Ale
= +2.36 X 10_11 + 2% pa_ 1
and for transverse modes d 'AW
- --6.45 × 10 -1~ -+ 0.6% Pa -1 .
Now the pressure derivative of the shear modulus ~ is given by (0Wt~
,
,
,
1 C~441
i
,
,
,
,
f
'
'
'
0.8
0.9
1.0
1.1
I-2
-1.0
x
-2.0
~-4.o
~i "5"0 ~ ~
-6.0
o7.0
-BO
OI
0.2
0,3
G4
0"5
O~
'
0.7
P r e s s u r e I k bar,
Fig. 2. Relative natural velocity change as a function of hydrostatic pressure for 10 MHz transverse waves in (Fe2Oa)o.as(P2Os)0.62.
M.P. Brassington et al. / Third order elastic constants
162
where the second term on the RHS is a correction for the reduction in dimensions of the sample with compression. Since
(aWtl
= Wt ~
,
\ aP ]e=-o
(4)
e=-o
and for shear waves (Wt)~_ o is 3.04 X 103 ms -1, we have (Ola/aP)~__o = - 0 . 1 6 , assuming that B T ~ B s. The pressure derivative of the bulk modulus B s is given by = ~2{ Po IZW1 (~)WI] - 4Wt [[~Wt~I']+_@TC1SI_4 V)J aC,4) }
(aBSl
\ OP/~o
\ OP /
~--o
(5)
and 0 2XW
(6)
Using ( W I ) ~ o = 5.25 X 103 ms -1 and assuming that B T ~ B s, we find that (~Bs/~P)p=_o = +4.73. The relative change in "natural" wave velocity as a function of applied uniaxial stress was measured for two shear modes; results for shear waves polarised in a
I
g
K
-5
>, •~
-10
~d g -is ,/I
.~- - 2 0
>~ - 25
0
Applied
stress
/ bar
Fig. 3. Relative natural velocity change as a function of uniaxial stress applied parallel to the polar\sat\on but perpendicular to the propagation direction of 10 MHz transverse waves in (Fe 2Oa)0.38 (P2Os)0.62.
M.P. Brassington et el. / Third order elastic constants i
!
w
163
1
0
-2,0 q - 40
"~.
• •
•
•
- (3.0
~o
"~
- 8-0
C
-10.0
•
•
•
•
-12.0
- 14.0
- lS.O
-18.0 10
20
30 Pressure I
40
50
60
70
bar
Fig. 4. Relative natural velocity change as a f u n c t i o n of uni a xi a l stress for 10 MHz transverse waves where the stress, wave propagation and polarisation directions are m u t u a l l y perpendicular.
direction parallel to the stress are given in fig. 3 and those for the polarisation normal to be applied stress in fig. 4:
is found to be -1.045 X 10 -11 + 7% Pa -1 and -2.81 × 10 -12 -+ 6% Pa -1, respectively, It can be seen that the results for the uniaxial stress experiments (figs. 3 and 4) display greater scatter than those for hydrostatic pressure (figs. 1 and 2). This is because the fragility of the sample limits the allowed uniaxial stress, and thus the consequent velocity change is very much smaller (by a factor of "'40), so that the errors and scatter are proportionately larger. For example, when the uniaxial stress was applied parallel to the shear wave polarisation direction (fig. 3) for the maximum applied stress of 14.8 bar the frequency change was only 2.5 parts in lO s . For uniaxial stress applied perpendicular to the polarisation direction, the frequency
164
M.P. Brassington et al. / Third order elastic constants
change was even less (1.75 parts in 10 s) for the maximum applied stress of 61.7 bar. Those are very much smaller changes in frequency induced by stress than could be obtained in the hydrostatic pressure experiments [8 parts in 104 for a hydrostatic pressure of 0.35 kbar (fig. 1)]. The scatter has been taken into account statistically and appears in the errors quoted for the pressure derivatives of natural velocity and consequently in the TOEC (table 1). An isotropic material such as a glass has three independent third order elastic stiffness constants, which may be taken as C123, C144 and C4s6 and given the symbols v~, v2 and v3 respectively. Then C112 = vl + 2v2 , C15s = v: + 2v3
(7)
and C1~1 = vl + 6v2 + 8va. The three independent constants may be obtained from hydrostatic and uniaxial stress derivatives of second order elastic stiffness constants. Thurston and Brugger [ 12] have derived equations relating the quantity (p'oW
P=o
to the three independent constants vl, v2 and va for various stress configurations. Now
~O
ZW
thus from the results shown in figs. 1 to 4 and from the relevant equations of Thurston and Brugger four linear equations in the three unknowns vl, v2 and v3 may be written: 1 (2cSl + 3vl + 10v2 + 8v3) = 3.92 - 1 - 3B T
1 --1 -- 3B x (2C44 + 3v2 + 4va) = - 0 . 3 6 , 1 - ~ - [ - 2 C 4 4 + v2(2o T - 1) + 2v3(o T - 1)] = 0.58 and
1 [oT(2c44 + 4vz) + v:(2o T -- 1)] = --0.156 ET
I
(8)
M.P. Brassington et al. / Third order elastic constants
165
By making the approximations that BT = BS, ET = ES and oT = us and weighting each equation by the reciprocal of its estimated experimental error, we have solved these four equations for ~1, v2 and v3 using a least-square technique. The results are u1 = (-160 + 20) GPa, v2 = (-17 + 2) GPa, and va = (-22 f 3) GPa. The ultrasonic technique provides mixed TOEC (as defined by Brugger [ 131). In principle the measured CrlK can be converted to adiabatic TOEC cJK. However, this requires a knowledge of thermodynamic parameters such as the thermal expansion (Yand of the specific heat Cn. Taking reasonable estimates for these parameters it has been shown that (CnrJK - Crr,) is negligible compared with the experimental error (table 1) [4]. The third order elastic stiffness constants C~JK of (PezOa)o.ss(P2Os)o.62 glass are listed in table 1. A check on the internal consistency of these results if provided by [I41 = -(C111 + 6C112 + 2C12s)/9BT which (assuming BT = B’) gives (aBS/W>,, of t4.73 measured directly from hydrostatic
as t4.77, in good agreement with that experiments alone.
3. Pressure dependences of the elastic mod& and Griineisen parameters All the second and third order elastic stiffness tensor components have been obtained for this iron phosphate glass and are compared in table 1 with the other complete sets reported previously for other types of glass. Although the density of (Pe203)e.3s(P205)0.62 glass is low compared with the covalently bound amorphous element arsenic, the velocities at which longitudinal and shear ultrasonic waves are propagated are markedly higher. This behaviour is much more like that of fused silica; consequently the values of the SOEC of the phosphate glass are similar to those of fused silica and Pyrex. This suggests that the interatomic binding forces in these phosphate glasses are probably also mixed ionic-covalent in nature. That both phosphate and silica-based glasses have comparatively high Debye temperatures agrees with this suggestion; in contrast 02 for amorphous arsenic is about three times smaller (table 1). However, at the third order the situation is almost completely reversed; the TOEC of the phosphate glass are all negative, in marked contrast to those of the silica-based glasses which are anomalously positive, exceptions being Ce56 for fused silica and C1+, for Pyrex. The anharmonic properties, that is the nonlinearity of the interatomic binding forces with respect to atomic displacements, of the iron phosphate glass are similar in kind to those in amorphous arsenic and to those crystalline solids which do not show soft mode behaviour. An unusuaI feature of glasses such as silica or Pyrex is that the hydrostatic pressure derivatives of the elastic moduli (aBs/W),, and (a/.@P),, are negative and
M.P. Brassington et aL / Third order elastic constants
166
the temperature derivatives (OBS/OT)p and (O#/~T)p are positive. This behaviour is in complete contrast to that of crystalline solids but it should not be considered as characteristic of the glassy state: amorphous arsenic has the more normal positive pressure and negative temperature derivatives of the elastic moduli [4]. In fact anomalous pressure or temperature dependences of elastic moduli are observed in crystalline solids which have soft acoustic phonon modes, or, in what amounts to much the same thing, in crystals with low coordination numbers so that bending vibrations are allowed. In silica glasses the anomalous behaviour probably results from the low coordination number. Recently Sato and Anderson [6] observed a positive (~B/OP)T but a negative (O#/OP)r for an SiO2 (70.5 wt.%) - CaO (11.6 w t . % ) - Na20 (8.7 w t . % ) - K20 (7.7 wt.%) soda-lime-silica glass; as they point out, the presence of alkali metal ions in the interstices of the silica network inhibits the soft transverse vibrations. For the iron phosphate glass the temperature dependences of both longitudinal and shear wave velocities and thus (OBS/aT) and (a#/~T) have the more usual negative slope [9]. However, while (~BSl~p)~_o is strongly positive, (O#/aP)e=o is just negative. The elastic behaviour of this glass is similar in kind to that of the soda-lime-silica glass whose transverse vibations are inhibited by the presence of a network modifier. The small but negative value found for (~#/aP) implies that for the iron phosphate glass the structure is open enough to allow a degree of bond-bending vibration, but less so than in fused silica. Sato and Anderson [6] have gained useful insight into the effect of pressure on the vibrational frequencies in the long wavelength limit by considering the Grtineisen mode gammas. For an isotropic material there are only two contributions to the generalised elastic Graneisen parameter )'elastic, one associated with longitudinal modes 3'1 and the other with transverse modes 7t. These two mode gammas may be found from the elastic constants of the material [ 15] using ~'l,t -
1
6Wl,t
(3B "r + 2Wl,t + kl,t)
(10)
where w 1 = C11, wt = C44, kl = C111 + 2C112 and k t = l(Cl 11 - C12 a)From the data obtained above we fred that for (F%Oa)o.aa(P2Os)0.62 glass 3'1 = +1.1 and 7t = -0.3. Because the Debye temperature of this glass is 425 K it cannot be assumed that at room temperature all possible vibrational modes are excited to the same degree: it is not possible to derive readily a generalised elastic Griineisen parameter "/elastic from the individual mode gammas 71 and ")'t --such a derivation would require details of the relative excitation of the longitudinal and shear modes. However, the magnitudes and the difference in sign of 3'1 and ")'t indicate that ~[elastic is small. The mode gammas 3'1 and 7t, while both strongly negative for fused silica and Pyrex and both strongly positive for a-As, are in the intermediate case of iron phosphate glass opposite in sign as Sato and Anderson [6] found in their soda-lime-silica glass with high modifier content. At room temperature longitudinal modes stiffen but shear modes soften with applied hydrostatic pressure. A negative Grtineisen param-
M.P. Brassington et al. / Third order elastic constants
167
eter 7t is an indication of facile bond-bending vibrations and a small force constant for the corresponding long wavelength transverse waves.
4. Three-phonon interactions The anharmonicity of the vibrational states is responsible for the phonon-phonon interactions and thus for phonon damping of ultrasonic waves, for p h o n o n phonon contributions to the thermal resistivity and also for the thermal expansion. The measurements of the TOEC allow a quantitative assessment of phonon-phonon interactions in the long wavelength limit for a phosphate glass. The p h o n o n phonon interactions, which play an important part in dynamical properties, such as for example the initiation of steady state heat-flow, are determined by the anharmonic terms in the strain Hamiltonian. The disturbance produced as a phonon of wavevector k propagates strain modulates the medium (in a way which is measured by the TOEC) as seen by a second phonon of wavevector k2 so that phonon-phonon scattering can take place. For a phonon-phonon normal process the scattered phonon has a wave vector k3 such that k3 = kl + k2
(11)
and momentum hk is conserved. Taylor and Rollins [ 16] have shown how to calculate for normal scattering processes the displacement amplitude X3 of a scattered phonon created by the non-linear interaction of two incident phonons propagating in an isotropic medium. This method is based on use of the first-order time dependent quantum mechanical perturbation theory to calculate transition probabilities between the phonon states. Their result is that the displacement amplitude X3 of the scattered phonon beam may be written as X3 = X1X2 VAw31a(1 + a)/8nrpctcl
(I 2)
where X1 and X2 are the displacement amplitudes of each of the incident phonon beams, V is the volume of interaction between the incident beams and r is the distance from the point of observation to the point of origin of the scattered phonons. The angular frequencies of the incident phonons of wavevectors kl and k2 are COl and co2 respectively, and a is co2/6Ol. A is a function, not only of the specific type of interaction and the interaction geometry but also of the second and third order elastic properties of the medium; expressions for A can be found in table 1 of reference [16] for each of the five possible types of these phonon interactions which are allowed under conservation of energy (E3 = El + E2) and momentum (hk3 = h k l + hk2).
The aim is to calculate the displacement amplitude IX31 of the scattered phonon emitted at a angle 7 for two input phonons interacting at an angle ~. Therefore a normalising value of unity can be allocated to the term XiX2co~/8~r in eq. (12), while taking the interaction volume V to vary as cosec ~. By inserting the elastic
168
M.P. Brassington et aL / Third order elastic constants
constant data given in table 1 into the expressions given by Taylor and Rollins and then solving eq. (I 2), the dependence of the normalised values IX3 [ of the displacement amplitude of the scattered phonon resulting from phonon-phonon interactions has been computed for the iron phosphate glass. Results for the possible types of interaction are plotted as a function of the angle ~ of the incident phonon in fig. 5. Excluding collinear interactions, there are two general groups, termed a and/3, of interactions. The first (tx) group comprises the types I, where two incident transverse (T) phonons interact to give a longitudinal (L) phonon Type I: T(col) + T(w2) ~ L(~ol + w2) , and its converse V: Type V: L ( w ~ ) ~ T(co2) + T ( w l - 602). For such interactions, the scattering angles at which zeros or minima occur in the scattered wave displacement amplitude depend upon the elastic constants o f the materials. For the phosphate glass iX3[ and thus the physically significant intensity X~a, does not become zero but does exhibit a minimum for both types I and V (zeros could exist for different choices of polarisation angles Oi). The second group (/3) of interaction types includes the mixed longitudinal and
,.4 10
!
/,
{6
'
'
', / }
i, ~
2
ii
zo
t 0
50' Incident
10(3" 150" interaction angle (I).
200"
Fig. 5. Absolute displacement amplitude IX3 I of the scattered phonon beam produced from the interaction of two phonons incident at an angle ~ in (Fe203)o.38(P2Os)0.62. Results are shown for interaction types ] (polazisation angles e2 = e3 = 0°), H (03 = 0°), II| (02 = 0°), IV (e 2 = 0 °) and V (#2 = e3 = 0 °) for def'mitions o f e i see ref. [16].
M.P. Brassington et al. / Third order elastic constants
169
transverse interactions: Type 1I: L ( w l ) ~ L(c02) + T(wl - w 2 ) , Type III: L(w~) + T(c02) ~ L ( w l + c02), Type IV: L ( w l ) ~ T(w2) + L(col - co2). For these interactions the zeros in IX31 are of geometrical origin and are independent of the material properties. Each of the curves for IXa[ terminates at a critical angle be beyond which the conservation laws o f energy and momentum can no longer be satisfied so that phonons ks are not produced. These quantitative determinations from the TOEC data of the p h o n o n - p h o n o n scattering amplitudes are a prerequisite for further development of an understanding o f the dynamical behaviour o f phonons in phosphate glasses; in particular they should provide a physical insight into anharmonic properties such as ultrasonic attenuation and long wavelength phonon scattering contributions to the thermal resistivity. We are most grateful to J. Penfold and G.D. Pitt for allowing us to use the S.R.C. high pressure facilities.
References [1] [2] [3] [4]
E.H. Bogardus, J. Appl. Phys. 36 (1965) 2504. D.S. Hughes and J.L. Kelly, Phys. Rev. 92 (1953) 1145. N. Soga, M. Kunigi and R. Ota, J. Phys. Chem. Solids 34 (1973) 2143. M.P. Brassington, W.A. Lambson, A.J. Miller, G.A. Saunders and Y.K. Yo--gurt~u, Phil. Mag. B42 (1980) 127. [5] J.C. Thompson and K.E. Bailey, J. Non-Crystalline Solids 27 (1978) 161. [6] Y. Sato and O.L. Anderson, J. Phys. Chem. Solids 41 (1980) 401. [7] M. Sayer and A. Mansingh, Phys. Rev. B6 (1972) 4629. [8] W. Chomka, O. Gzowski, L. Murawski and S. Samatowicz, J. Phys. Cll (1978) 3081. [9] J.M. Farley and G.A. Saunders, Phys. Stat. Sol. (a) 28 (1975) 199. [10] E. Kittinger, Ultrasonics 15 (1977) 30. [11] Y.K. Yo~urtqu, E.F. Larnson, A.J. Miller and G.A. Saunders, Ultrasonics 18 (1980) 155. [12] R.N. Thurston and K. Brugger, Phys. Rev. 133 (1964) A1604. [13] K. Brugger, Phys. Rev. 133 (1964) A1611. [14] F. Birch, Phys. Rev. 71 (1947) 809. [15] K. Brugger and T.C. Fritz, Phys. Rev. 157 (1967) 524. [16] L.H. Taylor and F.R. Rollins, Phys. Rev. 136 (1964) A591.
157
THIRD ORDER ELASTIC CONSTANTS AND V I B R A T I O N A L ANHARMONICITY OF A SEMICONDUCTING IRON PHOSPHATE GLASS M.P. BRASSINGTON, A.J. MILLER, J, PELZL * and G.A. SAUNDERS
School of Physics, University of Bath, Claverton Down, Bath, BA2 7A Y, UK Received 16 May 1980 Revised manuscript received 8 August 1980
From measurements of changes in transit time of 10 MHz ultrasonic waves as a function of hydrostatic and uniaxial stresses, the third order elastic constant of (Fe203)o.38(P2Os)o.62 glass have been determined. The vibrational behaviour of this glass lies intermediate between those of the amorphous form of arsenic and the anomalies characteristic of the silica glasses. The pressure derivative of the bulk modulus (aBs/aP)p=_o is positive (+4.73) but that of the shear modulus (a~/aP)t,=_o is negative (-0.16): the Griineisen mode gamma Vl for longitudinal modes is +1.1 while "rt for transverse modes is -0.3 - under pressure the longitudinal modes stiffen in the usual way but the shear modes soften. To examine the effects of vibrational anharmonicity on the phonon dynamics of a phosphate glass, the displacement amplitudes for the scattered waves in the long wavelength limit have been computed for the possible types of three-phonon interactions.
1. Introduction The hydrostatic and uniaxial pressure derivatives o f ultrasonic wave velocities o f fused silica [1] and Pyrex glass [2] are negative. The peculiarity o f this behaviour becomes strikingly apparent when it is realised that this means that the pressure derivative o f the bulk modulus B is negative (OB/aP)e=o is - 6 . 3 for fused silica [1 ]) so that the compressibility actually increases with pressure. In contrast (~B/ aP)e=-o is positive for the amorphous forms o f the elements selenium (+8.5) [3] and arsenic (+6.42) [4] and also for the chalcogenide glasses [5]. The anharmonicity o f the acoustic vibrational modes o f amorphous arsenic (a-As) is also entirely different from that o f the silica-based glasses: the third order elastic constants (TOEC) are negative for the amorphous element (except C144) [4] but are anomalously positive for both fused silica (except C4s6) [1] and Pyrex (except C144) [2]. Such wide variations in the effect o f pressure on the dispersion curve slopes near wavevector k = 0 in the anharmonicity make it difficult to decide whether there is elastic * On sabbatical leave from: Institut ftir Experimentalphysik VI, Ruhr Universit/it, 4630 Bochum, FRG. 0 0 2 2 - 3 0 9 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 5 0 © North-Holland
158
M.P. Brassington et a L / Third order elastic constants
behaviour which can be said to be typical of a glass. Sato and Anderson [6] argue that the anomalous decreases with pressure of both the bulk and shear moduli of silica and soda-lime-silica glasses are not characteristic of the glassy state but rather that the low coordination of the structure permits large atomic vibrations. Data on other types of glass are needed to resolve this problem. The (Fe2Oa)0.3s(P2Os)0.62 glass studied here is chosen as an example of the semiconducting transition metal phosphate glasses. Conduction takes place in these glasses by electron transfer from low to high valence states of the transition metal ion (which can exist in more than one valence state). Although these conduction processes are of great interest, and have been extensively studied [7,8], little is known about the vibrational properties of such glasses, save for the second order elastic constants (SOEC) [9]. In the present work the effects of hydrostatic pressure and uniaxial stress on the velocity of ultrasonic waves propagating through (Fe2Oa)o.as(P2Os)o.62 have been measured. The results provide the first sets of pressure derivatives of the SOEC and also the TOEC for a glass of this type. The hydrostatic pressure derivatives of the SOEC given quantitative information of the vibrational mode dependences'upon volume. The TOEC are the cubic coefficients of the Hamiltonian with respect to strain and thus depend upon the anharmonicity of the interatomic forces, that is their nonlinearity. Usual practice is to consider the anharmonic effects in terms of the GrOneisen mode gammas and these are compared and contrasted here with those of other glasses. The study of the vibrational anharmonicity is then extended to a computation of the displacement amplitude of the scattered phonon from phonon-phonon interaction processes, to provide basic details of the phonon dynamics in this type of glass.
2. Experimental procedure and remits The phosphate glass was supplied by M. Sayer; details of its preparation have been given by Sayer and Mansingh [7]. While embedded in cold set resin, an ultrasonic sample was cut with a diamond saw into an orthogonal block having dimensions 1.0 X 0.8 X 0.5 cm. A series of aloxite and diamond pastes was then used to lap each surface to a mirror finish with parallelism between pairs of faces better than 10 -s rad. The density was determined as 3010 kg m -3. The SOEC, obtained from pulse echo overlap measurements of the propagation velocities of longitudinal and transverse 10 MHz ultrasonic waves, are listed in table 1 together with the earlier results obtained by Farley and Saunders [9]. The difference between the two sets of data is almost entirely accounted for by the inclusion in the present work of a transducer correction [10] amounting to a 4.6% increase in the longitudinal and a 3% increase in the shear velocity. Ultrasonically determined elastic constants are adiabatic; since neither the thermal expansion coefficient a nor the specific heat Cp are available for this glass, the isothermal constants
M.P. Brassington et al. / Third order elastic constants
Table 1 A comparison of the elastic properties of (Fe~Oa)o.aa (P~0s)o.e~ phous materials Property
(Fez03)o.38(Pzos)o.62
Present kgm-3 -1 ms
Pyrex
Fused silica
[41
[21
[II
191
3010
3010
4770
5120
2501
3043
2990 78.9
1396 29.8
25.1
11.3
62
GPa
83.0 21.3
c44
GPa
21.9
26.9
9.3
Bs
GPa
45.9
ES
GPa
69.5
43.0 66.8
17.5 23.1
0.24
0.25
ez
with those of other amor-
a-As
5253
crs
159
0.21
_ 69.5
13.7 21.9 41.8 84.1 0.16 _
2203 5965 3768 78.3 15.9 31.3 47.0 93.5 0.17
Cl11
K GPa
-450 f 60 a
-465
400
530
Cl12
GPa
-200 + 25 a
-
-33
30
240
cl23
GPa
-160 f 20 a
-
-162
260
50
_-120
90
156
425
GPa
-18+
2a
-
Cl55
GPa
-62t
8a
-
c456
GPa
-22*
3a
Cl44
afi -
(ap1
+64 -108
90
-86
105
495
70 -10
+4.73 a
+6.42
-4.72
-6.3
-0.16 a
+1.73
-2.39
-4.1
+2.34 +1.45
-1.74 a
-2.8 a
-1.50 a
-2.36 a
P=o
YI
+1.1 a
Yt
-0.3 a
-
a) The calculation of these parameters has involved one or more of the following assumptions: BT 2 Bs, ET 2 Es and cT 1 us.
could not be calculated, however, the difference between isothermal (T) and adiabatic (S) elastic constants is expected to be substantially smaller than 1%. For the shear modulus it is usual practice to consider the Lame shear constant /Afor which /_I’ and pT are identical and equal to C44. The Debye temperature 0g has been estimated from the mean sound velocity
+)/3]d’3)
cm(=[($ +
M.P. Brassington et al. / Third order elastic constants
160
using k\47r ]
Cm
(1)
where N, the volume density of vibrational units, is taken to be equal to the number of atoms per unit volume. The Debye temperature calculated from this model, which represents all the .vibrational modes, is 425 K. The hydrostatic pressure dependences of the ultrasonic wave velocities were measured at constant temperature in a piston-cylinder equipment using castor oil as the pressure medium. Uniaxial stresses were applied with a screw press, the force exerted being measured by a proving ring. The small changes in wave transit times t(p) were measured by an automatic frequency controlled, gated carrier, pulse superpos~tion apparatus [11]. At an applied pressure P the "natural" velocity W(p)
o
8.0
x
~-~ 7.0
i"
<1 6.0
g
c ~
.0
-~ 4 0
~ 3.0
~ 2.0
1.0
0
0
I
04
0!2 Pressure
013
0.4
I k bar
Fig. 1. Relative natural velocity change as a function of hydrostatic pressure for 10 MHz ]ongitudinaJ waves i n (Fe203)o.3s(P2Os)0.62. (1 kbax is 10 s Pa).
M.P. Brassington et al. / Third order elastic constants
161
of ultrasound in the sample is given by
W(p)= lo/t(e)
(2)
where lo is the unstrained path length. The measured relative change in this natural velocity under hydrostatic pressure [i.e. AW(p)/Wo=(W(p)- Wo)/Wo; where I¢o = (I¢(e))p:_o] is plotted for longitudinal waves in fig. 1 and for shear waves in fig. 2. When a pressure of 0.3 kbar was exceeded, non-reproducible, nonlinear effects were observed, especially for longitudinal waves, so the results reported here are restricted to the reproducible and reversible region up to 0.3 kBar. For longitudinal modes d Ale
= +2.36 X 10_11 + 2% pa_ 1
and for transverse modes d 'AW
- --6.45 × 10 -1~ -+ 0.6% Pa -1 .
Now the pressure derivative of the shear modulus ~ is given by (0Wt~
,
,
,
1 C~441
i
,
,
,
,
f
'
'
'
0.8
0.9
1.0
1.1
I-2
-1.0
x
-2.0
~-4.o
~i "5"0 ~ ~
-6.0
o7.0
-BO
OI
0.2
0,3
G4
0"5
O~
'
0.7
P r e s s u r e I k bar,
Fig. 2. Relative natural velocity change as a function of hydrostatic pressure for 10 MHz transverse waves in (Fe2Oa)o.as(P2Os)0.62.
M.P. Brassington et al. / Third order elastic constants
162
where the second term on the RHS is a correction for the reduction in dimensions of the sample with compression. Since
(aWtl
= Wt ~
,
\ aP ]e=-o
(4)
e=-o
and for shear waves (Wt)~_ o is 3.04 X 103 ms -1, we have (Ola/aP)~__o = - 0 . 1 6 , assuming that B T ~ B s. The pressure derivative of the bulk modulus B s is given by = ~2{ Po IZW1 (~)WI] - 4Wt [[~Wt~I']+_@TC1SI_4 V)J aC,4) }
(aBSl
\ OP/~o
\ OP /
~--o
(5)
and 0 2XW
(6)
Using ( W I ) ~ o = 5.25 X 103 ms -1 and assuming that B T ~ B s, we find that (~Bs/~P)p=_o = +4.73. The relative change in "natural" wave velocity as a function of applied uniaxial stress was measured for two shear modes; results for shear waves polarised in a
I
g
K
-5
>, •~
-10
~d g -is ,/I
.~- - 2 0
>~ - 25
0
Applied
stress
/ bar
Fig. 3. Relative natural velocity change as a function of uniaxial stress applied parallel to the polar\sat\on but perpendicular to the propagation direction of 10 MHz transverse waves in (Fe 2Oa)0.38 (P2Os)0.62.
M.P. Brassington et el. / Third order elastic constants i
!
w
163
1
0
-2,0 q - 40
"~.
• •
•
•
- (3.0
~o
"~
- 8-0
C
-10.0
•
•
•
•
-12.0
- 14.0
- lS.O
-18.0 10
20
30 Pressure I
40
50
60
70
bar
Fig. 4. Relative natural velocity change as a f u n c t i o n of uni a xi a l stress for 10 MHz transverse waves where the stress, wave propagation and polarisation directions are m u t u a l l y perpendicular.
direction parallel to the stress are given in fig. 3 and those for the polarisation normal to be applied stress in fig. 4:
is found to be -1.045 X 10 -11 + 7% Pa -1 and -2.81 × 10 -12 -+ 6% Pa -1, respectively, It can be seen that the results for the uniaxial stress experiments (figs. 3 and 4) display greater scatter than those for hydrostatic pressure (figs. 1 and 2). This is because the fragility of the sample limits the allowed uniaxial stress, and thus the consequent velocity change is very much smaller (by a factor of "'40), so that the errors and scatter are proportionately larger. For example, when the uniaxial stress was applied parallel to the shear wave polarisation direction (fig. 3) for the maximum applied stress of 14.8 bar the frequency change was only 2.5 parts in lO s . For uniaxial stress applied perpendicular to the polarisation direction, the frequency
164
M.P. Brassington et al. / Third order elastic constants
change was even less (1.75 parts in 10 s) for the maximum applied stress of 61.7 bar. Those are very much smaller changes in frequency induced by stress than could be obtained in the hydrostatic pressure experiments [8 parts in 104 for a hydrostatic pressure of 0.35 kbar (fig. 1)]. The scatter has been taken into account statistically and appears in the errors quoted for the pressure derivatives of natural velocity and consequently in the TOEC (table 1). An isotropic material such as a glass has three independent third order elastic stiffness constants, which may be taken as C123, C144 and C4s6 and given the symbols v~, v2 and v3 respectively. Then C112 = vl + 2v2 , C15s = v: + 2v3
(7)
and C1~1 = vl + 6v2 + 8va. The three independent constants may be obtained from hydrostatic and uniaxial stress derivatives of second order elastic stiffness constants. Thurston and Brugger [ 12] have derived equations relating the quantity (p'oW
P=o
to the three independent constants vl, v2 and va for various stress configurations. Now
~O
ZW
thus from the results shown in figs. 1 to 4 and from the relevant equations of Thurston and Brugger four linear equations in the three unknowns vl, v2 and v3 may be written: 1 (2cSl + 3vl + 10v2 + 8v3) = 3.92 - 1 - 3B T
1 --1 -- 3B x (2C44 + 3v2 + 4va) = - 0 . 3 6 , 1 - ~ - [ - 2 C 4 4 + v2(2o T - 1) + 2v3(o T - 1)] = 0.58 and
1 [oT(2c44 + 4vz) + v:(2o T -- 1)] = --0.156 ET
I
(8)
M.P. Brassington et al. / Third order elastic constants
165
By making the approximations that BT = BS, ET = ES and oT = us and weighting each equation by the reciprocal of its estimated experimental error, we have solved these four equations for ~1, v2 and v3 using a least-square technique. The results are u1 = (-160 + 20) GPa, v2 = (-17 + 2) GPa, and va = (-22 f 3) GPa. The ultrasonic technique provides mixed TOEC (as defined by Brugger [ 131). In principle the measured CrlK can be converted to adiabatic TOEC cJK. However, this requires a knowledge of thermodynamic parameters such as the thermal expansion (Yand of the specific heat Cn. Taking reasonable estimates for these parameters it has been shown that (CnrJK - Crr,) is negligible compared with the experimental error (table 1) [4]. The third order elastic stiffness constants C~JK of (PezOa)o.ss(P2Os)o.62 glass are listed in table 1. A check on the internal consistency of these results if provided by [I41 = -(C111 + 6C112 + 2C12s)/9BT which (assuming BT = B’) gives (aBS/W>,, of t4.73 measured directly from hydrostatic
as t4.77, in good agreement with that experiments alone.
3. Pressure dependences of the elastic mod& and Griineisen parameters All the second and third order elastic stiffness tensor components have been obtained for this iron phosphate glass and are compared in table 1 with the other complete sets reported previously for other types of glass. Although the density of (Pe203)e.3s(P205)0.62 glass is low compared with the covalently bound amorphous element arsenic, the velocities at which longitudinal and shear ultrasonic waves are propagated are markedly higher. This behaviour is much more like that of fused silica; consequently the values of the SOEC of the phosphate glass are similar to those of fused silica and Pyrex. This suggests that the interatomic binding forces in these phosphate glasses are probably also mixed ionic-covalent in nature. That both phosphate and silica-based glasses have comparatively high Debye temperatures agrees with this suggestion; in contrast 02 for amorphous arsenic is about three times smaller (table 1). However, at the third order the situation is almost completely reversed; the TOEC of the phosphate glass are all negative, in marked contrast to those of the silica-based glasses which are anomalously positive, exceptions being Ce56 for fused silica and C1+, for Pyrex. The anharmonic properties, that is the nonlinearity of the interatomic binding forces with respect to atomic displacements, of the iron phosphate glass are similar in kind to those in amorphous arsenic and to those crystalline solids which do not show soft mode behaviour. An unusuaI feature of glasses such as silica or Pyrex is that the hydrostatic pressure derivatives of the elastic moduli (aBs/W),, and (a/.@P),, are negative and
M.P. Brassington et aL / Third order elastic constants
166
the temperature derivatives (OBS/OT)p and (O#/~T)p are positive. This behaviour is in complete contrast to that of crystalline solids but it should not be considered as characteristic of the glassy state: amorphous arsenic has the more normal positive pressure and negative temperature derivatives of the elastic moduli [4]. In fact anomalous pressure or temperature dependences of elastic moduli are observed in crystalline solids which have soft acoustic phonon modes, or, in what amounts to much the same thing, in crystals with low coordination numbers so that bending vibrations are allowed. In silica glasses the anomalous behaviour probably results from the low coordination number. Recently Sato and Anderson [6] observed a positive (~B/OP)T but a negative (O#/OP)r for an SiO2 (70.5 wt.%) - CaO (11.6 w t . % ) - Na20 (8.7 w t . % ) - K20 (7.7 wt.%) soda-lime-silica glass; as they point out, the presence of alkali metal ions in the interstices of the silica network inhibits the soft transverse vibrations. For the iron phosphate glass the temperature dependences of both longitudinal and shear wave velocities and thus (OBS/aT) and (a#/~T) have the more usual negative slope [9]. However, while (~BSl~p)~_o is strongly positive, (O#/aP)e=o is just negative. The elastic behaviour of this glass is similar in kind to that of the soda-lime-silica glass whose transverse vibations are inhibited by the presence of a network modifier. The small but negative value found for (~#/aP) implies that for the iron phosphate glass the structure is open enough to allow a degree of bond-bending vibration, but less so than in fused silica. Sato and Anderson [6] have gained useful insight into the effect of pressure on the vibrational frequencies in the long wavelength limit by considering the Grtineisen mode gammas. For an isotropic material there are only two contributions to the generalised elastic Graneisen parameter )'elastic, one associated with longitudinal modes 3'1 and the other with transverse modes 7t. These two mode gammas may be found from the elastic constants of the material [ 15] using ~'l,t -
1
6Wl,t
(3B "r + 2Wl,t + kl,t)
(10)
where w 1 = C11, wt = C44, kl = C111 + 2C112 and k t = l(Cl 11 - C12 a)From the data obtained above we fred that for (F%Oa)o.aa(P2Os)0.62 glass 3'1 = +1.1 and 7t = -0.3. Because the Debye temperature of this glass is 425 K it cannot be assumed that at room temperature all possible vibrational modes are excited to the same degree: it is not possible to derive readily a generalised elastic Griineisen parameter "/elastic from the individual mode gammas 71 and ")'t --such a derivation would require details of the relative excitation of the longitudinal and shear modes. However, the magnitudes and the difference in sign of 3'1 and ")'t indicate that ~[elastic is small. The mode gammas 3'1 and 7t, while both strongly negative for fused silica and Pyrex and both strongly positive for a-As, are in the intermediate case of iron phosphate glass opposite in sign as Sato and Anderson [6] found in their soda-lime-silica glass with high modifier content. At room temperature longitudinal modes stiffen but shear modes soften with applied hydrostatic pressure. A negative Grtineisen param-
M.P. Brassington et al. / Third order elastic constants
167
eter 7t is an indication of facile bond-bending vibrations and a small force constant for the corresponding long wavelength transverse waves.
4. Three-phonon interactions The anharmonicity of the vibrational states is responsible for the phonon-phonon interactions and thus for phonon damping of ultrasonic waves, for p h o n o n phonon contributions to the thermal resistivity and also for the thermal expansion. The measurements of the TOEC allow a quantitative assessment of phonon-phonon interactions in the long wavelength limit for a phosphate glass. The p h o n o n phonon interactions, which play an important part in dynamical properties, such as for example the initiation of steady state heat-flow, are determined by the anharmonic terms in the strain Hamiltonian. The disturbance produced as a phonon of wavevector k propagates strain modulates the medium (in a way which is measured by the TOEC) as seen by a second phonon of wavevector k2 so that phonon-phonon scattering can take place. For a phonon-phonon normal process the scattered phonon has a wave vector k3 such that k3 = kl + k2
(11)
and momentum hk is conserved. Taylor and Rollins [ 16] have shown how to calculate for normal scattering processes the displacement amplitude X3 of a scattered phonon created by the non-linear interaction of two incident phonons propagating in an isotropic medium. This method is based on use of the first-order time dependent quantum mechanical perturbation theory to calculate transition probabilities between the phonon states. Their result is that the displacement amplitude X3 of the scattered phonon beam may be written as X3 = X1X2 VAw31a(1 + a)/8nrpctcl
(I 2)
where X1 and X2 are the displacement amplitudes of each of the incident phonon beams, V is the volume of interaction between the incident beams and r is the distance from the point of observation to the point of origin of the scattered phonons. The angular frequencies of the incident phonons of wavevectors kl and k2 are COl and co2 respectively, and a is co2/6Ol. A is a function, not only of the specific type of interaction and the interaction geometry but also of the second and third order elastic properties of the medium; expressions for A can be found in table 1 of reference [16] for each of the five possible types of these phonon interactions which are allowed under conservation of energy (E3 = El + E2) and momentum (hk3 = h k l + hk2).
The aim is to calculate the displacement amplitude IX31 of the scattered phonon emitted at a angle 7 for two input phonons interacting at an angle ~. Therefore a normalising value of unity can be allocated to the term XiX2co~/8~r in eq. (12), while taking the interaction volume V to vary as cosec ~. By inserting the elastic
168
M.P. Brassington et aL / Third order elastic constants
constant data given in table 1 into the expressions given by Taylor and Rollins and then solving eq. (I 2), the dependence of the normalised values IX3 [ of the displacement amplitude of the scattered phonon resulting from phonon-phonon interactions has been computed for the iron phosphate glass. Results for the possible types of interaction are plotted as a function of the angle ~ of the incident phonon in fig. 5. Excluding collinear interactions, there are two general groups, termed a and/3, of interactions. The first (tx) group comprises the types I, where two incident transverse (T) phonons interact to give a longitudinal (L) phonon Type I: T(col) + T(w2) ~ L(~ol + w2) , and its converse V: Type V: L ( w ~ ) ~ T(co2) + T ( w l - 602). For such interactions, the scattering angles at which zeros or minima occur in the scattered wave displacement amplitude depend upon the elastic constants o f the materials. For the phosphate glass iX3[ and thus the physically significant intensity X~a, does not become zero but does exhibit a minimum for both types I and V (zeros could exist for different choices of polarisation angles Oi). The second group (/3) of interaction types includes the mixed longitudinal and
,.4 10
!
/,
{6
'
'
', / }
i, ~
2
ii
zo
t 0
50' Incident
10(3" 150" interaction angle (I).
200"
Fig. 5. Absolute displacement amplitude IX3 I of the scattered phonon beam produced from the interaction of two phonons incident at an angle ~ in (Fe203)o.38(P2Os)0.62. Results are shown for interaction types ] (polazisation angles e2 = e3 = 0°), H (03 = 0°), II| (02 = 0°), IV (e 2 = 0 °) and V (#2 = e3 = 0 °) for def'mitions o f e i see ref. [16].
M.P. Brassington et al. / Third order elastic constants
169
transverse interactions: Type 1I: L ( w l ) ~ L(c02) + T(wl - w 2 ) , Type III: L(w~) + T(c02) ~ L ( w l + c02), Type IV: L ( w l ) ~ T(w2) + L(col - co2). For these interactions the zeros in IX31 are of geometrical origin and are independent of the material properties. Each of the curves for IXa[ terminates at a critical angle be beyond which the conservation laws o f energy and momentum can no longer be satisfied so that phonons ks are not produced. These quantitative determinations from the TOEC data of the p h o n o n - p h o n o n scattering amplitudes are a prerequisite for further development of an understanding o f the dynamical behaviour o f phonons in phosphate glasses; in particular they should provide a physical insight into anharmonic properties such as ultrasonic attenuation and long wavelength phonon scattering contributions to the thermal resistivity. We are most grateful to J. Penfold and G.D. Pitt for allowing us to use the S.R.C. high pressure facilities.
References [1] [2] [3] [4]
E.H. Bogardus, J. Appl. Phys. 36 (1965) 2504. D.S. Hughes and J.L. Kelly, Phys. Rev. 92 (1953) 1145. N. Soga, M. Kunigi and R. Ota, J. Phys. Chem. Solids 34 (1973) 2143. M.P. Brassington, W.A. Lambson, A.J. Miller, G.A. Saunders and Y.K. Yo--gurt~u, Phil. Mag. B42 (1980) 127. [5] J.C. Thompson and K.E. Bailey, J. Non-Crystalline Solids 27 (1978) 161. [6] Y. Sato and O.L. Anderson, J. Phys. Chem. Solids 41 (1980) 401. [7] M. Sayer and A. Mansingh, Phys. Rev. B6 (1972) 4629. [8] W. Chomka, O. Gzowski, L. Murawski and S. Samatowicz, J. Phys. Cll (1978) 3081. [9] J.M. Farley and G.A. Saunders, Phys. Stat. Sol. (a) 28 (1975) 199. [10] E. Kittinger, Ultrasonics 15 (1977) 30. [11] Y.K. Yo~urtqu, E.F. Larnson, A.J. Miller and G.A. Saunders, Ultrasonics 18 (1980) 155. [12] R.N. Thurston and K. Brugger, Phys. Rev. 133 (1964) A1604. [13] K. Brugger, Phys. Rev. 133 (1964) A1611. [14] F. Birch, Phys. Rev. 71 (1947) 809. [15] K. Brugger and T.C. Fritz, Phys. Rev. 157 (1967) 524. [16] L.H. Taylor and F.R. Rollins, Phys. Rev. 136 (1964) A591.