Temperature dependence of elastic moduli of β-(Mg, Fe)2SiO4

Norihito Mayama, Isao Suzuki and Toshiaki Saito
Department of Earth Sciences, Okayama University, Okayama, Japan.
Ichiro Ohno
Department of Earth Sciences, Ehime University, Matsuyama, Japan.
Tomoo Katsura and Akira Yoneda
Institute for Study of the Earth’s Interior, Okayama University, Tottori, Japan.

Abstract
The elastic moduli of polycrystalline wadsleyite, β-(Mg0.91Fe0.09)2SiO4, were measured
up to 470 K by means of the resonant sphere technique.

The adiabatic bulk (KS) and shear (µ)

moduli were found to be 165.72(6) and 105.43(2) GPa at room temperature.

The averageslopes (dKS/dT and dµ/dT) in the range were determined to be –0.0175(3) and –0.0159(1) GPa/K.
We estimated that the P-, S-wave velocity and density jumps for the α- to β-phase transformation
at the 410-km depth condition were 9.5, 11.2 and 5.4 %, respectively. These results suggest thatthe olivine component at the depth should be 52 and 42 volume % for P- and S-waves.

1. Introduction
The 410-km discontinuity in the upper mantle is thought to be caused by thetransformation of olivine (α-phase) to wadsleyite (β-phase) (Ringwood, 1975).
transform to garnet phase [cf. Irifune and Isshiki, 1998].

Pyroxene may

But this may affect slopes of velocities

and may be excluded in discussion based on the discontinuous change.

Comparing the velocity

jumps caused by this transformation with those from the seismological observations, we may
infer the volume ratio of the olivine component in the upper mantle. Therefore the elastic
moduli and their pressure and temperature derivatives for the α- and β-phases have beenmeasured in many studies [e.g., Kumazawa and Anderson, 1969, Sawamoto et al., 1984]. The
temperature derivatives of elastic moduli of the α-phase have been obtained by means of theresonance method in a wide range of temperatures [Isaak et al., 1989, Isaak, 1992].

Recently,

those of the β-phase were obtained with the ultrasonic interferometry [Li et al., 1998]. In ourforegoing paper [Katsura et al., 2001], we reported temperature derivatives of elastic moduli fora polycrystalline β-phase specimen in a rather narrow temperature range of 278 to 318 K.

But

the errors of the temperature derivatives for the β-phase in these reports were larger in one orderof magnitude than those for the α-phase and so, the elastic moduli at mantle temperature were
simply estimated by linear extrapolation. Katsura et al., [2001] did not discuss the mantle
composition.

For discussion of mantle constitution, we need measurements of elastic moduli ofthe β-phase in much wider temperature range and also a method other than linear extrapolation
to estimate the elastic moduli at high temperature. In the present paper, we report the elastic
moduli of the β-phase measured under more stable conditions by the resonance method [Suzuki
et al., 1992a] in wider temperature range of 298 to 470 K, and try to apply an equation which
represents better the temperature dependence of elastic moduli.

2. Measurement
The β-phase specimen is the same as that of Katsura et al. [2001]. The specimen was
synthesized and sintered from San Carlos olivine using a Kawai-type high-pressure apparatus.
Pressure, temperature and duration time for the synthesis and sintering were 14(1) GPa, 1400 K

and 1 hour. The polycrystalline specimen was confirmed to be uniform under microscope andgood in crystallinity from the narrow widths of x-ray diffraction peaks.

The electron

microprobe analyses were made for the specimen and the Fe/(Mg+Fe) ratio was found to be
0.091(3), which was the same as that of starting material.

After shaping, the diameter of thesphere specimen was 2.250(2) mm at room temperature. With the data of mass and volume,bulk density was calculated to be 3.60(1) g/cm3.

This bulk density is in good agreement with

X-ray density, 3.57(3) g/cm3 [Sinogeikin et al., 1998], suggesting that our specimen is nearlypore-free.
The resonant spectra of the specimen at room temperature were measured by the common
frequency scanning method, or the CW method [Suzuki et al., 1992b].
measurement was from 1.8 to 5.0 MHz with 10 Hz intervals.

The frequency range of

The resonant frequency shifts

caused by supporting force for the specimen were corrected by using the contacting spheretheory [Yoneda, 2002].
Data acquisition at high temperature was made for the damping waveform of thespecimen stimulated by high voltage pulses, which was generally called the FT method [Suzuki
et al., 1994].

The oscillatory waveform data of the specimen was stacked and averaged up over

five thousand times to reduce random noises from circumstances.

The resonant spectrum was

obtained by the FFT analysis, in which frequency resolution was 100 Hz.

Buffer rods wereused to protect transducers from high temperature [Goto and Anderson, 1988].

Example of theoscillatory waveforms is shown in Figure 1A, and the resonant spectrum in Figure 1B.

Thetemperature range was from 298 to 470 K, in which data acquisition was made almost at every
10 K.

Temperature fluctuation during measurements was within 1 K.

The diameter anddensity of the specimen at high temperature were corrected by thermal expansion data of Suzuki
et al. [1980].

The resonant frequencies obtained at high temperatures were reduced by thecorrection terms at the ambient condition. The elastic moduli (adiabatic bulk (KS) and shear (µ)
moduli) at every temperature were calculated from the resonant frequencies by the least squares
method.

3. Result
In the spectrum data at room temperature, we found the lower 13 resonant modes for the
specimen.

The modes were well assigned with isotropic oscillation.

resonant peaks were about 1 kHz.
found through buffer rods.

The half-widths of

At high temperatures, the lower 9 modes (Fig. 1B) were

Although higher frequency modes were not detected because of

their smaller amplitudes, we had a sufficient number of modes for reduction of the isotropic
elastic moduli.

Figure 2 shows examples of resonant frequency changes against temperature.

The differences of resonant frequencies between heating and cooling processes are very small tobe 0.05 % maximum.

Figure 3 shows the elastic moduli of the β-phase as a function oftemperature. The present results in Figure 3 show smooth changes of elastic moduli against
temperature. Their derivatives were obtained by fitting a linear function to the data between
298 and 470 K.

The errors of the temperature derivatives were estimated by taking account of

not only residuals from the linear fitting but also errors of elastic moduli being deduced fromobserved frequencies at every temperature.

The KS and µ and their temperature derivatives(dKS/dT and dµ/dT) of the β-phase are shown in Table 1, along with data from other sources [e.g.,
Sawamoto et al. 1984].
The elastic moduli of the β-phase at room temperature obtained in this study are about
2.5 % smaller than those obtained by Sinogeikin et al. [1998].

The reason is not well known

but differences in measuring technique, chemical composition and structure of the specimens etc.

are possible. Compared with our previous results [Katsura et al., 2001], the present slopes ofelastic moduli against temperature are larger and their errors are smaller in one order ofmagnitude.

This situation is probably due to the wider temperature range and smoothtemperature changes of elastic moduli in the present study as compared to those in previous one,in which the elastic moduli against temperature are somewhat scattered because the S/N ratio ofresonant spectra is small, due to the smaller force supporting the specimen and the vibration of
the temperature chamber.

The present value of dKS/dT is somewhat different from that for theend member β-phase obtained by Li et al. [1998]. This discrepancy may be not due to theeffect of different Fe content, when we remind that the difference is only 10 % in the α-phase[Isaak et al., 1989, Isaak, 1992]. The larger slope of dKS/dT than that of dµ/dT in the present
study is also found in the relative materials of the α-phase [Isaak et al., 1989, Isaak, 1992] andringwoodite (γ-phase) [Jackson et al., 2000, Sinogeikin et al., 2001] as well as alminate spinel
MgAl2O4 [Suzuki et al., 2000a], whereas the slopes in Li et al. [1998] show the opposite
tendency. Li et al. [1998] succeeded in difficult measurements of travel times through a small
specimen in a solid pressure medium and simultaneously determined the pressure andtemperature derivatives of elastic moduli from the set of travel time data.

On the other hand,the present measurements were made simply as a function of temperature.

Then, three-digit

significant figures of slopes are listed in Table 1 with errors.

4. Discussion
We calculated the seismic wave velocity and density jumps at 410-km depth (e.g., 13.8
GPa and 1673 K) for a hypothetical pure olivine mantle from the present results. The elastic
moduli at high temperatures were estimated by using not linear extrapolation but the modified

Wachtman’s equation [Anderson, 1966, Suzuki et al., 2000b], and the seismic wave velocity anddensity were obtained by the third-ordered finite strain-equation [Davies and Dziewonski, 1975],
referring to pressure derivatives of KS and µ [Zha et al., 2000].

On the other hand, it wassuggested that the Fe/(Mg+Fe) ratio in olivine might vary during the transformation from the αto β-phases around 410-km depth [Irifune and Isshiki, 1998]. Then we also calculated theseismic wave velocity and density jumps for this model, and compared results from the two Fe
content models.

The first model is that the Fe/(Mg+Fe) ratio in the phases is constant, to be 9mol%, the same as composition of the specimen used in the present study, and designated as the
Fe constant model hereafter.

The second one is that the Fe/(Mg+Fe) ratio varies from 7.5 to

11.5 mol% at the transition by Fe partitioning between the phases and garnet [Irifune and Isshiki,
1998], the Fe partitioned model.

In order to obtain the seismic velocities at the 410-km depth

conditions, the elastic moduli reported for single crystals were adopted as reference values andthe present data were used only as their temperature dependence.
and the results for the two Fe models are shown Table 2.

The data used in calculations

The jumps of compressional (P),shear (S) wave velocities and density for the α- to β-phase transformation at 410-km depth wereobtained to be 9.5, 11.2 and 5.4 % for the Fe constant model and 8.1, 8.8 and 7.3 % for the Fe
partitioned model, respectively.

The Fe partitioned model consists of two parts of velocity

increase, a steeply increasing part in a rather narrow pressure interval (say 0.2 GPa) and a
gradually increasing part.

If we apply the former part to the observed seismic velocity jump,

this may infer the maximum content of olivine component in that depth of the mantle.

When

we adopt velocity jumps of 4.9 and 4.6 % for P- and S-waves at the 410-km discontinuity [Walck,
1984, Grand and Helmberger, 1984], and assume the jumps are only due to the phasetransformation, then the olivine component from P- and S-waves are about 52(15) and 42(10)volume % for the Fe constant model and 61(15) and 54(10) volume % for the Fe partitioned
model, respectively.

The error bounds were estimated by the higher or lower limits of theelastic moduli and their slopes. It is found that olivine component around the 410-km depth
increases about 10 volume % in numerals by introducing Fe partitioning between the phases andgarnet, though Irifune and Isshiki [1998] suggested that olivine component should increase 4-5
volume % as estimated at ambient condition.
Shearer and Flanagan [1999] discussed the relationship between the jumps of density
and P- or S-wave velocity at the 410-km discontinuity from observation of reflected phases andshowed a possible range of jumps within the confidence ellipsoids.

Using the present data, theolivine component within the ellipsoids is estimated to be 45-55 volume % for P-wave and 45-60
volume % for S-wave, respectively, irrespective of the Fe models.

Therefore combining thedensity jump with velocity ones, the possible range of olivine component becomes 45-60
volume %.

Acknowledge
We thank K. Shouno, M. Sakai, M. Walter and H. Yamada for their suggestions anddiscussions.

This work was partially supported by Grant-in Aid for Scientific Research No.

12440121 from the Japan Society for the Promotion of Science.

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Figure caption
Figure 1. An example of the oscillatory waveform (A) at high temperature and its spectrum (B)for the β-phase specimen.

The symbols represent resonant isotropic oscillation modes.

Figure 2. Examples of temperature dependence of resonant frequencies for 0T2 and 0S2 modes ofthe β-phase specimen.

The open circles and squares represent heating and cooling processes,

respectively.

Figure 3. Temperature dependence of elastic moduli of the β-phase. The open circles andsquares represent adiabatic bulk and shear moduli, respectively.

Error bars are shown.

Table 1. Elastic moduli and their temperature derivatives of the β-phase.

The x, KS *, µ*, P

and T represent Fe/(Mg+Fe) ratio, dKS/dT, dµ/dT, pressure and temperature ranges inmeasurement, respectively.
x(mol%)

KS(GPa)KS*(GPa/K)µ*(GPa/K) P(GPa)µ (GPa)T(K)reference

0298

Sawamoto et al. [1984]

0-7

298-873

Li et al. [1998]

0298

Sinogeikin et al. [1998]

0

174

114

0

172(2)

8

170(3)

9

165.7(1)–0.016(3)105.66(3)–0.012(1)0278-318

Katsura et al. [2001]

9

165.72(6)–0.0175(3) 105.43(2)–0.0159(1)0298-470

this study–0.012(1)113(1)–0.017(1)108(2)

Table 2. The data used for calculations at 410-km depth conditions (13.8 GPa and 1673 K) andresults for the two Fe models.

The x, KS', µ', KS(T), µ(T), ρ, vP and vS represent Fe/(Mg+Fe)

ratio, the pressure derivatives of KS and µ, KS and µ at T=1673 K, density, the compressional andshear wave velocities, respectively.
at 410-km depth
Fe modelxKS

KS (T)

mol%

GPa

GPa

α

9.0

129.4a104.5gβ9.0

170.0bpartitioned α
model
β

7.5
11.5

constant
model

KS ’

µ (T)

µ

µ’

ρ

ρ

vP

vS

3

g/cm

km/s

km/s g/cm3

GPa

GPa

4.2c

78.1a

59.1g

1.4c

3.353a

8.59

4.65

3.543

141.4h4.3c108.0b

82.0h

1.4c

3.602e

9.41

5.17

3.736

129.4a104.5g

4.2c

79.0a

59.4g

1.4c

3.330a

8.62

4.68

3.519

b

h

c

d

h

c

f

9.32

5.09

3.775

170.0

141.4

4.3

106.0

80.0

1.4

3.640

a: Isaak [1992], b: Sinogeikin et al. [1998], c: Zha et al. [2000], d: Li and Liebermann [2000], e: the present data, f:
modified for x being 11.5 % from Hazen et al. [1990] data, g: estimated value by the modified-Wachtman’s equation
fitted to the data of Isaak [1992], h: The present data extrapolated by the modified-Wachtman’s equation.

Amplitude (mV)

Amplitude (mV)

5

A

0

-5

0

1
2
Sampling time (ms)

20

3

B

S2

0

15
10
5

T4

0

T2

0

0

0S 3
S 1 0T 3

1

2

S4
1S 2
0

S0

0

2.5
3
3.5
Frequency (MHz)4

Fig. 1Frequency (MHz)

2.00
1.95

2.05

0

S2

1.95

1.90
0

1.85
250

Fig. 2

2.00

T2

300 350 400 450
Temperature (K)

1.90
500

107

165

106

164

105

163

104

162

103

161
250

300

350

400

Temperature (K)Fig. 3

450

102
500

Shear modulus (GPa)

Adiabatic bulk modulus (GPa)166