Ph.D. thesis (pdf) - dirac

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146 Boson Peak We are mainly interested in studying the vibrational density of states obtained from the inelastic signal. The theoretical expression for the inelastic one phonon single contribution to S inc (Q, ω) is given by equation 4.3.35, which reads S inel,inc (Q, ω) = 1 ( 2M exp − 〈u2 〉Q 2 ) Q 2n(ω) g(ω). (8.1.2) 3 ω where M is the mass of the molecule. Here the Q dependence and the energy dependence is totally disentangled. This means that the density of state (in arbitrary units) can be found from the energy dependence of the scattering function at any given constant Q. We primarily consider the data in the form g(ω)/ω 2 . This quantity is obtained directly from equation 8.1.2 by correcting for the bose factor, that is dividing by n(ω)ω. In the study of the pressure dependence of the boson peak in PIB3850, we compare the intensity of the boson peak at different pressures; this experiment is performed with a constant volume of the sample in the beam, however the number of scatterers is proportional to the pressure dependent number density. Moreover, the Debye Waller factor is also pressure dependent because 〈u 2 〉 is pressure dependent. These effects are eliminated by normalizing to the elastic intensity, which is given by the number of scatterers and the Debye Waller factor: I el,inc (Q) = N σ ( inc 4π exp − 〈u2 〉Q 2 ) . (8.1.3) 3 The resolution function measured at 2 K had a small wing up to about 2 meV, and has therefore been subtracted, when considering the inelastic signal of the low temperature data. The effect of this last correction is minor as soon as the temperature approaches T g of the sample because the intensity of inelastic and quasi-elastic scattering increases. In figure 8.2 we show the rDOS (g(ω)/ω 2 ) of PIB3580 obtained at a constant Q as well as the result obtained by summing over the angles. The data are in both cases corrected for the bose factor and normalized to the elastic intensity as described above. The data are finally adjusted by one pressure independent factor to make the data overlap. It is clearly seen that the results are in agreement, not only the shape and position of the boson peak, but also the pressure dependence of the intensity found by the two methods is the same.
8.1. Time of flight 147 g(ω)/ω 2 [arb. unit] 1.6 1.4 1.2 1 0.8 0.6 Patm 0.41GPa 0.8GPa 1.4GPa 0.4 0.2 0 0 5 10 15 ω [meV] Figure 8.2: Spectra of PIB3580 at different pressures. The full lines give g(ω)/ω 2 obtained from S(Q, ω) measured at a fixed Q value (Q=1.7 Å −1 ). The dots give g(ω)/ω 2 based on the S(ω) calculated by summing over all angles. (See the text for details). The results are in agreement, not only the shape and position of the boson peak, but also the pressure dependence of the intensity found by the two methods is the same. The Q-dependence From equation 8.1.2 it follows that only the Debye Waller factor and the Q 2 -term are Q-depend. A Q independent quantity can therefore be found by dividing by these S inel,inc (Q, ω) ( exp − 〈u2 〉Q 2 3 ) = 1 Q 2 2M n(ω) g(ω). (8.1.4) ω In figure 8.3 we illustrate the above at 3 different Q values, with the Debye Waller factor taken from the measured elastic intensity. The three curves should overlap according to equation 8.1.4, but this is clearly not the case. However, the shape is the same and the curves can be brought to overlap by adjusting the Debye Waller factor (figure 8.4). Additionally we find that the pressure dependence of the intensity is the same at all Q-values. We therefore conclude that the measured frequency dependence shown in figure 8.2 gives a correct representation of the vibrational density of states. Other techniques for extracting DOS from equation 8.1.2 exploit the Q dependence. Rewriting equation 8.1.2 to ( ) Sinc (Q, ω)ω ln Q 2 = − 〈u2 〉Q 2 n(ω) 3 + ln(g(ω)) (8.1.5) it is seen that the left hand side can be obtained directly from the raw data. Fitting
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THÈSE présentée par Kristine NIS
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Acknowledgement First of all I woul
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Contents Abstract iii Acknowledgeme
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CONTENTS ix 9 Summarizing discussio
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2 Introduction fragility with other
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Résumé du chapitre 2 De manière
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8 Slow and fast dynamics glass. The
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10 Slow and fast dynamics energy in
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12 Slow and fast dynamics increasin
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14 Slow and fast dynamics (linear)
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16 Slow and fast dynamics The dynam
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18 Slow and fast dynamics T > T 2
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20 Slow and fast dynamics as theore
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22 Slow and fast dynamics and sugge
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24 Slow and fast dynamics The boson
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26 Slow and fast dynamics modulus 3
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Chapter 3 What we learn from pressu
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3.2. Empirical scaling law and some
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3.3. Correlations with fragility 37
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3.4. Temperature dependences 39 mea
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3.5. Summary 41 assumption that the
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Chapter 4 Experimental techniques a
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4.2. Dielectric spectroscopy 47 4.1
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4.2. Dielectric spectroscopy 49 fie
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4.3. Inelastic Scattering Experimen
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Résumé du chapitre 5 La relaxatio
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72 Alpha Relaxation the dielectric
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74 Alpha Relaxation Measuring the c
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76 Alpha Relaxation 4 2 0 This work
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78 Alpha Relaxation for the 1 s iso
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80 Alpha Relaxation log10(τ α ) 2
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82 Alpha Relaxation 2 1 0 216.4K th
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84 Alpha Relaxation function has a
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86 Alpha Relaxation 2.5 2.5 2 2 log
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88 Alpha Relaxation 0.4 log 10 Im(
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90 Alpha Relaxation keeping the rel
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92 Alpha Relaxation correlation bet
Page 105 and 106: Chapter 6 High Q collective modes I
Page 107 and 108: 6.1. Inelastic X-ray scattering 97
Page 109 and 110: 6.2. Sound speed and attenuation 99
Page 115 and 116: 6.3. Nonergodicity factor 105 where
Page 117 and 118: 6.3. Nonergodicity factor 107 1 0.9
Page 119 and 120: 6.3. Nonergodicity factor 109 high
Page 121 and 122: 6.4. Nonergodicity factor and fragi
Page 127 and 128: 6.5. Summary 117 d log (e(ρ))/ d l
Page 129: Résumé du chapitre 7 Dans ce chap
Page 132 and 133: 122 Mean squared displacement colle
Page 134 and 135: 124 Mean squared displacement 1.5 I
Page 136 and 137: 126 Mean squared displacement 1.4 1
Page 138 and 139: 128 Mean squared displacement a cen
Page 140 and 141: 130 Mean squared displacement cumen
Page 142 and 143: 132 Mean squared displacement / a
Page 144 and 145: 134 Mean squared displacement I P i
Page 146 and 147: 136 Mean squared displacement as be
Page 149: Résumé du chapitre 8 Le pic de bo
Page 152 and 153: 142 Boson Peak The FEC-model sugges
Page 154 and 155: 144 Boson Peak the Qout Q in factor
Page 158 and 159: 148 Boson Peak 1.5 x 10−3 ω [meV
Page 160 and 161: 150 Boson Peak comparison of the ev
Page 162 and 163: 152 Boson Peak In figure 8.6 we sho
Page 164 and 165: 154 Boson Peak Predictions/explanat
Page 166 and 167: 156 Boson Peak meaning that the Deb
Page 168 and 169: 158 Boson Peak g(ω)/ω 2 [arb. uni
Page 170 and 171: 160 Boson Peak are equivalent becau
Page 172 and 173: 162 Boson Peak m P /m ρ 2 1.8 1.6
Page 174 and 175: 164 Boson Peak S(ω) [arb.unit] S(
Page 176 and 177: 166 Boson Peak 3 2.5 S(ω) [arb. un
Page 178 and 179: 168 Boson Peak perature effect on t
Page 181 and 182: Chapter 9 Summarizing discussion Wh
Page 183 and 184: 173 temperature in the harmonic app
Page 185: Chapter 10 Perspectives In this wor
Page 188 and 189: 178 BIBLIOGRAPHY Angell, C. A. [199
Page 190 and 191: 180 BIBLIOGRAPHY Casalini, R. and R
Page 192 and 193: 182 BIBLIOGRAPHY Farago, B., Arbe,
Page 194 and 195: 184 BIBLIOGRAPHY Inamura, Y., Arai,
Page 196 and 197: 186 BIBLIOGRAPHY Monaco, A., Chumak
Page 198 and 199: 188 BIBLIOGRAPHY Patkowski, A., Gap
Page 200 and 201: 190 BIBLIOGRAPHY Sekula, M., Pawlus
Page 203 and 204: Appendix A Details on the samples T
Page 205 and 206: A.1. Cumene 195 the value found fro
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A.2. PIB 197 The temperature depend
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A.4. DBP 199 peak differently at di
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A.6. Other samples 201 NH 2 tempera
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Appendix B Data compilations B.1 Da
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B.1. Data compilation 205 DHIQ = De
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B.1. Data compilation 207 Triphenyl
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Appendix C Dielectric setup Figure
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Abstract The degree of departure fr
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