Ph.D. thesis (pdf) - dirac

More documents

Recommendations

Info

$Guide to Imfufa LaTeX - dirac$

50 Experimental techniques and observables A more general problem with dielectric spectroscopy is the so-called local field problem, which we shall shortly discuss in the following. The problem has minor importance for the present work, we therefore refer to textbooks for more details (e.g. [Böttcher, 1973]). The specific implications of the local field problem on the interpretation of dielectric relaxation in viscous liquids is moreover discussed in [Niss and Jakobsen, 2003; Fatuzzo and Mason, 1967; Daz-Calleja et al., 1993]. The microscopic response corresponding to the dielectric susceptibility is the polarizability, α, given by 3 p = αE l , (4.2.4) where p is polarization of a molecule and E l is the field strength applied over the dipole. The macroscopic polarization is given by the sum of the microscopic polarizations, ∑ p = P. However, the relation between the macroscopic field in the sample and the field “seen” by a molecule is more complicated. This is because the field of the molecule itself is part of the macroscopic field while it is not part of the local field. The most common description of the local field is the Lorentz field, which leads to the Clausius-Mossotti approximation. The Lorentz field is the field in a spherical imaginary vacancy in the liquid. By imaginary is meant that the polarization of the rest of the liquid is calculated as if the dipole was there. This description includes the polarization of the surroundings due to the dipole, and the fact that this polarization gives rise to a field acting back on the dipole. Only the field from the dipole itself is excluded from the calculation of the local field. The Lorentz field is given by E l = ( 1 + χ ) E m = ǫ + 2 3 3 E m. (4.2.5) and inserting this in equation 4.2.4, combined with ∑ p = P and equation 4.2.3 gives the Clausius-Mossotti relation χ χ + 3 = ǫ − 1 ǫ + 2 = Nα 3ǫ 0 . (4.2.6) In deriving the Lorentz field it is assumed that the macroscopic field and the polarization of each molecule are always parallel. A general field, without this assumption is given by Onsager [1936]. However the Onsager field is not adequate when the response is frequency dependent, because the derivation assumes that the field and 3 The polarizability should in general be a tensor quantity as field and polarization are not necessarily parallel. However, the average polarization 〈p〉 will in isotropic material be parallel to the average local field 〈E l 〉, and this leads to a meaningful definition of α as a scalar.
4.3. Inelastic Scattering Experiments 51 the polarization are in phase. Fatuzzo and Mason [1967] propose a solution to this situation. The most important problem with the conversion from microscopic polarizability to macroscopic susceptibility can already be anticipated from equation 4.2.6, namely that the two are not proportional. This has the consequence that the frequency dependence of characteristic time of α(ω) will not be the same as the characteristic time of χ(ω). Moreover, this difference will systematically depend on the strength of the dielectric relaxation, meaning that it will have a different consequence depending on the size of the molecular dipole moment. However, the differences are still smaller than the type of differences which are always found depending on which experimental probe one uses to study the liquid [Daz-Calleja et al., 1993; Niss and Jakobsen, 2003]. In this work we shall not attempt to extract microscopic information from the dielectric relaxation, rather we follow the convention of taking the macroscopic dielectric relaxation time as a signature of the structural alpha relaxation. 4.3 Inelastic Scattering Experiments Neutron and X-ray scattering have several common features and we therefore introduce them at the same time. We start from neutron scattering and generalize to X-ray scattering by comments at relevant places. 4 We start by describing the principle of an inelastic scattering experiment and continue by discussing some practical limitations, regarding the energy and Q-domain that can be accessed by different probes. We subsequently give some general results which we shall later use in the analysis of our data. More technical information on the experimental methods is given in the relevant chapters. 4.3.1 The basic principle The basic principle inelastic scattering experiment is to let an incoming beam of probe with a well defined energy and wave vector hit a sample and to measure the wave vector and energy of the scattered beam. The difference in momentum and energy between the incoming and the out coming beam has been lost to (or gained from) the sample. This exchange of energy and momentum will in the limit where the interaction between the probe and the sample is weak only depend on the properties of the sample (linear response). The property which is measured is called the cross 4 Standard references for neutron scattering are [Lovesey, 1984; Squires, 1978], while inelastic X-ray scattering, is relatively new technique which is not yet treated in text books.
Page 1:
THÈSE présentée par Kristine NIS
Page 5 and 6:
Acknowledgement First of all I woul
Page 7 and 8:
Contents Abstract iii Acknowledgeme
Page 9: CONTENTS ix 9 Summarizing discussio
Page 12 and 13: 2 Introduction fragility with other
Page 15: Résumé du chapitre 2 De manière
Page 18 and 19: 8 Slow and fast dynamics glass. The
Page 20 and 21: 10 Slow and fast dynamics energy in
Page 22 and 23: 12 Slow and fast dynamics increasin
Page 24 and 25: 14 Slow and fast dynamics (linear)
Page 26 and 27: 16 Slow and fast dynamics The dynam
Page 28 and 29: 18 Slow and fast dynamics T > T 2
Page 30 and 31: 20 Slow and fast dynamics as theore
Page 32 and 33: 22 Slow and fast dynamics and sugge
Page 34 and 35: 24 Slow and fast dynamics The boson
Page 36 and 37: 26 Slow and fast dynamics modulus 3
Page 39 and 40: Chapter 3 What we learn from pressu
Page 41 and 42: 3.2. Empirical scaling law and some
Page 47 and 48: 3.3. Correlations with fragility 37
Page 49 and 50: 3.4. Temperature dependences 39 mea
Page 51: 3.5. Summary 41 assumption that the
Page 55 and 56: Chapter 4 Experimental techniques a
Page 57 and 58: 4.2. Dielectric spectroscopy 47 4.1
Page 59: 4.2. Dielectric spectroscopy 49 fie
Page 63 and 64: 4.3. Inelastic Scattering Experimen
Page 79: Résumé du chapitre 5 La relaxatio
Page 82 and 83: 72 Alpha Relaxation the dielectric
Page 84 and 85: 74 Alpha Relaxation Measuring the c
Page 86 and 87: 76 Alpha Relaxation 4 2 0 This work
Page 88 and 89: 78 Alpha Relaxation for the 1 s iso
Page 90 and 91: 80 Alpha Relaxation log10(τ α ) 2
Page 92 and 93: 82 Alpha Relaxation 2 1 0 216.4K th
Page 94 and 95: 84 Alpha Relaxation function has a
Page 96 and 97: 86 Alpha Relaxation 2.5 2.5 2 2 log
Page 98 and 99: 88 Alpha Relaxation 0.4 log 10 Im(
Page 100 and 101: 90 Alpha Relaxation keeping the rel
Page 102 and 103: 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 111 and 112:
6.2. Sound speed and attenuation 10
Page 113 and 114:
6.2. Sound speed and attenuation 10
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 123 and 124:
Page 125 and 126:
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 156 and 157:
146 Boson Peak We are mainly intere
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
Page 207 and 208:
A.2. PIB 197 The temperature depend
Page 209 and 210:
A.4. DBP 199 peak differently at di
Page 211 and 212:
A.6. Other samples 201 NH 2 tempera
Page 213 and 214:
Appendix B Data compilations B.1 Da
Page 215 and 216:
B.1. Data compilation 205 DHIQ = De
Page 217 and 218:
B.1. Data compilation 207 Triphenyl
Page 219 and 220:
Appendix C Dielectric setup Figure
Page 222:
Abstract The degree of departure fr
show all

Ph.D. thesis (pdf) - dirac

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?