Preface ........................................................ ix
1 Scattering experiment and structure functions; particles
and the correlation function of small-angle scattering ..... 1
1.1 Elastic scattering of a plane wave by a thin sample ........ 3
1.1.1 Guinier approximation and Kaya's scattering
patterns ............................................ 7
1.1.2 Scattering intensity in terms of structure
functions .......................................... 14
1.1.3 Particle description via real-space structure
functions .......................................... 19
1.2 SAS structure functions and scattering intensity .......... 22
1.2.1 Scattering pattern, SAS correlation function and
chord length distribution density (CLDD) ........... 22
1.2.2 Indication of homogeneous particles by i(rA) ....... 26
1.3 Chord length distributions and SAS ........................ 30
1.3.1 Sample density, particle models and structure
functions .......................................... 32
1.4 SAS structure functions for a fixed order range L ......... 34
1.4.1 Correlation function in terms of the intensity
IL(h, L) ........................................... 38
1.4.2 Extension to the realistic experiment I(s), s <
smax ............................................... 39
1.5 Aspects of data evaluation for a specific L ............... 44
1.5.1 The invariant of the smoothed scattering pattern
IL ................................................. 53
1.5.2 How can a suitable order range L for L-smoothing
be selected from an experimental scattering
pattern? ........................................... 55
2 Chord length distribution densities of selected
elementary geometric figures .............................. 59
2.1 The cone case-an instructive example ...................... 60
2.1.1 Geometry of the cone case .......................... 61
2.1.2 Flat, balanced, well-balanced and steep cones ...... 66
2.1.3 Summarizing remarks about the CLDD of the cone ..... 68
2.2 Establishing and representing CLDDs ....................... 69
2.2.1 Mathematica programs for determining CLDDs? ........ 69
2.3 Parallelepiped and limiting cases ......................... 71
2.3.1 The unit cube ...................................... 73
2.4 Right circular cylinder ................................... 73
2.5 Ellipsoid and limiting cases .............................. 74
2.6 Regular tetrahedron (unit length case α = 1) .............. 78
2.7 Hemisphere and hemisphere shell ........................... 80
2.7.1 Mean CLDD and size distribution of hemisphere
shells ............................................. 81
2.8 The Large Giza Pyramid as a homogeneous body .............. 81
2.8.1 Approach for determining γ(r) and Aμ(r) ............ 82
2.8.2 Analytic results for small chords r ................ 83
2.9 Rhombic prism Y based on the plane rhombus X .............. 87
2.10 Scattering pattern I(h) and CLDD A(r) of a lens ........... 88
3 Chord length distributions of infinitely long cylinders ... 95
3.1 The infinitely long cylinder case ......................... 96
3.2 Transformation 1: From the right section of a cylinder
to a spatial cylinder ..................................... 97
3.2.1 Pentagonal and hexagonal rods ...................... 98
3.2.2 Triangle/triangular rod and rectangle/rectangular
rod ............................................... 100
3.2.3 Ellipse/elliptic rod and the elliptic needle ...... 100
3.2.4 Semicircular rod of radius R ...................... 102
3.2.5 Wedge cases and triangular/rectangular rods ....... 102
3.2.6 Infinitely long hollow cylinder ................... 102
3.3 Recognition analysis of rods with oval right section
from the SAS correlation function ........................ 104
3.3.1 Behavior of the cylinder CF for r → ∞ ............. 104
3.4 Transformation 2: From spatial cylinder С to the base X
of the cylinder .......................................... 107
3.5 Specific particle parameters in terms of chord length
moments: The case of dilated cylinders ................... 109
3.6 Cylinders of arbitrary height H with oval RS ............. 110
3.7 CLDDs of particle ensembles with size distribution ....... 114
4 Particle-to-particle interference - a useful tool ........ 115
4.1 Particle packing is characterized by the pair
correlation function g(r) ................................ 116
4.1.1 Explanation of the function g(r) and Ripley's К
function .......................................... 116
4.1.2 Different working functions and denotations in
different fields .................................. 118
4.2 Quasi-diluted and non-touching particles ................. 119
4.3 Correlation function and scattering pattern of two
infinitely long parallel cylinders ....................... 127
4.4 Fundamental connection between γ(r), с and g(r) .......... 132
4.5 Cylinder arrays and packages of parallel infinitely
long circular cylinders .................................. 148
4.6 Connections between SAS and WAS .......................... 155
4.6.1 The function FREQ(rk) describes all distances rk .. 155
4.6.2 Scattering pattern of an aggregate of N spheres
(AN) .............................................. 158
4.7 Chord length distributions: An alternative approach to
the pair correlation function ............................ 162
5 Scattering patterns and structure functions of Boolean
models ................................................... 169
5.1 Short-order range approach for orderless systems ......... 170
5.2 The Boolean model for convex grains - the set Ξ .......... 171
5.2.1 Connections between the functions γ(r) and
γ0(r) for arbitrary grains of density N = n ....... 172
5.2.2 The chord length distributions of both phases ..... 174
5.2.3 Moments of the CLDD for both phases of the BIII .... 176
5.2.4 The second moments of φ(l) and ƒ(m) fix с;
0 ≤ с < 1 ......................................... 177
5.2.5 Interrelated CLDD moments and scattering
patterns .......................................... 177
5.3 Inserting spherical grains of constant diameter .......... 179
5.4 Size distribution of spherical grains .................... 184
5.5 Chord length distributions of the Poisson slice model .... 188
5.6 Practical relevance of Boolean models .................... 192
6 The "Dead Leaves" model .................................. 193
6.1 Structure functions and scattering pattern of a PC ....... 195
6.2 The uncovered "Dead Leaves" model ........................ 205
7 Tessellations, fragment particles and puzzles ............ 207
7.1 Tessellations: original state and destroyed state ........ 209
7.2 Puzzle particles resulting from DLm tessellations ........ 210
7.3 Punch-matrix/particle puzzles ............................ 214
7.4 Analysis of nearly arbitrary fragment particles via
their CLDD ............................................... 220
7.5 Predicting the fitting ability of fragments from SAS ..... 227
7.6 Porous materials as "drifted apart tessellations" ........ 230
8 Volume fraction of random two-phase samples for a fixed
order range L from γ(r, L) ............................... 237
8.1 The linear simulation model .............................. 239
8.1.1 LSM for an amorphous state of an AlDyNi alloy ..... 247
8.1.2 LSM analysis of a VYCOR, glass of 33% porosity .... 249
8.1.3 Concluding remarks on the LSM approach ............ 250
8.2 Analysis of porous materials via ν-chords ................ 251
8.2.1 Pore analysis of a silica aerogel from SAS data ... 253
8.2.2 Macropore analysis of a controlled porous glass ... 255
8.3 The volume fraction depends on the order range L ......... 257
8.4 The Synecek approach for ensembles of spheres ............ 259
8.5 Volume fraction investigation of Boolean models .......... 261
8.6 About the realistic porosity of porous materials ......... 262
9 Interrelations between the moments of the chord length
distributions of random two-phase systems ................ 269
9.1 Single particle case and particle ensembles .............. 270
9.2 Interrelations between CLD moments of random particle
ensembles ................................................ 272
9.2.1 Connection between the three functions g, φ and
ƒ ................................................. 279
9.2.2 The moments Mi, li, mi in terms of Q(t), p(t),
q(t) .............................................. 280
9.2.3 Analysis of the second moment M2 = -Q"(0) ......... 280
9.3 CLD concept and data evaluation: Some conclusions ........ 284
9.3.1 Taylor series of Q(t) in terms of the moments Mn
of the function g(r) .............................. 286
9.3.2 Sampling theorem, the number of independent SAS
parameters, CLD moments and volume fraction ....... 288
10 Exercises on problems of particle characterization:
examples ................................................. 289
10.1 The phase difference in a point of observation P ......... 290
10.2 Scattering pattern, CF and CLDD of single particles ...... 292
10.2.1 Determination of particle size distributions
for a fixed known particle shape .................. 292
10.2.2 About the P1 plot of a scattering pattern ......... 293
10.2.3 Comparing single particle correlation functions ... 294
10.2.4 Scattering pattern of a hemisphere of radius R .... 295
10.2.5 The "butterfly cylinder" and its scattering
pattern ........................................... 296
10.2.6 Scattering equivalence of (widely separated)
particles ......................................... 299
10.2.7 About the significance of the SAS correlation
function .......................................... 299
10.2.8 The mean chord length of an elliptic needle ....... 301
10.3 Structure functions parameters of special models ......... 303
10.3.1 The first zero of the SAS CF ...................... 303
10.3.2 Different models, different scattering patterns ... 304
10.3.3 Boolean model contra hard single particles and
quasidiluted particle ensembles ................... 306
10.3.4 A special relation for detecting the с of a Bm .... 307
10.3.5 Properties of the SAS CF of a Bm .................. 308
10.3.6 DLm from Poisson polyhedral grains ................ 309
10.4 Moments of g(r), integral parameters and с ............... 310
10.4.1 Properties of the moment M2 of g(r) ............... 310
10.4.2 Tests of properties of M2 special model
parameters ........................................ 311
10.4.3 Volume fraction and integral parameters ........... 312
10.4.4 Application of Eq. (9.16) for a ceramic
micropowder ....................................... 313
References .................................................... 315
Index ......................................................... 333
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