Neutron Scattering

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In the case of single crystal diffraction a rotation of the crystal and therefore also of the corresponding reciprocal lattice (which is rigidly attached to the crystal) is often used to set the diffiaction conditions for the measurement of intensities I(H) . If IHI > 2/a, (then dhkr < ;U2) the reflection hkl cannot be observed. This condition deines the so called limiting sphere, with center at 000 and radius 2/; : only the points of the reciprocal lattice inside the limiting sphere can be rotated into diffraction positions . Vice versa if ~, > 2dma,, where dma, is the largest interplanar spacing of the unit cell, then the diameter of the Ewald sphere is smaller then IHI ,;. Under these conditions no node of the reciprocal lattice can intercept the Ewald sphere . That is the reason why diffraction of visible light (wavelength 5000 Â) can never be obtained from crystals . T; determines the amount of information available from a diffraction experiment . In ideal conditions ;,,; should be short enough to measure all points ofthe reciprocal lattice wich signifcant diffraction intensities . For a real crystal of limited perfection and size the infmitely sharp diffraction peaks (delta functions) are to be replaced by broadened line shapes . One reason can be the local variation of the orientation of the crystal lattice (mosaic spread) implying some angular splitting of the vector H. A spread of interplanar spacings Ad/d, which may be caused by some inhomogeneities in the chemical composition of the Sample, gives rise to a variation of its magnitude IHI . The ideal diffraction geometiy on the other hand is also to be modified . In a real experiment the primary beam has a finite divergence and wavelength spread. The detector aperture is also finite . A gain of intensity, which can be accomplished by increasing the angular divergence and wavelengths bandwidth, has to be paid for by some worsening of the resolution fanction and hence by a limitation of the ability to separate different Bragg reflections . All of these influences can be studied by the Ewald construction . The influence of a horizontal beam divergence on the experimental conditions for a measurement of Braggintensities of a single crystal is illustrated in Fig . 3 where strictly monochiomatised radiation (only one wavelength ;, with AA,/;, = 0) is assumed. A so-called ù)-scan, where the crystal is rotated around the Sample axis perpendicular to the diffraction plane, may be used for a reliable collection of integrated intensities in adapting the detector aperture A20 as a function of the scattering angle 20 . It is obvious that larger A20-values will give rise to a higher background and may lead to difficulties for the Separation of neighboured reflections wich similar H-vectors in the reciprocal lattice . It is shown by this example that a larger beam divergence with an increase in intensity can restrict the resolution conditions . 7- 5
Bragg-intensities of single crystals are recorded in general by (0/n .20-scans (0
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Forschungszentrum Jülich in der He
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Forschungszentrum Jülich GmbH Inst
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Contents . . 1 Neutron Sources H .
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1 . Neutron Sources Harald Conrad 1
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d) f, = v a, , q - ti = P a, - 0 .3
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Period Example Flux (D [1013 1950-6
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In stage 1 the primary proton knock
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get as heat, the rest is transporte
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down power and stronger absorption
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Appendix Neutron Sources - an overv
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led remotely. It is rauch more conv
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2 . Properties of the neutron, elem
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In the 60's the ferst high flux rea
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Hot neutrons in reactors are obtain
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elated values for the FRJ-2 reactor
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2 .5 Scattering amplitude and cross
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(A+k2 ) yr = u(Z:) yr V( u~r)= z "
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_du _ mn \2 A2-~2z r2~ I(k' IVI k)I
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E ô ô x z w 0 z _w U N Figure 2 .
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scattering is observed with the ave
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For a spherical electron distributi
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A more thorough introduction to neu
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3 . Elastie Seattering from Many-Bo
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Q = Q = k2 + k2 - 2kk' cos2B (3 .3)
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3 .2 Fundamental Scattering Theory
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The meaning of (3 .l4) is immediate
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1 . e. in the second approximation,
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The saure problem will bc dealt wit
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ALQ) = Y-bfe'Q'-rB(r-(n .a+m .b+p»
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du LQ) = ALQJ2 = e'Q .A' . * -i e-
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Tab . 3.1 : The scattering lengths
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We note immediately that we should
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These magnetic structures can be un
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M1LQ)=Q-MQ)xQ M(Q)= IM (r)e i Q " d
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Orbital magnetic spin angular ,,- m
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aQ . r 3 àQ . N . QR . aQ .t k _ML
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Schweika Analysis Polarization W .
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ducing a complex component and were
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ir v s ftft 3956 2 s 2 0.81[X] 80 c
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Some polarizers using Bragg reflect
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Different to the coherent nuclear s
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where QBG denotes the background .
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4 .4 Applications We now consider e
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Fig. 4 .3 .4 probably represents th
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. occur only in the non-spin flip c
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The results for the magnetization d
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a large solid angle with polarizers
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scattering to spin-flip scattering
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An experimental verification of thi
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Correlation Functions Measured by S
Page 111 and 112: In real liquids however usually an
Page 113 and 114: The term in big parentheses of equa
Page 115 and 116: 4 0 -5 0 5 10 15 20 ho) [meV] Figur
Page 117 and 118: Figure 5 .5 : The pair correlation
Page 119 and 120: Figure 5 .6 : Partial differential
Page 121 and 122: Then equation (5 .21) can be conver
Page 123 and 124: The route from expression (5 .27) t
Page 125 and 126: In addition it is often useful to d
Page 127 and 128: 2 . The pair correlation function h
Page 129 and 130: G(r, t) G(r, t) i(Q, t) r Q i(Q, t)
Page 131 and 132: Insertion of this result into (5 .6
Page 133 and 134: With this example we can also demon
Page 136 and 137: 6. Continuum description : Grazing
Page 138 and 139: Starting point is the exact atomic
Page 140 and 141: For specular reflectivity measureme
Page 142 and 143: v v z l 2r V(z) = 2 -2erfC~ with er
Page 144 and 145: F{dV(z)/dz ) to calculate the refle
Page 146 and 147: Equation (6 .13) also shows that th
Page 148 and 149: ut is completely reflected . The cr
Page 150 and 151: 0.4° . The full dynamical theory c
Page 152 and 153: 1) If no magnetic induction B (whic
Page 154: Diffractometer Gernot Heger
Page 157 and 158: select a special wavelengths band (
Page 159: " The direction of the reciprocal l
Page 163 and 164: monochromator, on the mosaic spread
Page 165 and 166: complex sample environments, such a
Page 167 and 168: the diffraction plane) two further
Page 170: Small-angle Scattering and Reflecto
Page 173 and 174: L= (1) ( k ) a e -(k/k T ) 2 Ak (8
Page 175 and 176: possible neutron wave lengths betwe
Page 177 and 178: figure and with neutrons of about 1
Page 179 and 180: 1 .0 Danvinkurve 0 .8 0 .6 0 .4 0 .
Page 181 and 182: tails. This reflection curve leads
Page 183 and 184: For the Nickel film one gets a thic
Page 185 and 186: I The rotor of a velocity selector
Page 187 and 188: multiplier are arranged in a quadra
Page 190 and 191: 9 . Crystal spectrometer : triple-a
Page 192 and 193: machine which will be followed by a
Page 194 and 195: 10 .3 Beaur shaping Due to the fact
Page 196 and 197: G.k=zGz . (11) This case is fulfill
Page 198 and 199: tribution of bisecting planes (ntos
Page 200 and 201: directions which is exploited for t
Page 202 and 203: Fig . 14) Dispersion surfaces for B
Page 204 and 205: whereby E and v,, denote energy and
Page 206: [1] B . N. Brockhouse, in Inelastic
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10 . Time-of-flight spectrometers M
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Tnloderator /K V7 2/m/s A/nm Aiw/eV
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10.2 .1 Interpretation of spectra A
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----~ Probe ' (Chopper 2) Zeit I Ch
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the similar intensity distribution
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is performed without Jacobian due t
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10.2 .6 Crystal monochromators The
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larger energy transfers . However i
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10.3 Inverted TOF-spectrometer In t
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constant offset to the TOF . To be
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sample size of cm and a detection p
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Contents 10 .1 Introduction . . . .
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11 . Neutron spin-echo spectrometer
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2 0 L Tr -0 L 2 ArDet Figure 11 .1
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initial velocity-reassemble at the
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distribution (current sheets) that
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esults, where il is an irrelevant c
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10000 - 8000 - co 6000 - C ô 4000-
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e0 oX n .5 1 .0 1 .5 so 0 .0 0 .0 1
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nel" coils) the required homogeneit
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Figure 11 .9 : S(Q, t)IS(Q) of a 2
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Contents 11 .1 Introduction . . . .
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12 . Structure determination G . He
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Important: Thc measured Bragg inten
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determination of accurate atomic pa
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The knowledge of the overall occupa
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(even anharmonic) effects. This com
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Fig . 4 . Coordination ofthe D20 mo
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fenoelectric phase of orthorhombic
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(c) and (d) calculated densities at
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13 Inelastic Neutron Scattering Pho
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Figurel Schematical drawing of a ge
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equilibrium position the atom is in
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s - ' f N .mwn .I )oo K L .~um..un
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esults from the quantum mechanics o
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y absorption . 13 .1 .3 Magnetic in
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Figure 7 The plane in reciprocal sp
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5 100s1 [OSl ] IOSSI [SSS y i 4 0 r
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dynamics of ionic compounds with th
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16 14 12 F.. . 10 ô 8 q 6 cr 4 G.
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21 [ooç 1 " - 19 18 17 16 "" 298 K
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n = v - cap[i(pga . - wt)] (13 .22)
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Y c 3 c Cc E Figure 16 Magnon dispe
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. . 2 H . Bilz and W . Kress, Phono
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14 . Soft Matter : Structure Dietma
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length density of a sample against
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2 P(Q) = 1 ~exp(-Ii - j'6 Q2 ) (14
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fraction. In the region of large Q
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with the partial structure factor S
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and which lead to the scattering cr
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For the intramolecular and intermol
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500 450 -400 U 0 ;, 350 42, 300 s~.
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scaling behavior of the susceptibil
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15 Polymer Dynamics Dieter Richter
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constraints due to the mutual inter
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(15 .3) T ß ( E)=T ô exp E k,T -
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This incoherent approximation does
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10 an 0 2 0 Figure 15.4 : Relaxatio
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In order to test Eq .[15 .9] the re
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Arrhenius temperature dependence wi
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1 3kBT ~ z 2 72 2 p2 a = f2 N2 P =W
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S (Q,t) = 12 fdu exp ~ -u_ (S2Rt) v
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We now use these data, in order to
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c 0 .8 0 .6 00 .055 x 0 .1 110 .14
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small but systematic deviations fro
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S(Q,t) is predicted . Such a platea
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only one single parameter, the tube
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confinement of the relaxation of la
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16 Magnetism Thomas Brückel
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the ground state of metallic magnet
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c~ ~ o 0 `,~i) g',, b o 0 Q! b o 0
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only "sec" this component and not t
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200 ¬ 400 500 600 700 E 3 QI J (h0
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Fie. 16.4: Scattering geometry for
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0 .8 ' D3 ° Stassis 3d spin - - 3d
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Here, J;j denotes the exchange cons
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Fig. 16.8 : Contour plot of the mag
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0.0001 0.001 0 .01 0.1 2 Fig. 16 .1
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17 Translation and Rotation M . Pra
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17.1 Introduction Atomic and molecu
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17 .2.2 Diffusion, microscopic appr
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0.6 H20 2 x2tÄ z ) Figure 17 .2 :
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The scattering function of a polycr
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e (2) 1 (j 1, 0 > - 0,1 >), respect
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leads to the eigenvalue problem ~q
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allow a good determination of the E
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m Eu W Figure 17 .7 : Eigenenergies
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-so o so Energy transfer (NeV) Ener
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Bibliography [1] T. Springer, Quasi
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18 . Texture in Materials and Earth
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Further important structure related
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The axes of the cartesian Kp coordi
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4.0 Experimental Texture Analysis T
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Fig. 18.14: Variation of reflection
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measurements can be performed in tr
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Apart from the global description o
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In the following, two experimental
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The neutron diffraction pole figure
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dependent pole densities, and (4) g
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Schriften des Forschungszentrums J
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Forschungszentrum Jülich in der He
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Neutron Scattering

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