Neutron Scattering

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Fig .15 .6 displays the elastic intensity observed for PIB as a function of Q . The data were corrected for multiple scattering and fitted with Eq .[15 .5] . This elastic incoherent structure factor (EISF) (see lecture Quasielastic <strong>Scattering</strong>) reveals a jump distance d = 2.7A . For comparison the solid lines display the prediction for methyl group rotation, which was invoked by NMR spectroscopy . Obviously the neutron data point into the direction ofa larger motional amplitude . The squares in Fig . 15 .4 display the neutron results for the /3 time scale . Within a factor of 2 they agree with the dielectric spectroscopy results . Since the underlying process has an amplitude of 2.7A and is also dielectrically active, it cannot be understood as due to a methylgroup rotation alone . A possible interpretation is a combined backbone and methyl motion which is also supported by simulation results . We now turn to the a-relaxation and ask, whether the sub linear diffusion argument is supported by quasielastic neutron scattering. Fig. 15 .7 displays Kohlrausch-William-Watts relaxation rates obtained for four different polymers, polyvinylether (PVE) at 340K, polyisobutylene (PIB) at 365K, polybutadiene (PB) at 280K and polyisoprene (PI) at 340K. PVE 40K u 0.2 0 .4 0.6 0 .8 1 pI 340K Figure 15 .7 : (zrWW)f for 4 different polymers as a function of Q. The solid lines display a Q' power law . 15-9
In order to test Eq .[15 .9] the relaxation rates have been exponentiated with the exponent /3, obtained from the stretching of the relaxation fonctions in these polymers in dielectric spectroscopy . According to Eq.[15 .9], z,6 should bc proportional to Q"' . The solid lines in Fig .15 .7 display this power law relation . As may be seen, in all cases within experimental error the experimental relaxation times again obtained by backscattering spectroscopy follow the predicted power law behaviour . Thus, the experimental evidence supports a sub linear diffusion process as underlying the a-relaxation . We remark that this result is in disagreement wich assertions that the stretchnd exponential relaxation fonction of the a-process originates from heterogeneous motional processes, where polymer segments in different parts of the sample would relax at different relaxation rates . 15 .2 .3 Pair correlation function The dynamic pair correlation fonction for polymer relaxation has been studied thoroughly on polybutadiene as a fonction of temperature and momentum transfer . Fig . 15 .8 gives a synopsis of these results . The dynamic data presented have been taken at the positions of the first and second peaks in the static structure factor of this polymer . As may be seen from the middle part of Fig .15 .8 the first peak of the static structure factor moves strongly with temperature . This peak originates from interchain correlations, where weak v . d. Waals interactions lead to thermal expansion. The second peak relates mainly to intrachain correlations as may be seen from the temperature independence of its position indicative for covalent bonds . The temperature dependent relaxation spectra were rescaled in their time dependence wich the characteristic time for viscosity relaxation z, (actually the time dependent monomeric friction coefficient was used, see next paragraph) . By this procedure the time correlation functions at the first peak assemble to a master culve, showing that the dynamics at the interchain distance follows the saine relaxational behaviour as the macroscopic flow . On the other hand as evidenced by the lower part of Fig. 15 .8 at the second structure peak, such a scaling does not reassemble the data points to a master curve . Obviously the dynamics at the second peak at higher Q follows different dynamics . 15- 10
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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
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In real liquids however usually an
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The term in big parentheses of equa
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4 0 -5 0 5 10 15 20 ho) [meV] Figur
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Figure 5 .5 : The pair correlation
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Figure 5 .6 : Partial differential
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Then equation (5 .21) can be conver
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The route from expression (5 .27) t
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In addition it is often useful to d
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2 . The pair correlation function h
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G(r, t) G(r, t) i(Q, t) r Q i(Q, t)
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Insertion of this result into (5 .6
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With this example we can also demon
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6. Continuum description : Grazing
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Starting point is the exact atomic
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For specular reflectivity measureme
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v v z l 2r V(z) = 2 -2erfC~ with er
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F{dV(z)/dz ) to calculate the refle
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Equation (6 .13) also shows that th
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ut is completely reflected . The cr
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0.4° . The full dynamical theory c
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1) If no magnetic induction B (whic
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Diffractometer Gernot Heger
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select a special wavelengths band (
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" The direction of the reciprocal l
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Bragg-intensities of single crystal
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monochromator, on the mosaic spread
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complex sample environments, such a
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the diffraction plane) two further
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Small-angle Scattering and Reflecto
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L= (1) ( k ) a e -(k/k T ) 2 Ak (8
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possible neutron wave lengths betwe
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figure and with neutrons of about 1
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1 .0 Danvinkurve 0 .8 0 .6 0 .4 0 .
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tails. This reflection curve leads
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For the Nickel film one gets a thic
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I The rotor of a velocity selector
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multiplier are arranged in a quadra
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9 . Crystal spectrometer : triple-a
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machine which will be followed by a
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10 .3 Beaur shaping Due to the fact
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G.k=zGz . (11) This case is fulfill
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tribution of bisecting planes (ntos
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directions which is exploited for t
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Fig . 14) Dispersion surfaces for B
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whereby E and v,, denote energy and
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[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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Page 283 and 284: esults from the quantum mechanics o
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Page 287 and 288: Figure 7 The plane in reciprocal sp
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Page 293 and 294: 16 14 12 F.. . 10 ô 8 q 6 cr 4 G.
Page 295 and 296: 21 [ooç 1 " - 19 18 17 16 "" 298 K
Page 297 and 298: n = v - cap[i(pga . - wt)] (13 .22)
Page 299 and 300: Y c 3 c Cc E Figure 16 Magnon dispe
Page 301 and 302: . . 2 H . Bilz and W . Kress, Phono
Page 304 and 305: 14 . Soft Matter : Structure Dietma
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Page 308 and 309: 2 P(Q) = 1 ~exp(-Ii - j'6 Q2 ) (14
Page 310 and 311: fraction. In the region of large Q
Page 312 and 313: with the partial structure factor S
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Page 329 and 330: This incoherent approximation does
Page 331: 10 an 0 2 0 Figure 15.4 : Relaxatio
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Page 337 and 338: 1 3kBT ~ z 2 72 2 p2 a = f2 N2 P =W
Page 339 and 340: S (Q,t) = 12 fdu exp ~ -u_ (S2Rt) v
Page 341 and 342: We now use these data, in order to
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Page 345 and 346: small but systematic deviations fro
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Page 354: 16 Magnetism Thomas Brückel
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Page 365 and 366: Fie. 16.4: Scattering geometry for
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Page 371 and 372: Fig. 16.8 : Contour plot of the mag
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Page 376: 17 Translation and Rotation M . Pra
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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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