Neutron Scattering

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S (Q, t) = Sa (Q, t) S ß (Q, t) / S (Q) S (Q) (15 .7) The approximation behind Eq .[15 .7] is called Vinyard approximation and approximates the proper pair correlation fonction by its self counter part . The self correlation fonction for a diffusive process relates directly to the mean square displacement . 2 Sse(f (Q,t) = exp 1-2 (r2 (t)" (15 .8) where (r 2(t» is the mean square displacement of the flowing particle . According to Eq .[15 .1] this should be described by a stretched exponential with the consequence (r2 (t)> =D tß and =QZ1aD-, lp (15 .9) the combination of Eq .[15 .8] and [15 .9] invokes sublinear diffusion of the polymer segments as the underlying reason for the stretched exponential behaviour . Its signature is a power law dependence ofthe Kohlrausch-William-Watts relaxation times T.,,, with an exponent 2/ß . 15.2 .2 Experimental results 15.2 .2 .1 Self correlation function We commence with the secondaiy relaxation taking polyisobutylene as an example. Fig . 15 .4 presents the relaxation map of PIB . The solid line corresponds to the dielectric ß-relaxation, the dashed line named S represerrts NMR results interpreted as a methyl group rotation. y and y' are theoretically predicted relaxation mechanisms . In the dynamic regime, where the (Xrelaxation is too slow to contribute, neutron backscattering has been employed, in order to unravel details of the ß-process in this polymer. Fig .15 .5 presents as an example a spectrom taken at T = 270K and Q = 1 .7Â-1 . The spectrum is characterized by a narrow peak, which nearly coincides with the instrumental resolution function (dashed line) and a broad foot revealing the relaxational behaviour. Such a spectral shape is typical for broad distributions of relaxation times, where only a part of it is resolved in the spectrum . 15-7
10 an 0 2 0 Figure 15.4 : Relaxation map for PIB . Y marks the a-trace, " relates to dielectric relaxation experiments on the fi-relaxation, " results from QENS-experiments . 3 4 5 6 7 8 1000 / T(K) 210' â Figure 15 .5 : Backscattering spectrum from PIB at 270K at a Q-value Q = 1 .7A. The dashed line gives the resolution function, while the solid line displays the fit with the model (sec text) . 0 -10 -5 0 5 10 E( ~teV) w 0 .9 0 .8 N a 0 .7 0 .6 Figure 15 .6 : EISF for PIB (1 : 250K, " : 270K) . Solid lines : EISF for methylgroup rotation, dashed lines : fit result for a jump distance d = 2.7A. 0 .5 0 0.5 1 1 .5 2 Q (À -1 ) 15-8
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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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Page 281 and 282: s - ' f N .mwn .I )oo K L .~um..un
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 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
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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 316 and 317: For the intramolecular and intermol
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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 369 and 370: Here, J;j denotes the exchange cons
Page 371 and 372: Fig. 16.8 : Contour plot of the mag
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Page 376: 17 Translation and Rotation M . Pra
Page 379 and 380: 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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