Neutron Scattering

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and which lead to the scattering cross section dE _ 1 N d~( Q ) Y~b V DaD(rn)+bHaH- (rn)+b0(1-aD(rn-)_ - 6H-(rn)))e'Q'~ > s 1 which after some calculation leads to the form z according to. The partial structure factors are given as : (bD - bo ~TD(rr)+(bx - bo)6x(Ei)+ b, o, e iQr; 2 > (14.26) -0- and, fmally, to a sum of partial structure factors weighted with conesponding contrast factors d~(Q)=V [4bDS DD +24bD4b .S DH +4bHS 1 (14.27) 5 D(H) D(H) D D SDD = (PnZ ZP(Q)+ (1) Zn2 Z Z W(Q) (14.28) SHH =(1- ~D )nZZP(Q)+ ~D Zn2ZZW(Q) (14.29) H S De = (D(1 - (D~2Z2 W(Q) (14.30) H D and fmally one delivers the following scattering law dl(Q)= V ~bD-bH) Z q)(1-d)nZ Z P(Q)+(br oiy -b [ o)Z nZ Z P(Q)+n 2 Z Z W(Q) ] } (14.31) 5 For this systems the Babinet principal is not valid as it contains polymers and solvent molecules . If one matches the scattering length of the solvents and the averaged one of the polymers (broiy =d)bD +( 1- ~)bH) according to broiy =b o (,,zero" contrast), the second term in Eq .(14 .31) does not contribute to the scattering and, consequently, one again determines the form factor of a single chain according to (Q) = c -1)(1-1) VP(Q)K (14.32) ddY-- 144 1
with c the volume fraction of polymers in solution and the contrast factor K . This again is a demonstration of the possibilities of contrast variation with neutrons allowing to measure the form factor P(Q) of a polymer chain in solution . 14 .5 <strong>Scattering</strong> from a Block Copolymer Figure 14 .6 : Presentation ofa Diblock Copolymer We now derive the scattering law of a linear polymer, consisting of the two blocks A" and B" of different polymers and which are symmetric with respect to the number of the monomers z = 2z D = 2z H . The scattering cross section is given as _dl __ 1 Ab2SDD dn Vs (14 .33) with the relationship SDD = SHH = -SDH because of assuming an incompressible melt. The various partial structure factors are given as : r 1 2 7Z ,2 z SDD =n PDD +n2~2 WDD (14 .34) (2, ) 2 z \2 2 z II DH PDH +n -nC2/ ~2j ,WDH (14 .35) SI, =nCzJ 2 2 2 P,, +n2(ZJ 2 WH (14 .36) The symbolic representation of the structure factor of diblock copolymers has the form 144 2
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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
Page 264 and 265: The knowledge of the overall occupa
Page 266 and 267: (even anharmonic) effects. This com
Page 268 and 269: Fig . 4 . Coordination ofthe D20 mo
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Page 272 and 273: (c) and (d) calculated densities at
Page 274: 13 Inelastic Neutron Scattering Pho
Page 277 and 278: 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
Page 285 and 286: y absorption . 13 .1 .3 Magnetic in
Page 287 and 288: Figure 7 The plane in reciprocal sp
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Page 291 and 292: dynamics of ionic compounds with th
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
Page 316 and 317: For the intramolecular and intermol
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Page 320 and 321: scaling behavior of the susceptibil
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Page 325 and 326: constraints due to the mutual inter
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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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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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