Neutron Scattering

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decreased barrier increases the probability for quantum mechanical tunnelling of the proton and thus accelerates the diffusion process . Diffusion in the presence of traps Hydrogen traps in a metal lattice can be created by chemical impurities . In a lattice with traps the proton dynamics splits into a local motion around the trap and a diffusion between traps . A corresponding two state model [3] yields a scattering function consisting of 2 Lorentzians . At small Q diffusion, at large Q jumps around the trap dominate scattering . Since jumps around a trap are qualitatively very similar to rotations of a molecule around ils center of mass the saure theory can be applied to get classical jump rates and local librations . 17.2 .5 Librations Usually the 3-dimensional potential of a proton at equilibrium site is expanded harmonically and completed by anharmonic terras consistent with symmetry requirements . It determines ils eigenenergies . Vice versa the librations allow to deduce the potential . This information refines the potential beyond the pure knowledge of the barrier height obtained from QNS . 17.2 .6 Translational tunnelling At low temperatures the proton localises in a pocket of the potential . If the barrier between such pockets is weak, the proton wavefunctions of neighbouring pockets overlap and the degenerate librational states split into tunnelling substates . This translational tunnelling is formally almost equivalent to the rotational tunnelling to be described below . NbOo .oo,Ho .ooi (Fig . 17 .5) represents an especially clear case . The oxygen defect distorts the lattice locally and makes exactly two hydrogen sites - almost - equivalent . Almost : the presence of the particle itself in one minimum introduces an asymmetry . Thus one has to calculate the scattering function of an atom in an asymmetric double minimum [3] . Wave functions ~P are set up from basis functions I01 > and 110 > which describe the two possible proton sites . The two configurations can transform into each other by tunnelling due to a finite tunnel matrix element t. The corresponding Schrödinger equation H~P = EW in matrix form leads to the eigenvalues problem (symmetric case assumed for simplicity!) t \ t The characteristic polynom yields eigenvalues 1~ 1,2 = ±t . They are connected with the totally symmetric eigenvectorr e( l ) = 1~2 (I 1, 0 > + 10,1 >) and the antisymmetric eigenvector 17- 9
e (2) 1 (j 1, 0 > - 0,1 >), respectively . Under the assumption of a special shape of the double minimum potential, e .g . [4], (17 V (x) = x4 - ax e .24) one can relate the phenomenological tunnel matrix element t with the parameters of the potential . The observed tunnel transition is hw = 2t . The full calculation yields for a polycrystalline sample a scattering function si,~(Q, üj ) = 1(1- 9o(Qd»F 1 (w _ 2 ) 2 + FZ (17 .25) Here F is a complex expression of the order 1, which takes into account the different and temperature dependent populations of the two minima in the asymmetric potential . This scattering Figure 17 .5 : Left: Tunnel spectrum of H trapped by O in Nb(OH) o .0o2 . T=0 .1K . Instrument: IN6, ILL . Top : superconducting, bottom : normal conducting state . Right : Possible hydrogen-sites around an oxygen-defect (a) . Tunnelling can occur between each equivalent sites, e .g . o . function is almost identical with that of an 0 - H group which can assume two equilibrium orientations . It is more or les5 a semantic question to call a tunnel process translational or rotational . With the outlined matrix technique il is also possible to get the tunnelling sublevel structure of librational states of more complex potential geometries . 17- 1 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
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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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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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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 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
Page 381 and 382: 17 .2.2 Diffusion, microscopic appr
Page 383 and 384: 0.6 H20 2 x2tÄ z ) Figure 17 .2 :
Page 385: The scattering function of a polycr
Page 389 and 390: leads to the eigenvalue problem ~q
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Page 393 and 394: m Eu W Figure 17 .7 : Eigenenergies
Page 395 and 396: -so o so Energy transfer (NeV) Ener
Page 397 and 398: Bibliography [1] T. Springer, Quasi
Page 400 and 401: 18 . Texture in Materials and Earth
Page 402 and 403: Further important structure related
Page 404 and 405: The axes of the cartesian Kp coordi
Page 406 and 407: 4.0 Experimental Texture Analysis T
Page 408 and 409: Fig. 18.14: Variation of reflection
Page 410 and 411: measurements can be performed in tr
Page 412 and 413: Apart from the global description o
Page 414 and 415: In the following, two experimental
Page 416 and 417: The neutron diffraction pole figure
Page 418 and 419: dependent pole densities, and (4) g
Page 420 and 421: Schriften des Forschungszentrums J
Page 423: Forschungszentrum Jülich in der He
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Neutron Scattering

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