Neutron Scattering

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2 .5 <strong>Scattering</strong> amplitude and cross section, the Born approximation We now consider the scattering event by a fixed nucleus . The geometry for such a scattering process is sketched in Fig .2 .6 . The incoming neutrons are described by plane waves e'krr travelling in z -direction . At the target this plane wave interacts with a nucleus and is scattered into a solid angle S2 df. e ikr Incident neutrons r = (0, 0, Z) Figure 2.6 : <strong>Scattering</strong> geometiy for an incident plane wave scattered at a target . The partial cross section is defmed by _d6 _ current of scattered neutrons into (52, A2 current of incident neutrons dÇ2) (2 .5) Quantum mechanically the current is given by 2mi (Y/*V y/-VvV/ *) (2 .6) For an incident plane wave yr= ezkr Eq.[2 .6] leads immediately to j = h~ , where V is n the normalization volume . The scattered spherical wave has the form 1 f(Ç2)e"" therej(~ r 2-9
describes the solid angle dependent scattering amplitude. Inserting this form for the scattered wave into Eq .[2 .6] for large r J, _ hk If( )Iz ~2 mn dS2 2 .7) is obtained. Finally, inserting the incoming and scattered cunents into Eq .[2 .5] we obtain for the cross section d6 - I hk'I ml If (Ç2)12 dÇ2 - I f (sa)~z dS2 Ihklml dn (2 .8) du is also called differential cross section . We realize, that a scattering experiment delivers information on the absolute value of the scattering amplitude, but not on f(n) itself. Informations on the phases are lost. The total cross section is obtained by an integration over the solid angle (T = f d2 du (2 .9) 12 M Our next task is the derivation off(n) in the so called Bom approximation . We start with the Schrödinger equation ofthe scattering problem z - -4+V(r) 2m yr = EV/ (2 .10) where V(r) is the scattering potential . We note, that for scattering on a free nucleus the mass term in the kinetic energy has to be replaced by the reduced mass li = M mn . For large mn +M distances (r -> oo), V= 0 and we have E = h2k2l2m,, . Inserting into Eq.[2 .10] leads to the wave equation 2- 10
Page 1 and 2: Forschungszentrum Jülich in der He
Page 4 and 5: Forschungszentrum Jülich GmbH Inst
Page 6: Contents . . 1 Neutron Sources H .
Page 10 and 11: 1 . Neutron Sources Harald Conrad 1
Page 12 and 13: d) f, = v a, , q - ti = P a, - 0 .3
Page 14 and 15: Period Example Flux (D [1013 1950-6
Page 16 and 17: In stage 1 the primary proton knock
Page 18 and 19: get as heat, the rest is transporte
Page 20 and 21: down power and stronger absorption
Page 22 and 23: Appendix Neutron Sources - an overv
Page 24: led remotely. It is rauch more conv
Page 28 and 29: 2 . Properties of the neutron, elem
Page 30 and 31: In the 60's the ferst high flux rea
Page 32 and 33: Hot neutrons in reactors are obtain
Page 34 and 35: elated values for the FRJ-2 reactor
Page 38 and 39: (A+k2 ) yr = u(Z:) yr V( u~r)= z "
Page 40 and 41: _du _ mn \2 A2-~2z r2~ I(k' IVI k)I
Page 42 and 43: E ô ô x z w 0 z _w U N Figure 2 .
Page 44 and 45: scattering is observed with the ave
Page 46 and 47: For a spherical electron distributi
Page 48: A more thorough introduction to neu
Page 52 and 53: 3 . Elastie Seattering from Many-Bo
Page 54 and 55: Q = Q = k2 + k2 - 2kk' cos2B (3 .3)
Page 56 and 57: 3 .2 Fundamental Scattering Theory
Page 58 and 59: The meaning of (3 .l4) is immediate
Page 60 and 61: 1 . e. in the second approximation,
Page 62 and 63: The saure problem will bc dealt wit
Page 64 and 65: ALQ) = Y-bfe'Q'-rB(r-(n .a+m .b+p»
Page 66 and 67: du LQ) = ALQJ2 = e'Q .A' . * -i e-
Page 68 and 69: Tab . 3.1 : The scattering lengths
Page 70 and 71: We note immediately that we should
Page 72 and 73: These magnetic structures can be un
Page 74 and 75: M1LQ)=Q-MQ)xQ M(Q)= IM (r)e i Q " d
Page 76 and 77: Orbital magnetic spin angular ,,- m
Page 78 and 79: aQ . r 3 àQ . N . QR . aQ .t k _ML
Page 80: Schweika Analysis Polarization W .
Page 83 and 84: ducing a complex component and were
Page 85 and 86: ir v s ftft 3956 2 s 2 0.81[X] 80 c
Page 87 and 88:
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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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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