Neutron Scattering

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2 P(Q) = 1 ~exp(-Ii - j'6 Q2 ) (14 .7) z ;; and alter some calculation one gets the well known Debye formula with x = R g 2 Q 2 . PDebye(Db) (Q) -2[x - 1 + exp(-x) ] (14 .8) x In Figure14 .3 the form factor according to Debye's formula has been plotted. 0.8 a 0.6 0.4 0.2 0.0 0 2 4 6 8 10 x = (RgQ)2 Figure 14.3 : Debyeformfactor ofa linear chain The Symbol beside Eq .(14 .6) represents the form factor of a linear chain determined exclusively by intramolecular phase factors . Such symbols should help to make transparent the meaning of scattering laws ofmore complex polymers . There are expressions for the form factor in Eq.(14 .6) being approximately valid in the regions of small and large scattering vector if compared with the inverse size of the polymer ; they have a much simpler form and can easily be used for the analysis of the scattering data. So in the region of small Q, e.g . Q« 1/Rg one finds PDb -1-3 Rg Q2 or the Zimm approximation : Ppb -1 + 3Rg Q2 (14.9) as depicted in Figure 14 .4a. From this plot one gets the radius ofgyration and the scattering in forward direction according to Eq .(14 .9), delivering the polymer molar volume and volume 14-5
2.5 2.0 0.0 0.5 1 .0 1 .5 2.0 2.5 3.0 Figure 14 .4a : So-called Zimm representation : Within the Zimm approximation the plot of 1/P versus QZ gives a straight for the Debye form factor. The slope of the straight line is proportional to R g . 0.5 Kratkyauftragung 0.0 0 2 4 6 8 10 RQ Figure 14 .4b : "Kratky plot" of Debye's form factor :At large Q the representation of Qz . p(Q) versus Q gives a constant value. 14- 6
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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 . . . .
Page 258 and 259: 12 . Structure determination G . He
Page 260 and 261: Important: Thc measured Bragg inten
Page 262 and 263: 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
Page 279 and 280: equilibrium position the atom is in
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
Page 289 and 290: 5 100s1 [OSl ] IOSSI [SSS y i 4 0 r
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 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
Page 318 and 319: 500 450 -400 U 0 ;, 350 42, 300 s~.
Page 320 and 321: scaling behavior of the susceptibil
Page 322: 15 Polymer Dynamics Dieter Richter
Page 325 and 326: constraints due to the mutual inter
Page 327 and 328: (15 .3) T ß ( E)=T ô exp E k,T -
Page 329 and 330: This incoherent approximation does
Page 331 and 332: 10 an 0 2 0 Figure 15.4 : Relaxatio
Page 333 and 334: In order to test Eq .[15 .9] the re
Page 335 and 336: Arrhenius temperature dependence wi
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
Page 343 and 344: c 0 .8 0 .6 00 .055 x 0 .1 110 .14
Page 345 and 346: small but systematic deviations fro
Page 347 and 348: S(Q,t) is predicted . Such a platea
Page 349 and 350: only one single parameter, the tube
Page 351 and 352: confinement of the relaxation of la
Page 354: 16 Magnetism Thomas Brückel
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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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Neutron Scattering

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