Neutron Scattering

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Fig . 4 . Coordination ofthe D20 molecules in Na2S -9D20 . A B C A-B B-C A-C Z-BAC Z-ABC ZBAB' ZCAC' L A-L ~ LA L' 0(l) -D(1)_ ..S -D(V) ---S 0(2) -D(21).~ .0(5) -D(22). . . S 0(3) -D(31). . .S -D(32). . .0(4) 0(4) -D(41) . . .S -D(42), .,S' 0(5) -D(51), . .0(3) -D(52) . . .S mean values mean values 0-D .. .O mean values 0-D . . .S 0.9617) 2 .359(5) 3.319(5) 1,4(4) 178.0(6) 106.3(7) 0.961(7) 2,359(5) 3.319(5) 1 .4(4) 178A6) 0.964(7) 1 .793(7) 2.752(7) 1W4) 172.4(6) 106 .l(7) 0.962(7) 2.550(6) 3.5M5) 5-2s) 172.8(6) 0,977(7) 131l(5) 3.284(5) 4-7(4) 173,3(5) 10T5(7) 0.953(7) 1 .797{7) 2,730(7) 9.6(4) 169.3(6) 0,983(7) 2.294(5) M74(4) 3.4(4) 17-9.l(5) 104 .6(6) O.973(7) 1359(5) 3 .333(5) 03M 179.6(5) 0 .949(7) 1,ffl(7) 2,768(7) 9 .2(4) 166 .l(6) 105 .5(6) 0 .967(7) 14o(5) 3,401(5) 5Y(o 172,1(5) 0 .965 106,0 0,955 1309 2,750 1679 0970 1386 3353 1751 103M2) Na(1) 2.411(4) 116.1(2) Mai lois) 111 .5(2) Na(2) Na(T) 1588(5) 976(2) 2,380(5) 116 .9(2) Na(1) 0(5) 1397(5) 104 .8(2) 2 .768(7) 104 .1(2) Na(2) 0(3) 1418(51 105 .5(2) 2 .730(7) 103 .4(2) Na(1) 0(2) 1485(5) 101 .7(2) 2,752(7) 107,9 2,447 Table 1 . hiteratomic distances (Â) and angles (') for the hydrogen bonds and the ligands te, the water molecules in Na2S-9D20. imil
Remember: The scattering lengths ofthe proton and the deuteron are bx = -3 .74 fin and bD = +6 .67 fin, respectively . Their magnitudes are comparable to the average of all b; magnitudes and, therefore, H/D can be considered as "normal" atoms for neutron diffraction . The différent signs of bu and bD may be of interest in Fourier maps for contrast reasons . Experimental conditions like background and effective absorption are strongly affected by thé huge and exceptional incoherent neutron scattering cross-section of hydrogen (ß;n,(H) = 79 .7 barns as compared to 6; c(D) = 2 .0 barns).Very often deuterated compounds are preferred in order to profit from thé larger bD value, but mainly to reduce thé background from incoherent scattering . This volume-dépendent background may become crucial for neutron powder diffraction experiments, for which normally sample volumes of more than 1 cm3 are required . 12 .4 .2 Example ofa study ofH/D ordering : Ferroelectric phase transition in KH 2PO4 (KDP) The hydrogen problem is of special importance for structural phase transitions driven by proton ordering . As a well known example thé fenoelectric transition in KH2P04 (KDP) is presented . A characteristic feature of its crystal structure consists ofthé P0 4 groups linked by strong hydrogen bonds (see Fig . 5). At room temperature KDP crystallises in a tetragonal phase (space group : 1 42d), where thé protons in thé 0 . . .H...0 bonds are dynamically disordered according to a double-well potential . At T c = 122 K, KDP transforms to a Fig . 5 . Crystal structure of KH2P0 4 . 1 2-12
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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
Page 218 and 219: the similar intensity distribution
Page 220 and 221: is performed without Jacobian due t
Page 222 and 223: 10.2 .6 Crystal monochromators The
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Page 226 and 227: 10.3 Inverted TOF-spectrometer In t
Page 228 and 229: constant offset to the TOF . To be
Page 230 and 231: sample size of cm and a detection p
Page 232: Contents 10 .1 Introduction . . . .
Page 236 and 237: 11 . Neutron spin-echo spectrometer
Page 238 and 239: 2 0 L Tr -0 L 2 ArDet Figure 11 .1
Page 240 and 241: initial velocity-reassemble at the
Page 242 and 243: distribution (current sheets) that
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Page 248 and 249: e0 oX n .5 1 .0 1 .5 so 0 .0 0 .0 1
Page 250 and 251: nel" coils) the required homogeneit
Page 252 and 253: Figure 11 .9 : S(Q, t)IS(Q) of a 2
Page 254: Contents 11 .1 Introduction . . . .
Page 258 and 259: 12 . Structure determination G . He
Page 260 and 261: Important: Thc measured Bragg inten
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Page 264 and 265: The knowledge of the overall occupa
Page 266 and 267: (even anharmonic) effects. This com
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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 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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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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