Neutron Scattering

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determination of accurate atomic parameters (positional and thermal parameters, site occupations) of lighter elements in the presence ofheavy ones . The contrast in conventional X-ray diffraction is directly related to the ratio ofthe number of electrons Zj of thé différent atoms or ions j involved . The atomic scattering factor f in thé structure-factor formula, which represents thé Fourier transform of thé atomic electron density distribution, is proportional to Zj (f = Zj for sin9/A, = 0) . Standard X-ray techniques can hardly differentiate between atoms/ions of a similar number of electrons, and only an average structure - including a total occupation probability of mixed occupied sites - may be obtained in such cases . For neutrons thé atomic scattering factor f is replaced by thé nuclear scattering length (or coherent scattering amplitude) bj, which is ofthé same order of magnitude for all nuclei but varies from nucleus to nucleus in a non-systematic way . b,, values, which can be either positive or negative, depend on thé isotopes and nuclear spin states of thé element j . A nucleus of an isotope with spin I may have two différent neutron scattering lengths : one for thé combined spin state J = I + %2 and one with J = I - '/z . An important and fundamental example is provided by thé simplest of all nuclei, thé proton with spin I = 1/2 . The two spin states, J = 1 (triplet) and J = 0 (singlet), with statistical weights 3/4 and 1/4 respectively, have thé scattering lengths for afree proton : b SH=-23 .7 fin, b'H=+5 .38 fin, bfre,H = 1/4b'H + 3/4b'H = -1 .89 lin (with 10 -15 m = 1 lin). The value for thé bound proton in a ciystal structure, which is to be used in thé structure factor calculations, amounts to bl, = 2 -bfreeFl = -3 .741 fm. The natural isotope mixture and a statistical spin-state distribution lead to thé commonly used general formula bj = with thé sum of thé différent isotope fractions a+13+y+. . . = 1 (boa, bop, bj,, being thé individual scattering lengths of thé différent isotopes of thé element j) . The natural nickel isotopes, for instance, have extremely différent coherent scattering amplitudes : b( 58Ni) = +14 .4 fin, b(6°Ni) = +3 .0 fin, b(61Ni) = +7 .6 fm,b(62Ni) = -8 .7 fin, b(64Ni) = -0.37 fin resulting in an overall scattering length bNi = +10 .34 fm . <strong>Neutron</strong> experiments frequently make use of compounds containing single isotope elements, like fully deuterated samples . Incoherent scattering due to a statistical distribution of isotopes and nuclear spin states is not discussed here . It may influence thé effective absorption and thé background conditions ofneutron diffraction studies . 12- 5
12 .3 .1 Example ofcontrast variation : Crystal structure and magnetic ordering of (Mnl-%Cr%) 1+sSb A special possibility of contrast variation, the combination of X-ray and neutron diffraction information, is demonstrated for the example ofthe intermetallic compounds (Mnl_XCr,)1 + sSb, with 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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Page 214 and 215: 10.2 .1 Interpretation of spectra A
Page 216 and 217: ----~ Probe ' (Chopper 2) Zeit I Ch
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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
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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
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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
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
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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 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
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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
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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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Neutron Scattering

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