Neutron Scattering

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machine which will be followed by an optimization of the physics involved (one sided in a certain sense) in the shape of the backscattering spectrometer. 10 .2 Principle of the crystal spectrometer with three variable axes Fig. 1) shows the principle layout of a triple-axis spectrometer. <strong>Neutron</strong>s of a defmed wavelength Â are selected from an incident "white" beam by means of a single crystal (monochromator M, ferst axis) under the associated angle 20, according to eq . (1) . The sample S under inspection is positioned in the diffracted beam. The secondary spectrometer moves around the sample thereby selecting neutrons scattered by an angle 20, (second axis) which are further sorted with regard to their energy via the angular setting 20, of an analyzer crystal A (third axis) and counted by detector D . The scattering vector 2 as well as the energy transfer ho) are determined by the angles 20, and 20, for given incident wave vector k (see also Fig . 2) . Fig. 1) schematic layout of a triple-axis spectrometer with : collimators a ,, monochromator M, sample S, analyzer A, detector D, incident (scattered) wave vector k (k Thereby, the wave vectors k and k' are connected with the variables Q and ho) by the conservation of momentum and energy . In the resulting scattering triangle (Fig. 2) the deviation of the foot F from Y Q determines the energy transfer hco . The sense of this deviation determines whether the neutron bas gained (k < k, E-E'=hco < 0) or lost energy (k > k, hco > 0) by the scattering process . In order to influence the divergence of the neutron beam, collimators a are inserted before and alter the crystals by which neutrons are selected . Detection of neutrons is by materials with an exceptionally large absorption cross section . The involved nuclear processes are e.g . 1OB(n,a), 6Li(n,a), and 3He(n,p) . The most common type of detectors is conceived as proportional counter filled with 3He having an absorption cross section of 5300 barn for neutrons with 25 meV . With the typical active length of about 5 cm and a filling pressure of about 5 bar, the counting probability reaches >95% for thermal neutrons (the absorption cross section is reciprocal to the speed of the neutron) . It is meaningful to insert a neutron monitor at the exit of the primary spectrometer . This device is principally similar to a counter absorbing and detecting, however, a small fraction of the incident neutrons, only . The measured spectra may thus be normalized to the number of incident neutrons which is essential for long measuring periods since the neutron flux may 9- 3
undergo appreciable fluctuations . In addition, for a reactor of the type DIDO, the average neutron flux increases by about 10% during a normal cycle . The term triple-axis spectrometer indicates already the versatility of this instrument (not only in real space) which distinguishes this concept front other types of spectrometers . As long as the radius of action is not barred by the radiation shielding, arbitrary points in Q, _w - space may be selected. For fixed incident wave vector, those points belong to a parabolic surface which is defmed by momentum- and energy conservation as shown in Fig . 3) . Q=k-k' AE=hco=Z~~(kZ-k f z) . AE=ti(k-kZ)/2m 1/2Q = 1/2(k-k`) 1/2 hoo Q h2 Q3 / 2m Fig . 2) Geometrical relation between wave vectors k and k' on the one hand side to momentum- h Q and energy transfer hco on the other hand . Since (k k~ represents 6 dimensions, yet (_Q, 0) ) 4 dimensions, only, arbitrary solutions of vector P on the plane perpendicular to Q result for a given (_Q, OJ )- point. Fig .3 ) representation of the theoretical measuring range of a neutron spectrometer with fixed incident wave vector k. 9- 4
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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
Page 142 and 143: v v z l 2r V(z) = 2 -2erfC~ with er
Page 144 and 145: F{dV(z)/dz ) to calculate the refle
Page 146 and 147: Equation (6 .13) also shows that th
Page 148 and 149: ut is completely reflected . The cr
Page 150 and 151: 0.4° . The full dynamical theory c
Page 152 and 153: 1) If no magnetic induction B (whic
Page 154: Diffractometer Gernot Heger
Page 157 and 158: select a special wavelengths band (
Page 159 and 160: " The direction of the reciprocal l
Page 161 and 162: Bragg-intensities of single crystal
Page 163 and 164: monochromator, on the mosaic spread
Page 165 and 166: complex sample environments, such a
Page 167 and 168: the diffraction plane) two further
Page 170: Small-angle Scattering and Reflecto
Page 173 and 174: L= (1) ( k ) a e -(k/k T ) 2 Ak (8
Page 175 and 176: possible neutron wave lengths betwe
Page 177 and 178: figure and with neutrons of about 1
Page 179 and 180: 1 .0 Danvinkurve 0 .8 0 .6 0 .4 0 .
Page 181 and 182: tails. This reflection curve leads
Page 183 and 184: For the Nickel film one gets a thic
Page 185 and 186: I The rotor of a velocity selector
Page 187 and 188: multiplier are arranged in a quadra
Page 190 and 191: 9 . Crystal spectrometer : triple-a
Page 194 and 195: 10 .3 Beaur shaping Due to the fact
Page 196 and 197: G.k=zGz . (11) This case is fulfill
Page 198 and 199: tribution of bisecting planes (ntos
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Page 202 and 203: Fig . 14) Dispersion surfaces for B
Page 204 and 205: whereby E and v,, denote energy and
Page 206: [1] B . N. Brockhouse, in Inelastic
Page 210 and 211: 10 . Time-of-flight spectrometers M
Page 212 and 213: Tnloderator /K V7 2/m/s A/nm Aiw/eV
Page 214 and 215: 10.2 .1 Interpretation of spectra A
Page 216 and 217: ----~ 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
Page 224 and 225: larger energy transfers . However i
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
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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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