Neutron Scattering

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is the Hamiltonian of a free atom and p the momentum operator . time dependent operator exp(iQ-r(t)) shall be calculated . of motion follows . In the following the This is done using the equation (5 ihdt exp(iQ - r(t» = [exp(iQ - r(t», H] . .78) Here, the bracket on the right hand side is the commutator . In analogy to equation (5 .34) holds . 1 [exp(iQ - r(t», pz] = 1 exp(itH) [exp(iQ - r(t », pz] exp(-itH) (5 .79) 2msc2msc The commutator at equal times on the right hand side of the last equation can easily be calculated . For this purpose one uses the coordinate representation p = -ihV . By calculation of the derivatives [exp(iQ _ r(t», pz] = -h exp(iQ - r(0» (hQ2 + 2Q - p) (5 .80) Since p commutes with the Hamiltonian in the equation of motion the Hamiltonian can be applied directly on r(0) : ilid exp(iQ - r(t» = - h exp(iQ - r(0» (hQ2 + 2Q - dt 2msc p) (5 .81) This differential equation can be solved immediately and one obtains for the time dependent operator : exp(iQ - r(t» = exp(iQ - r(0» exp it (2m ., (hQ 2 + 2Q p)l . (5 .82) With this result the correlator in the scattering function (5 .39) can be calculated : (5 z (exp(-iQ . r(0» exp(iQ . r(t»> =exp (exp ihQ ihQ , p (2m sc ) ( 2msc / ~ . .83) As in equation (5 .72) Insertion into (5 .83) gives the average is taken by using the Boltzmann factor yielding p expp 2msc / - exp p . ~ - 2msc (5 .84) (exp(-iQ . r(0» exp(iQ . r(t»> = exp ~- Q z (tzkBT - iht) ~ . (5 .85) 2m sc identical to equation (5 .74) . Thus using the Van Hove correlation function we obtain the same result as was originally derived directly from the definition of the scattering function . 5- 23
With this example we can also demonstrate that neglecting the operator character of the position vectors leads to a wrong result . classical expression (5 .60) If we write the scattering function using the S cl (Q' w) - 27r t dtexp(-iwt) Cexp (-iQ - (r(t) - r(0») > (5 .86) is obtained . For a free atour we have r(t) = r(0) + tp/msc . Inserting this expression into (5 .86) and averaging leads to : 1 _ W2msc l Sc'MW) 2kBTQ2l - 47rErkBT exp ( . Comparison with (5 .73) shows that this result is wrong by neglecting the recoil energy term . Instead of being centred at Er the expression (5 .87) is symmetric with respect to W = 0 . (5 .87) References [1] G . E . Bacon : "Neutron Diffraction" (Clarendon Press, Oxford, 1975) . [2] G . E . Bacon (ed .) : "Fifty Years of Neutron Diffraction : The Advent Of Neutron Scattering" (Adam Hilger, Bristol, 1986) . [3] S . W . Lovesey : "Theory of Neutron Scattering from Condensed Matter" (Clarendon Press, Oxford, 1984)- [4] G . L . Squires : "Introduction to the theory of thermal neutron scattering" (Cambridge University Press, Cambridge, 1978) . [5] M . Bée : "Quasielastic neutron scattering" (Adam Hilger, Bristol, 1988) . [6] G . Placzek, Phys . Rev . 86 377 (1952) . [7] J . L . Yarnell, M . J . Katz, R . G . Wenzel, S . H . Koenig, Phys . Rev . A 7 2130 (1973) . [8] G . D . Wignall, D . G . H . Ballard, J . Schelten, Eur . Polym . J . 10 861 (l973) . 5-24
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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 .
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
Page 89 and 90: Different to the coherent nuclear s
Page 91 and 92: where QBG denotes the background .
Page 93 and 94: 4 .4 Applications We now consider e
Page 95 and 96: Fig. 4 .3 .4 probably represents th
Page 97 and 98: . occur only in the non-spin flip c
Page 99 and 100: The results for the magnetization d
Page 101 and 102: a large solid angle with polarizers
Page 103 and 104: scattering to spin-flip scattering
Page 105 and 106: An experimental verification of thi
Page 108: Correlation Functions Measured by S
Page 111 and 112: In real liquids however usually an
Page 113 and 114: The term in big parentheses of equa
Page 115 and 116: 4 0 -5 0 5 10 15 20 ho) [meV] Figur
Page 117 and 118: Figure 5 .5 : The pair correlation
Page 119 and 120: Figure 5 .6 : Partial differential
Page 121 and 122: Then equation (5 .21) can be conver
Page 123 and 124: The route from expression (5 .27) t
Page 125 and 126: In addition it is often useful to d
Page 127 and 128: 2 . The pair correlation function h
Page 129 and 130: G(r, t) G(r, t) i(Q, t) r Q i(Q, t)
Page 131: Insertion of this result into (5 .6
Page 136 and 137: 6. Continuum description : Grazing
Page 138 and 139: Starting point is the exact atomic
Page 140 and 141: 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
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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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Neutron Scattering

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