Neutron Scattering

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Chapter 17 Translation and Rotation M . Prager Contents 17 .1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 17.1 .1 Gaussian approximation . . . . . . . . . . . . . . . . . . . . . . . 2 17 .2 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 17.2 .1 Macroscopic diffusion . . . . . . . . . . . . . . . . . . . . . . . . 3 17.2 .2 Diffusion, microscopie approach : Langevin equation . . . . . . . . 4 17.2 .3 Jump diffusion on a Bravais lattice . . . . . . . . . . . . . . . . . . 5 17.2 .4 More complex cases . . . . . . . . . . . . . . . . . . . . . . . . . 8 17.2 .5 Librations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 17.2 .6 Translational tunnelling . . . . . . . . . . . . . . . . . . . . . . . 9 17 .3 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 17.3 .1 Jump rotation : methyl group in a 3-fold potential . . . . . . . . . . 11 17.3 .2 Rotational tunnelling : single particle mode] . . . . . . . . . . . . . 14 The ld rotor: solution based on 'free rotor' functions . . . . . . . . 15 The ld rotor: pocket state formalisme . . . . . . . . . . . . . . . . 17 Structural information . . . . . . . . . . . . . . . . . . . . . . . . 18 17.3 .3 Multidimensional tunnelling . . . . . . . . . . . . . . . . . . . . . 18 17 .4 Calculation of potentials 'ab-initio' . . . . . . . . . . . . . . . . . . . . 19 17- 1
17.1 Introduction Atomic and molecular motions in liquids and solids are driven by the thermal energy of the sample . Fluctuations may concentrate kinetic energy on one atom, which then is able to cross a potential barrier into a new site . give rise to quasielastic scattering . Such transport or orientational jumps occur randomly and At low temperature the classical motion dies out on the timescale of neutron spectrometers . The (classical) potentials are still prescrit, however . They now characterise the quantummechanical excitations of the lattice object : librations and tunnelling . Theories used are mostly single particle or mean field theories . By studying both, classical quasielastic scattering and quantum excitations a detailed information on the shape and the strength of potential barriers cari be obtained since neutron properties allow a resolution in space and time . If the crystal structure of a material is known one cari calculate the potentials from fundamental intermolecular interactions . The concept of "transferable pair interactions" may finally allow to predict potentials of new materials . Stochastic motions occur in many materials some of which attract technical interest. Hydrogen in metals is used for energy storage, microporous framework structures as zeolithes offer catalytically active surfaces, polymers cari aggregate to secondary structures like micelles with sometimes technically interesting properties . They mix or phase separate by diffusion. Adsorbates, intercalates, molecular and liquid crystals, matrix isolated species and liquids may bc studied this way. It was especially the invention of high resolution neutron scattering instruments (since -1972) which gave an impact to this topic which still holds . 17 .1 .1 Gaussian approximation The scattering function of a rare gas can be calculated exactly on the basis of plane wave functions arid transition matrix elements . It happens to have the shape of a Gaussian . With ß = k. B-1-1 and the recoil energy Er = 2M ( 4Er )2 CXP( - 4E, (hw - ET)2) Fouriertransformation in space and time yields the correlation function (Chapter 5) : r 2 Gs z (r,t) _ (2~Q2(t)) 2e :rp(-2Q2(t)) (17 .2) with a2(t) = t(t - i~,ß) / lll,ß (17 .3) 17-2
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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 . . . .
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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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Page 333 and 334: In order to test Eq .[15 .9] the re
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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
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Page 365 and 366: Fie. 16.4: Scattering geometry for
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Page 369 and 370: Here, J;j denotes the exchange cons
Page 371 and 372: Fig. 16.8 : Contour plot of the mag
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Page 376: 17 Translation and Rotation M . Pra
Page 381 and 382: 17 .2.2 Diffusion, microscopic appr
Page 383 and 384: 0.6 H20 2 x2tÄ z ) Figure 17 .2 :
Page 385 and 386: The scattering function of a polycr
Page 387 and 388: e (2) 1 (j 1, 0 > - 0,1 >), respect
Page 389 and 390: leads to the eigenvalue problem ~q
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Page 393 and 394: m Eu W Figure 17 .7 : Eigenenergies
Page 395 and 396: -so o so Energy transfer (NeV) Ener
Page 397 and 398: Bibliography [1] T. Springer, Quasi
Page 400 and 401: 18 . Texture in Materials and Earth
Page 402 and 403: Further important structure related
Page 404 and 405: The axes of the cartesian Kp coordi
Page 406 and 407: 4.0 Experimental Texture Analysis T
Page 408 and 409: Fig. 18.14: Variation of reflection
Page 410 and 411: measurements can be performed in tr
Page 412 and 413: Apart from the global description o
Page 414 and 415: In the following, two experimental
Page 416 and 417: The neutron diffraction pole figure
Page 418 and 419: dependent pole densities, and (4) g
Page 420 and 421: Schriften des Forschungszentrums J
Page 423: Forschungszentrum Jülich in der He
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Neutron Scattering

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