Neutron Scattering

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The Id rotor : pocket state formalisme Pocket states are useful basis functions for stronger potentials . They represent a single orientation of a molecule . Thus =I 123 > means thatproton 1 of the methyl group is at position 1, 2 at 2, 3 at 3 . A rotation is represented by cyclic permutation . We consider the groundstate only . As outlined for translational tunnelling pocket states are no eigenstates of the problem . They overlap and thus can transform into each other. Since wave functions decay exponentially into a potential wall (Gamow factor) the overlap or tunnelling matrix element is very sensitive to the strength of the potential . The eigenvalue matrix obtained from the Hamiltonian is 1123> 1231> ~123> t ~231> t A 312 > t t The characteristic polynom yields a unique eigenvalue A = 21, related to the totally symmetric A groundstate and a doubly degenerate eigenvalue A = -t related to the right and left handed E states, respectively . The matrix is formally identical to a jump matrix . The meaning of the eigenvalues is very different, however. A tetrahedron like methane requires 12 pocket states . The 9 eigenvalues are partially degenerate depending on the environmental symmetry . The mathematics becomes more complicated . To obtain the scattering function including intensities of transitions the influence of proton spins via the Pauli principle bas to be taken into account . The complete theory with inclusion of spin wavefunctions is found in ref.[8] . The resulting scattering function is normalized te, the number of protons in the rotor si ,(Q, ~) = (1 + 29o(Qd»s(~) + (3 - 3 j o(Qd»s( ~,j ) 2 2 + (3 - 3.7o(Qd))(s( ~,i + wc) + s( üj - üi c)) (17 .44) The first term represents purely elastic scattering . Its intensity is called elastic incoherent structure factor (EISF) . The second term is due to transitions between different but degenerate E- states . Finally there are inelastic A -- E transitions between tunnelling substates . The latter terms are ô-functions only at low temperature . By coupling to phonons they broaden and shift [10] until they merge into the single classical quasielastic Lorentzian. The width of tunnelling lines can be interpreted as a lifetime broadening due to transitions into the first excited librational level Eol of the same symmetry. Eol acts as activation energy and can be obtained from an Arrhenius plot. The fig .17 .8 shows this transition for acetamide CH3C0_NH2 , the 17-17
-so o so Energy transfer (NeV) Energy transfer . (pev) Energy transfer (peV) Figure 17 .8 : High resolution spectra of acetamide at 3 temperatures : transition from methyl rotational tunnelling to classical jump reorientation . most simple molecule containing the biologically important peptide group . an especially nice example of Bohr's correspondance Principle . The transition is Here the question arises, why the tunnelling energy itself does not appear as activation energy. This is a remarquable consequence of the Pauli principle : with change of the spatial symmetry the spin state symmetry has to change too to conserve the symmetry of the total wave function. Thus A groundstate and E tunnel level show different total spin. The spinless phonons cannot induce this transition (spin conservation) . Thats the reason why the very small tunnel splittings are not smeared out at 1BT » hwt . Structural information Tunnel spectra of materials with many methyl groups may show many tunnelling transitions due to the different rotational potentials . Like in Raman spectroscopy conclusions may be drawn on structural properties as molecules per unit cell or site symmetries . 17 .3.3 Multidimensional tunnelling Not always a rotation is a pure mode . It might couple to other degrees of freedom. Correspondingly the single particle model is no longer applicable . Well established is so far a combined rotation of the molecule and ils center of mass [11] . This type of dynamics is called rotationtranslation-coupling . It is a special case ofmany possible types ofmultidimensional tunnelling . 17-18
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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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(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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Page 354: 16 Magnetism Thomas Brückel
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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 379 and 380: 17.1 Introduction Atomic and molecu
Page 381 and 382: 17 .2.2 Diffusion, microscopic appr
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Page 385 and 386: The scattering function of a polycr
Page 387 and 388: e (2) 1 (j 1, 0 > - 0,1 >), respect
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Page 393: m Eu W Figure 17 .7 : Eigenenergies
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
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Page 423: Forschungszentrum Jülich in der He
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Neutron Scattering

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