Neutron Scattering

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aQ . r 3 àQ . N . QR . aQ .t k _MLQ) = fMS (N)e- d r =le-'Sik=je - -t1e -' 'S ik (3 .53) ik i k To calculate the scattering cross section, we now have to determine the expectation value of this operator for the quantum mechanical state of the sample averaged over the thermodynamic ensemble . This leads to M(Q) =- 2 PB - fm(Q) -7,eiQ .Rt - Si (3 .54) The single differential cross section for elastic scattering is thus given by : 2 _du = eiQRi ~ (Tnro)2 f.LQ)L Si1 (3 .55) M i Here, fm(Q) denotes the form factor, which is connected wich the spin density of the atour via a Fourier transform : .fm (Q)= fps(?")eiQ .rd3r (3 .56) Atom With the form (3 .55), we have expressed the cross section in simple atomic quantities, such as the expectation values of the spin moment at the various atoms . The distribution of the spin density within an atom is reflected in the magnetic form factor (3 .56) . For ions with spin and orbital angular momentum, the cross section takes a significantly more complicated foira [4, 5] . Under the assumption that spin- and orbital- angular momentum of each atour couple to the total angular momentum J (L/S-coupling) and for rather small momentum transfers (the reciprocal magnitude of the scattering vector has to be small compared to the size of the electron orbits), we can give a simple expression for this cross section in the so-called dipole approximation : _ 2 J 1 Q' ,I2 dS2-(Ynro) ' .Îm(QJ~Ji1ea-R . i (3 .57) 3-27
Here the magnetic form factor writes : fm(Q) = (jo(Q)+c2(j2(Q) (3 .58) gJ denotes the Lande g-factor, CZ = ? -1 and gJ (jl (Q) = 4z 1jl(Qr)R2 (r)r 2dr (3 .59) 0 are the spherical transforms of the radial density distributions R(r) with the spherical Bessel functions ji(Qr) . For isolated atoms, the radial part R(r) has been determined by Hartree-Fockcalculations and the fonctions (j0(Q)~ and (j2(Q)) in (3 .58) have been tabulated [6] . After having introduced the principles of magnetic scattering, we will discuss applications in chapter 16 . References [1] J . M. Cowley "Diffraction Physics" North-Holland, Amsterdam 1990 [2] B . W . Batterman, H . Cole "Dynamical Diffraction of X Rays by Perfect Crystals", Reviews of Modern Physics 36 3, 681 - 717 (1964) [3] V . F . Sears "Neutron Scattering Length and Cross Sections", Neutron News 3 (1992), 26 - 37 [4] G . L . Squires "Introduction to the theory of thermal neutron scattering" Cambridge University Press, Cambridge (1978) [5] S . W . Lovesey "Theory ofneutron scattering from condensed matter" Volume 2 : Polarisation effects and magnetic scattering, Clarendon Press, Oxford (1987) [6] A . J . Freeman & R. E . Watson, Acta Cryst. 14 (1961), 231 [7] J . Strempfer, Th . Brückel et al Eur . Phys . J. B 14 (2000), 63 [8] J . Strempfer, Th . Brückel et al Physica B 267-268 (1999), 56 3-28
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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
Page 28 and 29: 2 . Properties of the neutron, elem
Page 30 and 31: In the 60's the ferst high flux rea
Page 32 and 33: Hot neutrons in reactors are obtain
Page 34 and 35: elated values for the FRJ-2 reactor
Page 36 and 37: 2 .5 Scattering amplitude and cross
Page 38 and 39: (A+k2 ) yr = u(Z:) yr V( u~r)= z "
Page 40 and 41: _du _ mn \2 A2-~2z r2~ I(k' IVI k)I
Page 42 and 43: E ô ô x z w 0 z _w U N Figure 2 .
Page 44 and 45: scattering is observed with the ave
Page 46 and 47: For a spherical electron distributi
Page 48: A more thorough introduction to neu
Page 52 and 53: 3 . Elastie Seattering from Many-Bo
Page 54 and 55: Q = Q = k2 + k2 - 2kk' cos2B (3 .3)
Page 56 and 57: 3 .2 Fundamental Scattering Theory
Page 58 and 59: The meaning of (3 .l4) is immediate
Page 60 and 61: 1 . e. in the second approximation,
Page 62 and 63: The saure problem will bc dealt wit
Page 64 and 65: ALQ) = Y-bfe'Q'-rB(r-(n .a+m .b+p»
Page 66 and 67: du LQ) = ALQJ2 = e'Q .A' . * -i e-
Page 68 and 69: Tab . 3.1 : The scattering lengths
Page 70 and 71: We note immediately that we should
Page 72 and 73: These magnetic structures can be un
Page 74 and 75: M1LQ)=Q-MQ)xQ M(Q)= IM (r)e i Q " d
Page 76 and 77: Orbital magnetic spin angular ,,- m
Page 80: 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
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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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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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