Neutron Scattering

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Figures 16 .10 and 16.11 show the magnetic diffuse neutron scattering of MnS2 at different temperatures above TN and the magnetic Bragg peak at 4 .9 K (topmost figure). We can clearly observe, how with decreasing temperature the diffuse scattering becomes shaiper in reciprocal space and how the peak intensity increases strongly. However, for scans along (1,k,0) the diffuse scattering is not centred at the low temperature Bragg position, while it is centred for the perpendicular scans in the (h,k,0) plane. The magnetic short range order is "incommensurate" wich the lattice . This means that the periodicity observed in the diffuse magnetic scattering is not just a simple rational multiple of the chemical unit cell periodicity . Figure 16 .9 shows the temperature variation of the incommensurate component of the vector at which the diffuse scattering is centred . Note the jump characteristic for a first order transition . Figure 16 .9 demonstrates that we can understand the paramagneticantiferromagnetic phase transition in MnS2 as a transition from incommensurate short range order to commensurate long range order . Now it is well established that such "lock-intransitions" are of first order, which explains the unusual behaviour of MnS2 . The problem remains which interaction leads to the shift of the diffuse peak as compared to the Bragg reflection. This question can be solved with model calculations, such as the ones depicted in figure 16 .8 [14] . It tums out that an anisotropy term in the Hamiltonian can give rise to the observed effect . Finally we want to show an example for a truc "classical" second order transition, the PM-AF transition in MnF2 . In this case, we have performed the measurements with high energy synchrotron x-rays due to the better reciprocal space resolution as compared to neutrons [15] . Figure 16 .12 shows a double logarithmic plot of the reduced sublattice magnetisation m (m = 1VI/Ms, where Ms is the saturation value of the magnetisation) versus the reduces temperature ti, defmed in eq. (16.15) . In this plot, the data points nicely line up along a straight line, corresponding to a power law behaviour as expected from (16 .7) . The critical exponent ß of the sub-lattice magnetisation can be obtained to great precision : ß = 0 .333 (3), corresponding roughly to the exponent expected for an Ising system (n=1, d=3) according to table 16.1 . However, the calculated and measured value do not quite coincide, at least to within two standard deviations, which demonstrates that the precise values of the critical exponents are still not very well established. 1 6-17
0.0001 0.001 0 .01 0.1 2 Fig. 16 .12 : Magnetic Bragg diffraction from MnF2 . The inset shows the temperature dependence of the intensity of the magnetic 300 refection from 5 to 80K. In the main graph is plotted the reduced sub-lattice magnetisation as a function of reduced temperature on a double logarithmic scale together with a fit employing a potiver-law function. 16 .5 Suinmary We have given a few examples of the applications of neutron scattering in magnetism. We have seen how neutrons can be used to investigate the magnetisation density distribution on an atomic level . Besides the rather new technique of magnetic x-ray scattering, no other method can provide the saine information on magnetic structure and magnetisation density . <strong>Neutron</strong>s are ideally suited to study magnetic phase transitions, which are model examples of co-operative phenomena in many body systems . Unfortunately, we were not able to cover other subjects, such as the important fields of magnetic excitations or thin film magnetism. <strong>Neutron</strong> scattering is the technique to measure spin wave dispersion relations used to detennine magnetic interaction parameters (exchange interaction, anisotropy) - see chapter on excitations. In itinerant systems, the transition frein collective spin wave like excitations to single particle like "Stoner" excitations could bc observed with neutrons . Currently, more "exotic" excitations are in the centre of attention, such as the "resonance peak" in high 16- 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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Page 325 and 326: constraints due to the mutual inter
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Page 331 and 332: 10 an 0 2 0 Figure 15.4 : Relaxatio
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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: Fig. 16.8 : Contour plot of the mag
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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 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
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Forschungszentrum Jülich in der He
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