Neutron Scattering

More documents

Recommendations

Info

15.3 .1 Entropie forces - the Rouse model As the simplest model for chair relaxation, the Rouse model considers a Gaussian chair in a heat bath. The building blocks of such a Gaussian chair are segments consisting of several monomers, so that their end to end distance follows a Gaussian distribution. Their conformations are described by vectors ami , = r, - r,+1 along the chair. Thereby r is the position vector of the segment "n" . The chair is described by a succession of freely connected segments of length f . We are interested in the motion ofthese segments on a length scale f < r < R e, where R e"2 = n P Z is the end to end distance ofthe chair. The motion is described by a Langevin equation ~o dn =V "F (rn) + J n(t) , where ;o is the monomerc friction coefficient . For the stochastic force f(t) we have (fn(t»=0 and (fna(t)fmß( 0» =2kBT a, and /3 denote the Cartesian components of r . F(r,d is the free energy of the polymer chair . The force terni in Eq .[15 .10] is dominated by the conformational entropy of the chair S=k,~n W({rn }) where W (lLnl) is the probability for a chair conformation lrn} of a Gaussian chair of n- segments . 3/2 expp {117-7 } j -3 (Li 2 Ë Z With the boundary conditions offorce free ends Eq .[15 .10] is readily solved by cosine Fourier transformation, resulting in a spectrum of normal modes . These solutions are similar to e.g . the transverse vibrational modes of a linear chair except that relaxational motions are involved instead of periodic vibrations . The dispersion ofthe relaxation rates 11r is qua dratic in the number ofknots p along the chair. 15-13
1 3kBT ~ z 2 72 2 p2 a = f2 N2 P =WN2P p 0 =aR where rR is the Rouse time - the longest time in the relaxations spectrum - and W is the elementary Rouse rate . The mode conelation function for the Rouse modes is obtained as (xp (t) ~ N~ z xq R (0» =Saßspe 6,zpz exp (-tlrp ) (15 .13) (xo xô (0» = Saß N t 0 Thereby xp is the a-component of the number p normal mode and xô is the centre of mass coordinate . In order to study Brownian motion, the segment conelation functions in the real space 0 nn, (t) = « n, (t) - n ( 0»Zl are required. They are obtained by retransformation ofthe normal coordinates leading to 4r,2,,(t) =6DRt+In-ml Ë2 4N~Z N 1 r pz m ~ + z z cos Z p=, P ~ N p R t~~ in Eq.[15 .14] we use the tact that the mean square displacement of the centre of mass provides the diffusion constant. For the special case of the self conelation function (n - m) Ar,,, (t) reveals the mean square displacement of a polymer segment. We obtain 3kTt 11/2 In contrast to normal diffusion Ar,2, does not grow linearly, but wich the square route oftime . For the translational diffusion coefficient DR =kBT/N;o is obtained. DR is inversely proportional to the number of friction performing segments . 15- 14
Page 1 and 2:
Forschungszentrum Jülich in der He
Page 4 and 5:
Forschungszentrum Jülich GmbH Inst
Page 6:
Contents . . 1 Neutron Sources H .
Page 10 and 11:
1 . Neutron Sources Harald Conrad 1
Page 12 and 13:
d) f, = v a, , q - ti = P a, - 0 .3
Page 14 and 15:
Period Example Flux (D [1013 1950-6
Page 16 and 17:
In stage 1 the primary proton knock
Page 18 and 19:
get as heat, the rest is transporte
Page 20 and 21:
down power and stronger absorption
Page 22 and 23:
Appendix Neutron Sources - an overv
Page 24:
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 78 and 79:
aQ . r 3 àQ . N . QR . aQ .t k _ML
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
Page 129 and 130:
G(r, t) G(r, t) i(Q, t) r Q i(Q, t)
Page 131 and 132:
Insertion of this result into (5 .6
Page 133 and 134:
With this example we can also demon
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
Page 183 and 184:
For the Nickel film one gets a thic
Page 185 and 186:
I The rotor of a velocity selector
Page 187 and 188:
multiplier are arranged in a quadra
Page 190 and 191:
9 . Crystal spectrometer : triple-a
Page 192 and 193:
machine which will be followed by a
Page 194 and 195:
10 .3 Beaur shaping Due to the fact
Page 196 and 197:
G.k=zGz . (11) This case is fulfill
Page 198 and 199:
tribution of bisecting planes (ntos
Page 200 and 201:
directions which is exploited for t
Page 202 and 203:
Fig . 14) Dispersion surfaces for B
Page 204 and 205:
whereby E and v,, denote energy and
Page 206:
[1] B . N. Brockhouse, in Inelastic
Page 210 and 211:
10 . Time-of-flight spectrometers M
Page 212 and 213:
Tnloderator /K V7 2/m/s A/nm Aiw/eV
Page 214 and 215:
10.2 .1 Interpretation of spectra A
Page 216 and 217:
----~ Probe ' (Chopper 2) Zeit I Ch
Page 218 and 219:
the similar intensity distribution
Page 220 and 221:
is performed without Jacobian due t
Page 222 and 223:
10.2 .6 Crystal monochromators The
Page 224 and 225:
larger energy transfers . However i
Page 226 and 227:
10.3 Inverted TOF-spectrometer In t
Page 228 and 229:
constant offset to the TOF . To be
Page 230 and 231:
sample size of cm and a detection p
Page 232:
Contents 10 .1 Introduction . . . .
Page 236 and 237:
11 . Neutron spin-echo spectrometer
Page 238 and 239:
2 0 L Tr -0 L 2 ArDet Figure 11 .1
Page 240 and 241:
initial velocity-reassemble at the
Page 242 and 243:
distribution (current sheets) that
Page 244 and 245:
esults, where il is an irrelevant c
Page 246 and 247:
10000 - 8000 - co 6000 - C ô 4000-
Page 248 and 249:
e0 oX n .5 1 .0 1 .5 so 0 .0 0 .0 1
Page 250 and 251:
nel" coils) the required homogeneit
Page 252 and 253:
Figure 11 .9 : S(Q, t)IS(Q) of a 2
Page 254:
Contents 11 .1 Introduction . . . .
Page 258 and 259:
12 . Structure determination G . He
Page 260 and 261:
Important: Thc measured Bragg inten
Page 262 and 263:
determination of accurate atomic pa
Page 264 and 265:
The knowledge of the overall occupa
Page 266 and 267:
(even anharmonic) effects. This com
Page 268 and 269:
Fig . 4 . Coordination ofthe D20 mo
Page 270 and 271:
fenoelectric phase of orthorhombic
Page 272 and 273:
(c) and (d) calculated densities at
Page 274:
13 Inelastic Neutron Scattering Pho
Page 277 and 278:
Figurel Schematical drawing of a ge
Page 279 and 280:
equilibrium position the atom is in
Page 281 and 282:
s - ' f N .mwn .I )oo K L .~um..un
Page 283 and 284:
esults from the quantum mechanics o
Page 285 and 286: y absorption . 13 .1 .3 Magnetic in
Page 287 and 288: Figure 7 The plane in reciprocal sp
Page 289 and 290: 5 100s1 [OSl ] IOSSI [SSS y i 4 0 r
Page 291 and 292: dynamics of ionic compounds with th
Page 293 and 294: 16 14 12 F.. . 10 ô 8 q 6 cr 4 G.
Page 295 and 296: 21 [ooç 1 " - 19 18 17 16 "" 298 K
Page 297 and 298: n = v - cap[i(pga . - wt)] (13 .22)
Page 299 and 300: Y c 3 c Cc E Figure 16 Magnon dispe
Page 301 and 302: . . 2 H . Bilz and W . Kress, Phono
Page 304 and 305: 14 . Soft Matter : Structure Dietma
Page 306 and 307: length density of a sample against
Page 308 and 309: 2 P(Q) = 1 ~exp(-Ii - j'6 Q2 ) (14
Page 310 and 311: fraction. In the region of large Q
Page 312 and 313: with the partial structure factor S
Page 314 and 315: and which lead to the scattering cr
Page 316 and 317: For the intramolecular and intermol
Page 318 and 319: 500 450 -400 U 0 ;, 350 42, 300 s~.
Page 320 and 321: scaling behavior of the susceptibil
Page 322: 15 Polymer Dynamics Dieter Richter
Page 325 and 326: constraints due to the mutual inter
Page 327 and 328: (15 .3) T ß ( E)=T ô exp E k,T -
Page 329 and 330: This incoherent approximation does
Page 331 and 332: 10 an 0 2 0 Figure 15.4 : Relaxatio
Page 333 and 334: In order to test Eq .[15 .9] the re
Page 335: Arrhenius temperature dependence wi
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
Page 343 and 344: c 0 .8 0 .6 00 .055 x 0 .1 110 .14
Page 345 and 346: small but systematic deviations fro
Page 347 and 348: S(Q,t) is predicted . Such a platea
Page 349 and 350: only one single parameter, the tube
Page 351 and 352: confinement of the relaxation of la
Page 354: 16 Magnetism Thomas Brückel
Page 357 and 358: the ground state of metallic magnet
Page 359 and 360: c~ ~ o 0 `,~i) g',, b o 0 Q! b o 0
Page 361 and 362: only "sec" this component and not t
Page 363 and 364: 200 ¬ 400 500 600 700 E 3 QI J (h0
Page 365 and 366: Fie. 16.4: Scattering geometry for
Page 367 and 368: 0 .8 ' D3 ° Stassis 3d spin - - 3d
Page 369 and 370: Here, J;j denotes the exchange cons
Page 371 and 372: Fig. 16.8 : Contour plot of the mag
Page 373 and 374: 0.0001 0.001 0 .01 0.1 2 Fig. 16 .1
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
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
Page 391 and 392:
allow a good determination of the E
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
show all

Neutron Scattering

Create successful ePaper yourself

Delete template?

Save as template?