Neutron Scattering 1

Transcription

1 Neutron Scattering 1

2 Cross Section in 7 easy steps 1. Scattering Probability (TDPT) 2. Adiabatic Switching of Potential 3. Scattering matrix (integral over time) 4. Transition matrix (correlation of events) 5. Density of states 6. Incoming flux 7. Thermal average 2

3 1. Scattering Probability Probability of final scattered state, when evolving under scattering interaction P scatt = hf U I (t) ii 2 Time-dependent perturbation theory (Dyson expansion) 3

4 2.Adiabatic Switching Slow switching of potential time [-, ] Particle V Scattering Medium t V is approximately constant 4

5 3. Scattering Matrix Propagator for time t=- t= is called the scattering matrix hf U I (t i = -1,t f = 1) ii 2 = hf S ii 2 S is expanded in series: hf S (1) ii = iv fi hf S (2) ii = hf Z m 1 1 e i! fit dt = 2 i (! f! i ) V fi! X V mihm V ii Z 1 1 dt 1 e i! fmt 1 Z t1 1 dt 2 e i! mit 2 5

6 3. Scattering Matrix Be careful with integration (ɛ...) First and second order simplify to hf S (1) ii = -2 i6(! f -! i ) hf V ii hf S (2) ii = -2 i8(! f -! i ) X hf V mihm V ii! i -! m m 6

7 4.Transition Matrix The scattering matrix is given by the transition matrix hf S ii = -2 i8(! f -! i ) hf T ii which has the following expansion hf V mihm V ii V fm V mn V ni hf T ii = hf V ii + X + X +...! i -! m (! i -! m )(! i -! n ) m m,n 7

8 4.Transition Matrix Scattering probability P s =4 2 hf T ii (! f -! i )=2 t hf T ii 2 5(! f -! i ) (not well defined because of t ) Scattering rate W S =2 hf T ii 2 5(! f -! i ) 8

9 4.Transition Matrix Target is left in one (of many possible) state. Radiation is left in a continuum state Separate the two subsystems (no entanglement prior and after the scattering event) and rewrite the transition matrix 9

10 4.Transition Matrix Target: m k i, k Radiation: ki,! k Scattering rate W fi =2 hf T ii 2 (E f E i ) go back to definition, using explicit states W fi = hm f,k f T m i,k i i 2 Z 1 1 e i(! f + f! i i )t 10

11 4.Transition Matrix Work in Schrodinger pict. for radiation and Interaction pict. for target: i(! f -! i )t i( f - i )t hm f,k f T It I r m i,k i i = hm f,k f T St S r m i,k i i e e i(! f -! i )t i(! f -! i )t = hm f,k f T It S r (t) m i,k i i e = hm f T kf,k i (t) m i i e Scattering rate is then a correlation 1 1 Z W fi = e i(! f -! i )t hm i T k ~ 2-1 f,k i (0) m f ihm f T kf,k i (t) m i i NOTE:Time evolution of target only (e.g. lattice nuclei vibration) 11

12 5. Density of states # of states P k n(e k) R d 3 n(e) ny Plane wave in a cubic cavity dn 3 2 L k x = n x! d 3 n = d 3 k L 2 n (E)dEd = (k)d 3 k = L 3 2 k 2 dkd nx 12

13 5. Density of states dk de 3 3 L E 2 L! 2 k (E) =2 =2 2 ~ 3 c 3 2 ~c 3 Photons, k = E/~c! =1/~c Massive particles, E = ~2 k 2 2m 3 L k L 2mE 3 p (E) = = 2 ~ 2 2 ~ 3 13

14 6. Incoming Flux # scatterer per unit area and time < = # = v, since t = L/v, A = L 2 At L 3 Photons, < = c/l 3 ~k Massive particles, = ml 3 L 14

15 7.Thermal Average Average over initial state of target W S (i! + d,e + de) = (E) X P i X Wfi i f Scattering Cross Section Z 1 d 2 1 X D E = dte i! fit T if (0)T fi (t) d de ~ 2 th f 1 (E) 15

16 Neutrons Using < inc and (E) for massive particles, the scattering cross section is: d 2 1 ml 3 2 Z 1 D E kf = e i! fi t T (0)T fi (t) d d! 2 2 ~ 2 k if i -1 16

17 Neutrons Evaluate T for neutrons, with states k i,f i! k (r) = e ik r /L 3/2 we obtain T kf k i (t, Q) with Q = k f k i hk f T (t, r) k i i = Z L 3 d 3 r (r)t (r, t) k f k i (r) 1 Z = L 3 d 3 re iq r T (r, t) L 3 17

18 NeutronTransition Matrix We still need to take the expectation value with respect to the target states, T fi = 1 Z d 3 re iq r hm f T (r, t) m i i L 3 L 3 18

19 Fermi Potential T is an expansion of the interaction potential, here the nuclear potential analyze at least first order... 19

20 Fermi Potential Nuclear potential is very strong (V0~30MeV) And short range (r0 ~ 2fm) Not good for perturbation theory! Fermi approximation What is important is the product a / V 3 0 r 0 (a = scattering length) if kr

21 Fermi Potential Replace nuclear potential with weak, long range pseudo-potential neutron wavefunction Still, short range compared to wavelength Delta-function potential! V (r) = 2 ~ 2 21 µ a (r)

22 Scattering Length Free scattering length a, 2 ~ 2 2 ~ 2 V (r) = a (r)! b (r) µ m n bound scattering length b (include info about isotope and spin) m n A +1 b = a a µ A Mm n (reduced mass: µ = ) M + m n 22

23 Transition Matrix To first order, the transition matrix is just the potential T fi = 1 Z d 3 re iq r hm f V (r, t) m i i L 3 L 3 Using the nuclear potential for a nucleus at a position R, we have 1 Z iq r 2 ~ 2 2 ~ 2 T fi = d 3 re b(r) (R)T (r, t) = b(r)e L 3 L m 3 n m n iq R(t) 23

24 Transition Matrix To first order, for many scatter at position ri T fi (t) = 2 ~ 2 X m n i b i e iq r i (t) The scattering cross section becomes d 2 Z 1 k 1 f i! fi t X D E = e b`b iq r`(0) iq r j (t) j e e d d! 2 k i th 1 `,j 24

25 Scattering Lengths The bound scattering length depends on isotope and spin We need to take the average b`b j! b`b j - j = `, b`b j = b 2-2 j 6= `, b`b j = b Finally, b`b j = b 2 j,` + b 2 (1 j,`) Coherent/Incoherent scattering length: b`b j =(b b ) j,` + b = b 2 i + b 2 25 c

26 Scattering lengths Coherent scattering length b c = b Correlations in scattering events from the same target (scale-length over which the incoming radiation is coherent in a QM sense) Simple average over isotopes and spins 26

27 Scattering Lengths Incoherent scattering length b 2 2 i =(b 2 - b )c j,` Correlation of scattering events between different targets Variance of the scattering length over spin states and isotopes 27

28 Cross-section Averaging over the scattering lenght d 2 Z 1 k 1 f i! fi t X D E = e b`b iq r`(0) iq r j (t) j e e d d! 2 k i th 1 `,j we obtain S S (Q,! ) d 2 Z 1 k 1 f = e i! fit X (b 2 + b 2 ) d d! 2 k i c i 1 `,j D E e iq r`(0) e iq r j (t) th S(Q,! ) 28

29 Cross-section Using the dynamic structure factors, we can write the cross section as d 2 d d! = N k f k i 2 bi S s (Q,! )+ b 2 cs(q,! ) These functions encapsulate the target characteristics, or more precisely, the target response to a radiation of energy! and wavevector Q ~ 29

30 Structure Factors Self dynamic structure factor (incoherent) Z * X S i! fi t S (Q,! ) = e e e 2 N 1 ` iq r`(0) iq r`(t) Full dynamic structure factor (coherent) + 1 Z * 1 1 i! fi t X S(Q,! ) = e e -iq r`(0) e iq r j (t) 2 N -1 `,j + 30

31 Intermediate Scattering Functions Self dynamic structure factor 1 1 Z i! fi t F s S S (Q,! ) = e (Q, t) 2 1 * 1 X F s (Q, t) = e N Full dynamic structure factor 1 1 Z S(Q,! ) = e i! fit F (Q, t) 2 1 F (Q, t) = * 31 ` 1 X e e N `,j iq r`(0) iq r`(t) e iq r j (0) iq r`(t) + +

32 These functions are the Fourier transform (wrt time) of the structure factors They contain information about the target and its time correlation. Examples: - Lattice vibrations (phonons) - Liquid/Gas diffusion 32

33 Crystal Lattice D P`,j e Position in F (Q, t) is the nuclear lattice position E iq x j (0) iq x`(t) e Model as 1D quantum harmonic oscillator q position: ~ x = 2M! 0 (a + a ) Hamiltonian (phonons) M! 2 H = p x 2 = ~! 0 (a a + 1 ) 2M

34 Crystal Lattice Assumption: 1D, 1 isotope, 1 spin state Self-intermediate structure function: F S (Q, t) = e iq x(0) iq x(t) e Note: [x(0),x(t)] =0 6 (but it s a number) Use BCH formula: eae B = ea+b e [A,B]... D E iq [x(0) x(t)] [Q x(0),q x(t)] F S (Q, T ) = e e 34

35 Simplify using (Bloch) formula: D e a+ a E = e h( a+ a ) 2 i we get F S (Q, t) = e -Q2 hdx 2 i/2 e [Q x(0),q x(t)] with x 2 =2 x 2 +2 hx(0)x(t)i-h[x(0),x(t)]i F S (Q, t) = e Q 2 hx 2 i e Q 2 hx(0)x(t)i 35

36 The crystal is usually in a thermal state. Calculate F(Q,t) for a number state and then take a thermal average over Boltzman distribution ~ hn x 2 ni = (2n + 1) 2M! 0 hn x(0)x(t) ni = ~ [2n cos(! i! 0 t 0 t)+ e ] 2M! 0 Replace n! hni 36

37 Phonon Expansion Low temperature hni 0 x 2 = hx(0)x(t)i = e ~ ~ i! 0 t 2M! 0 2M! 0 Expand in series of ~ 2 Q 2 and calculate the dynamic structure factor 2M /(~! 0)= E kin /E bind S S (Q,! ) = F(F (Q, t) S S (Q,! ) e ~Q 2 h Q 2 2M! 0 ~Q 2 (!)+ (!! 0 )+ 2 1 ~Q 2 2 2M! M! 0 (! 2! 0 )+...

38 Phonon Expansion Neutron/q.h.o. energy exchange Zero-phonon = no excitation (elastic scattering) S S (Q,! ) e ~Q 2 h Q 2 2M! 0 ~Q 2 one-phonon = 1 quantum of Energy (!)+ (!! 0 )+ 2 1 ~Q 2 2 2M! 0 2M! 0 (! 2! 0 )+... two-phonons = 2 quantum of Energy n-phonons 38

39 High Temperature We find a classical result, where F cl = F[G cl (x, t)] S s and the space-time self correlation (classical) function G cl s (x, t)dx is the probability of finding the h.o. at x, at time t, if it was at the origin at time t=0. F cl (Q, T ) = e (k btq 2 /M! 2 0)[1 cos(! 0 t)] z 39

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