1 / 16

Scattering Theory - Invariant

Scattering Theory - Invariant. (Porod (1951)) Q = ∫ I( S ) d S = (1/(2π) 3 ) ∫ I( q ) d q = total scattering over the whole of reciprocal space. Scattering Theory - Invariant. (Porod (1951)) Q = ∫ I( S ) d S = (1/(2π) 3 ) ∫ I( q ) d q

gavivi
Download Presentation

Scattering Theory - Invariant

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Scattering Theory - Invariant (Porod (1951)) Q =∫I(S) dS = (1/(2π)3)∫I(q) dq = total scattering over the whole of reciprocal space

  2. Scattering Theory - Invariant (Porod (1951)) Q =∫I(S) dS = (1/(2π)3)∫I(q) dq = total scattering over the whole of reciprocal space Q = p (0) = 〈2〉 V (see derivation in Roe, 1.5)

  3. Pair distribution fcn Single atomic species (Roe, Sect. 4.1) Short range order Atom environments vary - can only get an avg picture Define PDF g(r) to describe structure Can get PDF from scattering data

  4. Pair distribution fcn Single atomic species (Roe, Sect. 4.1) Short range order Atom environments vary - can only get an avg picture Define PDF g(r) to describe structure # of atoms, on avg, in dr = n2 (r) dr, and g(r) = n2 (r)/ 〈n〉 where 〈n〉 = avg # density of atoms atom dr r

  5. Pair distribution fcn Single atomic species (Roe, Sect. 4.1) Short range order g(r) = radial distribution fcn when material is amorphous (isotropic)

  6. Pair distribution fcn Single atomic species I(q) = | A(q) | 2 = (b exp(iqrj) )(bexp(-iqrk)) = N b2 + b2  exp(-iqr jk) rjk = rj- rk = N b2 + N b2 ∫n2 (r) exp(-iqrk) dr N N j=1 k=1 N j=1 j≠k V

  7. Pair distribution fcn Single atomic species I(q) = | A(q) | 2 = (b exp(iqrj) )(bexp(-iqrk)) = N b2 + b2  exp(-iqr jk) rjk = rj- rk = N b2 + N b2 ∫n2 (r) exp(-iqrk) dr = N b2 + N b2 ∫[n2 (r) - 〈n〉]exp(-iqrk) dr + N b2 ∫〈n〉exp(-iqrk) dr = N b2 + N b2 〈n〉∫[g(r) - 1]exp(-iq rk) dr + N b2 〈n〉(q) N N j=1 k=1 N j=1 j≠k V

  8. Pair distribution fcn Single atomic species I(q) = | A(q) | 2 = (b exp(iqrj) )(bexp(-iqrk)) = N b2 + b2  exp(-iqr jk) rjk = rj- rk = N b2 + N b2 ∫n2 (r) exp(-iqr) dr = N b2 + N b2 ∫[n2 (r) - 〈n〉]exp(-iqr) dr + N b2 ∫〈n〉exp(-iqr) dr = N b2 + N b2 〈n〉∫[g(r) - 1]exp(-iqr) dr + N b2 〈n〉(q) unobservable - ignore Define interference fcn = i(q) = (I(q) - N b2)/ N b2 N N j=1 k=1 N j=1 j≠k V

  9. Pair distribution fcn Single atomic species = N b2 + N b2 〈n〉∫[g(r) - 1]exp(-iqr) dr + N b2 〈n〉(q) unobservable - ignore Define interference fcn = i(q) = (I(q) - N b2)/ N b2 Then i(q) = 〈n〉∫[g(r) - 1]exp(-iqr) dr And get g(r) - 1 from inverse Fourier transform of i(q) [g(r) - 1] = 1/〈n〉∫i(q)exp(iqr) dq V V V

  10. Pair distribution fcn Simple polymer (Cs & Hs) Need gCC( r), gHH(r), & gCH(r) gCH(r) = nCH(r) /〈nH〉 〈nH〉 = avg # density of H atoms

  11. Pair distribution fcn Simple polymer (Cs & Hs) Need gCC( r), gHH(r), & gCH(r) gCH(r) = nCH(r) /〈nH〉 〈nH〉 = avg # density of H atoms gCH(r) = gHC(r)

  12. Pair distribution fcn Simple polymer I(q) = N b2 + N bb∫n(r) exp(-iqr) dr m different types of atoms (for C & H, m=2) (= 1….m) denote atom types N = # of  m m m =1 =1 =1 V

  13. Pair distribution fcn • Simple polymer • I(q) = N b2 + N bb∫n(r) exp(-iqr) dr • m different types of atoms (for C & H, m=2) • (= 1….m) denote atom types • N = # of  • N =N x 〈n〉  〈n〉x;x of  • Then • I(q) = N x b2 + N 〈n〉x x bb∫[g(r) -1] exp(-iqr) dr m m m =1 =1 =1 V m m m V =1 =1 =1

  14. Pair distribution fcn • Simple polymer • I(q) = N x b2 + N 〈n〉x x bb∫[g(r) -1] exp(-iqr) dr • m different types of atoms (for C & H, m=2) • (= 1….m) denote atom types • N = # of  • N =N x 〈n〉  〈n〉x;x of  • Then • i(q) = 〈n〉 ∫[g(r) - 1]exp(-iqr) dr • And get g(r) - 1 from inverse Fourier transform of i(q) • [g(r) - 1] = 1/〈n〉∫i(q)exp(iqr) dq m m m V =1 =1 =1 V

  15. Pair distribution fcn Simple polymer I(q) = N x b2 + N 〈n〉x x bb∫[g(r) -1] exp(-iqr) dr i(q) = ∫[g(r) - 1]exp(-iqr) dr If: w = xb/  xb then: ∫g(r) exp(-iqr) dr =  ww∫g(r) exp(-iqr) dr m m m =1 =1 V =1 V m =1 m m V V =1 =1

  16. Pair distribution fcn Simple polymer I(q) = N x b2 + N 〈n〉x x bb∫[g(r) -1] exp(-iqr) dr i(q) = ∫[g(r) - 1]exp(-iqr) dr If: w = xb/  xb then: ∫g(r) exp(-iqr) dr =  ww∫g(r) exp(-iqr) dr Can get g(r), but not separate gN. m m m =1 =1 V =1 V m =1 m m V V =1 =1

More Related