Neutron Powder Diffraction Theory and Instrumentation

Transcription

1 NTC, Taiwen Aug. 31, 212 Neutron Powder Diffraction Theory and Instrumentation Qingzhen Huang NIST Center for Neutron Research (

2 Definitions E: energy; k: wave vector; v: velocity; : wavelength. k 2 ; h mv ; E h mv 2 2 2m, where h is Planck s constant and m is the mass of the neutron. Approximate conversions are as follows: E[ mev] 82 ( [ ]), 2 Å v[ mm / s] 4. [ Å]

3 Neutron and X-ray powder diffraction Position: 2dsinq = n, where is the incident beam wavelength, d and q are the distance between successive hkl planes and Bragg angles of reflections, respectively. Intensity: I = C F hkl 2, where F hkl is the amplitude of the diffracted X-ray or neutron hkl reflection. X-ray: F hkl = f j exp(2 i (hx + ky + lz)) e -2W, where f j is the X-ray atomic scattering factor of atom j for X-ray. Neutron: F hkl = b j exp(2 i (hx + ky + lz)) e -2W, where b j is the neutron scattering length for atom j. Magnetic: F hkl = q j f Mj exp(2 i (hx + ky + lz)) e -2W, where q j and f Mj are the magnetic interaction vector and the magnetic form factor for atom j, respectively.

4 f (C) f (Co 3+ ) Comparison of f(a), b(n), and f(m) f(c)-atomic scattering factor for Carbon f(co 3+ )-Magnetic form factor for Cobalt b(c)=.6648 cm -12 -Neutron scattering length for Carbon sin q? -.2

5 Neutron Scattering Amplitudes b (cm -12 ) Neutron Scattering Amplitudes N Be C D F O B He Fe Cr Ni Cu Zn Y Sr Ba Nd Co V Li H Mn Atomic Number Z

6 Intensity Intensity MgC x Ni 3 Neutron Constant neutron scattering length gives stronger intensities in higher angle range theta (deg) X-ray theta (deg) Decreasing of the x-ray atomic scattering factor decreases the intensities of reflections in higher angle range.

7 I(C 1. )-I(C.9 ) (%) Differences in Intensities MgC x Ni 3 Neutron X-ray Position theta (deg)

8 I(YD x )-I(YD ) 12 D in YDx compound b(d)/b(y)=.667/.772=.864, Ne(D)/Ne(Y)=1/39=.26;.864/.26=33 8 x=.1, Neutron x=2, X-ray Position theta (deg)

9 Intensity C in MgC x Ni 3 Superconductor b(mg):b(c):b(ni)=.538:.665:1.3, Ne(Mg):Ne(C):Ne(Ni)=12:6: MgC x Ni 3 x=1. (Neut.) x=.95 ( Neut.) x=.9 (Neut.) x=1. (x-ray) x=.95 (x-ray) x=.9 (x-ray) 1.14% 2.12% theta (deg)

10 Transition Metals Atom Atomic # b Ti V Cr Mn Fe Co Ni Cu Zn 3.568

11 Why X-ray, not Neutron? Cost Resolution Intensity X-ray Lower Higher Higher Why Neutron, not X-ray? Light Magnetic High angle T-atom elements structure data Mixture Neutron Better Yes Better Better X-ray No

12 Magnetic symmetry and symmetry operation and Reflection conditions

13 Definition of the vectors relevant in the evaluation of the magnetic structure factor. e and k are unit vectors in the directions of the scattering and magnetic moment, respectively. The magnetic interaction vector q is always perpendicular to the scattering vector.

14 Counts NPD pattern of GeCo 2 O 4 at 8 K Nuclear AF Magnetic theta (deg)

15 Counts Counts GeCo 2 O K 15 K 19 K 21 K 25 K T (K) theta (deg) 1: (1/2,,1/2) 2: (1/2,1,1/2) 3: (1/2,1,3/2) 4: (1/2,,5/2) (3/2,1,1/2) 5: (1/2,1,5/2) (1/2,2,1/2) 6: (3/2,,5/2) (1/2,2,3/2) 7: (1/2,,7/2) (3/2,1,5/2) (3/2,2,1/2) Antiferromagnetic peaks observed below 25 K.

16 Schematic representation of selected commensurate ferro- and antiferromagnetic ordering. Arrows show the magnitudes and directions of the moments. The crystallographic primitive unit cell is indicated by solid lines and the dot lines show a possible magnetic unit cell. a N = b N are the lattice parameters for nuclear structure and the a M and b M are those of the magnetic unit cell. (a) Ferromagnetic order; (b) a ferrimagnetic order; (c) a canted ferromagnetic order; (d) a collinear antiferromagnetic order; (e) a canted noncollinear antiferromagnetic order where both M x and M y direction have antiparallel components; (f) an example of a triangular antiferromagnetic structure in hexagonal and trigonal crystals.

17 Types of magnetic lattices The first row are the five white lattices P, C, A, I, and F. The other W & B lattices can be constructed by combination of a white with its corresponding black lattice having the origin at face center, or edge center, or body center of the white lattice (indicated by the subscript, see (e)).

18 Crystal systems, conventional coordinate systems and 36 magnetic lattices in three dimensions. System Symbol Conventional coordinate system Symbols of the 36 magnetic lattices Restrictions on cell parameters Parameters to be determined White White & Black Triclinic a a, b, c, a, b, g ap ap s Monoclinic m a g = 9, a, b, c, b mp, mc mp b, mp a, mp C, mc c, mc a Orthorhombic o a b g = 9 a, b, c op, oi, of (oc, oa) op I, of s, oi c, (op C, op A ), (oc A, oa C ) (op c, op a ), (oc c, oa a ), (oc a, oa c ) Tetragonal t a = b, a b g = 9 a = b c tp, ti tp c, tp C, tp I, ti c Trigonal h a = b = c, a b g a, a hr hr I Hexagonal h a = b, a b = 9, g = 12 a, c hp hp c Cubic c a = b = c, a b g = 9 cp, cf, ci cp I, cf s 14 (white) + 22 (white&black) = 36 magnetic lattice Ref. N.V.Belov, N.N.Neronora, and T.S.Smirnova, Soviet Physics, Crystallography, Vol. 2, #3, (1957)

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