Polarized Neutrons Intro and Techniques. Ross Stewart

Transcription

1 Polarized Neutrons Intro and Techniques Ross Stewart

2 Polarized neutron beams Each individual neutron has spin s=½ and an angular momentum of ±½ħ Each neutron has a spin vector and we define the polarization of a neutron beam as the ensemble average over all the neutron spin vectors, normalised to their modulus If we apply an external field (quantisation axis) then there are only two possible orientations of the neutrons: parallel and anti-parallel to the field. The polarization can then be expressed as a scalar: +½ħ B Spin-up "i -½ħ Spin-down #i where there are N + neutrons with spin-up and N - neutrons with spin-down

3 History of Polarized Neutrons 1932 Discovery of the neutron Chadwick (Proc Roy Soc A ) 1937 Theory of neutron polarization by a ferromagnet Schwinger (Phys Rev, 51, 544) 1938 Partial polarization of a neutron beam by passage through iron Frisch et al (Phys Rev 53, 719), Powers (Phys Rev 54, 827) 1940 Magnetic moment of the neutron determined by polarization analysis Alvarez and Bloch (Phys Rev 57,111) Theory of magnetic neutron scattering Halpern and Johnson (Phys Rev 51, 992; 52, 52; 55, 898) 1951 Polarizing mirrors, proof of the neutron s μ.b interaction Hughes and Burgy (Phys Rev, 81, 498)

4 History of Polarized Neutrons 1951 Polarizing crystals (magnetite Fe 3 O 4, Co92Fe8) Shull et al (Phys Rev 83, 333; 84, 912) 1959 First polarized beam measurements (of magnetic form factors of Ni and Fe) Nathans et al (Phys Rev Lett, 2, 254) 1963 General theory of neutron polarization analysis Blume (Phys Rev 130, 1670) Maleyev (Sov. Phys.: Solid State 4, 2533) 1969 First implementation of neutron polarization analysis, Oak Ridge, USA Moon, Riste and Koehler (Phys Rev 181, 920)

5 History of Polarized Neutrons 1972 Invention of neutron spin echo (IN11, ILL) Mezei (Z Phys Rev 255, 146) 1982 XYZ polarization analysis on a multidetector spectrometer (D7, ILL) Schärpf (AIP Conf. Proc. 89, 175) 1987 Invention of neutron resonance spin-echo (leading to SESANS, MIEZE,...) Golub and Gähler (Phys. Lett. A 123, 43) 1988 Development of neutron polarimetry measurements with CRYOPAD Tasset et. al. (J. Appl. Phys. 63, 3606 ) 2000 Routine use of 3 He neutron spin-filters for polarizing neutrons

6 Polarized neutrons today Single crystal diffraction Diffuse scattering Inelastic scattering (3-axis and TOF) Reflectometry (on and off-specular) SANS - magnetic and non-magnetic Neutron Spin-Echo Neutron Resonance Spin-Echo SESANS Larmor Diffraction Neutron Depolarization Polarized Neutron Tomography...

7 Polarized neutron beams What we often would like to do in polarized neutron experiments is measure the scalar polarization of the beam. Where F = N + N scattering experiment P = N + N N + N ( ) 1 ( ) +1 = N + /N N + /N = F 1 F +1 is called the Flipping Ratio and is a measurable quantity in a This description of a polarized beam is OK for experiments in which a single quantisation axis is defined: Longitudinal Polarization Analysis The technique of 3-dimensional neutron polarimetry, however is termed: Vector (or Spherical) Polarization Analysis

8 A Uniaxial PA experiment flipper 1 B flipper 2 polarizer "i! "i analyser detector First attempted by Moon, Riste and Koehler (Oak Ridge 1969) Phys Rev. 181 (1969) 920

9 A Uniaxial PA experiment flipper 1 B flipper 2 polarizer "i! "i #i! "i analyser detector First attempted by Moon, Riste and Koehler (Oak Ridge 1969) Phys Rev. 181 (1969) 920

10 A Uniaxial PA experiment flipper 1 B flipper 2 polarizer "i! "i #i! "i "i! #i analyser detector First attempted by Moon, Riste and Koehler (Oak Ridge 1969) Phys Rev. 181 (1969) 920

11 A Uniaxial PA experiment flipper 1 B flipper 2 polarizer "i! "i #i! #i #i! "i "i! #i analyser detector First attempted by Moon, Riste and Koehler (Oak Ridge 1969) Phys Rev. 181 (1969) 920

12 } } A Uniaxial PA experiment B flipper 1 flipper 2 polarizer analyser detector "i! "i #i! #i #i! "i "i! #i Non-spin-flip Spin-flip First attempted by Moon, Riste and Koehler (Oak Ridge 1969) Phys Rev. 181 (1969) 920

13 Polarizers Stern-Gerlach experiment (1922) Cu 2 MnAl (Heusler) crystal grown at ILL λ c λ 1 λ 2 λ 3λ 4 Supermirror systems (eg Co/Ti, Fe/Si etc) } } } } d 1 d 2 d 3 d 4 3He spin-filter

14 Flippers Dabbs Foil: current sheet Drabkin flipper: useful for white beams of limited size BG BG I BF B in B out P in P out - BG d z B G B G B F y B F B C Mezei Flipper: current sheet AFP Flipper: adiabatic fast passage

15 Uniaxial Polarization Analysis

16 Neutron polarization and scattering We start with the (elastic - k i = k f ) scattering cross-section dσ dω = m n 2π 2 Where the spin-state of the neutron S is either spin-up k ʹ S ʹ V ks 2 = 1 0 or spin down = 0 1 For nuclear scattering (no spin) V is the Fermi pseudopotential, and the matrix element is ʹ S b S = b ʹ S S = b 0 Non-spin-flip Spin-flip where we have used the fact that the spin states are orthogonal and normalised = = 0, = =1

17 Neutron polarization and magnetic scattering V is the magnetic scattering potential given by V m (Q) = γ r n 0 σ M (Q) = γ r n 0 σ ζ M ζ (Q) 2µ B 2µ B ζ (see e.g. Squires) where ζ = x, y, z. Here M (Q) represents the component of the Fourier transform of the magnetisation of the sample, which is perpendicular to the scattering vector Q - i.e. the neutron sensitive part. σ ζ are the Pauli spin matrices σ x = 0 1, σ 1 0 y = 0 i, σ i 0 z = Substitution of these into the magnetic potential gives us the matrix elements S ʹ V m (Q) S = γ nr 0 2µ B M z (Q) M z (Q) M x (Q) im y (Q) M x (Q) + im y (Q) Non-spin-flip Spin-flip

18 Magnetic scattering rule The non-spin-flip scattering is sensitive only to those components of the magnetisation parallel to the neutron spin The spin-flip scattering is sensitive only to those components of the magnetisation perpendicular to the neutron spin NB This is one of those points that you should take away with you. It is the basis of all magnetic polarization analysis techniques

19 Neutron polarization and nuclear scattering In general a bound state is formed between the nucleus and the neutron during scattering with either spins antiparallel (spin-singlet) or spins parallel (spin-triplet). The scattering lengths for these situations are different and are termed b - and b +. The scattering length operator is ˆ b = A + Bσ I A = (I +1)b + + Ib 2I +1, B = b + b 2I +1 (see e.g. Squires, p173) The calculation of the matrix elements now proceeds analogously to the case of magnetic scattering S ʹ b ˆ S = A + BI z A BI z B(I x ii y ) B(I x + ii y ) Since the nuclear spins are (normally) random Non-spin-flip Spin-flip I x = I y = I z = 0 b Therefore with the coherent scattering amplitude proportional to, we can write b = A i.e. the coherent scattering is entirely non-spin-flip

20 Moon-Riste-Koehler Equations Bringing all this together, we get = b γ nr 0 2µ B M z + BI z = b + γ n r 0 2µ B M z BI z = γ r n 0 ( M x im y ) + B( I x ii y ) 2µ B = γ r n 0 ( M x + im y ) + B( I x + ii y ) 2µ B Remember that: ~M? ( Q)= ~ ˆQ ~M( Q) ~ ˆQ = M( ~ Q) ~ Moon, Riste and Koehler (Phys Rev 181 (1969) 920) ~M( Q) ~ ˆQ ˆQ If the polarization is parallel to the scattering vector, then the magnetisation in the direction of the polarization will not be observed since the magnetic interaction vector is zero. i.e. all magnetic scattering will be spin-flip

21 Spin-incoherent scattering Now, let s take another look at the nuclear incoherent scattering. We know that this is given by b 2 b ( ) 2 Applying this to the b 2 = transition, and neglecting magnetic scattering, we get ( b + BI z ) 2 = ( b ) 2 + B 2 2 I z + 2 b BI z Now, for a randomly oriented distribution of nuclei of spin I, we have I = I(I +1) = I x 2 + I y 2 + I z 2 Therefore we can write I 2 x = I 2 y = I 2 z = 1 I(I +1) 3 since the distribution is isotropic b 2 ( b ) 2 = ( b ) 2 ( b ) B2 I(I +1) Isotope incoherent scattering spin incoherent scattering The other transitions are dealt with in a similar way

22 Finally, we get Moon-Riste-Koehler II = b γ nr 0 2µ B M z + b II b SI = b + γ n r 0 2µ B M z + b II b SI = γ r n 0 ( M x im ) y + 2 2µ B 3 b SI = γ r n 0 ( M x + im ) y + 2 2µ B 3 b SI where b II = ( b ) 2 ( b ) 2 b SI = B 2 I(I +1) The details of the magnetic scattering will in general depend on the direction of the neutron polarization with respect to the scattering vector, and also on the nature of the orientation of the magnetic moments

23 Scientific Examples

24 Polarized magnetic diffraction For a ferromagnetic sample aligned in a field perpendicular to the scattering vector we have ~M? ( ~ Q)= ~ M( ~ Q) ~M( Q) ~ ˆQ ˆQ = M( ~ Q) ~ and M has no component in the xy-plane, so that the spin-flip scattering is zero. This implies that we don t need to analyse the neutron spin, it will always end up in the same direction it started in. Therefore dσ dω = [ F N (Q) F M (Q)] 2 dσ dω = [ F N (Q) + F M (Q)] 2 where for neutrons polarized parallel to the field for neutrons polarized antiparallel to the field i F N (Q) = b i exp(iq r i ) i F M (Q) = γ n r 0 g J i J i f i (Q)exp(iQ r i ) Notice that to simulate an unpolarized measurement, we simply average the two polarized cross sections dσ dω = 1 2 F (Q) F (Q) N M = F 2 N (Q) + F 2 M (Q) NB we have neglected incoherent scattering here [( ) 2 + ( F N (Q) + F M (Q)) 2 ]

25 Polarized magnetic diffraction Using a spin flipper to access these two polarized cross sections we can determine the flipping ratio, R, of a particular Bragg reflection: R = ( dσ dω) dσ dω [ = F (Q) + F (Q) N M ] 2 ( ) [ F N (Q) F M (Q)] = 1+ γ 2 1 γ with γ = F M (Q) F N (Q) 2 Ni So, for example, in the case of Ni we measure a flipping ratio of 1.7 at the (111) reflection and 1.1 at the (400) reflection After all the reflections have been measured, FM(Q) can be deduced (assuming careful measurements of FN(Q) have been taken - at low fields/high temps). Then FM(Q) can be inverse Fourier transformed to get the real-space magnetisation density

26 D3, ILL Hot neutron 2-axis diffractometer at the ILL Crystal mounted on lowtemperature goniometer to access reflections out of the equatorial plane Other PND instruments at LLB, Oak Ridge, SINQ, FRM-II...

27 Form factor measurements Crystal structure P Javorsky, et. al., Phys. Rev. B (2003) P. Javorsky et al., Phys. Rev. B 67 (2003) ! UPtAl! crystal structure! Derived from Bragg peak positions/intensities Magnetisation density Derived from inverse Fourier transform of Fm(Q)

28 Spin density measurements SrFe2As2 Y Lee, et. al., Phys Rev B 81, R (2010)

29 Polarization analysis - paramagnets It can be shown (see Squires p 179) that in the case of a fully disordered paramagnet these expressions reduce to dσ dω dσ dω ζ NSF ζ SF = 1 3 = 1 3 γr 0 2 γr g 2 f 2 (Q)J(J +1) 1 P ˆ Q ˆ ( ) 2 g 2 f 2 (Q)J(J +1) 1+ P ˆ Q ˆ ( ) 2 where we have replaced the z-direction with the general direction ζ = x, y, or z Therefore, the scalar polarization becomes is given by h1 ( ˆP ˆQ) 2 i h1+(ˆp ˆQ) 2 i P = h1 ( ˆP ˆQ) 2 i + h1+(ˆp ˆQ) 2 i This is easily simplified to give the Halpern-Johnson Equation P 0 = ˆQ (P ˆQ) first derived in 1939, and valid for all paramagnetic and disordered magnets

30 Halpern-Johnson Equation P 0 = ˆQ (P ˆQ) where P` is the scattered polarization direction and P is the incident polarization direction We can immediately see that setting the polarization direction along the scattering vector has the desired effect of rendering all the magnetic scattering in the spin-flip cross-section. Now we suppose that we have a multi-detector in the x-y plane. In this case the unit scattering vector is z cosα where α is the angle between ˆ Q = sinα Q and an arbitrary x-axis 0 - the Schärpf angle α x y Q

31 The Schärpf Equations Substituting this unit scattering vector into the Halpern-Johnson Equation, and directing P in three orthogonal directions, x, y and z, leads to six cross sections (3 non-spin flip and 3 spinflip) Including the nuclear coherent, isotope incoherent and spin-incoherent terms we have dσ dω dσ dω dσ dω dσ dω NSF X SF X NSF Y dσ dω dσ dω SF Y NSF Z SF Z = 1 2 sin2 α dσ dω Schärpf and Capellmann Phys Stat Sol A135 (1993) 359 mag + 1 dσ 3 dω SI + dσ dω = 1 ( 2 cos2 α +1) dσ + 2 dσ dω mag 3 dω SI = 1 2 cos2 α dσ + 1 dσ + dσ dω 3 dω dω = 1 2 sin2 α +1 mag dω ( ) dσ = 1 dσ 2 dω = 1 dσ 2 dω mag mag mag + 1 dσ 3 dω + 2 dσ 3 dω SI + 2 dσ 3 dω SI SI SI + dσ dω nuc +II nuc +II nuc +II x-direction y-direction z-direction

32 D7, ILL Cold neutrons (to avoid too much Bragg scattering) Can be used as a diffuse scattering diffractometer or a cold time-of-flight spectrometer JRS, J. Appl. Cryst 42 (2009) 69 Other wide angle polarized instruments at FRM-II (DNS) NIST (Macs) - others coming soon

33 Supermirrors Supermirror bender analyser array on D7, ILL. There is over 250 m2 of supermirror in the full analyser array. (c.f. doubles tennis court is 260 m2) D7, ILL JRS, J. Appl. Cryst 42 (2009) 69

34 Polarized magnetic diffraction - powders A difference map between parallel and antiparallel cross-sections leaves the nuclear-magnetic interference term d d = d d * d d + =4F N (Q)F M (Q)) In the case of ferrimagnets, where some Bragg reflections are due to entirely one sublattice, this can lead to positive and negative peaks in the difference pattern. Fe3S4 - Greigite (NB can t warm above Tc) Chang, et. al., J Geophysical Res. 114 B07101 (2009)

35 Diffuse scattering examples Spin-Ice Frustrated ferromagnetic system with strong CF anisotropy Ho2Ti2O7 monopoles in H Fig. 2. Diffuse scattering maps from spin ice, Ho Ti O. Experiment [(A) to(c)] versus theory [(D) to T. Fennell, et al. Science 326, 415 (2009)

36 Polymer diffraction PI-h8 PI-d8 PI-d5 PI-d3 Polyisoprene: (CH 2 CH = C(CH 3 )CH 2 ) n Alvarez, et. al. Complete separation of SI scattering Internal normalisation (inc. D-W factor) Careful analysis of multiple scattering Close comparison with MD simulations Q (Å -1 ) Alvarez, et.al., Macromolecules 36 (2003) 238

37 Inelastic magnetic scattering Pyrochlore - Tb2Sn2O7 Rule, et. al., Phys. Rev. B (2007)

38 Science with Polarized Neutrons S(Q,t)/S(Q,0) Glass transition in polymer-glass, polybutadiene A. Arbe et al Phys. Rev. E 54, 3853 (1996) Magnetic slow-relaxation in spin-ice S(Q,t)/S(Q,0) Ehlers et al, J Phys: Condens Matter 18, R231 (2006)