Neutron Interferometry F. E. Wietfeldt Fundamental Neutron Physics Summer School 2015 June 19, 2015 Michelson Interferometer Mach-Zender Interferometer The Perfect Crystal Neutron Interferometer Perfect Crystal LLL Neutron Interferometer Bragg condition: nλ = 2d sinθ d = lattice spacing Perfect Crystal LLL Neutron Interferometer Perfect Crystal LLL Neutron Interferometer Perfect Crystal LLL Neutron Interferometer Nuclear Phase Shift Nuclear Phase Shift Nbλ 2 index of refraction: n = 1− 2π relative phase shift: D Δχ = k0l − nk0l = Nbλ cosθ Question: Why, in transparent matter, is the index of refraction of light usually n > 1 and for neutrons usually n < 1?! Interferogram Interferogram O beam: IO = A[1+ f cos(χ2 − χ1 )] H beam: I H = B − A f cos(χ2 − χ1 ) C − C contrast f = max min (O-beam) Cmax + Cmin Precision Phase Shift Measurement D Δχ = Nbλ cosθ Example: aluminum sample, λ = 2.70A& , 111 reflection: D = 100 µm ⇒ Δχ = 2π Non-Dispersive Geometry D path length l = sinθ Δχ = 2NbdD independent of λ Question: Why, in transparent matter, is the index of refraction of light usually n > 1 and for neutrons usually n < 1?! Question: Why, in transparent matter, is the index of refraction of light usually n > 1 and for neutrons usually n < 1?! Answer: Special relativity! light! neutron! E = pc E = p2c2 + m2c4 dω dE E ω v = = = = dω dE pc2 pc2 dk dp p k v = = = ≈ = h k dk dp E mc2 m 1 v ∝ v ∝ k k k n ≡ k0 net phase shift: Θ(ε0 ,γ 0 ) = 248π + 0.455(7) radians bcoh = 4.15041(21) fm Skew-Symmetric Neutron Interferometer NIST perfect crystal silicon interferometers S18 Neutron Interferometer at the Institut Laue-Langevin Kinematic Bragg Diffraction Kinematic Bragg Diffraction Kinematic Bragg Diffraction Kinematic Bragg Diffraction Dynamical Diffraction Theory r H = Bragg vector r 2πn H = d r K0 = internal forward scattered wave r K = internal Bragg scattered wave H Bragg condition: r r r K H − K0 = H Solve Schrödinger Eqn. inside crystal: 2 2 r r r (∇ + k0 )Ψ(r ) = v(r )Ψ(r ) r r with v(r ) = 4π b δ r − r = v eiHn ⋅r ∑ i ( i ) ∑ Hn i n Dynamical Diffraction Theory 2 2 2 2 2 Dispersion Equation: (K − K0 )(K − K H ) = vH 2 vH quadratic equation approximate: K − K K − K = ( 0 )( H ) 2 2 solutions for K 4k0 0 Dynamical Diffraction Theory Dynamical Diffraction Theory r α r r β r r α r r β r internal wave function: Ψ(r ) = ψ α eiK0 ⋅r + ψ β eiK0 ⋅r + ψ α eiKH ⋅r + ψ β eiKH ⋅r 0 0 H H Dynamical Diffraction Theory 1 ⎡ y ⎤ ψ α = ⎢1− ⎥ A 0 2 2 0 ⎣⎢ 1+ y ⎦⎥ 1 ⎡ y ⎤ ψ β = ⎢1+ ⎥ A 0 2 2 0 ⎣⎢ 1+ y ⎦⎥ 1 ⎡ 1 ⎤ ψ α = − ⎢ ⎥ A H 2 2 0 ⎣⎢ 1+ y ⎦⎥ 1 ⎡ 1 ⎤ ψ β = + ⎢ ⎥ A H 2 2 0 ⎣⎢ 1+ y ⎦⎥ k sin2θ y = 0 B δθ 2ν H misset parameter r α r r β r r α r r β r internal wave function: Ψ(r ) = ψ α eiK0 ⋅r + ψ β eiK0 ⋅r + ψ α eiKH ⋅r + ψ β eiKH ⋅r 0 0 H H Dynamical Diffraction Theory r r r r r ik0 ⋅r ikH ⋅r Transmitted wave: Ψtrans (r ) = ψ tr 0 e + ψ tr H e ⎡ iy ⎤ ν D ν D i(φ1 −φ0 ) 0 H ψ = ⎢cosΦ − sinΦ⎥e A φ0 = , φ1 = tr 0 2 0 cosθ cosθ ⎣⎢ 1+ y ⎦⎥ B B ⎡ ⎤ with ⎛ 1 ⎞ D −iy −i(φ +φ ) 1 0 Φ = ν H ψ tr H = ⎢ sinΦ⎥e A0 ⎜ 2 ⎟ 2 1+ y cosθB ⎣⎢ 1+ y ⎦⎥ ⎝ ⎠ 2 2 2 ⎡ 2 y 2 ⎤ I0 = ψ tr 0 = A0 ⎢cos Φ + 2 sin Φ⎥ ⎣ 1+ y ⎦ Transmitted intensities: 2 2 ⎡ 1 2 ⎤ I H = ψ tr H = A0 ⎢ 2 sin Φ⎥ ⎣1+ y ⎦ Transmitted Intensities For the (111) reflection in Si at λ=2.70 Å: y = 1 → 0.9 arcsec Some Consequences of Dynamical Diffraction • Pendellösung interference ⎛ 1 ⎞ D Φ = ν ⎜ H 2 ⎟ ⎝ 1+ y ⎠ cosθB • Anomalous transmission • Angle amplification Angle Amplification Ω For small δ (~10-3 arcsec): ≈ 106 δ Practical Neutron Interferometer 4π Rotational Symmetry of Spinors i r − αnˆ⋅S h Rotation operator: Rnˆ (α) = e α ˆ r r 1 −i n⋅σ Spin-1/2 particle: r 2 S = 2 hσ so Rnˆ (α) = e ⎛ e−iα /2 0 ⎞ Rotations about z-axis: Rz (α) = ⎝⎜ 0 eiα /2 ⎠⎟ Rz (2π )χ = −χ Symmetry: Rz (4π )χ = χ Larmor precession phase: Δφ = ±2πµ m λB / 2 n n l h Quantum Phase Shift Due To Gravity (COW Experiments) 2πλgA Δφ = m m m = neutron inertial mass h2 in grav in m = neutron gravitational mass A = H = area of parallelogram grav l test of weak equivalence principle at the quantum limit 2πλgA Δφ = 0 m m sinα = qsinα grav h2 in grav A0 = area of parallelogram at α = 0 a measured: q = 54.3 theory: q = 59.6 Systematic Effects in the COW Experiments 1 2 q = ⎡ q (1+ ε) + q + q2 ⎤ 2 COW ⎣( grav bend ) Sagnac ⎦ ’ dynamical bending of Earth s rotation diffraction interferometer correction 2min r r Sagnac effect: ΔφSagnac = Ω ⋅ A due to Earth's rotating frame h bending effect: repeat experiment with x rays, different wavelengths Littrell, et al. (1997) results: q theory q meas. discrepancy experiment COW COW [rad] [rad] (%) SS, 440 50. 97(5) 50.18(5) -1. 6 data from Werner, et al. (1988) SS, 220 100. 57(10) 99. 02(10) -1. 5 LLL, 440 113. 60(10) 112. 62(15) -0. 9 LLL, 220 223. 80(10) 221. 85(30) -0. 9 Layer and Greene (1991): x rays do not fill the Borrmann fan as completely as neutrons Possible improvement (A. Zeilinger, S. Werner, FEW, et al.): Suspend interferometer inside chamber filled with ZnBr2+D2O (floating COW) Precision Neutron Interferometric Measurements of Few-Body Neutron Scattering Lengths R. Haun, C. Shahi, F.E. Wietfeldt Tulane University M. Arif, W.C. Chen, T. Gentile, M. Huber, D.L. Jacobson, D. Pushin, S.A. Werner, L. Yang NIST T. C. Black University of North Carolina, Wilmington H. Kaiser, K. Schoen University of Missouri-Columbia W. M. Snow Indiana University Semi-phenomological nucleon-nucleon potential model AV18 Great success with NN scattering lengths, but unable to predict 3He, T binding energies Data from Wiringa et al., Phys. Rev. C 51, 38 (1995) NN Potential Models π π π π π Motivation •Precision few-body neutron scattering lengths provide an additional challenge for nuclear potential models. •Few body nuclear effective field theories (EFT) require precision experimental measurements to constrain short-range mean field potentials. Precision neutron interferometric measurement of the n-D coherent scattering length at NIST (2003) bc = 6.6727 ± 0.0045 fm Schoen, et al., Phys. Rev. C 67, 044005 (2003) Precision neutron interferometric measurement of the n-3He coherent scattering length at NIST (2004) bc = 5.8572 ± 0.0072 fm Huffman, et al., Phys. Rev. C 70, 014004 (2004) n-3He Scattering Lengths 3.8 ILL (2004) 3.6 ILL (2002) NIST (2004) AV18 3.4 (fm) 1 a ILL (1979) AV18+UIX R-matrix 3.2 3.0 7.0 7.2 7.4 7.6 7.8 a0 (fm) A measurement of the n-3He spin-incoherent scattering length at NIST (2008-2014) Spin-dependent neutron scattering 2bi r r total scattering length: b = bc + I ⋅σ n I(I + 1) I + 1 I coherent: b = b + b c 2I + 1 + 2I + 1 − I(I + 1) incoherent: b = b − b i 2I 1 ( + − ) + Polarized 3He gas target: Spin Exchange Optical Pumping Spin is transferred from optically polarized alkali atoms to 3He nuclei via the hyperfine interaction in collisions. The cell is polarized offline and then transferred to the neutron interferometer. Polarized 3He Cells Target cells: •Boron-free GE-180 glass •4 mm flat windows •40 mm long, 25 mm dia. 3 •1.5 atm He (with 4% N2) Two cells: “Pistachio” (115 hours) “Cashew” (35 hours) Measuring the Scattering Length 2(φ ↑ − φ ↓ ) b − b = − + − N λz P 3 3 Measuring the Scattering Length 2(φ ↑ − φ ↓ ) b − b = − + − N λz P 3 3 Measuring the Scattering Length 2(φ ↑ − φ ↓ ) b − b = − + − N λz P 3 3 measured simultaneously from the asymmetry in counter C4 Measuring N λz P 3 3 Measuring N3λz P3 ↑ ↓ 1 N − N (1+s)Pn tanhx C4 asymmetry = = 2 N ↑ N ↓ 1 + 1+ 2 (1− s)Pn tanh x ⎛ σ − σ ⎞ x = 0 1 N λz P ⎜ 4λ ⎟ 3 3 ⎝ th ⎠ Measuring N3λz P3 3 P3 = He polarization Pn = neutron polarization (flipper off) P (flipper on) s = n Pn (flipper off) ↑ ↓ 1 N − N (1+s)Pn tanhx C4 asymmetry = = 2 N ↑ N ↓ 1 + 1+ 2 (1− s)Pn tanh x ⎛ σ − σ ⎞ x = 0 1 N λz P ⎜ 4λ ⎟ 3 3 ⎝ th ⎠ Measuring N3λz P3 3 P3 = He polarization Pn = neutron polarization (flipper off) P (flipper on) s = n P (flipper off) n 3 He (n, p) cross section: 1 3 σ = σ + σ = 5333(7) barns th 4 0 4 1 ↑ ↓ 1 N − N (1+s)Pn tanhx C4 asymmetry = = 2 σ N ↑ N ↓ 1 1 ≈ 0 − 2 × 10−3 + 1+ 2 (1− s)Pn tanh x σ 0 (Hofmann and Hale, 2003) ⎛ σ − σ ⎞ x = 0 1 N λz P the dominant ⎜ 4λ ⎟ 3 3 ⎝ th ⎠ systematic error in this experiment Neutron Polarimetry Use optically thick (Nσz ~ 3) 3He cell, with known polarization, in place of neutron interferometer. Measure neutron count rate with both flip states and both directions of P3 Pn = 0.9291 ± .0008 s = .9951 ± .0003 Neutron Polarization Correction ⎛ sin Δφ ⎞ ⎛ η↑ sin Δφ ⎞ Δφ = arctan − arctan meas.