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Neutron Interferometry Neutron Interferometry F. E. Wietfeldt Summer School on Fundamental Neutron Physics June 22-26, 2009 Michelson Interferometer Mach-Zender 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 = k0 − nk0 = Nbλ cosθ 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 m 2 = µ ⇒ Δχ = π Non-Dispersive Geometry D path length = sinθ Δχ = 2NbdD independent of λ Perfect Crystal LLL Neutron Interferometer 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 H = Bragg vector 2πn H = d K0 = internal forward scattered wave K = internal Bragg scattered wave H Bragg condition: K − K = H H 0 Solve Schrödinger Eqn. inside crystal: ∇2 + k 2 Ψ(r ) = v(r )Ψ(r ) ( 0 ) 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 α β α β internal wave function: (r ) α eiK0 ⋅r β eiK0 ⋅r α eiKH ⋅r β eiKH ⋅r Ψ = ψ 0 + ψ 0 + ψ H + ψ 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 α β α β internal wave function: (r ) α eiK0 ⋅r β eiK0 ⋅r α eiKH ⋅r β eiKH ⋅r Ψ = ψ 0 + ψ 0 + ψ H + ψ H Dynamical Diffraction Theory ik ⋅r ik ⋅r Transmitted wave: Ψ (r ) = ψ e 0 + ψ e H trans tr 0 tr H ⎡ 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 − αnˆ⋅S Rotation operator: Rnˆ (α) = e α ˆ 1 −i n⋅σ Spin-1/2 particle: S = σ so R (α) = e 2 2 nˆ ⎛ 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: 2 m B / 2 Δφ = ± πµn nλ Quantum Phase Shift Due To Gravity (COW Experiments) 2πλgA Δφ = m m m = neutron inertial mass h2 in grav in mgrav = neutron gravitational mass A = H = area of parallelogram 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 Sagnac effect: ΔφSagnac = Ω ⋅ A due to Earth's rotating frame 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 Upcoming new effort (H. Kaiser, S. Werner, FEW, et al.): Suspend interferometer inside chamber filled with ZnBr2+D2O (floating COW) Measuring the Neutron's Mean Square Charge Radius Using Neutron Interferometry F. E. Wietfeldt, M. Huber Tulane University, New Orleans, USA M. Arif, D. L. Jacobson, S. A. Werner National Institute of Standards and Technology, Gaithersburg, USA T. C. Black University of North Carolina, Wilmington, USA H. Kaiser Indiana University, Bloomington, USA neutron: neutral but consists of charged quarks neutron mean square charge radius: r2 = ρ(r) r2d 3r n ∫ expected to be negative (positive core, negative skin): Fermi and Marshall, 1947 Neutron Electric Scattering Form Factor n 2 = Fourier transform of neutron charge density (Breit frame) GE (Q ) Expanding in momentum transfer Q2 : 1 G n (Q2 ) = q − r 2 Q2 + ... E n 6 n In the low Q2 limit: dG (Q2 ) r2 = −6 E n 2 dQ 2 Q =0 2 2 rn constrains the slope of GE(Q ) in electron scattering experiments and theory (e.g. Bates, Jefferson Lab) V. Ziskin Ph.D. thesis, 2005 Neutron-Atom Coherent Scattering Length bcoh = bN + Z [1- f (q)]bne Fourier transform of charge density 1 iq ⋅ r 3 f (q) = e ρatom (r)d r 2π ∫ bne = neutron-electron scattering length In 1st Born approximation: 2 ⎛ me ⎞ rn = 3a0 ⎜ ⎟ bne = (86.34 fm)bne ⎝ mn ⎠ Foldy Scattering Length 2 γ e −3 bF = − 2 = −1.468 × 10 fm from neutron's magnetic moment 2mec Incorrect interpretation: bne (meas.) = bintrinsic + bF Correct interpretation: The experimentally measured value of bne is entirely due to the static charge distribution in the neutron. [N. Isgur, Phys. Rev. Lett. 83, 272 (1999)] Previous Experiments Neutron Interferometer Experiment off Bragg: bcoh = bN + Z [1− f (0)]bne = bN near Bragg: b b Z [1 f (H )]b coh = N + − 111 ne Dynamical Phase Shift Through Bragg ν ΔΦ = H y ± 1+ y2 D D = crystal thickness dyn ( ) cosθB k sin2θ scaled misset angle y = B 2ν H F111λ 32λ ν H = = bcoh Vcell Vcell near Bragg: b b Z [1 f (H )]b coh = N + − 111 ne What we must measure: 1. Net dynamical phase shift through Bragg b Z[1 f (H )]b → νH → N + − 111 ne to ~10-5 The maximum slope is ~ 88π / arcsec so we need 0.01 arcsec angular precision to detect every 2 π of phase shift -5 2. Forward phase shift off Bragg → bN to ~10 and subtract 3. Neutron wavelength to ~10-3 4. Calculate f ( H ) to ~10-3 111 2 This will give bne, and hence 〈rn 〉, to < 1% Tulane-NIST neutron charge radius experiment 10 cm lever with nylon flexure bearing Physik Instrumente P-753 PZT nanopositioner 25 mm range 1.0 nm precision (.002 arcsec) Four Micro-E mercury rotation encoders .010 arcsec precision Preliminary Data: These data were taken at NIST in September 2005 Precision Neutron Interferometric Measurements of Few-Body Neutron Scattering Lengths F.E. Wietfeldt, M. Huber, P. Hao Tulane University D.L. Jacobson, M. Arif, T. Gentile, W.C. Chen, D. Pushin, P.R. Huffman, S.A. Werner NIST T. C. Black University of North Carolnia, 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 A new measurement of the n-3He spin-incoherent scattering length at NIST (2008) Spin-dependent neutron scattering 2bi 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.
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