Introduction to neutron reflection Adrian Rennie Outline - PowerPoint PPT Presentation

Introduction to neutron reflection Adrian Rennie

Outline Inteference of waves Refractive index Critical angle, total reflection

Reflection Light oil water

Reflection Light oil water

Reflection and Refraction: Snell’s Law For specular reflection: Optical Notation  i =  r  Beam  i   r  n 1 Transmitted beam is refracted:  t  n 2 n 2 sin  t = n 1 sin  i n is refractive index

Reflection and Refraction: Snell’s Law For specular reflection: Neutron Reflection Notation  i =  r Beam Transmitted beam is n 1  i   r  refracted:  t  n 2 n 2 cos  t = n 1 cos  i n is refractive index

Reflection – measured quantities Reflected beam Reflection deflected:  Reflectivity I 0 (  ) I R R(Q) = I R /I 0 (  ) Momentum transfer   Q = (4  /  ) sin 

Demonstration Calculations www.ncnr.nist.gov/instruments/magik/calculators/reflectivity-calculator.html www.ncnr.nist.gov/instruments/magik/calculators/magnetic-reflectivity-calculator.html

Critical Angle and Below (critical wavelength and above) Density difference between two bulk phases determines the critical Reflectivty - linear scale momentum transfer/angle, Q c or  c 1.2 1 Any variation in intensity below critical 0.8 R(Q) 0.6 angle is probably telling you about the 0.4 experiment rather than the interface 0.2 0 0.00 0.01 0.02 0.03 0.04 R (Q) = 1 for  <  c is often used as a -1 Q / Å calibrant R(Q) ~ 1/Q 4 for sharp interface Total reflection below critical angle  cos  = n 2 /n 1

Calculating Refractive Index Neutrons n = 1 – (  2  i b i /V / 2  ) λ is the wavelength  i b i is the sum of scattering lengths in volume V b is known for most stable nuclei  i b i /V

Scattering Lengths of Nuclei Nucleus Scattering Length / fm 1 H -3.741 2 H (or D) 6.675 C 6.648 O 5.805 Si 4.151 Cl 9.579 Source: H. Rauch & W. Waschkowski

Properties of Common Materials Material Scatt. Length Density Refractive / 10 -6 Å -2 index at 10 Å H 2 O -0.56 1.000009 D 2 O 6.35 0.999899 Si 2.07 0.999967 Air 0 1.000000 Polystyrene 1.4 0.999971

Contrast in a Thin Film 1.0E+01 1.0E+00 Calculation for Neutrons 1.0E-01 1.0E-02 100 Å layer with  =1, 3 & 5 x 10 -6 Å -2 Reflectivity 1.0E-03 on Si (  =2.07 x 10 -6 Å -2 ) 1.0E-04 1.0E-05 1.0E-06 Increasing contrast changes visibility of 1.0E-07 fringes 0.00 0.05 0.10 0.15 0.20 0.25 Q / A -1 Phase change makes large difference 6 5 Fringes (Kiessig fringes) – spacing 4  (z) 1e-6 A -2 indicates film thickness for a single layer. 3 2 1 0 -100 -50 0 50 100 150 200 Z / A

Roughness Reflectivity from rough surfaces is decreased. 1.0E+00 1.0E-04 1.0E-05 1.0E-01 1.0E-06 1.0E-02 1.0E-07 R(Q) 1.0E-03 1.0E-08 0.1 0.15 0.2 0.25 1.0E-04 Smooth 1.0E-05 4A Rough 1.0E-06 0.00 0.05 0.10 0.15 0.20 Q / A -1 L. Nevot, P. Crocé J. Phys. Appl. 15 , T61 (1980)

Intensity of Reflected Signal Waves interfere constructively for 2 d sin  =  2  3  ... (Bragg’s law) Measured reflectivity will depend on angle and wavelength. Total reflection for angles less than critical angle,  c = arccos(n 1 /n 2 )

Useful Physical Ideas Models for complex interfaces can be constructed from multiple thin layers of different refractive index, n or scattering length density,  .

Useful Physical Ideas Isotopes (e.g. D/H substitution) can be used to label particular species or alter contrast Neutrons have spin – effectively a field dependent contribution to scattering length

Abeles Optical Matrix Method     i i e r e   1  1 1 j j   j  r     j   i i r e e    1   1 1 j j j        sin ( 2 / ) sin p n n d j j j The picture can't be displayed. j j j j     ( ) /( ) [ ][ ]...[ ] r p p p p M M M M    1 1 1 2 1 j j j j j R n  ( ) * / * R Q M M M M 21 21 11 11

Magnetic Contrast B b m =  0 e 2 S  / 4  m e e, electronic charge Neutron m e , electron mass S, spin b tot = b nuclear + b m  0, Permeability of free space  , gyromagnetic ratio, 1.913 b tot = b nuclear ± b m

Magnetic Contrast B b m =  0 e 2 S  / 4  m e e, electronic charge Neutron m e , electron mass b tot = b nuclear - b m S, spin  0, Permeability of free space  , gyromagnetic ratio, 1.913 b tot = b nuclear ± b m

Scattering and Reflection  (Q) is Fourier transform of the  2 16 2   scattering length ( ) ( ) R Q Q 2 Q density distribution normal to the interface,   (z)      iQz ( ) Q ( ) z e dz  For sharp interface: R(Q) ~ 1/Q 4

Partial Structure Factors Interface consists of distinct components: 1, 2, 3     16 16 16 16 2 2 2 2         ( ( ( ( ) ) ) ) | | | | ( ( ( ( ) ) ) ) | | | | 2 2 2 2 iQz iQz iQz iQz R R R R Q Q Q Q z z z z e e e e dz dz dz dz     2 2 2 2 Q Q Q Q                 ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) ) ( ( ( ( ) ) ) ) z z z z b b b b n n n n z z z z b b b b n n n n z z z z b b b b n n n n z z z z 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 2 2 2    16 16 16 2 2 2 2 2 2 2 2 2                   2 2 2 ) ) ) ( ( ( ) ) ) ( ( ( 2 2 2 2 2 2 b b b b b b h h h R R R Q Q Q b b b h h h b b b b b b h h h b b b h h h b b b b b b h h h b b b h h h 3 1 31 11 11 11 1 1 1 2 2 2 12 12 12 22 22 22 2 2 2 3 3 3 23 23 23 33 33 33 3 3 1 1 31 31 1 1 1 2 2 2 3 3 3 2 2 2 Q Q Q h ij are transforms of n i n j – pair correlation functions h ij are transforms of n i n j – pair correlation functions Lu, J. R.; Thomas, R. K.; Penfold, J. Adv. Coll. Inter. Sci. 2000 , 84, 143-304.

Practical Aspects of Neutron Reflection How to Collect Data Adrian R. Rennie

Reflection – measured quantities Reflected beam deflected: Reflection   Reflectivity I 0 (  ) I R R(  ) = I R (  ) /I 0   Momentum transfer Q = (4  /  ) sin 

Best Sources of Neutrons ILL reactor continuous Thermal Flux 1.5 x 10 15 n cm -2 s -1 SNS, ORNL 60 Hz, 300  s 5 x 10 17 n cm -2 s -1 (Peak)

Neutrons: Speed & Wavelength Velocity, v, from de Broglie relation v λ = 3956 m s -1 Å i.e. 10 Å has 400 m s -1 Gravity is significant, separate wavelengths mechanically

Using a Pulsed Source Time Source Sample Detector  = 1 / f Distance Detection time (after source pulse) gives wavelength Choppers can select a wavelength

D17 Reflectometer FLIPPER SLIT FILTER MONOCHROMATOR SLIT ATTENUATORS FOCUSING GUIDE CHOPPER CASING

Practical Issues Reflectivity drops quickly with increasing Q (or angle). Signal is easily ‘lost’ in background. To observe fringes it will be necessary to measure over an appropriate range of Q and to have sufficient resolution (  Q small enough).

Reflection from a Thin Film Model calculation on 1.0E+01 1.0E+00 smooth surface. 1.0E-01 1.0E-02 Reflectivity 1.0E-03 Fringe spacing depends 1.0E-04 on thickness 1.0E-05 1.0E-06 1.0E-07 Fringe spacing ~ 2  /d 0.00 0.05 0.10 0.15 0.20 0.25 Q / A -1 Model layer with  = 5 x 10 -6 Å 2 on Si (2.07 x 10 -6 Å -2 ) Blue 30 Å, Pink 100 Å. No roughness.

Resolution in Q Q = (4  /  ) sin  Depends on  and  s 1 s 2 Angle resolution,  , depends n on collimation (slits)  Wavelength resolution depends on monochromator or time resolution in measuring neutron d pulse Higher Resolution = Lower Flux (  Q/Q) 2 = (  ) 2 + (  ) 2

Effects of Resolution 0 1% 3%  Q/Q -1 5% 7% log 10 R -2 -3 -4 0.00 0.05 0.10 Q / Å -1 Silicon substrate: film thickness 1500 Å (150 nm) scattering length density 6.3 × 10 −6 Å -2

Sample Holder D17 reflectometer ILL, France Temperature Sample sensor outlet n PTFE sample holder 5 cm Aluminium cell holder Back Reflection surface surface (silicon) Sample (silicon or inlet sapphire)

Alignment Rotation table must have centre on beam axis Sample must be centred on rotation (half obscure the direct beam) – eucentric mount Determine  from the position of beam on a detector

Aligning a Sample Detector z  i  r x  Design mount with surface at centre of rotation of  Eucentric mount. Put centre of surface on the line through axis of rotation (x direction) The rotation  stage must be centred on the incident beam.