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Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method

Crystallography using X-Ray based methods

Jakob Odersky Simon L¨

• we

EPFL

December 19, 2012

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Table of Contents

1

Basic properties of X-Rays Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

2

Identifying a substance using a polycrystalline method

3

Identifying a substance using a monocrystalline method

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Why X-Rays?

Why X-Rays? Penetration of “soft matter” Stopped by “hard matter“ Wavelength same order as atom dimension (≃ ˚ A) → Cristal diffraction

Figure: http://www.2mcctv.com/blog/wp-content/uploads/2012/10/electromagnetic-spectrum.jpg,

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Production of X-Rays

Production of X-Rays Emission of electrons by a heated cathode (e.g. Tungstene) Acceleration due to an external potential V0 Induction of electron transitions in anode (Cu, Mo, ...) Emission of X-Rays

Figure: http://en.wikipedia.org/wiki/File:WaterCooledXrayTube.svg, 15.12.12

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Study of the spectrum

Two types of spectra Continuous spectrum → Strong deceleration of the electrons in the anode results in emission of Bremsstrahlung → Duane-Hunt law yields λmin(˚ A) = 12.398

V0(kV)

Discrete spectrum → Incoming electrons induce transition from lower energy state E1 to higher energy state E2 → Deexcitation process emits photon of wavelength λ =

hc E2−E1 = hc ∆E

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Study of the spectrum

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Elastic scattering from a single electron

Figure: Taken from the Petr Leiman’s lectures.

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Elastic scattering from a single electron

Electromagnetic wave scattering Given an incident wave for electron at re: E(re, t) = E0 exp(2πi(k0re − νt)) Scattered spherical wave at R for R ≫ re: E(R, t) = Eor exp(−2πiSre) where S = k − k0 the scattering vector Generalization to an extended object: F(S) = F[ρ](S) where ρ the electron density → the recorded diffraction pattern is an image of the reciprocal space

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Elastic scattering from a single electron

Figure: Taken from the Petr Leiman’s lectures.

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Diffraction

Diffraction Diffraction is scattering dominated by interference effects

Destructive interference in most directions Constructive interference in well defined directions

→ Problem: what are these directions? Appears when there is internal order in the sample → crystals

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

What is a crystal?

Crystal A crystal is characterized by : homogeneity invariance of the sides microscopic periodicity anisotropy of physical properties

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Crystal Structure

Figure: http://4.bp.blogspot.com/-NB-CjJtE2Zg/Tja397z4-II/AAAAAAAAAKY/qxOrzU2cQuY/s1600/

space-lattice-unit-cell-represenatation.jpeg, 17.12.12 Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Diffraction conditions

Laue Conditions and Bragg’s Law One can show that the diffraction condition are: Sa = h Sb = k Sc = l where h,k,l are the Miller indexes Furthermore, it is easy to show that the reciprocal lattice vectors correspond to the scattering vectors which can give rise to diffraction S = H ⇒ H = 2 sin θ

λ

and H = n(ha∗ + kb∗ + lc∗) ⇒ H = nrhkl =

n dhkl

→ This yields Bragg’s Law: 2dhkl sin θ = nλ

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Ewald Sphere

Figure: Taken from the Petr Leiman’s lectures.

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

A first application

Determining the thickness of a nickel sheet Absorption of X-Rays in matter: I(x) = I0 exp(−µx) where µ depends on the matter and the wavelength Knowing µ one can determine the thickness: dNi =

ln I0

I

• µ

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

A first application

Determining the thickness of a nickel sheet Absorption of X-Rays in matter: I(x) = I0 exp(−µx) where µ depends on the matter and the wavelength Knowing µ one can determine the thickness: dNi =

ln I0

I

• µ

For the Kβ1 transition of copper, the mass attenuation coefficient is µm ≃ 2.79 · 102[cm]2[g]−1 ⇒ µ = ρCuµm ≃ 2483.1[cm]−1 This yields: dthick = 25[µm] dthin = 6.39[µm]

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Production of X-Rays Study of the spectrum (X-Ray) Diffraction Crystals Determining the thickness of a nickel sheet

Determining the thickness of a nickel sheet

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Optical study of the substance Powder diffraction

Table of Contents

1

Basic properties of X-Rays

2

Identifying a substance using a polycrystalline method Optical study of the substance Powder diffraction

3

Identifying a substance using a monocrystalline method

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Optical study of the substance Powder diffraction

Powder diffraction

Experimental setup A polycrystalline substance, commonly called ”powder“ Powder is much simpler to synthesize It is easier to prepare

Figure: Notice TP, Diffraction des rayons X par les poudres, EPFL

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Optical study of the substance Powder diffraction

Optical study of the substance

Optical microscopy The first basic informations can be gathered visually: size (100 − 200[µm]) visual isotropy (not polarized → isotropic) form (cubic system)

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Optical study of the substance Powder diffraction

Powder diffraction

Calibration Calibration by mixing in a substance with well known spectral lines (Si) determining the spectral lines of Silicon → Knowing the lattice spacing and the Kα transition wavelength of the cathode, one obtains:

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method Optical study of the substance Powder diffraction

Systematic absences

Identifying systematic absences The structure factor is given by: F(h, k, l) = fj exp(2πi(hxj + kyj + lzj)) For cubic structure, this implies: h,k,l all even or all odd The invariance under 90◦ rotation plus 1

4c translation, implies

l = 4n Calculating the lattice spacing In the specific case of a cubic system, dhkl =

a √ h2+k2+l2

The experiment yields a number of dhkl Using the fact that

a2 d2

hkl = h2 + k2 + l2 ∈ N, one obtains the

lattice spacing → a = 12.1503˚ A → AlK(SO4)2(H2O)12

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method The precession chamber Determining the lattice parameters Determining the space group using the systematic absences Determining the chemical composition of the substance Determining the structure of the crystal

Table of Contents

1

Basic properties of X-Rays

2

Identifying a substance using a polycrystalline method

3

Identifying a substance using a monocrystalline method The precession chamber Determining the lattice parameters Determining the space group using the systematic absences Determining the chemical composition of the substance Determining the structure of the crystal

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method The precession chamber Determining the lattice parameters Determining the space group using the systematic absences Determining the chemical composition of the substance Determining the structure of the crystal

The precession chamber

Basic principle aim: record as many reciprocal lattice nodes as possible position monocrystal in X-Ray beam record diffraction pattern whilst precessing crystal and film → the rotation enables several nodes to pass the Ewald sphere → record several reciprocal lattice nodes

Figure: pixeltech-dz.com

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method The precession chamber Determining the lattice parameters Determining the space group using the systematic absences Determining the chemical composition of the substance Determining the structure of the crystal

The precession chamber

Determining the lattice parameters a and b geometry yields a∗ = a∗

λf

a = 1

a∗

repeat for b

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method The precession chamber Determining the lattice parameters Determining the space group using the systematic absences Determining the chemical composition of the substance Determining the structure of the crystal

The precession chamber

Determining the lattice parameters a and b

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method The precession chamber Determining the lattice parameters Determining the space group using the systematic absences Determining the chemical composition of the substance Determining the structure of the crystal

The precession chamber

Determining the lattice parameter c

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method The precession chamber Determining the lattice parameters Determining the space group using the systematic absences Determining the chemical composition of the substance Determining the structure of the crystal

The precession chamber

Determining the space group using the systematic absences Furthermore, results obtained in precession chamber show systematic absences → between (hk0) and (hk1) planes symmetries → space group, tabulated information → help in positioning atoms

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method The precession chamber Determining the lattice parameters Determining the space group using the systematic absences Determining the chemical composition of the substance Determining the structure of the crystal

Scanning electron microscopy

Determining the chemical composition of the substance Sodium periodate / NaIO4

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method The precession chamber Determining the lattice parameters Determining the space group using the systematic absences Determining the chemical composition of the substance Determining the structure of the crystal

Determining the structure of the crystal

Number of molecules per cell (n) atomic mass M known density ρ known volume of cell V known ⇒ n = V ρNA

M

≈ 3.46 → 4 Position of the atoms using the group space and the Patterson method symmetries imply fixed positions for Na and I atoms need to determine position of O but distance between I and O is known using Patterson method, position of Os may be computed

Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods

Basic properties of X-Rays Identifying a substance using a polycrystalline method Identifying a substance using a monocrystalline method The precession chamber Determining the lattice parameters Determining the space group using the systematic absences Determining the chemical composition of the substance Determining the structure of the crystal

Results

lattice parameters:

a = b = 5.3˚ A c = 11.72˚ A

composition: NaIO4 structure:

position I: (0, 1

4, 1 8)

position Na: (0, 1

4, 5 8)

vector IO: (0.24, 0.1, 0.05) Figure: Generated by GrystalOgraph, by Nicolas

Schoeni and Gervais Chapuis, EPFL Jakob Odersky, Simon L¨

• we

Crystallography using X-Ray based methods