1 AP 5301/8301 Instrumental Methods of Analysis and Laboratory Lecture 5 X ray diffraction Prof YU Kin Man E-mail: kinmanyu@cityu.edu.hk Tel: 3442-7813 Office: P6422
Lecture 5: Outline Review on crystallography Lattice and crystal structure Miller indices Diffraction Braggs law Reciprocal lattice X-ray diffraction X-ray source and characteristic x-ray Diffraction intensity structure factor Powder diffraction analysis Phase identification Grain size and strain Texture Rocking curve Advanced XRD techniques Grazing incident High resolution Other advanced x-ray techniques 2
Review: crystal lattice 3 An infinite array of points in space. Each point has identical surroundings to all others. Arrays are arranged exactly in a periodic manner. y B b O α C a A x D E A crystal lattice is a set of infinite, arranged points related to each other by transitional symmetry.
Review: lattices & lattice translation vectors Lattice translation vectors a 1, a 2 describe how to move around a crystal A translation by any combination of the vectors will lead to another equivalent point and leaves the lattice unchanged translation symmetry a 1, a 2 are the basis vectors and the choice of the basis vector is not unique. Equivalent points have the same environment in the same orientation. 2 dimension (2D): r = r + u 1 a 1 + u 2 a 2 u 1, u 2 are integers a 2 a 1 a 2 b 2 r 4 a 1 a n, a n are primitive translation vectors (defined a minimum area) while b n are not b 1 3 dimension (3D): r r = r + u 1 a 1 + u 2 a 2 + u 3 a 3 For n dimensions: r =r+ n u n a n a 3 r a 2 a 1 or r = r + u 1 a + u 2 b + u 3 c Lattice translation vectors a n are primitive if there is no other cell of volume < a 1 a 2 a 3 that can build the lattice
Review: Bravais Lattice 5 In 1850, Auguste Bravais showed that there are only 14 different ways of arranging identical points in 3D space so that the points are equivalent in their surroundings. These arrangement are later called the Bravais Lattices. A Bravais Lattice is defined as an infinite array of points which appears exactly the same when viewed from any one of the lattice points. A Bravais Lattice consists of all points with position vector r of the form r = u 1 a+ u 2 b + u 3 c where a, b, c are any three non-coplanar primitive translation vectors and u i range through all integer values. There are only 7 different shapes of unit cell which can be stacked together to completely fill a 3 dimensional space without overlapping. This gives the 7 crystal systems with 14 Bravais lattices in which all crystal structure can be classified. The systems are defined according to the relationship between the 6 lattice constants: a, b, c, ( or a 1, a 2 a 3 )a, b, and g.
Crystal systems and Bravais lattices 6 Crystal System Conventional Unit Cell Bravais Lattices Triclinic Monoclinic Orthorhombic Tetragonal Cubic Trigonal/ Rhombohedral Hexagonal a b c α β γ a b c α = β = 90º γ a b c α = β = γ = 90º a = b c α = β = γ = 90º a = b = c α = β = γ a = b = c 120º > α = β = γ 90º a = b c α = β = 90º, γ = 60º Primitive (P) Primitive, Base-centered (C) Primitive, Base-centered, Bodycentered (I), Face-centered (F) Primitive, Body-centered Primitive, Body-centered, Facecentered Primitive Primitive
14 Bravais lattices 7 Triclinic Monoclinic Cubic Trigonal/ rhombodedral Orthorhombic Tetragonal Hexagonal P Simple/Primitive I Body Centered F Face Centered C Base Centered
Crystal structure=lattice+basis 8 Each lattice point can be an atom, group of atoms and molecules in the crystal basis. In a real crystal the lattice point is replaced by a basis and every basis is identical in composition, arrangement and orientation b a + 2 D lattice basis crystal r j If r j for the element B is 0, position of element A r j = x j a + y j b In 3D: r j = x j a + y j b + z j c where 0 x j, y j, z j 1
Crystal Structure 9 The atoms do not necessarily lie at lattice points but basis must be in the same orientation 3D + lattice basis Crystal structure
Crystal planes: 3-D 10 In a 3D crystal lattice we can identify multiple sets of equally spaced parallel planes Each group of planes forms a set and are a property of the lattice How do we define the planes?
Miller indices for planes 11 Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes. Method If the plane passes through the origin, select an equivalent plane or move the origin. Determine the intercepts of the plane along each of the three crystallographic directions. Take the reciprocals of the intercepts (if the plane does not intersect one of the axes, the intercept is at infinity and the inverse is zero). (1, 0, 0) Miller Indices (1 1 1) Enclose in parentheses ( ) X Y Z Intercepts 1 1 1/2 Reciprocals 1 1 2 x z (0, 0, 1/2) (0, 1,0) y
Family of planes In some situation when the unit cell has rotational symmetry, nonparallel planes may be equivalent by virtue of this symmetry. These plane are grouped together and called a family of planes and is expressed by a curly bracket {hkl} Examples in a cubic system: 100 : 100, 010, 001, 100, 0 10, 00 1 111 : 111, 11 1, 1 11, 111, 1 1 1, 1 11, 11 1, 1 1 1 (0 1 0) y x z z 12 (010) y (111) (111) y Two successive planes of indices (hkl) make intercepts na/h, nb/k, nc/l and (n+1)a/h, (n+1)b/k, (n+1)c/l, respectively, where n is an integer. The perpendicular distance between successive planes, or interplanar spacing d hkl, can be shown to be given by d hkl = 1 h A+kB+l C, h A + kb + l C 2 = h A + kb + l C h A + kb + l C x For an orthorhombic crystal: α = β = γ = 90 o ; A = 1, B = 1, C = 1 a b c 1 2 = h2 d hkl a 2 + k2 b 2 + l2 c 2
13 Plane separation d hkl Crystal System Unit Cell Characteristics d hkl Cubic Tetragonal Orthorhombic Hexagonal Monoclinic Trigonal/ Rhombohedral a = b = c 1 α = β = γ = 90 o d 2 = h2 + k 2 + l 2 a 2 a = b c 1 α = β = γ = 90 o d 2 = h2 + k 2 a 2 + l2 c 2 a b c α = β = γ = 90 o 1 d 2 = h2 a 2 + k2 b 2 + l2 c 2 a = b c 1 α = β = 90 o, γ = 60 o d 2 = 4 3 a b c α = β = 90º γ a = b = c 120 o > α = β = γ 90 o 1 d 2 = 1 sin 2 β h 2 + hk + k 2 a 2 + l2 c 2 h 2 a 2 + k2 sin 2 β b 2 + l2 c 2 2hlcosβ ac 1 d 2 = h2 + k 2 + l 2 sin 2 α + 2 hk + kl + hl cos 2 α cosα a 2 1 3cos 2 α + 2cos 3 α
Review: diffraction Diffraction is a wave phenomenon in which the apparent bending and spreading of waves occur when they meet an obstacle. Diffraction occurs with all waves including electromagnetic waves, such as light and radio waves as well as sound waves, water waves and matter waves. The simplest demonstration of diffraction is the double-slit diffraction experiment. 14 Interference pattern (fringes) Wider separation between the fringes with narrower slit separation.
Diffraction from a particle and solid 15 Single Particle The particle scatters the incident beam in all directions. Solid (a matrix of particles) For a crystalline solid, the scattered beams may add in a few directions to produce unique pattern. scattered f e r i( k r t) f = atomic form factor (scattering power of atom) =amplitude
Electromagnetic spectrum 16 X-rays are well suited to probe crystal structure
X-ray crystallography: a short history 17 Sir William Henry Bragg, FRS (1862-1942) Wilhelm Röntgen (1845-1923) Max von Laue (1879-1960) Sir William Lawrence Bragg, FRS (1890-1971) 1895: Röntgen discovered X-rays and led to the development of the field of X-ray diffraction crystallography (Röntgen was the first recipient of the Nobel prize in physics, in 1901) Max von Laue developed the use of X- ray diffraction (Nobel Prize, 1914) Typical X-ray wavelength ~ 0.1nm which is similar to interatomic spacing in crystals 1913-1914 :The Braggs together developed the principles for the analysis of crystal structure by means of X-rays and shared the 1915 Nobel Prize in physics
Diffraction When a wave incident on a crystal (periodic arrays of atoms) 18 a a a When l<2a: wave either passes through crystal (k unchanged) or for a particular incident angle will be diffracted
Bragg s law 19 When a monochromatic x-ray beam is incident on the surface of a crystal, the reflection takes places only when the angle of incidence has certain values. Bragg considered crystals as a set of parallel planes of atoms. The incident beam is reflected partially at each of these planes. The reflected rays are collected by a detector at a distance. According to physical optics, the interference is constructive only if the difference between the paths of any two consecutive rays is an integral multiple of the wavelength. The path difference D D nl n 1,2,3,..integers 2d sin θ = nλ
Bragg s law d 1 q 1 q 1 q 2 d 2 sin θ1 n 1 l d 2 sin θ2 n 2 Note: The smaller the spacing d, the higher the angle of reflection q. l The incident beam, the normal to the reflection plane, and the diffracted beam are always coplanar. The angle between the diffracted beam and the transmitted beam is always 2q (usually measured). sin θ cannot be more than unity; this requires nλ < 2d, for n = 1, λ < 2d This is why we cannot use visible light. 20 The diffracted beams from any set of lattice planes can only occur at the angles predicted by the Bragg law. The set of lattice planes is then represented by the diffracted beam, or diffracted spots.
The reciprocal lattice 21 Bragg s Law shows that there is a reciprocal relationship between the plane spacing d and the diffraction angle q, we can therefore relate the diffraction pattern to the crystal lattice by a mathematical construct, the reciprocal lattice. The reciprocal lattice is a set of imaginary points in which the direction of a vector from one point to another corresponds to a direction normal to a plane in a real lattice We can view the pattern created from X- ray diffraction as a new lattice that we can use to gain information about the crystal lattice. d = nλ 2 sin θ X-ray diffraction pattern for a single alum crystal - wiki.brown.edu
The Reciprocal Lattice Crystal planes (hkl) in the real-space or direct lattice are characterized by the normal vector n hkl and the interplanar spacing d hkl. x z nˆhkl d hkl y 22 In the reciprocal lattice, the position of the lattice point in the reciprocal space is given by the vector This vector is parallel to the [hkl] 2 G direction but has magnitude 2/d hkl, hkl nˆ hkl dhkl which is a reciprocal distance The reciprocal lattice is composed of all points lying at positions G hkl from the origin, so that there is one point in the reciprocal lattice for each set of planes (hkl) in the real-space lattice. So why do we need the reciprocal lattice? The reciprocal lattice simplifies the interpretation of diffraction data from crystals. The reciprocal lattice facilitates the calculation of wave propagation in the crystal (lattice vibrations, electron waves, etc.)
2D Reciprocal Lattice 23 Direct (Real) Space Reciprocal Space n 01 d 11 d 01 n 11 n 10 (01) planes (01) 2π d 01 o 2π (10) (11) d 10 d 10 pick some point as an origin lay out the normal to every family of parallel planes in the direct lattice set the length of each normal equal to 2X the reciprocal of the interplanar spacing ( 2π ) for its particular set of planes d place a point at the end of each normal
3D Reciprocal Lattice Reciprocal lattice vector can be written as The reciprocal lattice base vectors are defined: There are simple dot product relationships between reciprocal and directspace lattice vectors: Here, c b a c b A 2 c b a a c B 2 c b a b a C 2 2 c C b B a A b C a C a B c B c A b A 0 π l c G k b G h a G hkl hkl hkl 2 ; 2 ; 2 Reciprocal lattice direct lattice G = XA + YB + Z C Area of a plane in the unit cell in the direction plane Volume of real unit cell 24
The Laue Condition 25 For constructive interference, the scattering vector S must be a reciprocal lattice vector. S 2 k sin θ B Shkl G hkl We also know that: G hkl = 2π d hkl k 0 k Laue Equations: 2 2π λ sin θ hkl = 2π d hkl S a = 2πh S b = 2πk S c = 2πl Shkl G hkl q B k k 0 k 0 S=k-k 0 2 l Elastic Scattering λ = 2d hkl sin θ hkl Bragg s Law S hkl = 2 2π λ sin θ hkl
X-ray diffraction (XRD) 26 The lattice structure determines the position of the lines. The basis determines the relative intensity.
X-ray source 27 A typical Coolidge tube X-rays are produced whenever high-speed electrons collide with a metal target. A source of electrons hot W filament, a high accelerating voltage (30-50kV) between the cathode (W) and the anode, which is a water-cooled block of Cu or Mo containing desired target metal. The x-ray output is the shape characteristic x-ray lines of the anode metal on a continuum of bremsstrahlung radiation
X-ray production 28 As some e - approach the nucleus and are slowed down and pulled into a new direction, consequently some energy is released in the form of X-rays called Bremsstrahlung.
X-ray source: characteristic x-rays 29 Characteristic x-ray line energy= E final E initial Relative intensities of major x-ray lines K α1 = 100 L α1 = 100 M α1,2 = 100 K α2 = 50 L α2 = 50 M β = 60 K β1 = 15 30 L β1 = 50 K β2 = 1 10 L β2 = 250 K β3 = 6 15 L β3 = 1 6 L β4 = 3 5 L γ1 = 1 10 If an incoming electron has sufficient kinetic energy for knocking out an electron of the K shell (the inner-most shell), it may excite the atom to an high-energy state (K state). One of the outer electron falls into the K- shell vacancy, emitting the excess energy as a x-ray photon. Characteristic x-ray energy: E x ray = E final E initial
Characteristic x-ray and filters 30 Element K a1 (Å) K a2 (Å) K b (Å) K absorption edge (Å) Ag 0.56084 0.56380 0.49707 0.4859 25.52 Mo 0.7093 0.71359 0.632288 0.632288 20.0 Cu 1.54056 1.54439 1.392218 1.392218 8.98 Ni 1.65791 1.66175 1.50014 1.4881 8.33 Co 1.78896 1.79285 1.62079 1.6082 7.71 Excitation potential (kv) Accurate XRD measurements requires a single x-ray line as the probe. For example, for a Cu anode, Kb line can be eliminated by a Ni filter
Diffraction intensity Diffraction intensity: I hkl F hkl 2 F hkl - Structure Factor 31 F hkl = basis f j exp 2πi(hu j + kv j + lw j ) where f j is the atomic scattering factor, and is dependent on atomic number u j,v j, w j are the fractional distances within the unit cell h, k, l is the Miller indices of the plane Atomic scattering factor: λ f θ sin θ Z 2 where Z is the atomic number of the atom Ag Br Fe Denser atoms scatter with greater intensity Intensity decreases as the scattering angle increases C
Structure factor: FCC crystal 32 D Four atoms at positions, (uvw): A(0,0,0), B(½,0,½), C(½,½,0), D(0,½,½) F hkl = basis f j exp 2πi(hu j + kv j + lw j ) F hkl = f = f exp 2πi 0 + exp 2πi basis h 2 + l 2 exp 2πi(hu j + kv j + lw j ) + exp 2πi h 2 + k 2 = f 1 + e iπ(h+l) + e iπ(h+k) + e iπ(k+l) + exp 2πi k 2 + l 2 F hkl = 4f h, h, l all odd or all even 0 h, k, l mixed e.g. (111), (220), (222), etc. (210), (112), (320), etc.
Structure factor 33 BCC Al For monatomic BCC crystals: F hkl = 2f if h + k + l is even 0 if h + k + l is odd For FCC crystals: F hkl = 4f h, h, l all odd or all even 0 h, k, l mixed
X-ray powder diffraction 34 X-ray powder diffraction (XRD) is a rapid analytical technique primarily used for phase identification of a crystalline material and can provide information on unit cell dimensions. Samples can be powder, sintered pellets, thin film coatings on substrates, etc. The sample holder and the x-ray detector are mechanically linked. If the sample holder turns q, the detector turns 2q, so that the detector is always ready to detect the Bragg diffracted beam. x-ray detectors (e.g. Geiger counters) are used instead of the film to record both the position and intensity of the x-ray peaks Bragg-Brentano geometry
XRD: single crystal case A single crystal specimen in a Bragg-Brentano diffractometer would produce only one family of peaks in the diffraction pattern 35
XRD: polycrystal case A polycrystalline sample should contain thousands of crystallites. Therefore, all possible diffraction peaks should be observed. 36
XRD patterns (diffractogram) 37 A cone along the sphere corresponds to a single Bragg angle 2-theta The tens of thousands of randomly oriented crystallites in an ideal sample produce a Debye diffraction cone. The linear diffraction pattern is formed as the detector scans through an arc that intersects each Debye cone at a single point; thus giving the appearance of a discrete diffraction peak.
XRD: common applications 38 Phase Composition of a Sample Quantitative Phase Analysis: determine the relative amounts of phases in a mixture by referencing the relative peak intensities Unit cell lattice parameters and Bravais lattice symmetry Index peak positions Lattice parameters can vary as a function of, and therefore give you information about, alloying, doping, solid solutions, strains, etc. Residual Strain (macrostrain) Epitaxy/Texture/Orientation Crystallite Size and Microstrain Indicated by peak broadening Other defects (stacking faults, etc.) can be measured by analysis of peak shapes and peak width
XRD: Phase Identification 39 The most common use of XRD is for phase identification since the diffraction pattern for every phase is as unique as your fingerprint Phases with the same chemical composition can have drastically different diffraction patterns Use the position and relative intensity of a series of peaks to match experimental data to the reference patterns in the database By accurately measuring peak positions over a long range of 2q, you can determine the unit cell lattice parameters of the phases in your sample Effects such as alloying, substitutional doping, temperature and pressure, etc. can create changes in lattice parameters that you may want to quantify.
XRD powder patterns: JCPDS Card Joint Committee on Powder Diffraction Standards (JCPDS) collected over 300,000 diffraction patterns Quality of data 40 1. file number; 2.three strongest lines; 3. lowest-angle line; 4. chemical formula and name; 5. data on diffraction method used; 6. crystallographic data; 7. optical and other data; 8. data on specimen ; 9. data on diffraction pattern.
XRD: alloy composition analysis 41 (0002) Diffraction peaks of ZnO 1-x S x alloy Increasing x ZnO is alloyed with ZnS to form ZnO 1-x S x alloy Wurtzite ZnO (c=0.52 nm) and ZnS (c=0.626 nm) As x increases (more S substituting in O sublattice), the lattice parameter increases Bragg law: λ = 2d sin θ increasing d means decreasing θ. Vegard's law: lattice parameter of a solid solution of two constituents is approximately equal to a rule of mixtures of the two constituents' lattice parameters c ZnO1 x S x = xc ZnS + (1 x)c ZnO Composition x can be derived from the measured lattice parameter c.
XRD: crystallite size 42 Crystallites smaller than ~120nm create broadening of diffraction peaks this peak broadening can be used to quantify the average crystallite size of nanoparticles using the Scherrer equation contributions due to instrument broadening should be known by using a standard sample (e.g. a single crystal) Scherrer equation: B 2θ = Kλ L cos θ where B is the 2θ FWHM peak broadening in radian, λ is the wavelength of the x- ray used, L is the grain size and K~0.9
XRD: lattice strain 43 d o No Strain 2q d 1 Uniform Strain: (d 1 -d o )/d o Peak moves, no shape changes Dq a Dd a strain 2q Non-uniform Strain d 1 constant Peak broadens Dd Broadeing b D2q 2 tanq d 2q
Preferred orientation (texture) 44 In common polycrystalline materials, the grains may not be oriented randomly (not the grain shape, but the orientation of the unit cell of each grain, ). This kind of texture arises from all sorts of treatments, e.g. casting, cold working, annealing, etc. If the crystallites (or grains) are not oriented randomly, the diffraction cone will not be a complete cone. texturing may affect the properties due to anisotropic nature. Grain Random orientation Preferred orientation
Texture in materials 45 Also, if the direction [u 1 v 1 w 1 ] is parallel for all regions, the structure is like a single crystal However, the direction [u 1 v 1 w 1 ] is not aligned for all regions, the structure is like a mosaic structure, also called as Mosaic Texture [uvw] i.e. perpendicular to the surface of all grains is parallel to a direction [uvw]
Preferred orientation 46 Preferred orientation of crystallites can create a variation in diffraction peak intensities that can be qualitatively analyzed using a 1D diffraction pattern (powder pattern) quantitatively analyzed by a pole figure which maps the intensity of a single peak as a function of tilt and rotation of the sample
Preferred orientation 47 Texture PbTiO 3 (001) MgO (001) highly c-axis oriented PbTiO 3 (PT) simple tetragonal X-ray diffraction scan patterns from (a) PbTiO3 (101) and (b) MgO (202) reflections X-ray diffraction q-2q scan profile of a PbTiO 3 thin film grown on MgO (001) at 600 C.
X-ray rocking curve 48 Performs a θ-2θ scan and fix the θ-2θ geometry where a strong diffraction peak is observed A rocking curve scan is then acquired by varying the orientation of the sample by an angle Δω around its equilibrium position (rocking), while keeping the detector position fixed. The width of this peak W (FWHM) will be determined by several factors: The mean spread in orientation for the set of planes belonging to the chosen Bragg reflection. the lateral size of the crystalline domains, similar to the Scherrer broadening described for θ-2θ scans, but depends here on the lateral size of the crystallites. X-ray tube s 2q Detector
Rocking curve (omega scan) 49 [400] s [400] s [400] s 2q 2q 2q A perfect crystal will produce a very sharp peak, observed only when the crystal is properly tilted so that the crystallographic direction is parallel to the diffraction vector s The RC from a perfect crystal will have some width due to instrument broadening and the intrinsic width of the crystal material Defects like mosaicity, dislocations, and curvature create disruptions in the perfect parallelism of atomic planes This is observed as broadening of the rocking curve The center of the rocking curve is determined by the d-spacing of the peaks
Grazing incident XRD (GIXRD) Grazing incidence X-ray diffraction (GID, GIXRD) uses small incident angles for the incoming X-ray (<5 o ), so that diffraction can be made surface sensitive. It is used to study surfaces and layers because wave penetration is limited. Distances are on the order of nanometers. 50 kept constant ~0.2 to 4 o Conventional Bragg-Brentano configuration: 2q- scans probe only grains aligned parallel to the surface Parallel-beam glazing incidence configuration: 2q scans probe grains in all directions
GIXRD: x-ray penetration 51 2q Grain orientation Directions surface Various directions Glancing angle 2q Depth resolution Constant, tens of mm Few nm-mm, depth profiling possible by varying Surface sensitive Ideal for ultra thin films (nm) Best configuration Bragg-Brentabo Parallel beam Parallel beam
GIXRD: example 52
High resolution XRD (HRXRD) 53
HRXRD: applications 54 Commonly used for measuring single crystals, epitaxial films, heterostuctures, superlattices, quantum dot, etc: Lattice distortions within 10-5. Rocking curve analysis Strain relaxation and lattice parameter measurements. Alloy composition and superlattice periods. Interface smearing in heterostructures (dynamical simulation).
HRXRD 55 Example: strained In x Ga 1-x As on GaAs (001) substrate Strain, layer thickness and composition x can be obtained
Other x-ray techniques 56 Reciprocal space mapping Accurate lattice parameters in and out of plane Strain and composition gradients Strain relaxation Mosaic size and rotation Misfit dislocation density Nanostructure dimensions, Lattice disorder and diffuse scattering. X-ray absorption (XAS): Typically require a synchrotron radiation source Measures the local bonding environment of a certain element in a material Information such as bond angle, bond length, coordination number, bind type, bond distortion, etc. can be obtained
Synchrotron radiation 57
Interference of photoelectron waves 58 Interference of outgoing and incoming part of photoelectron modulates absorption coefficient 2.3 hω > E abs Correction factor=c/c exp 2.2 2.1 2.0 1.9 1.8 0 2 4 6 8 10 12 14 k (Å) Advantages: Atomic-species specific, local bonding environment (length, angle, disorder, neighbor) Disadvantages: very short range (<~5-6 Å), sensitive to multiple scattering, overlapping edges...
md ck 2 EXAFS 2.5 2.0 1.5 1.0 0.5 0.0 4 2 0-2 -4-6 Cu K-edge for Cu metal foil X-ray absorption near edge structure (XANES) m 0 d Extended x-ray absorption fine structure (EXAFS) 9000 9200 9400 9600 photon energy (ev) 2 4 6 8 10 12 14 photoelectron wave number (Å -1 ) fk j)( Rk k ksn j j kr j jek e j 22 2 2/)(2 )( 0 sin( 2( )) l 2 kr c FT(ck 2 ) 8 6 4 2 j j For each coordination shell j: R j, N j, 2 j are the sought distance, coord. number and variance of distance (disorder) f j (k)= f j (k) e i j (k) λ is the electron free path, is the scattering amplitude S 0 2 accounts for many-electron excitations. First nearest neighbor 2 nd NN 3 rd NN 4 th NN 0 0 2 4 6 8 distance (Å) 59