Milleri indeksite kalkulaator kristalltasandite tuvastamiseks

Arvutage Milleri indeksid kristalltasandi lõikepunktide põhjal selle lihtsa tööriista abil. Oluline kristallograafias, materjaliteaduses ja tahke aine füüsikas.

Milleri indeksite kalkulaator

Kristallitasandi katkestused

Sisestage kristallitasandi katkestused x-, y- ja z-telgedega. Kasutage '0', kui tasand on teljega paralleelne (lõpmatu katkestus).

X-telje katkestus

Sisestage number või 0 lõpmatuse jaoks

Y-telje katkestus

Sisestage number või 0 lõpmatuse jaoks

Z-telje katkestus

Sisestage number või 0 lõpmatuse jaoks

Milleri indeksid

Selle tasandi Milleri indeksid on:

(1,1,1)

Kopeeri lõikelauale

Visualiseerimine

Mis on Milleri indeksid?

Milleri indeksid on märkisüsteem, mida kasutatakse kristallograafias tasandite ja suundade määramiseks kristallivõre.

Milleri indeksite (h,k,l) arvutamiseks katkestustest (a,b,c):

1. Võtke katkestuste pöördväärtused: (1/a, 1/b, 1/c) 2. Muutke need kõige väiksemaks täisarvude kogumiks sama suhtega 3. Kui tasand on teljega paralleelne (katkestus = lõpmatus), on vastav Milleri indeks 0

Negatiivsed indeksid on tähistatud numbri kohal oleva joonega, nt (h̄,k,l)
Märkistus (hkl) tähistab konkreetset tasandit, samas kui {hkl} tähistab ekvivalentsete tasandite peret
Suunainvesteeringud on kirjutatud nurksulgudesse [hkl], ja suundade pered on tähistatud <hkl>

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Dokumentatsioon

Miller Indices Calculator

Introduction

The Miller Indices Calculator is a powerful tool for crystallographers, materials scientists, and students to determine the Miller indices of crystal planes. Miller indices are a notation system used in crystallography to specify planes and directions in crystal lattices. This calculator allows you to easily convert the intercepts of a crystal plane with the coordinate axes into the corresponding Miller indices, providing a standardized way to identify and communicate about specific crystal planes.

Miller indices are fundamental to understanding crystal structures and their properties. By representing planes with a simple set of three integers (h,k,l), Miller indices enable scientists to analyze X-ray diffraction patterns, predict crystal growth behaviors, calculate interplanar spacing, and study various physical properties that depend on crystallographic orientation.

What Are Miller Indices?

Miller indices are a set of three integers (h,k,l) that define a family of parallel planes in a crystal lattice. These indices are derived from the reciprocals of the fractional intercepts that a plane makes with the crystallographic axes. The notation provides a standardized way to identify specific planes within a crystal structure.

Visual Representation of Miller Indices

x y z

a=2 b=3 c=6

(3,2,1) Plane

Miller Indices (3,2,1) Crystal Plane

A 3D visualization of a crystal plane with Miller indices (3,2,1). The plane intercepts the x, y, and z axes at points 2, 3, and 6 respectively, resulting in Miller indices (3,2,1) after taking reciprocals and finding the smallest set of integers with the same ratio.

Formula for Calculating Miller Indices

To calculate the Miller indices (h,k,l) of a crystal plane, follow these mathematical steps:

Determine the intercepts of the plane with the x, y, and z crystallographic axes, giving values a, b, and c.
Take the reciprocals of these intercepts: 1/a, 1/b, 1/c.
Convert these reciprocals to the smallest set of integers that maintain the same ratio.
The resulting three integers are the Miller indices (h,k,l).

Mathematically, this can be expressed as:

$h : k : l = \frac{1}{a} : \frac{1}{b} : \frac{1}{c}$

Where:

(h,k,l) are the Miller indices
a, b, c are the intercepts of the plane with the x, y, and z axes, respectively

Special Cases and Conventions

Several special cases and conventions are important to understand:

Infinity Intercepts: If a plane is parallel to an axis, its intercept is considered infinity, and the corresponding Miller index becomes zero.
Negative Indices: If a plane intercepts an axis on the negative side of the origin, the corresponding Miller index is negative, denoted with a bar over the number in crystallographic notation, e.g., (h̄kl).
Fractional Intercepts: If the intercepts are fractional, they are converted to integers by multiplying by the least common multiple.
Simplification: Miller indices are always reduced to the smallest set of integers that maintain the same ratio.

Step-by-Step Guide to Using the Calculator

Our Miller Indices Calculator provides a straightforward way to determine the Miller indices for any crystal plane. Here's how to use it:

Enter the Intercepts: Input the values where the plane intersects the x, y, and z axes.
- Use positive numbers for intercepts on the positive side of the origin.
- Use negative numbers for intercepts on the negative side.
- Enter "0" for planes that are parallel to an axis (infinity intercept).
View the Results: The calculator will automatically compute and display the Miller indices (h,k,l) for the specified plane.
Visualize the Plane: The calculator includes a 3D visualization to help you understand the orientation of the plane within the crystal lattice.
Copy the Results: Use the "Copy to Clipboard" button to easily transfer the calculated Miller indices to other applications.

Example Calculation

Let's walk through an example:

Suppose a plane intercepts the x, y, and z axes at points 2, 3, and 6 respectively.

The intercepts are (2, 3, 6).
Taking the reciprocals: (1/2, 1/3, 1/6).
To find the smallest set of integers with the same ratio, multiply by the least common multiple of the denominators (LCM of 2, 3, 6 = 6): (1/2 × 6, 1/3 × 6, 1/6 × 6) = (3, 2, 1).
Therefore, the Miller indices are (3,2,1).

Use Cases for Miller Indices

Miller indices have numerous applications across various scientific and engineering fields:

Crystallography and X-ray Diffraction

Miller indices are essential for interpreting X-ray diffraction patterns. The spacing between crystal planes, identified by their Miller indices, determines the angles at which X-rays are diffracted, following Bragg's law:

$n\lambda = 2d_{hkl}\sin\theta$

Where:

$n$ is an integer
$\lambda$ is the wavelength of the X-rays
$d_{hkl}$ is the spacing between planes with Miller indices (h,k,l)
$\theta$ is the angle of incidence

Materials Science and Engineering

Surface Energy Analysis: Different crystallographic planes have different surface energies, affecting properties like crystal growth, catalysis, and adhesion.
Mechanical Properties: The orientation of crystal planes influences mechanical properties such as slip systems, cleavage planes, and fracture behavior.
Semiconductor Manufacturing: In semiconductor fabrication, specific crystal planes are selected for epitaxial growth and device fabrication due to their electronic properties.
Texture Analysis: Miller indices help characterize preferred orientations (texture) in polycrystalline materials, which affect their physical properties.

Mineralogy and Geology

Geologists use Miller indices to describe crystal faces and cleavage planes in minerals, helping with identification and understanding formation conditions.

Educational Applications

Miller indices are fundamental concepts taught in materials science, crystallography, and solid-state physics courses, making this calculator a valuable educational tool.

Alternatives to Miller Indices

While Miller indices are the most widely used notation for crystal planes, several alternative systems exist:

Miller-Bravais Indices: A four-index notation (h,k,i,l) used for hexagonal crystal systems, where i = -(h+k). This notation better reflects the symmetry of hexagonal structures.
Weber Symbols: Used primarily in older literature, particularly for describing directions in cubic crystals.
Direct Lattice Vectors: In some cases, planes are described using the direct lattice vectors rather than Miller indices.
Wyckoff Positions: For describing atomic positions within crystal structures rather than planes.

Despite these alternatives, Miller indices remain the standard notation due to their simplicity and universal applicability across all crystal systems.

History of Miller Indices

The Miller indices system was developed by British mineralogist and crystallographer William Hallowes Miller in 1839, published in his treatise "A Treatise on Crystallography." Miller's notation built upon earlier work by Auguste Bravais and others, but provided a more elegant and mathematically consistent approach.

Prior to Miller's system, various notations were used to describe crystal faces, including the Weiss parameters and the Naumann symbols. Miller's innovation was to use the reciprocals of intercepts, which simplified many crystallographic calculations and provided a more intuitive representation of parallel planes.

The adoption of Miller indices accelerated with the discovery of X-ray diffraction by Max von Laue in 1912 and the subsequent work of William Lawrence Bragg and William Henry Bragg. Their research demonstrated the practical utility of Miller indices in interpreting diffraction patterns and determining crystal structures.

Throughout the 20th century, as crystallography became increasingly important in materials science, solid-state physics, and biochemistry, Miller indices became firmly established as the standard notation. Today, they remain essential in modern materials characterization techniques, computational crystallography, and nanomaterial design.

Code Examples for Calculating Miller Indices

1import math
2import numpy as np
3
4def calculate_miller_indices(intercepts):
5    """
6    Calculate Miller indices from intercepts
7    
8    Args:
9        intercepts: List of three intercepts [a, b, c]
10        
11    Returns:
12        List of three Miller indices [h, k, l]
13    """
14    # Handle infinity intercepts (parallel to axis)
15    reciprocals = []
16    for intercept in intercepts:
17        if intercept == 0 or math.isinf(intercept):
18            reciprocals.append(0)
19        else:
20            reciprocals.append(1 / intercept)
21    
22    # Find non-zero values for GCD calculation
23    non_zero = [r for r in reciprocals if r != 0]
24    
25    if not non_zero:
26        return [0, 0, 0]
27    
28    # Scale to reasonable integers (avoiding floating point issues)
29    scale = 1000
30    scaled = [round(r * scale) for r in non_zero]
31    
32    # Find GCD
33    gcd_value = np.gcd.reduce(scaled)
34    
35    # Convert back to smallest integers
36    miller_indices = []
37    for r in reciprocals:
38        if r == 0:
39            miller_indices.append(0)
40        else:
41            miller_indices.append(round((r * scale) / gcd_value))
42    
43    return miller_indices
44
45# Example usage
46intercepts = [2, 3, 6]
47indices = calculate_miller_indices(intercepts)
48print(f"Miller indices for intercepts {intercepts}: {indices}")  # Output: [3, 2, 1]
49

1function gcd(a, b) {
2  a = Math.abs(a);
3  b = Math.abs(b);
4  
5  while (b !== 0) {
6    const temp = b;
7    b = a % b;
8    a = temp;
9  }
10  
11  return a;
12}
13
14function gcdMultiple(numbers) {
15  return numbers.reduce((result, num) => gcd(result, num), numbers[0]);
16}
17
18function calculateMillerIndices(intercepts) {
19  // Handle infinity intercepts
20  const reciprocals = intercepts.map(intercept => {
21    if (intercept === 0 || !isFinite(intercept)) {
22      return 0;
23    }
24    return 1 / intercept;
25  });
26  
27  // Find non-zero values for GCD calculation
28  const nonZeroReciprocals = reciprocals.filter(val => val !== 0);
29  
30  if (nonZeroReciprocals.length === 0) {
31    return [0, 0, 0];
32  }
33  
34  // Scale to integers to avoid floating point issues
35  const scale = 1000;
36  const scaled = nonZeroReciprocals.map(val => Math.round(val * scale));
37  
38  // Find GCD
39  const divisor = gcdMultiple(scaled);
40  
41  // Convert to smallest integers
42  const millerIndices = reciprocals.map(val => 
43    val === 0 ? 0 : Math.round((val * scale) / divisor)
44  );
45  
46  return millerIndices;
47}
48
49// Example
50const intercepts = [2, 3, 6];
51const indices = calculateMillerIndices(intercepts);
52console.log(`Miller indices for intercepts ${intercepts}: (${indices.join(',')})`);
53// Output: Miller indices for intercepts 2,3,6: (3,2,1)
54

1import java.util.Arrays;
2
3public class MillerIndicesCalculator {
4    
5    public static int gcd(int a, int b) {
6        a = Math.abs(a);
7        b = Math.abs(b);
8        
9        while (b != 0) {
10            int temp = b;
11            b = a % b;
12            a = temp;
13        }
14        
15        return a;
16    }
17    
18    public static int gcdMultiple(int[] numbers) {
19        int result = numbers[0];
20        for (int i = 1; i < numbers.length; i++) {
21            result = gcd(result, numbers[i]);
22        }
23        return result;
24    }
25    
26    public static int[] calculateMillerIndices(double[] intercepts) {
27        double[] reciprocals = new double[intercepts.length];
28        
29        // Calculate reciprocals
30        for (int i = 0; i < intercepts.length; i++) {
31            if (intercepts[i] == 0 || Double.isInfinite(intercepts[i])) {
32                reciprocals[i] = 0;
33            } else {
34                reciprocals[i] = 1 / intercepts[i];
35            }
36        }
37        
38        // Count non-zero values
39        int nonZeroCount = 0;
40        for (double r : reciprocals) {
41            if (r != 0) nonZeroCount++;
42        }
43        
44        if (nonZeroCount == 0) {
45            return new int[]{0, 0, 0};
46        }
47        
48        // Scale to integers
49        int scale = 1000;
50        int[] scaled = new int[nonZeroCount];
51        int index = 0;
52        
53        for (double r : reciprocals) {
54            if (r != 0) {
55                scaled[index++] = (int) Math.round(r * scale);
56            }
57        }
58        
59        // Find GCD
60        int divisor = gcdMultiple(scaled);
61        
62        // Convert to smallest integers
63        int[] millerIndices = new int[reciprocals.length];
64        for (int i = 0; i < reciprocals.length; i++) {
65            if (reciprocals[i] == 0) {
66                millerIndices[i] = 0;
67            } else {
68                millerIndices[i] = (int) Math.round((reciprocals[i] * scale) / divisor);
69            }
70        }
71        
72        return millerIndices;
73    }
74    
75    public static void main(String[] args) {
76        double[] intercepts = {2, 3, 6};
77        int[] indices = calculateMillerIndices(intercepts);
78        
79        System.out.println("Miller indices for intercepts " + 
80                           Arrays.toString(intercepts) + ": " + 
81                           Arrays.toString(indices));
82        // Output: Miller indices for intercepts [2.0, 3.0, 6.0]: [3, 2, 1]
83    }
84}
85

1' Excel VBA Function for Miller Indices Calculation
2Function CalculateMillerIndices(x As Double, y As Double, z As Double) As String
3    Dim recipX As Double, recipY As Double, recipZ As Double
4    Dim nonZeroCount As Integer, i As Integer
5    Dim scale As Long, gcdVal As Long
6    Dim scaledVals() As Long
7    Dim millerH As Long, millerK As Long, millerL As Long
8    
9    ' Calculate reciprocals
10    If x = 0 Then
11        recipX = 0
12    Else
13        recipX = 1 / x
14    End If
15    
16    If y = 0 Then
17        recipY = 0
18    Else
19        recipY = 1 / y
20    End If
21    
22    If z = 0 Then
23        recipZ = 0
24    Else
25        recipZ = 1 / z
26    End If
27    
28    ' Count non-zero values
29    nonZeroCount = 0
30    If recipX <> 0 Then nonZeroCount = nonZeroCount + 1
31    If recipY <> 0 Then nonZeroCount = nonZeroCount + 1
32    If recipZ <> 0 Then nonZeroCount = nonZeroCount + 1
33    
34    If nonZeroCount = 0 Then
35        CalculateMillerIndices = "(0,0,0)"
36        Exit Function
37    End If
38    
39    ' Scale to integers
40    scale = 1000
41    ReDim scaledVals(1 To nonZeroCount)
42    i = 1
43    
44    If recipX <> 0 Then
45        scaledVals(i) = Round(recipX * scale)
46        i = i + 1
47    End If
48    
49    If recipY <> 0 Then
50        scaledVals(i) = Round(recipY * scale)
51        i = i + 1
52    End If
53    
54    If recipZ <> 0 Then
55        scaledVals(i) = Round(recipZ * scale)
56    End If
57    
58    ' Find GCD
59    gcdVal = scaledVals(1)
60    For i = 2 To nonZeroCount
61        gcdVal = GCD(gcdVal, scaledVals(i))
62    Next i
63    
64    ' Calculate Miller indices
65    If recipX = 0 Then
66        millerH = 0
67    Else
68        millerH = Round((recipX * scale) / gcdVal)
69    End If
70    
71    If recipY = 0 Then
72        millerK = 0
73    Else
74        millerK = Round((recipY * scale) / gcdVal)
75    End If
76    
77    If recipZ = 0 Then
78        millerL = 0
79    Else
80        millerL = Round((recipZ * scale) / gcdVal)
81    End If
82    
83    CalculateMillerIndices = "(" & millerH & "," & millerK & "," & millerL & ")"
84End Function
85
86Function GCD(a As Long, b As Long) As Long
87    Dim temp As Long
88    
89    a = Abs(a)
90    b = Abs(b)
91    
92    Do While b <> 0
93        temp = b
94        b = a Mod b
95        a = temp
96    Loop
97    
98    GCD = a
99End Function
100
101' Usage in Excel:
102' =CalculateMillerIndices(2, 3, 6)
103' Result: (3,2,1)
104

1#include <iostream>
2#include <vector>
3#include <cmath>
4#include <numeric>
5#include <algorithm>
6
7// Calculate GCD of two numbers
8int gcd(int a, int b) {
9    a = std::abs(a);
10    b = std::abs(b);
11    
12    while (b != 0) {
13        int temp = b;
14        b = a % b;
15        a = temp;
16    }
17    
18    return a;
19}
20
21// Calculate GCD of multiple numbers
22int gcdMultiple(const std::vector<int>& numbers) {
23    int result = numbers[0];
24    for (size_t i = 1; i < numbers.size(); ++i) {
25        result = gcd(result, numbers[i]);
26    }
27    return result;
28}
29
30// Calculate Miller indices from intercepts
31std::vector<int> calculateMillerIndices(const std::vector<double>& intercepts) {
32    std::vector<double> reciprocals;
33    
34    // Calculate reciprocals
35    for (double intercept : intercepts) {
36        if (intercept == 0 || std::isinf(intercept)) {
37            reciprocals.push_back(0);
38        } else {
39            reciprocals.push_back(1.0 / intercept);
40        }
41    }
42    
43    // Find non-zero values
44    std::vector<double> nonZeroReciprocals;
45    for (double r : reciprocals) {
46        if (r != 0) {
47            nonZeroReciprocals.push_back(r);
48        }
49    }
50    
51    if (nonZeroReciprocals.empty()) {
52        return {0, 0, 0};
53    }
54    
55    // Scale to integers
56    const int scale = 1000;
57    std::vector<int> scaled;
58    for (double r : nonZeroReciprocals) {
59        scaled.push_back(std::round(r * scale));
60    }
61    
62    // Find GCD
63    int divisor = gcdMultiple(scaled);
64    
65    // Convert to smallest integers
66    std::vector<int> millerIndices;
67    for (double r : reciprocals) {
68        if (r == 0) {
69            millerIndices.push_back(0);
70        } else {
71            millerIndices.push_back(std::round((r * scale) / divisor));
72        }
73    }
74    
75    return millerIndices;
76}
77
78int main() {
79    std::vector<double> intercepts = {2, 3, 6};
80    std::vector<int> indices = calculateMillerIndices(intercepts);
81    
82    std::cout << "Miller indices for intercepts [";
83    for (size_t i = 0; i < intercepts.size(); ++i) {
84        std::cout << intercepts[i];
85        if (i < intercepts.size() - 1) std::cout << ", ";
86    }
87    std::cout << "]: (";
88    
89    for (size_t i = 0; i < indices.size(); ++i) {
90        std::cout << indices[i];
91        if (i < indices.size() - 1) std::cout << ",";
92    }
93    std::cout << ")" << std::endl;
94    
95    // Output: Miller indices for intercepts [2, 3, 6]: (3,2,1)
96    
97    return 0;
98}
99

Numerical Examples

Here are some common examples of Miller indices calculations:

Example 1: Standard Case
- Intercepts: (2, 3, 6)
- Reciprocals: (1/2, 1/3, 1/6)
- Multiply by LCM of denominators (6): (3, 2, 1)
- Miller indices: (3,2,1)
Example 2: Plane Parallel to an Axis
- Intercepts: (1, ∞, 2)
- Reciprocals: (1, 0, 1/2)
- Multiply by 2: (2, 0, 1)
- Miller indices: (2,0,1)
Example 3: Negative Intercepts
- Intercepts: (-1, 2, 3)
- Reciprocals: (-1, 1/2, 1/3)
- Multiply by 6: (-6, 3, 2)
- Miller indices: (-6,3,2)
Example 4: Fractional Intercepts
- Intercepts: (1/2, 1/3, 1/4)
- Reciprocals: (2, 3, 4)
- Already in integer form
- Miller indices: (2,3,4)
Example 5: Special Plane (100)
- Intercepts: (1, ∞, ∞)
- Reciprocals: (1, 0, 0)
- Miller indices: (1,0,0)

Frequently Asked Questions

What are Miller indices used for?

Miller indices are used to identify and describe planes and directions in crystal lattices. They provide a standardized notation that helps crystallographers, materials scientists, and engineers communicate about specific crystal orientations. Miller indices are essential for analyzing X-ray diffraction patterns, understanding crystal growth, calculating interplanar spacing, and studying various physical properties that depend on crystallographic orientation.

How do I handle a plane that is parallel to one of the axes?

When a plane is parallel to an axis, it never intersects that axis, so the intercept is considered to be at infinity. In Miller indices notation, the reciprocal of infinity is zero, so the corresponding Miller index becomes zero. For example, a plane parallel to the y-axis would have intercepts (a, ∞, c) and Miller indices (h,0,l).

What do negative Miller indices mean?

Negative Miller indices indicate that the plane intercepts the corresponding axis on the negative side of the origin. In crystallographic notation, negative indices are typically denoted with a bar over the number, such as (h̄kl). Negative indices represent planes that are equivalent to their positive counterparts in terms of physical properties but have different orientations.

How do Miller indices relate to crystal structure?

Miller indices directly relate to the atomic arrangement in a crystal structure. The spacing between planes with specific Miller indices (dhkl) depends on the crystal system and lattice parameters. In X-ray diffraction, these planes act as reflecting planes according to Bragg's law, producing characteristic diffraction patterns that reveal the crystal structure.

What is the difference between Miller indices and Miller-Bravais indices?

Miller indices use three integers (h,k,l) and are suitable for most crystal systems. Miller-Bravais indices use four integers (h,k,i,l) and are specifically designed for hexagonal crystal systems. The fourth index, i, is redundant (i = -(h+k)) but helps maintain the symmetry of the hexagonal system and makes equivalent planes more easily recognizable.

How do I calculate the angle between two crystal planes?

The angle θ between two planes with Miller indices (h₁,k₁,l₁) and (h₂,k₂,l₂) in a cubic crystal system can be calculated using:

$\cos\theta = \frac{h_1h_2 + k_1k_2 + l_1l_2}{\sqrt{(h_1^2 + k_1^2 + l_1^2)(h_2^2 + k_2^2 + l_2^2)}}$

For non-cubic systems, the calculation is more complex and involves the metric tensor of the crystal system.

What is the relationship between Miller indices and d-spacing?

The d-spacing (interplanar spacing) for planes with Miller indices (h,k,l) depends on the crystal system. For a cubic crystal with lattice parameter a, the relationship is:

$d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$

For other crystal systems, more complex formulas apply that incorporate the specific lattice parameters.

Can Miller indices be fractions?

No, by convention, Miller indices are always integers. If the calculation initially yields fractions, they are converted to the smallest set of integers that maintain the same ratio. This is done by multiplying all values by the least common multiple of the denominators.

How do I determine the Miller indices of a crystal face experimentally?

Miller indices of crystal faces can be determined experimentally using X-ray diffraction, electron diffraction, or optical goniometry. In X-ray diffraction, the angles at which diffraction occurs are related to the d-spacing of crystal planes through Bragg's law, which can be used to identify the corresponding Miller indices.

What are the Miller indices of common crystal planes?

Some common crystal planes and their Miller indices include:

(100), (010), (001): Primary cubic faces
(110), (101), (011): Diagonal faces in cubic systems
(111): Octahedral face in cubic systems
(112): Common slip plane in body-centered cubic metals

References

Miller, W. H. (1839). A Treatise on Crystallography. Cambridge: For J. & J.J. Deighton.
Ashcroft, N. W., & Mermin, N. D. (1976). Solid State Physics. Holt, Rinehart and Winston.
Hammond, C. (2015). The Basics of Crystallography and Diffraction (4th ed.). Oxford University Press.
Cullity, B. D., & Stock, S. R. (2014). Elements of X-ray Diffraction (3rd ed.). Pearson Education.
Kittel, C. (2004). Introduction to Solid State Physics (8th ed.). Wiley.
Kelly, A., & Knowles, K. M. (2012). Crystallography and Crystal Defects (2nd ed.). Wiley.
International Union of Crystallography. (2016). International Tables for Crystallography, Volume A: Space-group symmetry. Wiley.
Giacovazzo, C., Monaco, H. L., Artioli, G., Viterbo, D., Ferraris, G., Gilli, G., Zanotti, G., & Catti, M. (2011). Fundamentals of Crystallography (3rd ed.). Oxford University Press.
Buerger, M. J. (1978). Elementary Crystallography: An Introduction to the Fundamental Geometrical Features of Crystals. MIT Press.
Tilley, R. J. (2006). Crystals and Crystal Structures. Wiley.

Try our Miller Indices Calculator today to quickly and accurately determine the Miller indices for any crystal plane. Whether you're a student learning crystallography, a researcher analyzing material structures, or an engineer designing new materials, this tool will help you identify and understand crystal planes with ease.

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Milleri indeksite kalkulaator kristalltasandite tuvastamiseks

Milleri indeksite kalkulaator

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Milleri indeksid

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Mis on Milleri indeksid?

Dokumentatsioon

Miller Indices Calculator

Introduction

What Are Miller Indices?

Visual Representation of Miller Indices

Formula for Calculating Miller Indices

Special Cases and Conventions

Step-by-Step Guide to Using the Calculator

Example Calculation

Use Cases for Miller Indices

Crystallography and X-ray Diffraction

Materials Science and Engineering

Mineralogy and Geology

Educational Applications

Alternatives to Miller Indices

History of Miller Indices

Code Examples for Calculating Miller Indices

Numerical Examples

Frequently Asked Questions

What are Miller indices used for?

How do I handle a plane that is parallel to one of the axes?

What do negative Miller indices mean?

How do Miller indices relate to crystal structure?

What is the difference between Miller indices and Miller-Bravais indices?

How do I calculate the angle between two crystal planes?

What is the relationship between Miller indices and d-spacing?

Can Miller indices be fractions?

How do I determine the Miller indices of a crystal face experimentally?

What are the Miller indices of common crystal planes?

References

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