Kikokoto cha Indices za Miller kwa Utambuzi wa Ndege za Kijamii
Hesabu indices za Miller kutoka kwa kukutana kwa ndege za kijiometri kwa kutumia chombo hiki rahisi. Muhimu kwa crystallography, sayansi ya vifaa, na matumizi ya fizikia ya hali thabiti.
Kihesabu cha Miller Indices
Mikondo ya Kijivu ya Kristali
Ingiza mikondo ya kijivu ya kristali na mhimili wa x, y, na z. Tumia '0' kwa mikondo inayolingana na mhimili (mhimili wa milele).
Ingiza nambari au 0 kwa milele
Ingiza nambari au 0 kwa milele
Ingiza nambari au 0 kwa milele
Miller Indices
Miller indices za ndege hii ni:
Uonyeshaji
Miller Indices ni nini?
Miller indices ni mfumo wa alama unaotumika katika crystallography kuashiria ndege na mwelekeo katika lattice za kristali.
Ili kuhesabu Miller indices (h,k,l) kutoka kwa mikondo (a,b,c):
1. Chukua reciprocals za mikondo: (1/a, 1/b, 1/c) 2. Badilisha kuwa seti ndogo zaidi ya nambari nzito zikiwa na uwiano sawa 3. Ikiwa ndege inaelekea kwenye mhimili (mhimili = milele), index yake ya Miller ni 0
- Indices hasi zinaonyeshwa kwa bar juu ya nambari, mfano, (h̄,k,l)
- Alama (hkl) inawakilisha ndege maalum, wakati {hkl} inawakilisha familia ya ndege sawa
- Indices za mwelekeo zinaandikwa katika mabano ya mraba [hkl], na familia za mwelekeo zinaonyeshwa kwa <hkl>
Nyaraka
Miller Indices Calculator
Introduction
Miller Indices Calculator ni chombo chenye nguvu kwa wachunguzi wa crystallography, sayansi ya vifaa, na wanafunzi kubaini Miller indices za ndege za crystal. Miller indices ni mfumo wa alama unaotumika katika crystallography kubainisha ndege na mwelekeo katika lattice za crystal. Chombo hiki kinakuwezesha kwa urahisi kubadilisha kukutana kwa ndege ya crystal na aksisi za kuratibu kuwa Miller indices zinazofanana, na kutoa njia iliyokubalika ya kutambua na kuwasiliana kuhusu ndege maalum za crystal.
Miller indices ni msingi wa kuelewa miundo ya crystal na mali zao. Kwa kuwakilisha ndege kwa seti rahisi ya nambari tatu (h,k,l), Miller indices zinawawezesha wanasayansi kuchambua mifumo ya kutafakari X-ray, kutabiri tabia za ukuaji wa crystal, kuhesabu nafasi kati ya ndege, na kujifunza mali mbalimbali za kimwili zinazotegemea mwelekeo wa crystallographic.
What Are Miller Indices?
Miller indices ni seti ya nambari tatu (h,k,l) ambazo zin定义 familia ya ndege zinazofanana katika lattice ya crystal. Indices hizi zinatokana na reciprocals za kukutana kwa ndege ambayo ndege inafanya na aksisi za crystallographic. Alama hii inatoa njia iliyokubalika ya kutambua ndege maalum ndani ya muundo wa crystal.
Visual Representation of Miller Indices
Formula for Calculating Miller Indices
Ili kuhesabu Miller indices (h,k,l) za ndege ya crystal, fuata hatua hizi za kihesabu:
- Tambua kukutana kwa ndege na aksisi za x, y, na z, ukitoa thamani a, b, na c.
- Chukua reciprocals za kukutana hizi: 1/a, 1/b, 1/c.
- Badilisha reciprocals hizi kuwa seti ndogo zaidi ya nambari ambazo zinahifadhi uwiano sawa.
- Nambari tatu zinazotokana ni Miller indices (h,k,l).
Kihesabu, hii inaweza kuonyeshwa kama:
Ambapo:
- (h,k,l) ni Miller indices
- a, b, c ni kukutana kwa ndege na aksisi za x, y, na z, mtawalia
Special Cases and Conventions
Kuna kesi maalum na taratibu kadhaa muhimu kuelewa:
-
Infinity Intercepts: Ikiwa ndege inakaribia aksisi, kukutana kwake kunachukuliwa kuwa infinity, na Miller index inayohusiana inakuwa sifuri.
-
Negative Indices: Ikiwa ndege inakutana na aksisi upande hasi wa asili, Miller index inayohusiana ni hasi, ikionyeshwa kwa bar juu ya nambari katika alama ya crystallographic, mfano (h̄kl).
-
Fractional Intercepts: Ikiwa kukutana ni fractional, zinabadilishwa kuwa nambari nzima kwa kuzidisha na nambari ndogo zaidi ya pamoja.
-
Simplification: Miller indices kila wakati zinapunguzika hadi seti ndogo zaidi ya nambari ambazo zinahifadhi uwiano sawa.
Step-by-Step Guide to Using the Calculator
Miller Indices Calculator yetu inatoa njia rahisi ya kubaini Miller indices kwa ndege yoyote ya crystal. Hapa kuna jinsi ya kuitumia:
-
Ingiza Kukutana: Ingiza thamani ambapo ndege inakutana na aksisi za x, y, na z.
- Tumia nambari chanya kwa kukutana kwenye upande chanya wa asili.
- Tumia nambari hasi kwa kukutana kwenye upande hasi.
- Ingiza "0" kwa ndege ambazo ni za karibu na aksisi (infinity intercept).
-
Tazama Matokeo: Calculator itahesabu kiotomati na kuonyesha Miller indices (h,k,l) za ndege iliyoainishwa.
-
Onyesha Ndege: Calculator inajumuisha uonyeshaji wa 3D ili kukusaidia kuelewa mwelekeo wa ndege ndani ya lattice ya crystal.
-
Nakili Matokeo: Tumia kitufe cha "Nakili kwenye Clipboard" ili kwa urahisi kuhamasisha Miller indices zilizohesabiwa kwa programu nyingine.
Example Calculation
Hebu tufanye mfano:
Kiasi ndege inakutana na aksisi za x, y, na z katika maeneo 2, 3, na 6 mtawalia.
- Kukutana ni (2, 3, 6).
- Chukua reciprocals: (1/2, 1/3, 1/6).
- Ili kupata seti ndogo zaidi ya nambari zenye uwiano sawa, zidisha kwa nambari ndogo zaidi ya pamoja ya denominator (LCM ya 2, 3, 6 = 6): (1/2 × 6, 1/3 × 6, 1/6 × 6) = (3, 2, 1).
- Kwa hivyo, Miller indices ni (3,2,1).
Use Cases for Miller Indices
Miller indices zina matumizi mengi katika nyanja mbalimbali za kisayansi na uhandisi:
Crystallography and X-ray Diffraction
Miller indices ni muhimu kwa kutafsiri mifumo ya kutafakari X-ray. Nafasi kati ya ndege za crystal, zinazotambulika kwa Miller indices zao, zinatumika kuamua pembe ambazo X-rays zinatafakari, kufuata sheria ya Bragg:
Ambapo:
- ni nambari
- ni urefu wa mawimbi ya X-rays
- ni nafasi kati ya ndege zenye Miller indices (h,k,l)
- ni pembe ya kuingia
Materials Science and Engineering
-
Uchambuzi wa Nguvu ya Uso: Ndege tofauti za crystallographic zina nguvu tofauti za uso, zinazoathiri mali kama ukuaji wa crystal, katalisisi, na kushikamana.
-
Mali za Mekaniki: Mwelekeo wa ndege za crystal unaathiri mali za mekaniki kama mifumo ya kuteleza, ndege za cleavage, na tabia ya kuvunjika.
-
Utengenezaji wa Semiconductor: Katika utengenezaji wa semiconductor, ndege maalum za crystal huchaguliwa kwa ukuaji wa epitaxial na utengenezaji wa kifaa kutokana na mali zao za kielektroniki.
-
Uchambuzi wa Texture: Miller indices husaidia kuainisha mwelekeo uliochaguliwa (texture) katika vifaa vya polycrystalline, ambavyo vinaathiri mali zao za kimwili.
Mineralogy and Geology
Wajiolojia hutumia Miller indices kuelezea uso wa crystal na ndege za cleavage katika madini, kusaidia katika utambuzi na kuelewa hali za uundaji.
Educational Applications
Miller indices ni dhana za msingi zinazofundishwa katika masomo ya sayansi ya vifaa, crystallography, na fizikia ya hali thabiti, na kufanya chombo hiki kuwa zana muhimu ya elimu.
Alternatives to Miller Indices
Ingawa Miller indices ndiyo alama inayotumika zaidi kwa ndege za crystal, mifumo kadhaa mbadala inapatikana:
-
Miller-Bravais Indices: Mfumo wa alama nne (h,k,i,l) unaotumika kwa mifumo ya crystal ya hexagonal, ambapo i = -(h+k). Alama hii inawakilisha bora simetria ya miundo ya hexagonal.
-
Weber Symbols: Zinatumika hasa katika maandiko ya zamani, hasa kwa kuelezea mwelekeo katika crystals za cubic.
-
Vikundi vya Lattice vya Moja kwa Moja: Katika baadhi ya matukio, ndege zinaweza kuelezwa kwa kutumia vikundi vya lattice vya moja kwa moja badala ya Miller indices.
-
Msimamo wa Wyckoff: Kwa kuelezea nafasi za atomiki ndani ya muundo wa crystal badala ya ndege.
Licha ya hizi mbadala, Miller indices zinabaki kuwa alama ya kawaida kutokana na urahisi wao na matumizi yao ya ulimwengu mzima katika mifumo yote ya crystal.
History of Miller Indices
Mfumo wa Miller indices ulitengenezwa na mchunguzi wa madini na crystallographer wa Uingereza William Hallowes Miller mwaka 1839, uliochapishwa katika kitabu chake "A Treatise on Crystallography." Alama ya Miller ilijengwa juu ya kazi ya awali ya Auguste Bravais na wengine, lakini ilitoa njia ya kisasa na inayohusiana kwa kihesabu.
Kabla ya mfumo wa Miller, mifumo mbalimbali ilitumika kuelezea uso wa crystal, ikiwa ni pamoja na vigezo vya Weiss na alama za Naumann. Ubunifu wa Miller ulikuwa kutumia reciprocals za kukutana, ambayo ilirahisisha hesabu nyingi za crystallographic na kutoa uwakilishi rahisi wa ndege zinazofanana.
Kupitishwa kwa Miller indices kuliongezeka na ugunduzi wa kutafakari X-ray na Max von Laue mwaka 1912 na kazi inayofuata ya William Lawrence Bragg na William Henry Bragg. Utafiti wao ulionyesha matumizi ya vitendo ya Miller indices katika kutafsiri mifumo ya kutafakari na kubaini miundo ya crystal.
Katika karne ya 20, wakati crystallography ilipokuwa muhimu zaidi katika sayansi ya vifaa, fizikia ya hali thabiti, na biokemia, Miller indices zilijengwa kwa nguvu kama alama ya kawaida. Leo, zinabaki kuwa muhimu katika mbinu za kisasa za uainishaji wa vifaa, crystallography ya kompyuta, na kubuni nanomaterial.
Code Examples for Calculating Miller Indices
1import math
2import numpy as np
3
4def calculate_miller_indices(intercepts):
5 """
6 Hesabu Miller indices kutoka kwa kukutana
7
8 Args:
9 intercepts: Orodha ya kukutana tatu [a, b, c]
10
11 Returns:
12 Orodha ya Miller indices tatu [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 # Pata thamani zisizo sifuri kwa ajili ya 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 # Pata GCD
33 gcd_value = np.gcd.reduce(scaled)
34
35 # Badilisha tena kuwa nambari ndogo zaidi
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# Mfano wa matumizi
46intercepts = [2, 3, 6]
47indices = calculate_miller_indices(intercepts)
48print(f"Miller indices kwa kukutana {intercepts}: {indices}") # Output: [3, 2, 1]
491function 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 // Pata thamani zisizo sifuri kwa ajili ya 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 ili kuepuka matatizo ya floating point
35 const scale = 1000;
36 const scaled = nonZeroReciprocals.map(val => Math.round(val * scale));
37
38 // Pata GCD
39 const divisor = gcdMultiple(scaled);
40
41 // Badilisha kuwa nambari ndogo zaidi
42 const millerIndices = reciprocals.map(val =>
43 val === 0 ? 0 : Math.round((val * scale) / divisor)
44 );
45
46 return millerIndices;
47}
48
49// Mfano
50const intercepts = [2, 3, 6];
51const indices = calculateMillerIndices(intercepts);
52console.log(`Miller indices kwa kukutana ${intercepts}: (${indices.join(',')})`);
53// Output: Miller indices kwa kukutana 2,3,6: (3,2,1)
541import 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 // Hesabu 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 // Hesabu thamani zisizo sifuri
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 ili kuepuka matatizo ya floating point
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 // Pata GCD
60 int divisor = gcdMultiple(scaled);
61
62 // Badilisha kuwa nambari ndogo zaidi
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 kwa kukutana " +
80 Arrays.toString(intercepts) + ": " +
81 Arrays.toString(indices));
82 // Output: Miller indices kwa kukutana [2.0, 3.0, 6.0]: [3, 2, 1]
83 }
84}
851' 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 ' Hesabu 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 ' Hesabu thamani zisizo sifuri
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 ' Pata GCD
59 gcdVal = scaledVals(1)
60 For i = 2 To nonZeroCount
61 gcdVal = GCD(gcdVal, scaledVals(i))
62 Next i
63
64 ' Hesabu 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' Matumizi katika Excel:
102' =CalculateMillerIndices(2, 3, 6)
103' Matokeo: (3,2,1)
1041#include <iostream>
2#include <vector>
3#include <cmath>
4#include <numeric>
5#include <algorithm>
6
7// Hesabu GCD ya nambari mbili
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// Hesabu GCD ya nambari nyingi
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// Hesabu Miller indices kutoka kwa kukutana
31std::vector<int> calculateMillerIndices(const std::vector<double>& intercepts) {
32 std::vector<double> reciprocals;
33
34 // Hesabu 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 // Pata thamani zisizo sifuri
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 // Pata GCD
63 int divisor = gcdMultiple(scaled);
64
65 // Badilisha kuwa nambari ndogo zaidi
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 kwa kukutana [";
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 // Matokeo: Miller indices kwa kukutana [2, 3, 6]: (3,2,1)
96
97 return 0;
98}
99Numerical Examples
Hapa kuna mifano ya kawaida ya hesabu za Miller indices:
-
Mfano 1: Kesi ya Kawaida
- Kukutana: (2, 3, 6)
- Reciprocals: (1/2, 1/3, 1/6)
- Zidisha kwa LCM ya denominators (6): (3, 2, 1)
- Miller indices: (3,2,1)
-
Mfano 2: Ndege ya Karibu na Aksisi
- Kukutana: (1, ∞, 2)
- Reciprocals: (1, 0, 1/2)
- Zidisha kwa 2: (2, 0, 1)
- Miller indices: (2,0,1)
-
Mfano 3: Kukutana kwa Hasi
- Kukutana: (-1, 2, 3)
- Reciprocals: (-1, 1/2, 1/3)
- Zidisha kwa 6: (-6, 3, 2)
- Miller indices: (-6,3,2)
-
Mfano 4: Kukutana kwa Fractional
- Kukutana: (1/2, 1/3, 1/4)
- Reciprocals: (2, 3, 4)
- Tayari katika mfumo wa nambari nzima
- Miller indices: (2,3,4)
-
Mfano 5: Ndege Maalum (100)
- Kukutana: (1, ∞, ∞)
- Reciprocals: (1, 0, 0)
- Miller indices: (1,0,0)
Frequently Asked Questions
Miller indices zinatumika kwa ajili gani?
Miller indices zinatumika kutambua na kuelezea ndege na mwelekeo katika lattice za crystal. Zinatoa alama iliyokubalika ambayo husaidia wachunguzi wa crystallography, sayansi ya vifaa, na wahandisi kuwasiliana kuhusu mwelekeo maalum wa crystal. Miller indices ni muhimu kwa kuchambua mifumo ya kutafakari X-ray, kuelewa ukuaji wa crystal, kuhesabu nafasi kati ya ndege, na kujifunza mali mbalimbali za kimwili zinazotegemea mwelekeo wa crystallographic.
Nifanyeje na ndege ambayo ni karibu na moja ya aksisi?
Wakati ndege inakaribia aksisi, haikutani na aksisi hiyo, hivyo kukutana kwake kunachukuliwa kuwa kwenye infinity. Katika alama za Miller indices, reciprocals za infinity ni sifuri, hivyo index inayohusiana inakuwa sifuri. Kwa mfano, ndege inayokaribia aksisi ya y itakuwa na kukutana (a, ∞, c) na Miller indices (h,0,l).
Nini maana ya Miller indices hasi?
Miller indices hasi zinaashiria kwamba ndege inakutana na aksisi inayohusiana upande hasi wa asili. Katika alama ya crystallographic, indices hasi mara nyingi huonyeshwa kwa bar juu ya nambari, kama (h̄kl). Indices hasi zinawakilisha ndege ambazo ni sawa na wenzao chanya kwa maadili ya kimwili lakini zina mwelekeo tofauti.
Miller indices zina uhusiano gani na muundo wa crystal?
Miller indices zinahusiana moja kwa moja na mpangilio wa atomiki katika muundo wa crystal. Nafasi kati ya ndege zenye Miller indices maalum (dhkl) inategemea mfumo wa crystal na vigezo vya lattice. Katika kutafakari X-ray, ndege hizi hufanya kama ndege za kutafakari kulingana na sheria ya Bragg, zikitoa mifumo ya kutafakari inayoweza kufichua muundo wa crystal.
Ni tofauti gani kati ya Miller indices na Miller-Bravais indices?
Miller indices hutumia nambari tatu (h,k,l) na ni zinazofaa kwa mifumo mingi ya crystal. Miller-Bravais indices hutumia nambari nne (h,k,i,l) na zimetengenezwa mahsusi kwa mifumo ya crystal ya hexagonal. Index ya nne, i, ni ya ziada (i = -(h+k)) lakini inasaidia kudumisha symmetri ya mfumo wa hexagonal na kufanya ndege zinazofanana kuwa rahisi kutambulika.
Nifanyeje kuhesabu pembe kati ya ndege mbili za crystal?
Pembe θ kati ya ndege zenye Miller indices (h₁,k₁,l₁) na (h₂,k₂,l₂) katika mfumo wa crystal wa cubic inaweza kuhesabiwa kwa kutumia:
Kwa mifumo isiyo cubic, hesabu ni ngumu zaidi na inahusisha tensor ya metric ya mfumo wa crystal.
Je, Miller indices zinaweza kuwa fractions?
Hapana, kwa kawaida, Miller indices kila wakati ni nambari nzima. Ikiwa hesabu ya awali inatoa fractions, zinabadilishwa kuwa seti ndogo zaidi ya nambari ambazo zinahifadhi uwiano sawa. Hii inafanywa kwa kuzidisha thamani zote kwa nambari ndogo zaidi ya pamoja ya denominator.
Nifanyeje kubaini Miller indices za uso wa crystal kwa njia ya majaribio?
Miller indices za uso wa crystal zinaweza kubainishwa kwa njia ya majaribio kwa kutumia kutafakari X-ray, kutafakari elektroni, au goniometry ya kioo. Katika kutafakari X-ray, pembe ambazo kutafakari kunatokea zinahusiana na d-spacing ya ndege za crystal kupitia sheria ya Bragg, ambayo inaweza kutumika kutambua Miller indices zinazohusiana.
Nini Miller indices za ndege za crystal maarufu?
Baadhi ya ndege maarufu za crystal na Miller indices zao ni:
- (100), (010), (001): Uso wa msingi wa cubic
- (110), (101), (011): Uso wa diagonal katika mifumo ya cubic
- (111): Uso wa octahedral katika mifumo ya cubic
- (112): Ndege ya kawaida ya kuteleza katika metali za cubic zenye mwili
References
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Miller, W. H. (1839). A Treatise on Crystallography. Cambridge: For J. & J.J. Deighton.
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Ashcroft, N. W., & Mermin, N. D. (1976). Solid State Physics. Holt, Rinehart and Winston.
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Hammond, C. (2015). The Basics of Crystallography and Diffraction (4th ed.). Oxford University Press.
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Cullity, B. D., & Stock, S. R. (2014). Elements of X-ray Diffraction (3rd ed.). Pearson Education.
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Kittel, C. (2004). Introduction to Solid State Physics (8th ed.). Wiley.
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Kelly, A., & Knowles, K. M. (2012). Crystallography and Crystal Defects (2nd ed.). Wiley.
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International Union of Crystallography. (2016). International Tables for Crystallography, Volume A: Space-group symmetry. Wiley.
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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.
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Buerger, M. J. (1978). Elementary Crystallography: An Introduction to the Fundamental Geometrical Features of Crystals. MIT Press.
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Tilley, R. J. (2006). Crystals and Crystal Structures. Wiley.
Jaribu Miller Indices Calculator yetu leo ili kwa urahisi na kwa usahihi kubaini Miller indices kwa ndege yoyote ya crystal. Iwe wewe ni mwanafunzi unayejifunza crystallography, mtafiti anayechambua miundo ya vifaa, au mhandisi anayebuni vifaa vipya, chombo hiki kitakusaidia kutambua na kuelewa ndege za crystal kwa urahisi.
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