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X-Ray Diffraction Crystallography

Introduction, Examples and Solved Problems

Inst. Multidisciplinary Research for, Advanced Materials, Tohoku University, Sendai, Japan

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Graduate School of Engineering, Dept. Materials Science & Engineering, Kyoto University, Kyoto, Japan

Tutorial-like scientific monograph on X-ray diffraction analysis

Summarizes the complete knowledge on X-Ray diffraction crystallography and structure analysis of crystals for researchers and graduate students

Presents the crystallographic basics in a systematic way and fundamental properties of X-rays

Excellent book for newcomers with 90 exercises and solutions + 90 problems

Includes supplementary material:

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151 Citations

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Table of contents (9 chapters)

Front matter, fundamental properties of x-rays.

Geometry of Crystals

Scattering and diffraction, diffraction from polycrystalline samples and determination of crystal structure, reciprocal lattice and integrated intensities of crystals, symmetry analysis for crystals and the use of the international tables, supplementary problems (100 exercises), solutions to supplementary problems, back matter.

From the reviews:

Yoshio Waseda, Kozo Shinoda

Graduate School of Engineering, Dept. Materials Science & Engineering, Kyoto University, Kyoto, Japan

Eiichiro Matsubara

Book Title : X-Ray Diffraction Crystallography

Book Subtitle : Introduction, Examples and Solved Problems

Authors : Yoshio Waseda, Eiichiro Matsubara, Kozo Shinoda


Publisher : Springer Berlin, Heidelberg

eBook Packages : Chemistry and Materials Science , Chemistry and Material Science (R0)

Copyright Information : Springer-Verlag Berlin Heidelberg 2011

Hardcover ISBN : 978-3-642-16634-1 Published: 09 March 2011

Softcover ISBN : 978-3-642-44255-1 Published: 21 November 2014

eBook ISBN : 978-3-642-16635-8 Published: 18 March 2011

Edition Number : 1

Number of Pages : XI, 310

Topics : Characterization and Analytical Technique , Crystallography and Scattering Methods , Microsystems and MEMS , Condensed Matter Physics , Spectroscopy

Learning Objectives

By the end of this section, you will be able to:

Since X-ray photons are very energetic, they have relatively short wavelengths, on the order of 10 −8 10 −8 m to 10 −12 10 −12 m. Thus, typical X-ray photons act like rays when they encounter macroscopic objects, like teeth, and produce sharp shadows. However, since atoms are on the order of 0.1 nm in size, X-rays can be used to detect the location, shape, and size of atoms and molecules. The process is called X-ray diffraction , and it involves the interference of X-rays to produce patterns that can be analyzed for information about the structures that scattered the X-rays.

Perhaps the most famous example of X-ray diffraction is the discovery of the double-helical structure of DNA in 1953 by an international team of scientists working at England’s Cavendish Laboratory—American James Watson, Englishman Francis Crick, and New Zealand-born Maurice Wilkins. Using X-ray diffraction data produced by Rosalind Franklin, they were the first to model the double-helix structure of DNA that is so crucial to life. For this work, Watson, Crick, and Wilkins were awarded the 1962 Nobel Prize in Physiology or Medicine. (There is some debate and controversy over the issue that Rosalind Franklin was not included in the prize, although she died in 1958, before the prize was awarded.)

Figure 4.24 shows a diffraction pattern produced by the scattering of X-rays from a crystal. This process is known as X-ray crystallography because of the information it can yield about crystal structure, and it was the type of data Rosalind Franklin supplied to Watson and Crick for DNA. Not only do X-rays confirm the size and shape of atoms, they give information about the atomic arrangements in materials. For example, more recent research in high-temperature superconductors involves complex materials whose lattice arrangements are crucial to obtaining a superconducting material. These can be studied using X-ray crystallography.

Figure shows a white background with a pattern of black dotted lines. There is a bright white spot in the center surrounded by a grey ring. A white line goes up and left from the spot.

Historically, the scattering of X-rays from crystals was used to prove that X-rays are energetic electromagnetic (EM) waves. This was suspected from the time of the discovery of X-rays in 1895, but it was not until 1912 that the German Max von Laue (1879–1960) convinced two of his colleagues to scatter X-rays from crystals. If a diffraction pattern is obtained, he reasoned, then the X-rays must be waves, and their wavelength could be determined. (The spacing of atoms in various crystals was reasonably well known at the time, based on good values for Avogadro’s number.) The experiments were convincing, and the 1914 Nobel Prize in Physics was given to von Laue for his suggestion leading to the proof that X-rays are EM waves. In 1915, the unique father-and-son team of Sir William Henry Bragg and his son Sir William Lawrence Bragg were awarded a joint Nobel Prize for inventing the X-ray spectrometer and the then-new science of X-ray analysis.

In ways reminiscent of thin-film interference, we consider two plane waves at X-ray wavelengths, each one reflecting off a different plane of atoms within a crystal’s lattice, as shown in Figure 4.25 . From the geometry, the difference in path lengths is 2 d sin θ 2 d sin θ . Constructive interference results when this distance is an integer multiple of the wavelength. This condition is captured by the Bragg equation ,

where m is a positive integer and d is the spacing between the planes. Following the Law of Reflection, both the incident and reflected waves are described by the same angle, θ , θ , but unlike the general practice in geometric optics, θ θ is measured with respect to the surface itself, rather than the normal.

Figure shows atoms in a crystal as dots arranged in a grid. They are at a distance d from each other. Two parallel rays, labeled light rays in phase, strike one atom each from up and left, are deflected, and go up and right. The atoms concerned are labeled a and b, b being directly below a. The incident rays form an angle theta with the horizontal. Their extensions form an angle of 20 degrees with the deflected rays. A dotted line connects a and b. Another one connects a with the ray incident on b, making an angle theta with ab, thus forming a triangle. The side of the triangle along the ray incident on b is labeled d sine theta. The ray deflected from b is smaller than the ray deflected from a, by a distance 2d sine theta.

Example 4.7

X-ray diffraction with salt crystals, significance, check your understanding 4.6.

For the experiment described in Example 4.7 , what are the two other angles where interference maxima may be observed? What limits the number of maxima?

Although Figure 4.25 depicts a crystal as a two-dimensional array of scattering centers for simplicity, real crystals are structures in three dimensions. Scattering can occur simultaneously from different families of planes at different orientations and spacing patterns known as called Bragg planes , as shown in Figure 4.26 . The resulting interference pattern can be quite complex.

Figure shows two crystal lattices, with atoms shown as small circles, connected to each other by lines. In the first lattice, flat planes formed in the lattice are highlighted. In the second, slanted planes formed in the lattice are highlighted. In each case, the planes are seen as a combination of different atoms in the same lattice.

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Geological Society of South Africa logo

Combining XRF and XRD analyses and sample preparation to solve mineralogical problems

Maggi Loubser, Sabine Verryn; Combining XRF and XRD analyses and sample preparation to solve mineralogical problems. South African Journal of Geology 2008;; 111 (2-3): 229–238. doi:

Download citation file:

X-ray Fluorescence Spectroscopy (XRF) has reached the stage where it is classified as a mature analytical technique. The theoretical principles are well understood. In addition modern instrumentation demonstrates enhanced generator and temperature stability. High sensitivity is possible even for light elements and effective matrix correction software is available to the modern day spectroscopist. Apart from its continued applications in research and development, XRF has become a routine process control tool.

X-ray Powder diffraction (XRD), on the other hand, has with minor exceptions as in the cement industry, largely remained a research tool, despite being an older analytical technique than XRF. XRD has progressed significantly in the past decade from a mainly qualitative technique for the identification of crystalline materials to a quantitative tool with the advance of more powerful software packages. This software has improved instrument control, but also quantification and structure determination using the Rietveld method. Consequently, XRD is rapidly entering the process control environment.

In this paper the authors demonstrate, with practical examples from different industrial applications, how combined XRF and XRD use can provide truly quantitative phase analyses. XRF is used to verify XRD data and visa versa.

The data obtained in this study clearly illustrate the value that can be added to either technique if XRF and XRD data are used together, and indicate some applications in routine process co.


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Single-crystal X-ray Diffraction

Christine M. Clark, Eastern Michigan University Barbara L. Dutrow, Louisiana State University

What is Single-crystal X-ray Diffraction

Single-crystal X-ray Diffraction is a non-destructive analytical technique which provides detailed information about the internal lattice of crystalline substances, including unit cell dimensions, bond-lengths, bond-angles, and details of site-ordering. Directly related is single-crystal refinement, where the data generated from the X-ray analysis is interpreted and refined to obtain the crystal structure.

Fundamental Principles of Single-crystal X-ray Diffraction

Max von Laue, in 1912, discovered that crystalline substances act as three-dimensional diffraction gratings for X-ray wavelengths similar to the spacing of planes in a crystal lattice. X-ray diffraction is now a common technique for the study of crystal structures and atomic spacing.

X-ray diffraction is based on constructive interference of monochromatic X-rays and a crystalline sample. These X-rays are generated by a cathode ray tube, filtered to produce monochromatic radiation, collimated to concentrate, and directed toward the sample. The interaction of the incident rays with the sample produces constructive interference (and a diffracted ray) when conditions satisfy Bragg's Law ( n λ =2 d sin θ ). This law relates the wavelength of electromagnetic radiation to the diffraction angle and the lattice spacing in a crystalline sample. These diffracted X-rays are then detected, processed and counted. By changing the geometry of the incident rays, the orientation of the centered crystal and the detector, all possible diffraction directions of the lattice should be attained.

All diffraction methods are based on generation of X-rays in an X-ray tube . These X-rays are directed at the sample, and the diffracted rays are collected. A key component of all diffraction is the angle between the incident and diffracted rays. Powder and single-crystal diffraction vary in instrumentation beyond this.

Interpretation of data: Typical mineral structures contain several thousand unique reflections, whose spatial arrangement is referred to as a diffraction pattern. Indices ( hkl ) may be assigned to each reflection, indicating its position within the diffraction pattern. This pattern has a reciprocal Fourier transform relationship to the crystalline lattice and the unit cell in real space. This step is referred to as the solution of the crystal structure. After the structure is solved, it is further refined using least-squares techniques. This procedure is described fully on the single-crystal structure refinement (SREF) page.

Single-crystal X-ray Diffraction Instrumentation - How Does It Work?

X-ray diffractometers consist of three basic elements, an X-ray tube, a sample holder, and an X-ray detector. X-rays are generated in a cathode ray tube by heating a filament to produce electrons, accelerating the electrons toward a target by applying a voltage, and impact of the electrons with the target material. When electrons have sufficient energy to dislodge inner shell electrons of the target material, characteristic X-ray spectra are produced. These spectra consist of several components, the most common being K α and K β . K α consists, in part, of K α 1 and K α 2 . K α 1 has a slightly shorter wavelength and twice the intensity as K α 2 . The specific wavelengths are characteristic of the target material. Filtering, by foils or crystal monochrometers, is required to produce monochromatic X-rays needed for diffraction. K α 1 and K α 2 are sufficiently close in wavelength such that a weighted average of the two is used. Molybdenum is the most common target material for single-crystal diffraction, with MoK α radiation = 0.7107 Å . These X-rays are collimated and directed onto the sample. When the geometry of the incident X-rays impinging the sample satisfies the Bragg Equation, constructive interference occurs. A detector records and processes this X-ray signal and converts the signal to a count rate which is then output to a device such as a printer or computer monitor. X-rays may also be produced using a synchotron, which emits a much stronger beam.

Single-crystal diffractometers use either 3- or 4-circle goniometers. These circles refer to the four angles (2 θ , χ , φ , and Ω ) that define the relationship between the crystal lattice, the incident ray and detector. Samples are mounted on thin glass fibers which are attached to brass pins and mounted onto goniometer heads. Adjustment of the X, Y and Z orthogonal directions allows centering of the crystal within the X-ray beam.

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X-rays leave the collimator and are directed at the crystal. Rays are either transmitted through the crystal, reflected off the surface, or diffracted by the crystal lattice. A beam stop is located directly opposite the collimator to block transmitted rays and prevent burn-out of the detector. Reflected rays are not picked up by the detector due to the angles involved. Diffracted rays at the correct orientation for the configuration are then collected by the detector.

Modern single-crystal diffractometers use CCD (charge-coupled device) technology to transform the X-ray photons into an electrical signal which are then sent to a computer for processing.


Single-crystal X-ray diffraction is most commonly used for precise determination of a unit cell, including cell dimensions and positions of atoms within the lattice. Bond-lengths and angles are directly related to the atomic positions. The crystal structure of a mineral is a characteristic property that is the basis for understanding many of the properties of each mineral. Specific applications of single-crystal diffraction include:

Strengths and Limitations of Single-crystal X-ray Diffraction?


User's Guide - Sample Collection and Preparation

Goniometer head.

Samples are mounted on the tip of a thin glass fiber using an epoxy or cement. Care should be taken to use just enough epoxy to secure the sample without embedding it in the mounting compound. The fiber may be ground to a point to minimize absorption by the glass. This fiber is attached to a brass mounting pin, usually by the use of modeling clay, and the pin is then inserted into the goniometer head.

Sample Centering

The goniometer head and sample are then affixed to the diffractometer. Samples can be centered by viewing the sample under an attached microscope or video camera and adjusting the X,Y and Z directions until the sample is centered under the cross-hairs for all crystal orientations.

Data Collection, Results and Presentation

Data Collection Once the crystal is centered, a preliminary rotational image is often collected to screen the sample quality and to select parameters for later steps. An automatic collection routine can then be used to collect a preliminary set of frames for determination of the unit cell. Reflections from these frames are auto-indexed to select the reduced primitive cell and calculate the orientation matrix (which relates the unit cell to the actual crystal position within the beam). The primitive unit cell is refined using least-squares and then converted to the appropriate crystal system and Bravias lattice. This new cell is also refined using least-squares to determine the final orientation matrix for the sample.

After the refined cell and orientation matrix have been determined, intensity data is collected. Generally this is done by collecting a sphere or hemisphere of data using an incremental scan method, collecting frames in 0.1 ° to 0.3 ° increments (over certain angles while others are held constant). For highly symmetric materials, collection can be constrained symmetrically to reduce the collection time. Data is typically collected between 4 ° and 60 ° 2 θ for molybdenum radiation. A complete data collection may require anywhere between 6-24 hours, depending on the specimen and the diffractometer. Exposure times of 10-30 seconds per frame for a hemisphere of data will require total run times of 6-13 hours. Older diffractometers with non-CCD detectors may require 4-5 days for a complete collection run.

Corrections for Background, Absorption, etc.   After the data have been collected, corrections for instrumental factors, polarization effects, X-ray absorption and (potentially) crystal decomposition must be applied to the entire data set. This integration process also reduces the raw frame data to a smaller set of individual integrated intensities. These correction and processing procedures are typically part of the software package which controls and runs the data collection.

Phase Problem and Fourier Transformation   Once the data have been collected, the phase problem must be solved to find the unique set of phases that can be combined with the structure factors to determine the electron density and, therefore, the crystal structure. A number of different procedures exist for solution of the phase problem, but the most common method currently, due to the prevalence of high-speed computers, is using direct methods and least-squares, initially assigning phases to strong reflections and iterating to produce a refined fit.

Structure solution   Solution of the phase problem leads to the initial electron density map. Elements can be assigned to intensity centers, with heavier elements associated with higher intensities. Distances and angles between intensity centers can also be used for atom assignment based on likely coordination. If the sample is of a known material, a template may be used for the initial solution. More information about structure solution and refinement can be found on the single-crystal structure refinement page .

Structure Refinement   Once the initial crystal structure is solved, various steps can be done to attain the best possible fit between the observed and calculated crystal structure. The final structure solution will be presented with an R value, which gives the percent variation between the calculated and observed structures. The single-crystal structure refinement page provides further information on the processes and steps involved in refining a crystal structure.

The following literature can be used to further explore Single-crystal X-ray Diffraction

Related Links

Teaching Activities and Resources

Teaching activities, labs, and resources pertaining to Single-crystal X-ray Diffraction.

« Thermal Ionization Mass Spectrometry (TIMS)       Time-of-Flight Secondary Ion Mass Spectrometry (ToF-SIMS) »


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If Biden runs again, he’ll be 82 soon after the election. His age would make his vice presidential pick that much more important. Even though he’s likely to keep Harris on the ticket, Democratic operatives are increasingly concerned about how voters will react to her remaining a heartbeat away from the presidency for another four years. There’s nothing new about vice presidents being political liabilities, or getting dumped at reelection time. Replacing a running mate, nevertheless, is a delicate matter. 


Aaron Burr, who ran for vice president with Thomas Jefferson in 1800, secretly tried to get the Electoral College to elect him president instead. Four years later, Jefferson barred Burr from the ticket and later ordered his arrest as a traitor.

Abraham Lincoln rarely saw his vice president, Hannibal Hamlin, who oddly served as a guard and company cook during the last part of the Civil War. He was replaced on the ticket in 1864 by Andrew Johnson, who became president when Lincoln was assassinated less than six weeks into the new term.

Ulysses Grant’s first vice president was implicated in a scandal and defeated for renomination. Grover Cleveland ran three times for president, winning twice. In each race, he had a different running mate. 

Franklin Roosevelt’s selection of House Speaker John Nance Garner as vice president was the result of a deal for delegates at the 1932 Democratic convention. Never close to Roosevelt, Garner stayed on the ticket in 1936 but was shown the door in 1940 when Henry Wallace, the agriculture secretary, replaced him.

In 1944, a gravely ill Roosevelt sought a fourth term. Democratic bosses feared that Wallace, a left-wing mystic, would succeed to the presidency if he remained vice president. The wily FDR secretly allowed party leaders to sideline Wallace, throwing open the vice presidential nomination at the convention. The prize went to Sen. Harry Truman. After only 10 weeks as vice president, Truman became president when Roosevelt died.

The last vice president to be excluded from the ticket was Nelson Rockefeller in 1976. When Gerald Ford became president upon Richard Nixon’s resignation, he appointed the former New York governor to fill the second spot. But after a strong challenge from Ronald Reagan in GOP primaries, Ford needed to strengthen his right flank. He couldn’t do it with Rockefeller, a centrist. A hard-bitten realist, Rockefeller understood the game. He graciously offered to withdraw, and Ford accepted.

Would Harris step aside, as Rockefeller did? Would she do it to keep former President Donald Trump or Gov. Ron DeSantis (R-FL) from winning? It’s unlikely. First, her historic role as the nation’s first woman, African American, and Asian American vice president would make doing so more complicated. Second, Harris still wants to be president.

If Biden doesn’t run again, Harris likely will. But if her next presidential campaign is anything like the last one, she’d have an uphill climb. When Harris started her 2020 White House bid, pundits thought she had great potential. After spending $42 million, she dropped out before any votes were cast. Polls showed she lacked support from key constituencies, particularly women and black voters.

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Ron Faucheux is a nonpartisan political analyst, pollster, and writer. He publishes , a nationwide newsletter on polls and public opinion.

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  1. X-Ray Diffraction Crystallography: Introduction, Examples and Solved

    This is particularly important for beginners in X-ray diffraction crystallography. One aim of this book is to offer guidance to solving the problems of 90 typical substances. For further convenience, 100 supplementary exercises are also provided with solutions.

  2. 4.6 X-Ray Diffraction

    These can be studied using X-ray crystallography. Figure 4.24 X-ray diffraction from the crystal of a protein (hen egg lysozyme) produced this interference pattern. Analysis of the pattern yields information about the structure of the protein. (credit: "Del45"/Wikimedia Commons) Historically, the scattering of X-rays from crystals was used ...

  3. X-ray Diffraction-Solving Problems with Phase Analysis

    X-ray diffraction (XRD), in use for more than 100 years, can quickly distinguish between crystalline phases of a wide variety of materials such as active pharmaceutical ingredients, paints and pigments, and corrosion. 28 minutes. X ray Diffraction - Solving Problems with Phase Analysis Watch on Download the slides.

  4. PDF Solutions of Selected Problems and Answers

    Problem 1.13s It is more convenient for dimensional analysis to employ the G-CGS system (to get rid of ε0 and μ0). The skin depth will depend on: (a) The frequency ω (see the statement of the problem). (b) The velocity of light c (EM phenomenon). (c) The conductivity σ (we are dealing with a good conductor). But [σ]= e2/ aB =[t]−1. Hence ...

  5. X-ray Powder Diffraction (XRD)

    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. The analyzed material is finely ground, homogenized, and average bulk composition is determined. Fundamental Principles of X-ray Powder Diffraction (XRD)

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    Röntgen was investigating cathode rays in different types of evacuated glass tubes and trying to determine their range in air. He noticed that while the rays were being produced, a screen coated in fluorescent barium platinocyanide would glow. He was intrigued because the screen was too far from the tube to be affected by the cathode rays.

  7. XRD X-ray diffraction worked example problem

    XRD X-ray diffraction worked example problem - YouTube 0:00 9:00 MSE example problems tutorial XRD X-ray diffraction worked example problem Taylor Sparks 20.2K subscribers 998 70K views...

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    XRD data acquisition conditions were adjusted for sample size, Table 1. Zoom In Again, solid-state NMR could have been utilized in this work, although the smaller sample amounts (<5mg) might have been problematic. Zoom In Case 2: Stability of an Amorphous Drug Substance Poor water solubility of drugs poses a difficult challenge for formulation.

  9. Solved Problem 4: The XRD pattern of \( \mathrm{CsPbI} 3 \)

    Problem 4: The XRD pattern of CsPbI3 perovskite material obtained by using a Copper anode X -ray tube (λ = 1.542 A) is given below. The Bragg's angles of the main diffraction lines are the following : 1. Show that the crystalline structure is simple cubic 2. What are the Miller indices of the diffraction lines listed in the above table ? 3.

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    3 answers. Feb 5, 2023. I have a CeO2-based catalyst. After adding 1% of Ni, the XRD peak for CeO2 decreases. But then, when I add 5%, 10%, and 15%, the peak increases. But all the peaks are still ...

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    This problem was solved by using a pattern subtraction technique, which permitted selective subtraction of the XRD pattern of the constituents of the formulation from the overall XRD pattern.

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  14. Combining XRF and XRD analyses and sample preparation to solve

    X-ray Powder diffraction (XRD), on the other hand, has with minor exceptions as in the cement industry, largely remained a research tool, despite being an older analytical technique than XRF. XRD has progressed significantly in the past decade from a mainly qualitative technique for the identification of crystalline materials to a quantitative ...

  15. Single-crystal X-ray Diffraction

    Phase Problem and Fourier Transformation Once the data have been collected, the phase problem must be solved to find the unique set of phases that can be combined with the structure factors to determine the electron density and, therefore, the crystal structure. A number of different procedures exist for solution of the phase problem, but the ...

  16. Phase problem

    In physics, the phase problem is the problem of loss of information concerning the phase that can occur when making a physical measurement. The name comes from the field of X-ray crystallography, where the phase problem has to be solved for the determination of a structure from diffraction data. The phase problem is also met in the fields of imaging and signal processing.

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  18. [Solved] X-Ray Diffraction MCQ [Free PDF]

    nλ = 2d sinθ λ = Wavelength of the X-ray d = distance of crystal layers θ = incident angle (the angle between the incident plane and the scattering plane) n = integer (order) Calculation: Given: d = 4.5 × 10 -10 m, θ = 30°, n = 2 We know that, nλ = 2d sin θ ∴ 2λ = 2 × 4.5 × 10 -10 × sin 30° λ = 2.25 × 10 -10 m India's #1 Learning Platform

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