Constructing an Ewald Sphere
Summary
TLDRThe video script delves into the fundamental principles of X-ray diffraction in crystallography, highlighting Bragg's Law and its significance in understanding how diffraction patterns are formed. It explains the concept of planes separated by a distance 'd' interacting with X-rays of wavelength 'lambda' and the conditions for diffraction to occur. The script introduces the idea of constructive and destructive interference leading to the appearance of spots in a diffraction pattern, indicative of 3D diffraction. The Ewald sphere model is presented as a more comprehensive 3D representation, illustrating how to construct the sphere, the role of crystal orientation, and the impact of varying the X-ray wavelength. The reciprocal space concept is used to redefine Bragg's Law, and the video concludes by demonstrating how changes in wavelength and crystal orientation can alter the accessible reflections, providing a deeper insight into the practical aspects of crystallography.
Takeaways
- 📐 **Bragg's Law in Crystallography**: Bragg's Law is fundamental for understanding how x-ray diffraction works in crystallography, relating the interplanar distance (d), the wavelength of x-rays (λ), and the diffracting angle (θ).
- 🌟 **Constructive and Destructive Interference**: In a diffraction experiment, constructive interference of x-rays leads to bright spots on the detector, while destructive interference results in no spots, forming a diffraction pattern.
- 📊 **3D Diffraction Pattern**: Diffraction is actually a 3D phenomenon, and the evolved sphere model helps to visualize and understand the 3D nature of diffraction spots.
- 🌐 **Reciprocal Space**: The concept of reciprocal space is introduced where each spot on the evolved sphere represents constructive interference and can be indexed with Miller indices (h k l).
- 🔬 **Radius of the Reflecting Sphere**: The radius of the reflecting or Ewald sphere in reciprocal space is inversely proportional to the wavelength of the x-rays (1/λ).
- 📉 **Effect of Wavelength on Diffraction**: Shorter wavelengths increase the radius of the reflecting and limiting spheres, allowing access to more reflections, whereas longer wavelengths result in smaller spheres.
- ⛓ **Lattice Planes and Reflections**: The arrangement and orientation of atoms in a crystal determine the reflections that can be harvested during a diffraction experiment.
- 🔄 **Crystal Rotation and Reflection Access**: Rotating the crystal allows for the collection of different reflections by changing the orientation of the lattice planes relative to the x-rays.
- 🔍 **Data Collection and Analysis**: The evolved sphere construction is essential for understanding how to collect data and which reflections are accessible at a given wavelength.
- 🛠️ **Crystallographic Techniques**: The script outlines the process of deriving Bragg's Law from both real and reciprocal space, providing a comprehensive view of diffraction analysis.
- 💡 **Practical Application**: The knowledge of Bragg's Law and the evolved sphere model is applied to grow and study crystals, highlighting the practical use of these theoretical concepts.
Q & A
What is Bragg's Law and how is it relevant in crystallography?
-Bragg's Law is a fundamental principle in X-ray crystallography that describes the conditions under which diffraction of X-rays occurs in a crystal lattice. It states that constructive interference of X-rays happens when the path difference between the rays reflected from successive planes of atoms is an integer multiple of the wavelength. Mathematically, it is represented as nλ = 2d * sin(θ), where n is an integer, λ is the wavelength of the X-rays, d is the interplanar spacing, and θ is the angle of incidence.
What is the significance of the term 'inter-planar distance' in Bragg's Law?
-The term 'inter-planar distance' refers to the distance between successive planes of atoms in a crystal lattice. It is crucial in Bragg's Law because diffraction occurs when this distance is of the same order as the wavelength of the X-rays. The inter-planar distance (d) determines the angles at which constructive interference will take place.
How does the concept of constructive and destructive interference relate to the appearance of spots in a diffraction pattern?
-In a diffraction experiment, the spots observed on the detector are a result of the interference of diffracted X-rays. Constructive interference, where the waves are in phase, leads to bright spots or reflections on the detector. Conversely, destructive interference, where the waves are out of phase, results in no spots or a reduction in intensity at those positions.
What is the evolved sphere and how does it improve the understanding of diffraction conditions in 3D?
-The evolved sphere is a 3D model that represents the conditions for X-ray diffraction more accurately than Bragg's Law, which is primarily a 2D representation. It is a conceptual tool that helps visualize the diffraction spots in three dimensions, similar to a sphere where each point on the surface represents a constructive interference from atoms in real 3D space. This model is particularly useful for understanding how to collect data and which reflections are accessible at a given wavelength.
What is the Miller index and how is it used in crystallography?
-The Miller index is a set of three integers (h, k, l) that denote the orientation of a crystal plane in a 3D lattice. It is used to label the spots observed on the evolved sphere, indicating how far the spot is from the origin along the reciprocal axes. The Miller indices are derived from the intercepts that the plane makes with the axes when extended to the origin.
How does the reciprocal space concept help in understanding Bragg's Law?
-Reciprocal space is a mathematical construct that helps in visualizing and understanding the diffraction pattern in a more intuitive way. In reciprocal space, the radius of the reflecting sphere is inversely proportional to the wavelength of the X-rays (1/λ), which allows for a clear understanding of how different wavelengths affect the accessible reflections and the resulting diffraction pattern.
What is the relationship between the wavelength of X-rays and the size of the reflecting and limiting spheres?
-The size of the reflecting and limiting spheres in reciprocal space is inversely proportional to the wavelength of the X-rays. A shorter wavelength results in larger spheres, which means more reflections are accessible. Conversely, a longer wavelength results in smaller spheres, limiting the number of accessible reflections.
How does the orientation of the crystal affect the reflections that can be harvested?
-The orientation of the crystal, or the lattice planes within it, determines which reflections will satisfy Bragg's Law and thus be observable. By rotating the crystal, different sets of lattice planes are brought into the correct orientation to diffract the X-rays, allowing for the observation of different reflections.
What is the significance of the angle θ in the context of Bragg's Law?
-The angle θ, known as the diffracting angle, is the angle between the incident X-ray beam and the plane of atoms in the crystal. It is a critical parameter in Bragg's Law as it determines the path difference between the X-rays reflected from successive planes of atoms, which in turn affects whether constructive or destructive interference will occur.
How does the choice of X-ray wavelength affect the outcome of a diffraction experiment?
-The choice of X-ray wavelength directly impacts the size of the reflecting and limiting spheres in reciprocal space, and thus the range of reflections that can be observed. A shorter wavelength allows for a larger sphere and more accessible reflections, potentially providing higher resolution data. A longer wavelength results in a smaller sphere and fewer accessible reflections.
What is the role of the detector in a diffraction experiment?
-The detector in a diffraction experiment is used to measure the intensity and position of the diffracted X-rays. It records the pattern of bright spots (constructive interference) and dark regions (destructive interference), which together form the diffraction pattern. This pattern is then analyzed to determine the structure of the crystal.
How does the concept of the evolved sphere help in the data collection process during a diffraction experiment?
-The evolved sphere provides a visual and conceptual framework for understanding which reflections are accessible at a given wavelength and crystal orientation. By rotating the crystal, different reflections come into the accessible region defined by the limiting sphere, allowing for systematic data collection and a more comprehensive understanding of the crystal structure.
Outlines
🌟 Understanding Bragg's Law and Crystallography
This paragraph introduces Bragg's Law, a fundamental principle in crystallography that explains how x-rays interact with crystal structures. It discusses the concept of planes separated by a distance 'd' and how diffraction occurs when this distance is similar to the wavelength of the x-rays. The law is expressed as nλ = 2d sinθ, where 'n' is an integer, 'λ' is the wavelength, 'd' is the interplanar spacing, and 'θ' is the angle of diffraction. The paragraph also explains how constructive and destructive interference of x-rays on a detector leads to the formation of a diffraction pattern, which is a series of spots rather than continuous planes. The evolved sphere model, developed by Paul Peter Ewald in the 1920s, is introduced as a 3D representation of diffraction spots, with each spot on the sphere representing constructive interference and being assigned a Miller index (h k l). The reciprocal space concept is used to redefine Bragg's Law, and the effects of changing the x-ray wavelength and crystal orientation on the diffraction experiment are discussed.
🔬 The Impact of X-Ray Wavelength and Crystal Orientation
This paragraph delves into the practical implications of Bragg's Law and the evolved sphere model. It explains how the size of the reflecting and limiting spheres in reciprocal space is influenced by the wavelength of the x-rays used. For instance, moving from copper radiation (1.54 angstroms) to molybdenum radiation (0.71 angstroms) would double the size of these spheres. The paragraph illustrates how changing the x-ray wavelength or the orientation of the crystal lattice can alter the reflections that are accessible during a diffraction experiment. It also describes the process of constructing an evolved sphere and how rotating the crystal or the reflecting sphere within the limiting sphere can help access more reflections. The summary concludes with an encouragement to grow one's own crystals, indicating a hands-on approach to understanding these concepts.
Mindmap
Keywords
💡Bragg's Law
💡Diffraction
💡Crystallography
💡X-rays
💡Reciprocal Space
💡Evolved Sphere
💡Miller Indices
💡
💡Constructive Interference
💡Destructive Interference
💡Wavelength
💡Lattice Planes
Highlights
Bragg's law is fundamental in crystallography, describing the interaction of X-rays with crystal planes.
Diffraction occurs when the interplanar distance (d) is of the same order as the X-ray wavelength (lambda).
Bragg's law is expressed as n lambda = 2d sine theta, where n is an integer, lambda is the wavelength, d is the interplanar spacing, and theta is the diffracting angle.
Constructive interference of diffracted X-rays results in bright spots or reflections on the detector.
Destructive interference leads to the absence of spots, contributing to the diffraction pattern.
Diffraction is a 3D phenomenon, and Bragg's law is a 2D representation.
The evolved sphere model provides a 3D visualization of diffraction spots, enhancing understanding of diffracting conditions.
The origin in reciprocal space is on the surface of the evolved sphere, unlike real space where it's at the center.
Each spot on the evolved sphere represents constructive interference and can be assigned a Miller index (h k l).
The radius of the reflecting or evolved sphere in reciprocal space is the reciprocal of the X-ray wavelength (1/lambda).
The angle between the diffracted beam and the lattice plane is theta, which is crucial for understanding the geometry of diffraction.
The evolved sphere model helps in understanding how to collect data and which reflections are accessible at a given wavelength.
Rotation of the crystal or the reflecting sphere allows access to different reflections.
Shorter X-ray wavelengths increase the radius of the evolved and limiting spheres, allowing access to more reflections.
Longer wavelengths result in smaller spheres, limiting the accessible reflections.
Changing the arrangement or orientation of atoms in the crystal can alter the reflections harvested.
Bragg's law can be derived from both real space and reciprocal space, providing a comprehensive understanding of diffraction.
The practical application of these principles is demonstrated through the growth of crystals.
Transcripts
bragg's law is famous in crystallography
as it is an elegant and simple
understanding of how diffraction works
it uses the concept of planes separated
by a distance d
interacting with x-rays of a wavelength
lambda
diffraction occurs when the inter-planar
distance d
is the same order as that of the
wavelength of the x-rays
here using a little bit of pythagoras we
show how the x-rays must travel
2d sine theta longer for the bottom
plane
than the first for both to be in a
diffracting position
this gives our famous law n lambda
equals
2d sine theta where n must be an integer
number
lambda the wavelength and d are
interplane spacing
and theta is the diffracting angle
in a diffraction experiment what we
measure on the detector
is how the diffracted x-rays interfere
with each other
constructive interference leads to
bright spots
or reflections whereas destructive
interference does not
this gives us our diffraction pattern
composed of a range of spots
the appearance of spots leads on quite
nicely to show that we do not observe
planes of diffraction
but spots as diffraction is 3d
bragg's law is a great representation of
how diffraction works in 2d
however we can make a better 3d model to
represent diffracting conditions
this is the evolved sphere and we are
going to discuss
how it can be constructed how to know
our crystal is in a diffraction
orientation
and what effect the wavelength has on
our diffraction experiment
paul peter evolved in the 1920s
visualized the diffraction spots in 3d
much like this pudding a slice of this
pudding is like the 2d array of spots
that we observe on the detector
unlike a sphere in real space where the
origin is in the middle of the sphere
in reciprocal space the origin is on the
surface of the sphere
this is the point where the x-ray leaves
the reciprocal space
each spot on this evolved sphere which
represents constructive interference of
x-ray diffraction
from atoms in real 3d space can be
assigned an index
h k l this is called a miller index
and these values denote how far the spot
is from the origin on the surface of the
evol sphere
along the reciprocal axis h k and l
okay let's simplify it for you
if we introduce some x-rays entering the
crystal with a wavelength lambda
then in reciprocal space the radius of
sphere that the x-rays can see
will be one over lambda this is the
reflecting sphere
or the eval sphere let us now replace
the crystal with layers of atoms
after all that is what they are the
x-rays enter the reflecting sphere
through the point q
and exit through o which we know is the
origin
while doing so part of it gets
diffracted by the atom at c
and exit the sphere through a point p
highlighted here
the angle the diffracted beam cp makes
with the lattice plane
is theta now if we draw some imaginary
lines to connect point q
to p and o to p we have a triangle
basic geometry would tell us that angle
p q o
is also theta now for some basic
trigonometry
sine theta is the distance o p
over o q we know that the radius of a
reflecting sphere
is one over lambda which makes the
distance o
q two over lambda while the distance
between
o p is one over d that makes sine theta
to be one over d
over two over lambda that is lambda
over two d so now we have defined
bragg's law again
using the concept of reciprocal space
not real space
assume your crystal is a collection of
atoms evenly spaced
the reflections from it would also be a
collection of points in the reciprocal
space
let's draw an evolved sphere on it the
evil construction
is really useful in understanding how to
collect data
and what reflections are accessible at
that wavelength
if we rotate the reflecting sphere
around point o
which is the same as rotating our
crystal we can draw
another sphere this second sphere is the
limiting sphere
with a radius of 2 over lambda any
reflections beyond this sphere
are not accessible with this wavelength
let's try that again
now watch what happens to the
diffraction spots as we rotate the
reflecting
or evolved sphere within the limiting
sphere
now let's see how we might access more
reflections
if we move to a shorter wavelength of
x-ray our spheres will increase in
radius
due to the reciprocal relationship
between the radius of the spheres and
the wavelength
this means in going from copper
radiation
1.54 angstroms to molybdenum radiation
0.71 angstroms our sphere
will double in size we can highlight a
lattice spot here
so that you can observe the changes
better of course
the inverse is true we're going to
longer wavelengths will result in a
smaller reflecting and limiting sphere
of diffraction
now let's see what will happen if we
change the way atoms are arranged
or even simply just their orientation
like we would if we rotated the crystal
we harvest different reflections
we have now learnt how we can construct
an evolved sphere
and saw how the choice of the x-ray
wavelength and the orientation of the
lattice planes
can affect the reflections we can
harvest we also
derived the bragg's law from real space
and reciprocal space
now we know all of that it is time to
grow
our own crystals
[Music]
you
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