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I want to talk a little bit more about diffraction actually. And as a way of actually making

a transition to our next topic, this may seem a little odd way – our next topic is sampling

an interpolation. And going from diffraction to sampling interpolation may seem like a

little odd of way going but it’s – there’s an interesting connection here that I want to

exploit. The topic itself that I – the general areas of diffraction, and in particular what I

want to talk about today, is interesting in itself and it does make actually for a nice link,

so I want to talk about the problem of crystallography.

We’re gonna actually return to this when we have higher dimensional, when we talk

about higher-dimensional Fourier transforms. So today, I’m only gonna talk about the

one-dimensional case, which of course is not realistic but it has some essential ideas that

you find in the higher-dimensional case. And as I say, it makes a nice transition to the

next topic that we’re gonna be talking about.

Let me remind you about the headline from last time. So headline from last time, when

we talked about diffraction and the Fourier transform, is that diffraction patterns are

given by or determined by the Fourier transform of the apertures that cause the

diffraction. Of course, this simplifies things but that’s – if you’re looking for a quick

summary of what our chief conclusion was last time, this is it. Diffraction patterns are

determined by the Fourier transform of the apertures or the aperture function.

We had approximation; we talked about far-field diffraction, all the rest of that jazz,

never mind. That was all important, of course, but this is the main thing to be carried

away with it, and that’s what I’m gonna be using today also.

Now here is the setup for what was troubling people, what was puzzling people when xray

crystallography, x-ray diffraction was first invented or first brought to bear on certain

set of important problems. So the setup that I want to talk about it as follows: x-rays were

discovered in 1895 by Roentgen, of course, Roentgen, R-O-E-N-T-G-E-N, or some

approximation of that spelling. All right?

Matter of fact, I remember actually in 1995 everybody was celebrating the 100th

anniversary of x-rays, a very exciting time. And the question was what are they? Are they

waves? I mean their fundamental nature was not understand, so what are they, or what

were they. Are they waves, for example? It was a new phenomenon. If so, then certain

considerations led them to conclude that the wavelengths should be about ten the minus

eighth centimeters. All right?

If so, and the wavelength, which was too small to measure precisely, wavelength should

be around ten to the minus eighth centimeters. All right? So that’s too small to measure

by other means – by the means that we’re used to measuring different sorts of visible

light, say, other sorts of waves which were diffraction gratings, too small to measure. Let

me just say too small to measure, period, with any of the standard techniques, okay, for

example with diffraction gratings.

On the other hand, or a different set of – a different line of questioning was crystals; 2.)

Crystals. Been around for a long time and observed for a long time. What are they? In

particular, it was clear from people – to people who were cutting them open, chiseling

them around and making small crystals out of big crystals, that somehow the microscopic

a transition to our next topic, this may seem a little odd way – our next topic is sampling

an interpolation. And going from diffraction to sampling interpolation may seem like a

little odd of way going but it’s – there’s an interesting connection here that I want to

exploit. The topic itself that I – the general areas of diffraction, and in particular what I

want to talk about today, is interesting in itself and it does make actually for a nice link,

so I want to talk about the problem of crystallography.

We’re gonna actually return to this when we have higher dimensional, when we talk

about higher-dimensional Fourier transforms. So today, I’m only gonna talk about the

one-dimensional case, which of course is not realistic but it has some essential ideas that

you find in the higher-dimensional case. And as I say, it makes a nice transition to the

next topic that we’re gonna be talking about.

Let me remind you about the headline from last time. So headline from last time, when

we talked about diffraction and the Fourier transform, is that diffraction patterns are

given by or determined by the Fourier transform of the apertures that cause the

diffraction. Of course, this simplifies things but that’s – if you’re looking for a quick

summary of what our chief conclusion was last time, this is it. Diffraction patterns are

determined by the Fourier transform of the apertures or the aperture function.

We had approximation; we talked about far-field diffraction, all the rest of that jazz,

never mind. That was all important, of course, but this is the main thing to be carried

away with it, and that’s what I’m gonna be using today also.

Now here is the setup for what was troubling people, what was puzzling people when xray

crystallography, x-ray diffraction was first invented or first brought to bear on certain

set of important problems. So the setup that I want to talk about it as follows: x-rays were

discovered in 1895 by Roentgen, of course, Roentgen, R-O-E-N-T-G-E-N, or some

approximation of that spelling. All right?

Matter of fact, I remember actually in 1995 everybody was celebrating the 100th

anniversary of x-rays, a very exciting time. And the question was what are they? Are they

waves? I mean their fundamental nature was not understand, so what are they, or what

were they. Are they waves, for example? It was a new phenomenon. If so, then certain

considerations led them to conclude that the wavelengths should be about ten the minus

eighth centimeters. All right?

If so, and the wavelength, which was too small to measure precisely, wavelength should

be around ten to the minus eighth centimeters. All right? So that’s too small to measure

by other means – by the means that we’re used to measuring different sorts of visible

light, say, other sorts of waves which were diffraction gratings, too small to measure. Let

me just say too small to measure, period, with any of the standard techniques, okay, for

example with diffraction gratings.

On the other hand, or a different set of – a different line of questioning was crystals; 2.)

Crystals. Been around for a long time and observed for a long time. What are they? In

particular, it was clear from people – to people who were cutting them open, chiseling

them around and making small crystals out of big crystals, that somehow the microscopic

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