Question by Andy K: In semiconductor physics, what exactly is effective mass?
I have been reading about what is termed “effective mass” in terms of electrons and holes. I’m not sure if I’m understanding it correctly. Can anyone please explain in layman’s terms?
Answer by Schuyler
Effective mass compares the apparent energy of a free electron with that of one perturbed by a periodic potential (i.e. crystal lattice). An electron will be more or less sluggish in lattice depending on where its wavefunction (probability amplitude) exhibits peaks or nodes relative to the nuclei. For example, if the electron’s wavefunction in the crystal tends to place the electron close to the nuclei of the lattice, then the electron is effectively heavier due to the Couomb attraction! But including a new potential energy term is not easy to tabulate or compare for different materials, so instead this additional energy contribution is “absorbed” into the free-electron (kinetic) energy formula by allowing the mass to vary.
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Sine waves over Clam Beach
Image by kern.justin
Part of tomorrow’s post on thegoldensieve.com
On the road to find out.
I snapped off this image as a car swept past on the Avenue of the Giants. This is an old stretch of Highway 101 that weaves through the giant redwoods – sometimes darting dangerously close to these beautiful trees. Opportunity abounds to grab an interesting photograph or simply gawk at the stunning groves throughout its more than 30 miles of highway. There is an abiding joy in personal education, even if the object of this education is obscurity for the sake of personal entertainment. This photograph reminds me a bit of a song by Cat Stevens (like most of his work it reflected a search for meaning in the world and made said work very beautiful before he found a messianic religion that preaches a plethora of anachronistic moralities, and honestly, ceased to be interesting) "On the road to find out":
… So on and on I go, the seconds tick the time outThere’s some much left to know, and I’m on the road to find out.
Don’t you want to know how a (every)thing works?
I spent my free moments on Friday learning the underlying geometry behind how and why tilting the lens or image plane is capable of producing dramatically long depth of field. The principle behind this is named after one of the first to describe it properly, an Austrian photographer named Theodor Scheimpflug. What’s amazing is that today anyone with an internet connection and a high school-level understanding of geometry has access to all the tools they need to teach themselves anything they should want to know about the Scheimpflug Principle. It is common now to see photographs that use the tilt of the lens not to increase depth of field, but paradoxically, to dramatically shorten or truncate it, producing selective focus. I won’t go into the geometry here just now – instead I will save that for the (off) chance that I have access to a tilt/shift mechanism sometime in the future and/or I can make a series of photographs illustrating the principle clearly – but I will mention the principle by waving my hands a bit.
Sine waves (the sine function partly determines the Scheimpflug principle) are everywhere, here you see a form of one in the 2D projection of mixing clouds over the waves on Clam Beach in northern California:
An illustration of selective (shallow) focus created by a tilt mechanism (Nikon 45mm f/2.8 PC-E):
It should make intuitive since to begin with that without any tilt, your lens has a focal plane that is infinite and runs through a space parallel to the image plane. It isn’t infinitely deep, but it is infinitely wide and tall. This is so much the better as it means that we can point the camera at some distant object, say a mountain, and get it and the moon and the ground and clouds all in focus. But suppose altitude is less of an issue, and instead of imaging the moon, or some other object high in the sky, we want to take a picture of a long flat road and the mountains in the distance. Moving both the camera and lens to point at the road you have now taken that focal plane and set it an an angle to the Earth. If it is wide enough, it will contain both the road (which is just a few feet from your lens) and the distant mountain, but your perspective will have changed and what are normally straight lines will now spread apart or bend together as they recede, e.g. the mountains will look as though they are leaning backward. This is called convergence and though it may be corrected by stretching one or pinching the other side of an image in post-processing, it would be better to correct this error in the camera so as to avoid having to crop the image after stretching it in post. Sometimes we can use this effect to our advantage, and here you see I’ve pointed my 14mm lens upward so the buildings and lampposts lean dramatically into the frame, pulling your eye with them:
But sometimes we ought to correct convergence so that the sense of place is increased by more closely reproducing what one’s eyes see. Although your eye is capable of producing convergence as surely as a camera’s lens, your brain is also capable of correcting for it so that you know these lines are straight and their convergence is a trick of flattening the third dimension in the production of a 2D image. Without a 3D image, no photograph can produce the same visual cues and so we correct for convergence to create a heightened sense of place:
But why isn’t it enough to just use a tilt-shift or correct your image in post? Why understand?
Because, until you understand a thing, you will only ever be able to use it to the ends in which you were originally instructed, or those that you discover in accident. Understanding the theory behind something allows you to make connections between ideas which at first seem distant and unentangled. Correcting convergence in camera is achieved with a tilt-shift because of the principle I linked above. When the camera has a non-zero tilt off an axis orthogonal to the image plane, it has a wedge-shaped focal plane that is rotated when the lens is tilted through a point determined by the angle of tilt and focal length. Directly below the lens, the focal plane is infinitely shallow, but it grows on either side with a rate that depends on the focal length, f-number, circle of least confusion and a few other optical properties we will discuss in other posts. Suffice it to say, by tilting the lens off axis, you can place a very close object inside the focal plane and be assured that the depth of field will encompass very tall and distant objects because it grows in depth with distance.
I won’t pretend that it is essential that every person or even ever photographer should know how and why a tilt-shift lens works, but instead I will point out that I have a Faustian interest in knowing many things or at least grasping the principles behind them. If you watched the video on beauty and physics I posted earlier, you would have heard Gell-Mann quote Newton, "… nature is very consonant and conformable to her self." That is to say, natural laws are universal rules, and learning how a tilt-shift mechanism works reveals fundamental properties about how light is focused, how images are cast and how geometry determines perspective. Moreover, with the right computer power in hand you can use fixed properties like camera tilt to determine a number of physical properties of the thing being imaged. Neat huh?
Moreover, I think once you’ve struggled to understand a thing, you’ve changed fundamentally the way you see the world and gain a new angle, a new line of sight on things. Understanding lenses changes the way you see light and I think you and I are better photographers for our increased perception of the universe. For those of you who enjoy reading and understanding things like this – I highly recommend Craig Bohren’s "Clouds in a glass of beer." You’ll never want for an explanation of certain phenomena again, nor will you lack a starting point for understanding all sorts of things that before you never gave a second thought. Examples include – why do clouds on one side of the sky appear darker than those on the other side? Why are breaking waves white at their crests but blue or even green elsewhere?
Bohren’s book will continually play with light scattering and eventually will impress upon you how scattering, reflectance and reflectance are all at once intimately related. Here the top of our atmosphere catches, reflects and scatters the waning light of some further western nations’ sunsets into blue hour. Why blue hour is "blue" and not "red" will be the topic of another day’s post. For now, I will share with you Half Dome in Half Light, a two minute exposure from the rim of the Yosemite Valley.