3.4.pdf: Matter Waves Review - PDF

 1. Introduction (Completed) 1. Introduction...

 1. Introduction (Completed) 1. Introduction In the last topic, we approached the issue of Abbe's di9raction limit (d = λ/2NA) by using some clever tricks to get around it. What we haven't considered yet is using a shorter wavelength, which would give us a smaller minimum resolvable separation, d. We can do this by employing high-energy electrons instead of photons which, as we are about to see, have an associated wavelength much smaller than that of light used in optical microscopy. Matter Waves (Completed) Matter Waves You will likely have covered the concept of matter waves already in other modules (or will do so soon), but let's quickly recap the concept here. In 1924, Louis de Broglie asserted that "nature likes symmetry", so if waves could behave as particles (photons), then particles (e.g. electrons) should be able to behave as waves. He combined ideas from both relativity (E = mc2) and the photoelectric e9ect (E = hν ) to arrive at the following hypothesis: For a particle with a given momentum, p, the particle will behave like a wave of wavelength λ. The momentum is determined from the particle mass, m, and the velocity, v, by p = mv. This wavelength, λ, of a particle is often referred to as the de Broglie wavelength. h h λ= = p mv We can use this concept of matter waves to describe electrons as orbiting the nucleus like a standing wave. There is incomplete content above. You must complete this before you can proceed through the course. Recap: Matter Waves (Completed) Recap: Matter Waves    0:00 / 1:57  1x   There is incomplete content above. You must complete this before you can proceed through the course. Davisson-Germer Experiment (Optional) (Completed) Davisson-Germer Experiment (Optional) Note: this experiment will not be examinable, but you might ?nd it interesting nonetheless. In 1927, Clinton Davisson and Lester Germer concluded a set of experiments which demonstrated the wave nature of the electron, proving the de Broglie hypothesis was correct. Essentially, they applied the concept of Bragg di9raction (normally used for X-rays) to electron waves: if electrons do indeed behave as waves, they should show the same sort of di9raction behavior. If the lattice spacing, d, of a crystal is known, then Bragg's law can be used to determine the incoming wavelength, λ, at a di9raction maximum angle, θ, according to: nλ = 2d sin θ In the experiment, Davisson and Germer Yred a beam of electrons of varying energies (accelerating voltages) at a nickel crystal ( d = 2.15 Å), and measured the electron intensity coming out at an angle of 50∘. Assuming n = 1, this would lead to a maximum intensity at a wavelength λ = 2.15 sin 50∘ = 1.65 Å. A maximum intensity was measured with an accelerating voltage of 54 V, which corresponds to a de Broglie wavelength of λ = 1.67 Å (we will go over how to calculate this in a little bit). Additional peaks corresponding to n = 2, 3, 4, … were also measured, providing evidence for the wave nature of electrons. Source: Georgia State University There is incomplete content above. You must complete this before you can proceed through the course. Nobel Prize: de Broglie (Completed) Nobel Prize: de Broglie Only 5 years after his hypothesis, de Broglie won the 1929 Nobel Prize in Physics "for his discovery of the wave nature of electrons". This was an incredibly important discovery in the nascent Yeld of quantum mechanics. There is incomplete content above. You must complete this before you can proceed through the course. Electron Source (Completed) Electron Source In order to use electrons for imaging, we need to generate high-energy electrons (high speed, small λ); in the order of 1 − 40 keV for SEM and 30 − 2000 keV for TEM. Energies of electrons are frequently quoted in electronvolts, which is a convenient unit as 1 eV is the energy gained by an electron accelerated through an electric potential of 1 V. (For reference, 1 eV = 1.602 × 10−19 J as the elementary charge e = 1.602 × 10−19 C and 1 J = 1 C × 1 V). To achieve these sorts of energies, we use what is known as an "electron gun" to produce the correct energy electrons, in the desired direction. Below is an example of such an electron source. We can use this to easily determine an electron's de Broglie wavelength according to its accelerating voltage, as it can be assumed that all the electron's energy is kinetic energy. So: 1 h eV = mv2, ∴ λ = 2 √ 2meV where e is the elementary charge, m is the mass of an electron, h is Planck's constant, and V is the accelerating voltage. If we return to Abbe's di9raction limit, a 10 keV electron will have λ = 12 pm, which gives a resolution limit of approximately 7.5 Å ( sin θ tends to be quite small in electron microscopes, and thus d increases substantially compared to λ). Starting at the top, a current is run through a Ylament (e.g. W) causing it to heat up and emit electrons through a Wehnhelt cylinder (or grid cap) which has an aperture at the bottom. The voltage di9erence between this cap and the anode below accelerates electrons through a second aperture in the anode, forming a high-energy beam. Special lenses are used to control and direct this beam onto the sample. Source: Goldstein et al. There is incomplete content above. You must complete this before you can proceed through the course. Recap: Electron Source (Completed) Recap: Electron Source    0:00 / 5:17  1x   There is incomplete content above. You must complete this before you can proceed through the course. Electron Lenses (Completed) Electron Lenses As with optical microscopy, we need lenses to control the direction of electrons and focus them onto a desired point. To do this, we use → , inside a central circular aperture. special electromagnetic lenses that generate a magnetic Yeld, B This magnetic Yeld interacts with moving charges, applying a force to them according to: → = q(→ F → v × B) → is the force experienced by the particle, q is the particle charge (−e for an electron), and v where F → is the particle velocity. By controlling the magnetic Yelds inside the lens we are able to manipulate the forces acting upon the charge, and hence its direction. To visualize in which direction a moving charge has a force applied to it, you can use the right hand rule. If you take a right hand and extend the thumb, index, and middle Ynger outwards at 90∘ from each other (as shown below), you can assign the various vectors to this. So if a charge is moving along the index Ynger across a magnetic Yeld in the direction of the middle Ynger, it will experience an upwards force (along thumb). Source: AMMRF There is incomplete content above. You must complete this before you can proceed through the course. Below is a simple diagram showing how a magnetic electron lens (a cross-section is displayed) may a9ect the trajectory of an electron in comparison to an optical system. Source: AMMRF There is incomplete content above. You must complete this before you can proceed through the course. Recap: Electron Lenses (Completed) Recap: Electron Lenses There is incomplete content above. You must complete this before you can proceed through the course. Vacuum System (Completed) Vacuum System There is one Ynal consideration when using electrons for imaging. Unlike an optical microscope, we require a high vacuum ( < 10−6 mbar) to prevent electrons from being scattered by gaseous atoms and/or molecules. Additionally, this prevents arcing between the cathode and anode in the electron gun, as well as reducing the temperature required for high electron emission from the Ylament. Here we see a table of di9erent vacuum ranges, including their pressure range, number density (how many molecules are present in a given volume), and the mean free path of an electron. The mean free path is the average distance travelled by an electron before a collision event. As you can see, in ambient pressure this is only a few dozen nanometers, while it increases to several kilometers by the time we have an ultra-high vacuum. As we want most of our electrons to reach the sample and detector without being scattered by any gaseous molecules, we need as high a vacuum (i.e. low pressure) as possible. Vacuum range Pressure (mbar) Number density (molecules / m3) Mean free path Ambient pressure 1013 2.7 × 1025 68 nm Low vacuum 300 − 1 1025 − 1022 0.1 − 100 μm Medium vacuum 1 − 10−3 1022 − 1019 0.1 − 100 mm High vacuum 10−3 − 10−7 1019 − 1015 10 cm − 1 km Ultra-high vacuum 10−7 − 10−12 1015 − 1010 1 km − 105 km Extremely high vacuum < 10−12 < 1010 > 105 km There is incomplete content above. You must complete this before you can proceed through the course. 1982 Nobel Prize in Chemistry (Completed) 1982 Nobel Prize in Chemistry There are a number of Nobel Prizes directly related to electron microscopy. The Yrst one awarded was to Aaron Klug "for his development of crystallographic electron microscopy and his structural elucidation of biologically important nucleic acid-protein complexes". There is incomplete content above. You must complete this before you can proceed through the course. 1986 Nobel Prize in Physics (Completed) 1986 Nobel Prize in Physics Not long after Aaron Klug, Ernst Ruska was awarded the Nobel Prize in Physics "for his fundamental work in electron optics, and for the design of the Hrst electron microscope“. He had built the very Yrst electron microscope all the way back in 1931, and was able to achieve greater resolutions than via optical methods by 1933. Below is an image with one of his microscopes working alongside Max Knoll. Technically, he shared his Nobel prize with the inventors of the scanning tunneling microscope, which we will come to in Topic 4. There is incomplete content above. You must complete this before you can proceed through the course.   2. Scanning Electron Microscope (SEM) (Completed) 2. Scanning Electron Microscope (SEM) The scanning electron microscope (SEM) is the 8rst of two electron microscopes we will be looking at in this topic. Its principal mode of operation is in scanning a beam of electrons onto the sample and detecting secondary electrons coming out of it. Let's take a closer look at the basic operation of a typical SEM instrument. 1. Electrons from the electron gun pass through a set of condenser lenses which focus the electrons into a tighter beam (like in a light microscope). 2. An objective lens focuses the beam of primary electrons (PE) onto the sample. 3. Electrons either backscatter (BSE) or generate secondary electrons (SE), which are collected by separate detectors. 4. X-rays that are produced can also be detected to gain additional chemical information (EDX - see later). 5. The electron beam is scanned across the surface in x- and y- directions, building up an image based on the number of BSE and SE detected at each point. Source: Inkson et al. There is incomplete content above. You must complete this before you can proceed through the course. A scanning electron microscope is actually quite bulky, as it needs to incorporate a vacuum system (including a way to get samples in and out easily), electron sources and detectors, as well as the computer interface to control the entire equipment. These are sold as a combined unit, and can be seen below and to the left. The Lego recreation on the right unfortunately never made it to market, but you can 8nd some instructions on how to build one if you look carefully online. Source: Boise State University There is incomplete content above. You must complete this before you can proceed through the course. Video: SEM (Completed) Video: SEM Below is a short video that shows the basic working principle behind the SEM.    0:00 / 2:03  1x   There is incomplete content above. You must complete this before you can proceed through the course. Example: Pollen (Completed) Example: Pollen One of the great advantages of SEM is that it has an extraordinary depth of >eld, meaning large areas in diVerent focal planes will remain in focus. Here we can see an image of pollen grains from sunXower, morning glory, prairie hollyhock, oriental lily, evening primrose, and castor bean. Note: Electron microscope images are colorless, for an obvious reason, we are not using light! Source: NASA There is incomplete content above. You must complete this before you can proceed through the course. Backscattered and Secondary Electrons (Completed) Backscattered and Secondary Electrons Secondary electrons (left) are generated when a primary electron inelastically collides with an electron bound to an atom, knocking it out. SE usually have energies of and are accelerated to the detector using a charged grid (). Backscattered electrons (center) are elastically scattered (without losing energy) and reXected back out of the sample. Heavier atoms are able to backscatter electrons more easily, so these give chemical information about the surface. If we look at the 8nal diagram on the right, we can see that the majority of BSE (and X-rays) come from deep within a sample (up to a micron or so), while SE generally come from the 8rst few tens of nanometers near the surface. Source: Wikimedia There is incomplete content above. You must complete this before you can proceed through the course. Below we can see the diVerence that switching between BSE and SE detectors can make. At the bottom is the secondary electron image of a particular sample, while at the top is the same region but captured using backscattered electrons instead. Hopefully you can see the improved contrast of the top image, which could be used to understand the morphology of the sample to a greater extent. There is incomplete content above. You must complete this before you can proceed through the course. SEM Samples (Completed) SEM Samples As we are 8ring huge numbers of electrons at a surface, we need to make sure that the sample does not become negatively charged, as this would lead to electrons being de;ected away from our intended target! To combat this, we need to ensure our samples are conductive. This actually has several bene8ts: It prevents the build-up of charge on the sample by allowing it to be grounded. Without this, the primary electron beam will become deXected. Less likely for thermal damage to occur, as metals are good thermal conductors as well. We are hitting the sample with a constant stream of high-energy electrons (in a vacuum) over long periods of time. Metals show greater secondary electron emission, which in turn helps improve the contrast of the 8nal image. If a sample is non-conductive, we normally sputter on a thin coat of a conductive material (e.g. gold). The diagram here shows how a sputtering system works. This ant had to be sputter-coated before being imaged with an SEM. Heavy ions (e.g. ) are accelerated towards a metal, causing atoms to become dislodged (sputter) and coat the sample opposite. There is incomplete content above. You must complete this before you can proceed through the course. Example: Nanoparticles (Completed) Example: Nanoparticles Here are some SEM images of nanoparticle 8lms from this journal article, formed in diVerent conditions. (a) and (b) show silver and gold nanoparticle 8lms, respectively, formed at a water-air interface. In the case of silver, it has formed in a network structure, while the gold nanoparticles showed a much closer structure (c) and (d) show silver and gold nanoparticle 8lms, respectively, formed at a water-hexane interface. In this diVerent environment, the silver nanoparticles also become more packed. There is no need to remember this speci8c example, it is just here to demonstrate how we can use an SEM to deduce some information about the chemistry that is causing these changes in the structures formed. There is incomplete content above. You must complete this before you can proceed through the course. Recap: SEM (Completed) Recap: SEM    0:00 / 6:09  1x   There is incomplete content above. You must complete this before you can proceed through the course. Demo: Virtual SEM (Completed) Demo: Virtual SEM Below is a great little "virtual" SEM that you can play around with, created by Myscope. It will let you go through the various steps in loading a sample and turning on the electron beam, as well as focusing and adjusting image contrast. Don't worry about the term astigmatism in the controls, this just refers to the fact that in optical/electronic systems, there exist diVerent focal planes depending on the orientation of the incoming light/electrons. Note: it is highly recommended to use a tablet or laptop/desktop. Virtual SEM There is incomplete content above. You must complete this before you can proceed through the course. Energy-Dispersive X-ray Spectroscopy (EDX/EDS) (Completed) Energy-Dispersive X-ray Spectroscopy (EDX/EDS) In normal operation, an SEM will also generate X-rays characteristic of the speci8c atoms on the surface. This occurs due to the fact that secondary electrons which have been knocked out of inner shells will leave behind a hole. Electrons from an outer (higher-energy) shell can then drop down into this hole, emitting an X-ray of energy equal to the energy diVerence between these two levels. The energies of the X-rays are speci@c to each atom, and therefore allow for chemical analysis. In particular, heavier atoms can be quite sensitive to this. The technique of measuring these characteristic X-rays is known as energy-dispersive X-ray spectroscopy (EDX/EDS). Below is an EDX spectrum of K309 glass. Each peak corresponds not only to a speci8c element (e.g. Si, Al), but also to a speci8c transition as denoted by the Siegbahn notation (e.g. Kα, Lβ). There is incomplete content above. You must complete this before you can proceed through the course. Example: Pigment Analysis (Completed) Example: Pigment Analysis The following example is taken from the following article analyzing the various pigments and dating a particular 18th century painting. While not required for this module, I would recommend it for anyone to have a read through. EDX was used to determine the chemical compositions of various parts of the painting, which in turn allowed the author to determine the pigment used. Shown below is from the lower-left part of the painting (circled in red). There is incomplete content above. You must complete this before you can proceed through the course. Recap: EDX Recap: EDX    0:00 / 3:33  1x   There is incomplete content above. You must complete this before you can proceed through the course.   3. Transmission Electron Microscope (TEM) (Completed) 3. Transmission Electron Microscope (TEM) TEM (Completed) TEM The second instrument in this topic is the transmission electron microscope (TEM), which works a little di