Author: logancollins

PDF Version: Notes on the Drude Model

Assumptions of the Drude Model

• The Drude model provides a classical mechanics approach to describing conductivity in metals. This model makes several key assumptions (some of which are better approximations than others).
• Electrons in a metal behave much like particles in an ideal gas (no Coulombic interaction and no collisions between particles). This is called the independent electron approximation.
• Positive charges are located on immobile ions. The electrons do not experience coulombic interaction with the ions, but they do collide with the ions and can change direction and velocity.
• Electrons reach thermal equilibrium by collisions with the ions. Their mean kinetic energy within the lattice at equilibrium is given below. The mass of an electron is represented as m_e, the average velocity at a given temperature is v_T, the Boltzmann constant is k_B, and the temperature in Kelvin is T. At room temperature, v_T is about 10⁵ m/s.

• The average distance of an electron’s free movement between collisions is called the mean free path λ. For metals, the mean free path is typically estimated as 1 nm based on known ionic packing parameters. To calculate the mean time τ between collisions (called the relaxation time), the equation τ=λ/v_T is used.

Applying the Drude Model

• To apply the Drude model, the density of the “gas” formed by the free electrons must be known. This parameter is called the conduction electron density n (the number of free electrons per volume).
• The conduction electron density is computed by assuming that each atom contributes Z_V conducting electrons. Z_V represents the number of outer shell electrons for metal atoms in the ionic lattice. For instance, alkaline Earth metals have a Z_V value of 2. Given the density ρ_m in kg/m³ and the atomic mass M in kg per atom, the conduction electron density is Z_Vρ_m/M.
• Given the average time for a collision-free drift τ, the average drift velocity of an electron in a metal can be computed using the equation below. Here, e represents the charge of an electron in Coulombs and ℇ is electromotive force in volts (a vector quantity).

• The number of electrons passing through a given area per unit time J_e and the amount of charge passing through a given area per unit time J_c are given below. Here, n is the number of electrons.

• The current density j is computed by the equation below.

• This can be used to derive the following expression, which is equivalent to Ohm’s law V=IR.

• As such, the conductivity σ and the resistivity ρ are given by the equations below.

• The mobility μ of an electron in a lattice is given below. The mobility can be interpreted as the ratio of the drift velocity to the applied electric field.

• Using mobility, conductivity and resistivity can be computed by alternative formulas.

The Drude Model and the Hall Effect

• The Drude model explains the Hall effect, a phenomenon in which an electric field ℇ_H arises perpendicular to both the current density j_x (which points in the direction of electron movement) and the magnetic field B_z.
• The Hall effect occurs when a current flows through a conductor while under a magnetic field. As a result of the magnetic field, positive charges accumulate on one side of the conductor (and negative charges on the other side).

• For electrons to pass through the given region, the field ℇ_H must cancel the Lorenz force which acts in the opposite direction (the Lorenz force is the sum F_L=qE+qv×B of the magnetic force and electric force on a moving charged particle). The magnitude of the Hall field is given below. R_H is called the Hall coefficient and can be measured experimentally.

• As the value of R_H approaches one, the field ℇ_H more exactly cancels the Lorenz force. The value of R_H varies between types of metals, so some metals are better conductors than others.

The Drude Model and Optical Reflectivity of Metals

• Light can be described as an electromagnetic wave in the form of a transverse plane.
• The electric field for light in the propagating in the z direction is given by the equation below. ℇ₀ is the amplitude in the xy plane, λ₀ is the wavelength in a vacuum, n is the index of refraction, and κ is a parameter that accounts for attenuation of the light’s intensity inside of a given material.

• Alternatively, the equation for the electric field of light propagating in the z direction can be expressed using the dielectric constant of the given material.

• An electron within the electric field from an electromagnetic wave moves according to the following equation of motion (derived from F=ma). Note that the charge –e and the exponential are distinct.

• Solving the differential equation above gives the following result.

• As a result of the electric field from the electromagnetic wave, the electron undergoes positional displacement in a periodic manner. This leads to a changing dipole moment given by –ex(t).
• Dielectric functions describe the permittivity of given media over time. Permittivity is the amount of charge needed to generate a single unit of electric flux within the medium. Using the above results and some other known formulas, the dielectric function for a material can be derived.

• By setting a parameter ω_P² equal to ne²/m_eε₀, the dielectric function can be rewritten as below. Note that ω_P is called the plasma frequency.

• For ω<ω_P, the value of ε is real and negative, making the square root of ε purely imaginary. As such, the light does not transmit into the metal in this case. Since energy is conserved, the light is reflected instead.
• For ω>ω_P, the value of ε is real and positive. As such, the light does propagate into the metal in this case.
• Metals reflect low-frequency light and are transparent for high-frequency light. The transition occurs at the plasma frequency ω_P.
• The plasma frequency can be measured experimentally or calculated using the conduction electron density n.

The Drude Model and the Wiedemann-Franz Law

• Thermal conductivity is defined by the equation below where Q is the amount of heat transferred per time t, k is the thermal conductivity constant for a given material, A is the cross-sectional area, d is the thickness of the material, and ΔT is the difference in temperature across the material. Note that this definition only describes the simple 1-dimensional case, but analogous formulas can be used for more general situations.

• The Wiedemann-Franz law states that, for any metal at a given temperature, the ratio of thermal conductivity κ to electrical conductivity σ is a constant L. Furthermore, L is proportional to temperature as temperature varies. L is called the Lorenz number.

• The ideal gas-related equations from the Drude model can be used to generate an equivalent formula for LT. Despite some minor inconsistencies with experimental data, this theoretical calculation often gives strikingly accurate results.

Shortcomings of the Drude Model

• The Drude model does not take into account collisions between electrons themselves. It also does not consider electrostatic interactions between the electrons and the lattice ions.
• The de Broglie wavelengths of electrons with some thermal energy are on the nanometer scale. This means that electrons cannot be treated as classical particles (since they have substantial wave character) under the conditions of the Drude model.
• As mentioned, the Drude model underestimates conductivity of metals at low temperatures. This is because the assumption of a constant mean free path (based on atomic spacing) is incorrect. The mean free path varies greatly with temperature, particularly in pure crystalline substances.
• The Drude model cannot explain the conductivity of alloys. Even small impurities can drastically decrease the conductivity of metals in a way which is not predicted by the Drude model.
• From a classical mechanics perspective, the electrons should contribute greatly to the heat capacity of metals. But this result does not agree with experimental data (and using quantum mechanical models instead resolves the issue).

Reference: Hofmann, P. (2015). Solid State Physics: An Introduction. Wiley.

This piece started as a hand-drawn pencil sketch which I “doodled” during a class (yes, I also paid attention to the lecture). Afterwards, I took a picture of the sketch and cleaned it up using Paint.net, a program similar to Photoshop. I was quite pleased with the result!

     Most ideologies include methods for finding reasons to live. But after seeing historical tragedies in which such methods have led to tremendous suffering, some have given up on the idea of purpose. This is not an entirely unreasonable response, but I would argue that it still represents an unjustifiably pessimistic outlook. Although most systems of existential meaning suffer from crippling flaws, such flaws do not invalidate the possibility of purpose. Religious, political, and philosophical approaches have been applied towards resolving the question of existential purpose (Triandis, 2008). Each system possesses certain merits, but they are incomplete since their theoretical frameworks lacks rigor and data. In this essay, I will discuss a toolset and a general philosophical framework which may aid efforts to construct a meaningful world. I propose that existential meaning arises from specific computational structures encoded in matter which can be understood and engineered using the empirical and self-correcting process of scientific inquiry.

     Religious thinkers suggest that specific dogma associated with a given belief system provide meaning. They argue that the purpose of life involves fulfilling certain natural laws which they associate with deities, scriptures, and prophets. However, religious ideologies lack strong empirical support for their dogmas. This may arise from religious doctrines not including effective ways for them to evolve as new knowledge is acquired and circumstances change (Emerson & Hartman, 2006). It should be noted that some religions possess a greater level of flexibility than others. For instance, followers of Hinduism have demonstrated openness to their religion evolving as scientific knowledge provides new insights about the universe. Varadaraja Raman has written on deconstructing the schism between Hindu spirituality and science by reinterpreting Hindu ideas as metaphors for scientific concepts (Raman, 2009). In addition, some followers of the Vedanta philosophy subscribe to a nondualist view of the universe and have made attempts to reconcile Hindu beliefs with new scientific findings (Dorman, 2011). However, other Hindu philosophies have remained bound to tradition, rejected concepts like evolution by natural selection, and even embraced radical postmodernist characterizations of rationality as a tool for oppression (Raman, 2012). Since many religions have exhibited staticity in their interpretations of purpose when new data contradicts their dogma, I propose that existing and past religions have created approximate models for purpose, but fallen short of their ultimate goals.

     Political strategies for finding meaning often involve creating identity via tribal association with certain values while rejecting outgroups which deviate from said values. According to a meta-analysis performed by Jost et al., both conservatives and liberals possess existential motivations driven by specific fears associated with opposing values (Jost, Stern, Rule, & Sterling, 2017). For conservatives, the data indicated fear of governmental control, racial minorities, and social changes that conflict with traditional views. Among liberals, the data indicated that threatening circumstances trigger an increase in tendencies towards strong moralization of political issues. Across political affiliations, such tribalistic behaviors may correlate with the same psychologies which drive religious and philosophical approaches to uncovering existential meaning (Norris & Inglehart, 2011). This is not inherently wrong, but it does reframe politics as holding similarities to religion in that politics may only provide approximations in its pursuit of meaning. Furthermore, political techniques have so far failed to create stable utopian societies, though secular political systems have debatably realized improvements over theocratic systems (Lilla, 2008).

     Stoic philosophers such as Albert Camus argue that accepting absurdity may represent a superior way of living (Camus, 2013). Stoicism seeks to ignore value and so avoid existential dread (Nagel, 1971). But stoicism is paradoxical in that it denies the phenomenological existence of value (Block, 1995; D. J. Chalmers, 1995; Nagel, 1974). Stoics may claim to live passively, but they will always fail to exercise consistency in this doctrine since value is a fundamental constituent of reality. Any entity will possess at least some degree of motivation, regardless of whether the entity wants to possess said motivations. The universal existence of motivation can be illustrated by causality in natural systems. For this argument, I take the nondualist position that consciousness is identical to specific physical processes and does not possess a supernatural component. Consider a rock which rolls down a hill and strikes a second rock. In this scenario, the second rock cannot prevent itself from having a reaction. Biological organisms are no exception to causality. From atoms to galaxies, the cosmos is driven by change. As much as a stoic may try to condition him or herself against reaction, physical reality remains intrinsically causal, so this task will ultimately be in vain.

     To provide a contextual background for my proposal that existential meaning is encoded in matter, I will introduce a physical argument for panpsychism. Despite its past association with metaphysics, panpsychism is rekindling among contemporary thinkers as an explanation for consciousness (Strawson, 2006). Integrated information theory or IIT (Oizumi, Albantakis, & Tononi, 2014) is a mathematical formulation which seeks to quantify consciousness using the information arising from dynamical systems. Galen Strawson has argued that IIT implies panpsychism since all physical structures contain some amount of information (Strawson, 2006). Furthermore, Adam Barrett has proposed modifications to IIT which may help account for fundamental physical interpretations of the universe like quantum field theory (Barrett, 2014). Panpsychic descriptions of reality are reentering philosophical and scientific discourse as new data are acquired and new theoretical interpretations develop.

     But many still view the idea that inanimate objects may possess primitive qualia as ludicrous. To counter this presumption, consider a fragment of quartz resting on a ridge. As the sun rises, photons excite the atoms on the crystal’s surface, causing thermal oscillations to propagate into the quartz. This thermal diffusion is modulated by crystallographic defects, causing a heterogeneous distribution of heat inside the rock. As dusk falls, the quartz fragment begins to cool, emitting heat at varying rates across the surface. The particular rates are influenced directly by this quartz specimen’s pattern of internal defects. Next, consider a mouse, also located on the ridge. As the sun rises, photons excite the retinaldehyde molecules in the mouse’s eyes, triggering signal transduction via electrochemical systems. This signal moves into the mouse’s brain, where it propagates through a series of neural pathways, causing a heterogeneous distribution of neural activity. Soon, the signal’s interaction with preexisting brain structures is translated into a motor action; the mouse blinks and looks away from the bright illumination. The particular motor response is modulated by the structural organization of this mouse’s brain at the given time. The quartz and the mouse both receive sensory inputs, process them according to internal properties, and then give motor outputs. Although the rock’s “brain” is much more disorganized and chaotic than the mouse’s brain, it operates by the same basic principles and could plausibly experience a primitive form of consciousness. As such, the possibility of panpsychism cannot be readily dismissed as absurd or metaphysical.

     Another prominent objection to panpsychism arises from brain processes which occur in a subconscious fashion. For instance, activity in the primary visual cortex (V1) does not correlate with conscious visual experience except for a few special cases (Crick & Koch, 1995) (Boyer, Harrison, & Ro, 2005) (Boehler, Schoenfeld, Heinze, & Hopf, 2008). However, the presence of subconscious neural events does not necessarily indicate that the said events are subconscious from the viewpoint of their associated anatomies. Instead, anatomical structures like V1 may experience their own independent qualia. The full informational content of their perceptions may not be transmitted or translated into the brain areas like the prefrontal cortex (PFC) which can be identified with a patient’s sense of self (Mitchell, Banaji, & Macrae, 2005). Of course, some data does transfer into higher brain regions to facilitate processes like vision, but the information undergoes an extensive series of transformations before arriving at the PFC and other regions associated with conscious processing. For this reason, the “unconscious anatomies” objection is insufficient to invalidate panpsychism.

     Under the assumption of panpsychism or the similar description provided by protopanpsychism (Chalmers, 2015), cognitive states are identical to configurations of matter and energy. As emotions like joy, anger, wonder, sadness, and love emerge from functional circuits in the brain, the same emotions may emerge from informationally equivalent circuits in other substrates like classical computers equipped with biologically-realistic software models or neuromorphic semiconductor devices (Koene, 2013). This reasoning applies to feelings of existential meaning as well. If the emotions associated with fulfillment and purpose are encoded in physical matter, then they represent an intrinsic property of the universe. As such, the possibility of existential meaning cannot be dismissed on the grounds that it arises as an irrational psychological construction, though psychological biases still must be taken into consideration.

     Feelings of existential meaning come in many flavors and magnitudes, some of which may represent objectively “better” states than others. To explain this principle, I will describe several illustrative fictional scenarios. First, consider a venture capitalist named Aiko, a man who enjoys the thrill of financial competition. Aiko feels some level of fulfillment from his pursuits. Despite this, Aiko also experiences loneliness since he has not maintained close relationships with his family. Now consider Anastasia, a feminist intellectual who has made notable strides towards gender equality. By comparison to Aiko, Anastasia experiences a stronger sense of fulfillment from her work as well as lesser misgivings from the sacrifices she has made in other areas. In a somewhat fanciful exploration, consider an unbiased spirit which decides to inhabit Aiko’s mind and then Anastasia’s mind. This spirit is assumed to exhibit perfect rationality. Upon comparing the two people’s internal states, the spirit concludes that it would prefer Anastasia’s emotional experiences over Aiko’s emotional experiences. This account demonstrates the idea that purpose can manifest differently across individuals, that such differences can undergo objective comparison, and that some states may represent objectively superior qualia.

     Despite this, I would argue that the events which trigger emotional states associated with purpose are irrelevant to existential meaning. Distinct sets of events can still cause distinct emotional states, but only the resulting emotions themselves define purpose. To understand this, consider a factory worker called Akhmed, who was born with a mutation which influenced his neural development such that he easily gains feelings of fulfillment even in mundane situations. Also consider an influential political leader named Carlos, who was born with an unfortunate susceptibility to clinical depression. Akhmed’s feeling of purpose in his monotonous factory work represents an objectively better state than Carlos’s sensation of meaninglessness even as Carlos facilitates widespread social improvements. Note that this scenario focuses upon the emotional states of the specific physical subsystems corresponding to Akhmed and Carlos, not on the physical subsystems corresponding to the people who Carlos has helped. The qualia involved in an emotion, rather than the causes for said qualia, are necessary for explaining the emotion’s value. Although meaning can vary across individuals, informatic circuits which underlie meaning may provide a route for quantifying purpose’s value and for pursuing existential fulfillment with greater methodological precision.

     From a practical perspective, this proposal indicates that deliberate creation of positive emotional states associated with purpose is an intrinsically worthwhile pursuit, but also that this pursuit must undergo consideration in a larger context. I suggest that it may not always be ethical for a subsystem of the universe such as an individual human to create positive emotional states within herself if the process causes harm to others. Note that a full exploration of potential physical bases for morality is beyond the scope of this essay. But despite the conflicts raised by altering emotional states in a heterogeneously distributed manner, reconfiguring matter on larger scales towards positive emotional states represents a clearer imperative.

     In some ways, earlier approaches to the question of existential purpose have already suggested the goal of reconfiguring matter towards meaningful emotional states. But certain religious, political, and philosophical approaches have demonstrated more success than others. In cases like Islamic fundamentalism, Christofascism (Sölle, 1970), Nazism, and social Darwinism, their attempts have backfired and caused widespread harm rather than improvement. But as mentioned, such approaches have suffered from poor theoretical and empirical rigor in their development. I would argue that using evidence-based methods towards creating a better tomorrow will facilitate the creation of policies, technologies, and ethical guidelines which possess sufficient theoretical and empirical rigor to simultaneously maximize the likelihood of widespread beneficial results and minimize the likelihood of widespread harmful results. Existential meaning is tied to physical reality and so can be engineered. In order to create existential meaning without falling prey to the same mistakes observed through history, we must utilize empirical methods to guide the construction of a bright future in which positive emotions are recognized as physical expressions of purpose and effective strategies for maintaining such emotions are implemented.

References

Barrett, A. (2014). An integration of integrated information theory with fundamental physics. Frontiers in Psychology. Retrieved from https://www.frontiersin.org/article/10.3389/fpsyg.2014.00063

Block, N. (1995). On a confusion about a function of consciousness. Behavioral and Brain Sciences, 18(2), 227–247.

Boehler, C. N., Schoenfeld, M. A., Heinze, H.-J., & Hopf, J.-M. (2008). Rapid recurrent processing gates awareness in primary visual cortex. Proceedings of the National Academy of Sciences, 105(25), 8742 LP-8747. Retrieved from http://www.pnas.org/content/105/25/8742.abstract

Boyer, J. L., Harrison, S., & Ro, T. (2005). Unconscious processing of orientation and color without primary visual cortex. Proceedings of the National Academy of Sciences of the United States of America, 102(46), 16875 LP-16879. Retrieved from http://www.pnas.org/content/102/46/16875.abstract

Camus, A. (2013). The myth of Sisyphus. Penguin UK.

Chalmers, D. (2015). Panpsychism and panprotopsychism. Consciousness in the Physical World: Perspectives on Russellian Monism, 246.

Chalmers, D. J. (1995). Facing up to the problem of consciousness. Journal of Consciousness Studies, 2(3), 200–219.

Crick, F., & Koch, C. (1995). Are we aware of neural activity in primary visual cortex? Nature, 375(6527), 121–123.

Dorman, E. (2011). Hinduism and Science: The State of the South Asian Science and Religion Discourse. Zygon®, 46(3), 593–619. http://doi.org/10.1111/j.1467-9744.2011.01201.x

Emerson, M. O., & Hartman, D. (2006). The Rise of Religious Fundamentalism. Annual Review of Sociology, 32(1), 127–144. http://doi.org/10.1146/annurev.soc.32.061604.123141

Jost, J. T., Stern, C., Rule, N. O., & Sterling, J. (2017). The Politics of Fear: Is There an Ideological Asymmetry in Existential Motivation? Social Cognition, 35(4), 324–353. http://doi.org/10.1521/soco.2017.35.4.324

Koene, R. A. (2013). Uploading to Substrate‐Independent Minds. The Transhumanist Reader: Classical and Contemporary Essays on the Science, Technology, and Philosophy of the Human Future, 146–156.

Lilla, M. (2008). The Stillborn God: Religion, Politics, and the Modern West. Vintage Books. Retrieved from https://books.google.com/books?id=nfSMDQAAQBAJ

Mitchell, J. P., Banaji, M. R., & Macrae, C. N. (2005). The Link between Social Cognition and Self-referential Thought in the Medial Prefrontal Cortex. Journal of Cognitive Neuroscience, 17(8), 1306–1315. http://doi.org/10.1162/0898929055002418

Nagel, T. (1971). The absurd. The Journal of Philosophy, 68(20), 716–727.

Nagel, T. (1974). What Is It Like to Be a Bat? The Philosophical Review, 83(4), 435–450. http://doi.org/10.2307/2183914

Norris, P., & Inglehart, R. (2011). Sacred and Secular: Religion and Politics Worldwide. Cambridge University Press. Retrieved from https://books.google.com/books?id=ObwtZ36m1hwC

Oizumi, M., Albantakis, L., & Tononi, G. (2014). From the Phenomenology to the Mechanisms of Consciousness: Integrated Information Theory 3.0. PLOS Computational Biology, 10(5), e1003588. Retrieved from https://doi.org/10.1371/journal.pcbi.1003588

Raman, V. (2009). Truth and Tension in Science and Religion. Beech River Books. Retrieved from https://books.google.com/books?id=UggdrSiVvhsC

Raman, V. (2012). Hinduism and Science: Some Reflections. Zygon®, 47(3), 549–574. http://doi.org/10.1111/j.1467-9744.2012.01274.x

Sölle, D. (1970). Beyond Mere Obedience: Reflections on a Christian Ethic for the Future. Augsburg Publishing House. Retrieved from https://books.google.com/books?id=zbeCGwAACAAJ

Strawson, G. (2006). Realistic monism: why physicalism entails panpsychism.

Triandis, H. C. (2008). Fooling Ourselves: Self-Deception in Politics, Religion, and Terrorism: Self-Deception in Politics, Religion, and Terrorism. ABC-CLIO. Retrieved from https://books.google.com/books?id=GGdrumE1JuYC

PDF Version: Notes on Upconversion Nanoparticles – Logan Thrasher Collins

Background and Overview

• Photoluminescence spectroscopy (also called fluorescence spectroscopy) uses electromagnetic radiation to excite electrons in various materials. Photons are emitted from the materials as the electrons return to lower energy states. In biology and biomedicine, photoluminescence spectroscopy provides a noninvasive tool which allows for visualization of biological processes over a wide range of scales (subcellular to macroscale).
• Many biological applications of fluorescence spectroscopy require exogenous contrast agents like organic dyes, fluorescent proteins, quantum dots, and metal complexes. But such agents often possess limitations including (i) low signal-to-background ratio as a result of tissue autofluorescence and scattering when short wavelengths are used, (ii) low penetration depth when UV or visible light excitation is used as well as when UV or visible light are emitted, (iii) potential DNA damage from short wavelengths like UV, and (iv) potential toxicity of contrast agents containing heavy metals like cadmium and lead.
• Near-infrared (NIR) wavelengths provide a potentially superior alternative to other types of excitation for many situations. Biological tissue exhibits “optical transparency” in the NIR range of 700-1100 nm, allowing for deeper tissue penetration. In addition, NIR usually causes less autofluorescence and scattering. Some other alternatives exist, but these generally require expensive ultrashort pulsed lasers, so they are often less practical than NIR techniques.
• Lanthanide-doped upconversion nanoparticles (UCNPs) absorb two or more low-energy photons and emit a single photon with a higher energy. In this way, the long wavelengths of NIR can be converted to shorter wavelength radiation like visible light and UV upon emission.

Nanochemistry of UCNPs

• In lanthanide-doped UCNPs, trivalent lanthanide cations (elements with atomic numbers in the range of 57-71 which possess a +3 charge) are embedded in a chosen dielectric lattice that is 100 nm or less in diameter. Dielectric lattices are crystalline solids which possess a net polarity due to the contributions of individual atoms within the lattice. This polarization can be quantified by an optically-measureable lattice dielectric constant.
• Using appropriate lanthanide dopants allows for wavelength-selective upconversion. NIR excitation wavelengths are converted to specific emission wavelengths. For instance, NIR can be selectively converted to blue light by some UCNPs and to ultraviolet by other UCNPs.
• When UCNPs are excited by NIR light, electrons in the trivalent lanthanide cations undergo 4f-4f orbital electronic transitions. Since the 5s and 5p shells provide an electronic shielding effect, the transitions are sharp rather than broad. The 4f-4f transitions which occur in UCNPs would ordinarily be forbidden by quantum mechanical laws, but interactions of the lanthanide ions with the crystal lattice of the nanoparticle allow “orbital mixing” which facilitates such transitions.
• The upconversion photoluminescence intensity of UCNPs exhibits a nonlinear dependence of the number of excitation photons required to induce photoluminescence from the nanoparticles. Here, the number of excitation photons is given by n, K is a material-specific constant, and P is the power of the laser (called a pump laser) used to excite the nanoparticles. It should be noted that a saturation effect occurs at high excitation energy densities, leading to a decrease in the apparent value of n.

• Upconversion quantum yield (UCQY) is defined as the number of photons emitted per photon absorbed. It is proportional to the ratio of the emitted upconversion light intensity to the absorbed light intensity. Using these relations and the equation for upconversion photoluminescence intensity, the equation below can be derived. Here, α is an absorption constant that is specific to the material and the excitation wavelength.

Upconversion Mechanisms

• There are five basic upconversion mechanisms including excited-state absorption, energy transfer upconversion, cooperative sensitization upconversion, cross relaxation, and photon avalanche.
• Excited-state absorption: multiple photons are absorbed by a single lanthanide ion, exciting electrons up a ladder-like series of energy levels. It should be noted that only certain lanthanides possess the proper characteristics for this mechanism to occur (Ho³⁺, Er³⁺, Nd³⁺, and Tm³⁺) when commercially-available diode lasers are used (which usually have incident wavelengths of around 975 or 808 nm). The relevant energy levels are called G, E1, and E2 (for ground state, excited state 1, and excited state 2). Electrons in the E1 excited state have long lifetimes before they return to the ground state G. As such, after an electron is excited to E1, it remains there for long enough that it can be promoted to E2 by another photon. When the electron decays from E2 all the way to G, this large change in energy causes a photon with a shorter wavelength to be emitted (shorter than the photons involved in the excitation).
• Energy transfer upconversion: a lanthanide ion called the sensitizer absorbs a photon and is excited to E1, then transfers its energy to an ion called the activator. The sensitizer then decays back to G. This occurs twice, causing the activator to be excited to E1 and then to E2. When the electron decays from E2 all the way to G, this large change in energy again causes a photon with a shorter wavelength to be emitted. Some of the most efficient UCNPs used for biomedical purposes operate by energy transfer upconversion and involve sensitizer/activator ion pairs of Yb³⁺/Tm³⁺, Yb³⁺/Er³⁺, or Yb³⁺/Ho³⁺ with excitation wavelengths of about 975 nm. The Yb³⁺ ion does not have an E2 energy level, it can only be excited from G to E1. This makes the upconversion process more efficient since the two energy levels of Yb³⁺ cannot cause cross-relaxations (a phenomenon which has a deleterious effect on upconversion). In addition to Yb³⁺-based UCNPs that work by energy transfer upconverion, some single-lanthanide UCNPs have also been developed. In these, the single type of lanthanide ion acts as a sensitizer and an activator.
• Cooperative sensitization upconversion: two ions of the same type absorb excitation photons and are excited to E1 states. Next, both ions simultaneously transfer energy to a third ion (which can be of a different type) and excite the third ion to its own E1 state. In this case, the E1 state of the third ion is twice as energetic as the E1 states of the other two ions. When the third ion returns to its ground state, it releases a photon with a shorter wavelength than the incident photons. Cooperative sensitization upconversion is much less efficient than the other mechanisms, but it may have some advantages for high-resolution imaging applications. The cooperative sensitization mechanism has been reported to occur with Yb³⁺/Tb³⁺, Yb³⁺/Eu³⁺, and Yb³⁺/Pr³⁺ ion pairs.
• Cross relaxation: an ion is excited to E2 and then transfers its energy to a second ion as it returns to E1, exciting the second ion to E1. This mechanism depends on  ion-ion interactions, so it is closely associated with dopant ion concentration. It can cause quenching at high concentrations, but tuning ion concentrations allows modulation of emitted light colors via other mechanisms. Note that cross relaxation on its own does not cause upconversion, but that it can influence upconversion events as other steps occur.
• Photon avalanche: processes of excited-state absorption and cross-relaxation interact to create a positive feedback loop. Minimal upconversion occurs until a threshold of power input is passed, but the upconverted photoluminescence intensity vastly increases once the threshold has been exceeded. First, excited-state absorption causes an ion at E1 to transition into E2. Then this ion transfers its energy to a second ion, its electron goes down to E1, and the second ion’s electron is excited from G to E1 by cross relaxation. Last, the second ion transfers its energy to a third ion, resulting in two ions at the E1 state by the end of the loop. In this way, more and more ions are excited to the E1 state as the loop repeats (two, four, eight, etc.) Some of the ions in the E2 state (from the first step) will decay to G, causing upconverted photons to be released.

Reference: Chen, G., Qiu, H., Prasad, P. N., & Chen, X. (2014). Upconversion Nanoparticles: Design, Nanochemistry, and Applications in Theranostics. Chemical Reviews, 114(10), 5161–5214. http://doi.org/10.1021/cr400425h

PDF Version: Notes on Quantum Mechanics – Logan Thrasher Collins

Fundamentals

• The Rydberg equation gives the wavelength of the photon emitted when an electron moves from an excited state n_i to a lower energy level n_f. Here, R is the Rydberg constant, R = 1.09710⁷ m^-1.

• For electromagnetic waves the speed of light c, the frequency, and the wavelength are related by f=c/λ.
• Particles exhibit a wavelength called the de Broglie wavelength, which can be computed using Planck’s constant h = 6.62610^-34 m²kg/s over the momentum mv.

• The energy of a wave (in J) is given by the equation below.

The Schrӧdinger Equation

• Any particle can be described by a wave function ψ(x,y,z,t). In 1D, the wave function is ψ(x,t). The physical meaning of the wave function will be explained in the next section.
• The 1D Schrӧdinger equation relates a particle’s wave function ψ(x,t), its potential energy V, and its total energy. Solving this differential equation gives a formula for the wave function of a particle. The symbol ℏ represents the reduced Planck’s constant, h/2π.

• For the 3D case, the Laplacian operator ∇² (the sum of the second partial derivatives) is used.

Interpreting Wave Functions

• The wave function completely specifies the state of a quantum mechanical system. However, in many cases this state cannot be measured without altering the system, so the wave function can only be interpreted in terms of probability.
• To interpret a wave function, operators corresponding to measureable quantities must be used (i.e. position, momentum, energy, etc.)
• The expectation value represents the average value that a quantum mechanical system takes on for a given physical quantity. However, this “average” comes with a caveat. When a measurement of a quantum system is taken, a phenomenon known as wave function collapse occurs. This changes the system in a way which prevents the average of multiple measurements from approaching the expectation value. Instead, the expectation value can be thought of as representing the average value which would occur if the same measurement was taken from many identical quantum mechanical systems (but just a single time from each).
• To compute the expectation value given an operator â (which corresponds to some physical quantity a) over a 1D domain, the following integral is used. An analogous triple integral is used for the 3D case. ψ* represents the wave function’s complex conjugate (the sign of any imaginary part is reversed).

• This integral also requires that the wave function be multiplied by a normalization constant A. The normalization constant is essential since it adjusts the total probability that the particle’s physical quantity (i.e. position) will fall somewhere in the domain (-∞,∞). For position, this means that the particle exists somewhere in space. To compute the normalization constant, solve the following integral for A.

• Below, a table of operators corresponding to physical quantities is given for 1D and 3D cases.

Uncertainty in Quantum Mechanics

• Since measurement via light or any other mechanism will impact a quantum mechanical system, the momentum and position of a particle cannot be known simultaneously.
• The more precisely a particle’s momentum is known, the less precisely that particle’s position is known and vice versa. This is quantified by the Heisenberg uncertainty principle.

• To compute the standard deviation for some physical property (associated with an operator â) in a quantum mechanical system, the equation below can be used.

Solutions to the Schrӧdinger Equation

• Solutions to the time-independent Schrӧdinger equation (called stationary states) can be converted into solutions to the time-dependent Schrӧdinger equation using a complex exponential as shown below.

• When dealing with expectation values, the complex exponential term cancels since the time-dependent wave function times its complex conjugate gives the result ψ(x)²e^-iEt/ℏe^iEt/ℏ = ψ(x)²e ^{-iEt/ℏ+iEt/ℏ} = ψ(x)². As such, measureable quantities associated with the wave function’s stationary states are constant through time.
• The general solution to the Schrӧdinger equation follows the principle of superposition. For this reason, linear combinations of solutions are also valid solutions. Note that the coefficients can be complex.

• The time-independent Schrӧdinger equation can be solved using the method of separation of variables (a technique for partial differential equations) and solving for boundary conditions.
• The infinite set of solutions to the Schrӧdinger equation are orthonormal, a property expressed by the relation involving the Kronecker delta below.

• In most cases, any function f(x) can be expressed as a linear combination of the infinite solutions to the Schrӧdinger equation (this property is called completeness).

• The n^th coefficient for the Fourier series expansion of f(x) can be computed using the equation below.