Section 42–2 Thermionic emission

Idealizations / Approximations / Limitations

 

This section is about thermionic emission, historically termed the Edison effect, which occurs when a material is heated sufficiently to enable electrons the kinetic energy required to overcome the surface’s potential barrier. While Owen Willans Richardson was awarded the 1928 Nobel Prize in Physics for formalizing the relationship, Feynman simplified the complex equation into a toy model involving the Boltzmann factor. The principle can be used in the modern AI chip manufacturing, e.g., it governs the emission of electrons from tungsten tips enclosed within the Scanning Electron Microscope.

 

1. Idealizations:

“Then there would be a certain density of electrons at equilibrium which would, of course, be given by exactly the same formula as (42.1), where Va is the volume per electron in the metal, roughly, and W is equal to qeϕ, where ϕ is the so-called work function, or the voltage needed to pull an electron off the surface….. In other words, the answer is that the current of electricity that comes in per unit area is equal to the charge on each times the number that arrive per second per unit area, which is the number per unit volume times the velocity, as we have seen many times: I = q_env = (q_ev/Va)e^−q_eϕ/kT (Feynman et al., 1963, p. 42-4).”

 

Feynman’s equation I = q_env = (q_ev/V_a)e^−qeϕ/kT contains the essential Boltzmann factor in the exponential, but the prefactor q_ev/V_a embodies several idealizations. First, replacing the electron number density n with 1/V_a treats conduction electrons as if each occupies a fixed average volume in the metal, but thermionic emission is essentially a surface phenomenon. Second, the use of a single average speed v ignores the Fermi–Dirac velocity distribution and directional effects; in reality, only electrons with sufficiently large outward velocity components normal to the surface can escape. Third, the expression assumes that every electron reaching the surface with enough energy overcomes the barrier, neglecting reflection, surface scattering, and impurities. Thus, the prefactor functions as a simplified “attempt rate” — charge × available carriers × average speed — providing a toy model while omitting the quantum statistics that is incorporated in the Richardson–Dushman law.

 

“We may give another example of a very practical situation that is similar to the evaporation of a liquid—so similar that it is not worth making a separate analysis. It is essentially the same problem (Feynman et al., 1963, p. 42-4).”

 

Strictly speaking, evaporation and thermionic emission are not identical physical problem, even though both involve particles escaping from a surface. The Richardson-Dustman law describes electron emission from a metal and is characterized by a T² prefactor multiplied by a Boltzmann factor, reflecting the presence of a work function barrier. In contrast, Langmuir’s law (often associated with Hertz-Knudsen equation) describes the evaporation of neutral atoms and follows classical kinetic theory, yielding a 1/ÖT dependence in the flux expression, which is commonly written in pressure form without a Boltzmann factor. Although the two laws differ in detail—electrons versus neutral atoms, Fermi–Dirac statistics versus Maxwell–Boltzmann statistics—they share a similar foundation of surface-related processes.

 

It is also worth mentioning that Irving Langmuir had an exceptionally broad research program centered on surface phenomena. His investigations into surface adsorption, thin films, and tungsten-filament bulb naturally led him to study both evaporation and thermionic emission. Awarded the 1932 Nobel Prize in Chemistry in 1932 for his work in surface chemistry, Langmuir also contributed to the development of Child-Langmuir Law (for thermionic emission), which describes the space-charge limited current in vacuum tubes and was foundational for early electronics technology.

 

2. Approximations

“The filament of the tube may be operating at a temperature of, say, 1100 degrees, so the exponential factor is something like e⁻¹⁰; when we change the temperature a little bit, the exponential factor changes a lot. Thus, again, the central feature of the formula is the e^−qeϕ/kT (Feynman et al., 1963, p. 42-4).”

 

In Feynman’s equation, the central approximation lies in the use of the Boltzmann factor e^−q_eϕ/kT, where q_eϕ (or W) is the work function—the energy barrier electrons must overcome to escape the metal. At a filament temperature around 1100 K, the ratio W/kT may be about 10, making the emission proportional to e^−10, a very small number; because this exponential contains 1/T, even a slight increase in temperature significantly reduces the exponent and therefore produces a large increase in current. Interestingly, Richardson noted in his Nobel Lecture, it is experimentally difficult to distinguish between emission laws proportional to T^½e^-W/kT and T²e^-W/kT: the algebraic power of T changes slowly compared with the exponential term, and small adjustments in the constants can mask the difference. However, the essential physics of thermionic emission still lies in the exponential Boltzmann factor, which governs the fraction of electrons energetic enough to overcome the work-function barrier.

 

In his Nobel Lecture, Richardson (1929) mentions: “In 1901, I was able to show that each unit area of a platinum surface emitted a limited number of electrons. This number increased very rapidly with the temperature, so that the maximum current i at any absolute temperature T was governed by the law i=AT^½e^-W/kT …Eq.(1)……In 1911 as a result of pursuing some difficulties in connection with the thermodynamic theory of electron emission I came to the conclusion that i=AT²e^-W/kT … Eq.(2) was a theoretically preferable form of the temperature emission equation to Eq.(1) with, of course, different values of the constants A and w from those used with (1). It is impossible to distinguish between these two equations by experimenting. The effect of the T² or T^½ term is so small compared with the exponential factor that a small change in A and w will entirely conceal it. In fact, at my instigation K. K. Smith in 1915 measured the emission from tungsten over such a wide range of temperature that the current changed by a factor of nearly 10¹², yet the results seemed to be equally well covered by either (1) or (2).”

 

3. Limitations

As a matter of fact, the factor in front is quite wrong—it turns out that the behavior of electrons in a metal is not correctly described by the classical theory, but by quantum mechanics, but this only changes the factor in front a little. Actually, no one has ever been able to get the thing straightened out very well, even though many people have used the high-class quantum-mechanical theory for their calculations (Feynman et al., 1963, p. 42-5).

 

The Paradox of Feynman's Pessimism: Why Agreement and Disagreement Both Clarify the Truth

Feynman's seemingly pessimistic statements about the development of thermionic emission present a productive paradox: by simultaneously agreeing and disagreeing with him, we gain a richer understanding of how physics progresses.

 

1. The Humility of Agreeing: Surface Physics is Messy

Agreeing with Feynman is an exercise in intellectual humility. His prefactor is indeed “quite wrong” because it ignores Fermi–Dirac statistics. Even after the quantum mechanical refinement to the Richardson–Dushman law—where the Richardson constant (A = 4pm_ek²/h³) replaces the earlier empirical prefactor—experimental values often deviate from the theoretical prediction. Surface contamination, surface roughness, and space-charge effects all complicate ideal experimental conditions. In practice, while the Boltzmann factor remains robust and reliable, the prefactor is sensitive to the “messy” material-specific surface conditions that are difficult to control. Feynman’s pessimism is not cynicism but a methodological caution: theoretical elegance does not ensure experimental exactness.

 

2. The Optimism of Disagreeing: A Theoretical Achievement

Disagreeing with Feynman allows us to recognize what was genuinely “straightened out.” By the mid-1920s, Saul Dushman and others had used quantum statistics to transform the empirical prefactor into one that is derived from fundamental constants (m_e, q_e, k, and h). This was not yet a revolution but a revelation: thermionic emission is a manifestation of connections between Fermi–Dirac statistics, phase space, and electron behavior. When experimental values deviate from the Richardson constant, the discrepancy typically reflects imperfect surfaces rather than a breakdown of quantum theory. In this sense, much was “straightened out”, that is, there are corrections in the foundational physics, even if real materials introduce unavoidable complications.

 

3. Synthesis: Where Theory meets Reality

Holding both views simultaneously provides a mature scientific perspective. When designing thermionic energy converters or optimizing electron sources for AI chip fabrication, engineers may refine the Richardson–Dushman law*. Yet Feynman's skepticism may function like a craftsman's caliper, continually emphasizing the gap between theoretical predictions and real surfaces. The tension between theoretical completeness and experimental complexity is not evidence of failure; it is the engine of refinement in surface science. Thus, the “optimist” may still use the Richardson’s constant as a guide, while the “pessimist” accounts for the surface contamination and imperfections—and progress emerges from the dialogue between the two.

 

*In VLSI Technology, the formula related to thermionic emission is modified as shown below:

Source: VLSI Technology (Sze, 1983)

Note: In his 1928 Nobel Lecture, Richardson explicitly acknowledged the importance of Sommerfeld’s quantum-theoretical treatment of the electron gas in metals: “This great problem was solved by Sommerfeld in 1927. Following up the work of Pauli on the paramagnetism of the alkali metals, which had just appeared, he. showed that the electron gas in metals should not obey the classical statistics as in the older theories, such as that of Lorentz for example, but should obey the new statistics of Fermi and Dirac… The only clear exceptions which emerged were the magnitude of the work function in relation to temperature as deduced from the cooling effect and the calculation of the actual magnitude of the absolute constant A which enters into the AT²e-w/kT formula. As this contains Planck’s constant h its elucidation necessarily involved some form of quantum theory.”

 

Key Takeaways: Why Feynman Embraces the “Wrong” Formula

Feynman’s goal is not to provide a handbook for industrial engineering, but to illuminate the conceptual core of Statistical Mechanics.

• The Scaffolding: His central aim is to show that the Boltzmann Factor is the universal engine behind virtually all “escape” processes. Whether the subject is evaporation, thermionic emission, or chemical reaction rates, the exponential suppression associated with an energy barrier is the main physical principle.
• The Essence of Physics: From this perspective, the precise temperature dependence of the prefactor—whether it is ÖT or T² in the Richardson-Dushman law—is secondary. The exponential term governs the scale of the effect; the prefactor refines it.

Thus, the deeper lesson is this: “once you understand the 'thermal jiggle' and the 'energy hill,' you are 99% of the way to the truth. The last 1% is just coefficient-hunting.” The remaining refinements—coefficients, quantum statistics, and material-specific corrections—are crucial for quantitative precision, but they do not change the underlying physics. Feynman teaches us to see the forest first; the trees, however beautiful and necessary, can be examined later.

The Moral of the Lesson:

Placing aluminum foil inside a microwave oven (Magnetron) is effectively introducing a highly reflective conductor into an electromagnetic cavity. While the magnetron generates microwaves through thermionic emission, the oven chamber is designed to bounce those waves until they are largely absorbed by your food. When foil is added, it does more than “shield”: it alters the boundary conditions of the cavity. Used carefully, foil is a tool for selective shielding to prevent portions of food from overcooking; used carelessly, it transforms the oven into an uncontrolled discharge chamber.

Why Foil Sparks: Load Geometry, Not Heat

There is a fundamental difference between the thermionic emission inside the microwave oven and electric field concentration on the foil’s surface.

• Inside the Magnetron (The Source): Electrons are “boiled” off a cathode via thermionic emission. The frequency of microwaves depends on the geometry (size and shape) of the microwave oven and the strength of the magnetic field. This “Source Geometry” determines how fast the electrons move and oscillate.
• On the Foil (The Load): Conversely, arcing on the foil is not directly related to the thermal “boiling” of electrons; it is driven by electric field concentration. Here, the shape of the foil (Load Geometry) determines the electric field strength and the possibility of sparks or arcing.

 

The Physics of Field Enhancement

When microwaves (oscillating electromagnetic fields) strike a conductor like aluminum, they induce surface currents. The resulting electric field is governed by the geometry of the conductor.

• Smooth Surfaces: On a flat sheet of aluminum foil, the electric charge spreads evenly, and electric fields remain moderate.
• Sharp Edges/Points: According to the Gauss’ law, the surface charge density — and thus the local electric field — is inversely proportional to the radius of curvature. In short, sharper points or edges ® stronger electric field.

This is the same principle that makes lightning rods work. If the electric field at a sharp edge becomes strong enough to ionize air, it can create a visible spark or arc.

 

Practical Guidelines for Safe Foil Use

To use foil safely, you should manage both geometry and energy absorption:

1. The Anti-Crumple Rule: Never use crumpled foil. Every wrinkle creates numerous microscopic “points” that intensify the local electric field. Keep foil as flat and smooth as possible.

2. Maintain Clearance: Keep foil well away from the oven walls. If they are too close, the potential difference can produce a direct arc between them, damaging the interior.

3. “Round off” the corners: If you are shielding a turkey wing or delicate pastry, tuck or fold the edges of foil into curves. Curved shapes distribute charge more evenly, significantly reducing the risk of arcing compared to jagged edges.

4. Manage the Load: Absorption Matters

Microwaves need a "load" (something to absorb energy, like water or fat in food).

• If you wrap your food entirely in foil, you effectively create a “No-Load” condition. Thus, it reflects most of the energy rather than absorbing it. Excessive reflection may increase standing waves and stress internal components.
• Use only small patches of foil, allowing most of the food to absorb microwaves.  

 

iPhone Prank: The "Apple Wave" Catastrophe

In 2014, a notorious hoax originated on the internet claiming that a feature called “Apple Wave” allow users to charge an iPhone by “microwaving” it. This incident provides a good review on the contributions of geometry:

The Geometry Trap: A smartphone is a dense thicket of complex metal structures. The internal circuitry and components have thousands of sharp features that may trigger arcing or fire.

Thermal Runaway: The most dangerous component is the Lithium-Ion battery. Microwaves induce massive currents in the battery's conductive layers. This leads to thermal runaway—a state where the battery's internal temperature rises so fast it causes a self-sustaining fire, releasing flammable gases and potentially exploding.

Source: How Not To Charge Your iPhone: Users Fall For 'Apple Wave' Microwave Prank | IBTimes

Youtube: Microwaving iPhone Battery!! Don't Try this at Home!

Summary: Geometry is Destiny

In a microwave oven, aluminum foil is neither inherently safe nor inherently destructive, but it is a “passive” element whose safety is determined entirely by its shape. It only becomes potentially dangerous when its geometry—sharp points and narrow gaps—allows the electric field to break down the insulating properties of the air. Feynman would likely appreciate this contrast—while the exponential factor governs the birth of electrons in thermionic emission, but in the realm of microwave safety, geometry is destiny.

 

Review Questions:

1. Does Feynman suggest that thermionic emission is fundamentally similar to the evaporation of a liquid that a separate, specialized analysis is redundant?

2.Why is the Boltzmann factor regarded as the “central feature” of thermionic emission, while the pre-exponential factor is often treated as secondary? Does this distinction reflect deeper theoretical stability in the exponential term versus material sensitivity in the prefactor?

3. How would you evaluate Feynman’s claim that no one has ever been able to straighten out" thermionic emission despite the use of “quantum mechanics. In light of Saul Dushman’s derivation of the universal constant in Richardson-Dushman law, should this be interpreted as a failure of quantum mechanics, or as an acknowledgment of the complexity of real-world surface physics?

 

References:

Crowell, C. R. (1965). "The Richardson constant for thermionic emission in Schottky barrier diodes". Solid-State Electronics. 8(4), 395–399.

Dushman, S. (1930). Thermionic emission. Reviews of Modern Physics, 2(4), 381.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Richardson, O. W. (1929). Thermionic phenomena and the laws which govern them. Nobel Lecture, December, 12, 1929.

Sze, S.M. (1983). VLSI Technology. New York: McGraw-Hill.