Problem 96 - Fundamentals of Physics

Question

At the end of World War II, Dutch authorities arrested Dutch artist Hans van Meegeren for treason because, during the war, he had sold a masterpiece painting to the Nazi Hermann Goering. The painting, Christ and His Disciples at Emmaus by Dutch master Johannes Vermeer \((1632-1675),\) had been discovered in 1937 by van Meegeren, after it had been lost for almost 300 years. Soon after the discovery, art experts proclaimed that Emmaus was possibly the best Vermeer ever seen. Selling such a Dutch national treasure to the enemy was unthinkable treason. However, shortly after being imprisoned, van Meegeren suddenly announced that he, not Vermeer, had painted Emmaus. He explained that he had carefully mimicked Vermeer's style, using a 300 -year-old canvas and Vermeer's choice of pigments; he had then signed Vermeer's name to the work and baked the painting to give it an authentically old look. Was van Meegeren lying to avoid a conviction of treason, hoping to be convicted of only the lesser crime of fraud? To art experts, Emmaus certainly looked like a Vermeer but, at the time of van Meegeren's trial in 1947 , there was no scientific way to answer the question. However, in 1968 Bernard Keisch of Carnegie-Mellon University was able to answer the question with newly developed techniques of radioactive analysis. Specifically, he analyzed a small sample of white lead-bearing pigment removed from Emmaus. This pigment is refined from lead ore, in which the lead is produced by a long radioactive decay series that starts with unstable \({ }^{238} \mathrm{U}\) and ends with stable \({ }^{206} \mathrm{~Pb}\). To follow the spirit of Keisch's analysis, focus on the following abbreviated portion of that decay series, in which intermediate, relatively shortlived radionuclides have been omitted: $${ }^{230} \mathrm{Th} \frac{{ }_{75.4 \mathrm{ky}}}{\underline{\phantom{xx}}}^{226} \mathrm{Ra} \frac{{ }_{1.60 \mathrm{ky}}}{\underline{\phantom{xx}}}^{210} \mathrm{~Pb} \frac{{ }_{22.6 \mathrm{ky}}}{\underline{\phantom{xx}}}^{206} \mathrm{~Pb}$$ The longer and more important half-lives in this portion of the decay series are indicated. (a) Show that in a sample of lead ore, the rate at which the number of \({ }^{210} \mathrm{~Pb}\) nuclei changes is given by $$\frac{d N_{210}}{d t}=\lambda_{226} N_{226}-\lambda_{210} N_{210}$$ where \(N_{210}\) and \(N_{226}\) are the numbers of \({ }^{210} \mathrm{~Pb}\) nuclei and \({ }^{226} \mathrm{Ra}\) nuclei in the sample and \(\lambda_{210}\) and \(\lambda_{226}\) are the corresponding disintegration constants. Because the decay series has been active for billions of years and because the half-life of \({ }^{210} \mathrm{~Pb}\) is much less than that of \({ }^{226} \mathrm{Ra}\), the nuclides \({ }^{226} \mathrm{Ra}\) and \({ }^{210} \mathrm{~Pb}\) are in equilibrium; that is, the numbers of these nuclides (and thus their concentrations) in the sample do not change. (b) What is the ratio \(R_{226} / R_{210}\) of the activities of these nuclides in the sample of lead ore? (c) What is the ratio \(N_{226} / N_{210}\) of their numbers? When lead pigment is refined from the ore, most of the \(226 \mathrm{Ra}\) is eliminated. Assume that only \(1.00 \%\) remains. Just after the pigment is produced, what are the ratios (d) \(R_{226} / R_{210}\) and (e) \(N_{226} / N_{210} ?\) Keisch realized that with time the ratio \(R_{226} / R_{210}\) of the pigment would gradually change from the value in freshly refined pigment back to the value in the ore, as equilibrium between the \({ }^{210} \mathrm{~Pb}\) and the remaining \({ }^{226} \mathrm{Ra}\) is established in the pigment. If Emmaus were painted by Vermeer and the sample of pigment taken from it were 300 years old when examined in \(1968,\) the ratio would be close to the answer of (b). If Emmaus were painted by van Meegeren in the 1930 s and the sample were only about 30 years old, the ratio would be close to the answer of (d). Keisch found a ratio of \(0.09 .\) (f) Is Emmaus a Vermeer?

Step-by-Step Solution

Answer

Emmaus is not a Vermeer; it's a modern painting.

Step 1: Express the Rate of Change of Nuclei

The rate at which the number of \( ^{210}\mathrm{Pb} \) nuclei changes is given by the difference between the rate of creation of \( ^{210}\mathrm{Pb} \) nuclei from \( ^{226}\mathrm{Ra} \) and its rate of decay. This can be expressed as: \[ \frac{dN_{210}}{dt} = \lambda_{226} N_{226} - \lambda_{210} N_{210} \]Here, \( \lambda_{226} \) is the decay constant for \( ^{226}\mathrm{Ra} \) and \( \lambda_{210} \) is the decay constant for \( ^{210}\mathrm{Pb} \). Both \( N_{226} \) and \( N_{210} \) are the number of nuclei for \( ^{226}\mathrm{Ra} \) and \( ^{210}\mathrm{Pb} \) respectively.

Step 2: Find the Ratio of Activities

In equilibrium, the rate of decay of \( ^{226}\mathrm{Ra} \) equals the rate of production of \( ^{210}\mathrm{Pb} \). Therefore, activities (or decay rates) are related by:\[ R_{226} = R_{210} \]Expressing activities in terms of decay constants and number of nuclei gives:\[ \lambda_{226} N_{226} = \lambda_{210} N_{210} \]Thus, the ratio of these activities is:\[ \frac{R_{226}}{R_{210}} = 1 \]

Step 3: Determine the Ratio of Nucleic Numbers

From the equilibrium condition used in Step 2, we continue with:\[ \lambda_{226} N_{226} = \lambda_{210} N_{210} \]Thus, the ratio of the numbers of nuclei is:\[ \frac{N_{226}}{N_{210}} = \frac{\lambda_{210}}{\lambda_{226}} \]

Step 4: Calculate New Ratios After Refinement

After refining the pigment, 99% of \( ^{226}\mathrm{Ra} \) is eliminated, leaving only 1%:Which says:\[ N'_{226} = 0.01 N_{226} \]For activities just after refinement:\[ R'_{226} = \lambda_{226} N'_{226} \R'_{210} = \lambda_{210} N_{210} \]The new ratio of activities is:\[ \frac{R'_{226}}{R'_{210}} = \frac{0.01 \lambda_{226} N_{226}}{\lambda_{210} N_{210}} = 0.01 \]

Step 5: Determine the New Nucleic Numbers Ratio

The ratio of numbers of \( ^{226}\mathrm{Ra} \) to \( ^{210}\mathrm{Pb} \) nuclei after refinement is:\[ \frac{N'_{226}}{N_{210}} = \frac{0.01 N_{226}}{N_{210}} = 0.01 \times \frac{\lambda_{210}}{\lambda_{226}} \]

Step 6: Conclusion Based on Keisch's Finding

Using the found data: - If Emmaus were a Vermeer, the ratio in 1968 should be close to 1 (from Step 2). - If it were painted 30 years ago by van Meegeren, it should be close to 0.01 (from Step 4). Keisch found a ratio of 0.09, aligning more closely with the 30-year prediction. Therefore, Emmaus is likely a modern work and not a Vermeer.

Key Concepts

Radioactive Analysis

Radioactive analysis is a powerful scientific technique used to study the presence and amounts of radioactive substances within a sample. This method hinges on understanding decay series, where unstable isotopes transform into stable ones over time. In the context of the story of van Meegeren's painting, radioactive analysis was utilized to verify the age of the white lead pigment used in the painting. By measuring the ratio of different isotopes present, scientists can deduce how long ago the pigment was refined.
This method is based on the principle that certain radioactive isotopes decay at steady, predictable rates, known as half-lives. When scientists analyze these isotopes, they can create a timeline of how long the sample has existed in its present form. This was key to determining whether the painting was a genuine Vermeer or a modern forgery. In essence, radioactive analysis unlocks the clockwork of atomic particles, offering insights into both natural and man-made artefacts.

Nuclear Physics

Nuclear physics delves into the fundamental components of matter, primarily focusing on nuclei and their interactions. An essential aspect of nuclear physics is understanding radioactive decay, where unstable atomic nuclei lose energy by emitting radiation. This concept is integral to radioactive analysis as it allows scientists to track the changes in a substance over time.
In nuclear physics, we categorize isotopes by their stability—some are stable, while others are radioactive and decay into other elements over time. The subject isn't just about theoretical understanding; practical applications include dating ancient artefacts, as shown in van Meegeren's case. By assessing which isotopes are present and their proportions, scientists can paint a picture of a material’s history, much like piecing together a puzzle. This amalgamation of theory and practice demonstrates the power of nuclear physics in various fields, including art authentication and archaeology.

Half-Life Calculation

Calculating half-lives is a critical aspect of nuclear physics and radioactive analysis. A half-life is the time required for half of the radioactive nuclei in a sample to decay. This metric provides a clock for dating materials, which proved instrumental in examining van Meegeren's painting. By understanding the half-lives of isotopes in the lead pigment, scientists estimated its age.
The equation used involves the decay constant (\( \lambda \)) and the number of remaining radioactive atoms (\( N \)), outlined as:\[N(t) = N_0 e^{-\lambda t}\]
This equation helps calculate how many atoms remain over time and thus the elapsed time since the pigment's production. Half-life calculations were pivotal for Keisch in proving that Emmaus was likely a modern creation rather than a centuries-old masterpiece. This concept shows how precise mathematical tools blend with scientific inquiry to resolve historical mysteries.

Art Authentication

Art authentication combines science and art to verify the originality of artworks. It leverages scientific techniques like radioactive analysis to investigate materials used in artworks. The goal is to determine their age and origin, especially in distinguishing genuine pieces from forgeries. In the intriguing case of van Meegeren, art experts originally judged authenticity through style and aesthetics, but science provided a new lens—through radioactive analysis.
Scientific methods, including pigment analysis, enable experts to look beyond visual and stylistic evidence. It digs into the atomic details, revealing clues about when the materials were created. This is crucial in cases where stylistic evaluation alone is insufficient. For art historians and scientists alike, blending technical analysis with traditional methods leads to a holistic understanding of artworks, enhancing their ability to protect cultural heritage and prevent art fraud.