Question

The Krumbein phi scale is used in geology to classify sediments such as silt, sand, and gravel by particle size. The scale is modelled by the function \(D(\varphi)=2^{-\varphi}\) where \(D\) is the diameter of the particle, in millimetres, and \(\varphi\) is the Krumbein scale value. Fine sand has a Krumbein scale value of approximately \(3 .\) Coarse gravel has a Krumbein scale value of approximately -5 a) Why would a coarse material have a negative scale value? b) How does the diameter of fine sand compare with the diameter of coarse gravel?

Short answer

A negative scale value indicates larger particles. Fine sand has a diameter of \( \frac{1}{8} \text{ mm} \), while coarse gravel has a diameter of \( 32 \text{ mm} \).

Step-by-step solution

Understanding the Krumbein phi scale

The Krumbein phi scale function is given by \(D(\varphi) = 2^{-\varphi}\), where \(D\) is the diameter of the particle in millimetres and \(\varphi\) is the Krumbein scale value. This function helps classify sediment particles based on their size.

Analyze coarse material having a negative scale value

The negative value for coarse gravel indicates that as the Krumbein scale value \(\varphi\) decreases, the particle size increases due to the inverse relationship in the function \(D(\varphi) = 2^{-\varphi} \). So, a negative scale value corresponds to a larger particle size, implying that coarse gravel consists of larger particles compared to fine sand.

Calculate the diameter of fine sand

Given fine sand has a Krumbein scale value of approximately 3, substitute \(\varphi = 3\) into the function: \[ D(3) = 2^{-3} = \frac{1}{8} \text{ mm} \].

Calculate the diameter of coarse gravel

Given coarse gravel has a Krumbein scale value of approximately -5, substitute \(\varphi = -5\) into the function: \[ D(-5) = 2^{-(-5)} = 2^{5} = 32 \text{ mm} \].

Compare diameters

Fine sand has a particle diameter of \( \frac{1}{8} \text{ mm}\) and coarse gravel has a particle diameter of \( 32 \text{ mm} \). Clearly, coarse gravel has a significantly larger particle size than fine sand due to the scale function relationship.

Key concepts

Krumbein phi scale

The Krumbein phi scale is a logarithmic scale used in geology to classify the size of sediment particles. It is modeled by the function \(D(\varphi)=2^{-\varphi}\), where \(D\) represents the particle diameter in millimeters, and \(\varphi\) represents the phi value on the scale.
This scale helps geologists differentiate between sediments like silt, sand, and gravel.
A smaller phi value means a larger particle size, and vice versa. For example, the phi value for fine sand is about 3, while for coarse gravel, it is around -5.
Understanding the Krumbein phi scale allows us to solidly classify sedimentary particles based on their sizes, providing crucial insights into sedimentary environments and processes.

Sediment classification

Sediment classification categorizes sediment particles based on size. This classification is essential for understanding sedimentary processes and environments.
According to the Krumbein phi scale, different phi values correspond to different sediment classes:

• Clay has the highest phi values, indicating the smallest particle sizes.
• Silt particles are smaller than 0.0625 mm but larger than clay particles.
• Sand particles range from 0.0625 mm to 2 mm, with fine sand having a higher phi value than coarse sand.
• Gravel includes larger particles, with sizes exceeding 2 mm, and coarse gravel has a very low or negative phi value.

Understanding sediment classification provides us with important information about the depositional environment and transport mechanisms involved in sediment movement.

Inverse relationships in functions

In the context of the Krumbein phi scale, an inverse relationship exists between the phi value \(\varphi\) and the particle diameter \(D\).
The function given is \(D(\varphi) = 2^{-\varphi}\). This implies:

• A higher phi value results in a smaller particle size.
• A lower or negative phi value results in a larger particle size.

This inverse relationship means that as one quantity increases, the other decreases. For example, when \(\varphi = 3\), \(D(\varphi) = \frac{1}{8}\) mm, but when \(\varphi = -5\), \(D(\varphi) = 32\) mm.
Understanding these inverse relationships assists in interpreting geological data and predicting particle size distributions in sedimentary deposits.

Geological scales

Geological scales, like the Krumbein phi scale, are essential tools for measuring and classifying different geological phenomena.
The phi scale provides a systematic way to classify sediment particles based on their sizes, ranging from clay to boulders.
These scales help geologists and sedimentologists interpret past environments and evolving sedimentary processes. Besides the Krumbein phi scale for sediments, there are other geological scales for various measurements such as:

• Time scales like the geologic time scale for dating Earth's history.
• Pressure scales for understanding metamorphic environments.
• Temperature scales for geothermometry.

Each geological scale serves a specific purpose and provides insights critical to the study of geological processes and histories.