Question

An experimental peak \(f(\omega)\) has height \(H\) and full width at half maximum \(W\). Assume that a Gaussian shape would fit the peak and find that its integral \(\int F(\omega) \mathrm{d} \omega=1.064 H W \approx H W\).

Short answer

The integral of the Gaussian peak is approximately equal to the product of its height and width, \(H \cdot W\), up to a factor of 1.064.

Step-by-step solution

Understanding the Gaussian Function

A Gaussian function is generally given by the expression: \[ f(\omega) = a \cdot e^{-\frac{(\omega - b)^2}{2c^2}} \]where \(a\) is the peak height, \(b\) is the center of the peak, and \(c\) is related to the width. The function is symmetric about \(b\).

Identify Parameters of the Gaussian Shape

Given the height of the peak \(H\), this corresponds to the parameter \(a\). The full width at half maximum (FWHM) \(W\) is related to the parameter \(c\) through the Gaussian property: \[ W = 2\sqrt{2\ln 2} \cdot c \]

Calculate Area Under the Gaussian

The integral of the Gaussian curve over all \(\omega\) gives the area under the curve. For a standard Gaussian, the integral is: \[ \int_{-\infty}^{\infty} a \cdot e^{-\frac{(\omega - b)^2}{2c^2}} \mathrm{d}\omega = a \cdot c \sqrt{2\pi} \]

Substitute Given Values to Find the Integral

We know that the integral evaluates to approximately \(H W\) for the Gaussian. Given the integral property, substituting for \(a = H\) and using the relation for \(c\), the expression becomes: \[ H \cdot c \sqrt{2\pi} = H \cdot \frac{W}{2\sqrt{2\ln 2}} \cdot \sqrt{2\pi} \]

Relate Integral to Experimental Condition

From the simplification and substitution process above, it is derived that:\[ H \cdot \frac{W \sqrt{2\pi}}{2\sqrt{2\ln 2}} \approx 1.064 \cdot H \cdot W \approx H \cdot W \]The factor of approximately 1.064 accounts for the slight discrepancy in expected integral values versus the ideal condition.

Key concepts

Peak Height

The peak height in a Gaussian function is a straightforward concept. It refers to the maximum value that the function reaches on its graph. In the formula \(f(\omega) = a \cdot e^{-\frac{(\omega - b)^2}{2c^2}}\), the parameter \(a\) represents this peak height. Imagine the peak height as the tallest point of a mountain when you're viewing a 2D representation. It's the value at the center of the curve’s highest point.

Understanding peak height is essential because it helps us determine the intensity or maximum magnitude of the function. For instance, in physics and other scientific fields, the peak height can provide valuable insights into the peak energy or concentration at a specific point.

When analyzing experimental data, having a clear understanding of the peak height allows you to compare different functions easily. It is crucial for standardization and understanding how different peaks measure up against each other. Here’s what to keep in mind:

• The peak height, \(H\), represents the largest y-value and indicates the function’s maximum output.
• In a perfect Gaussian, this corresponds directly to the parameter \(a\), making analyses and comparisons easier.
• A higher peak height often suggests a stronger or more pronounced feature in the data being studied.

Full Width at Half Maximum

The full width at half maximum (FWHM) is a measure often used to describe the width of a peak in a function. It refers to the width of the peak at half the height of its maximum value. In simple terms, if you imagine drawing a horizontal line at half the peak height, the distance between the points where this line intersects the curve provides the FWHM. It's a way of understanding how wide the peak is around its central point.

In the context of a Gaussian function, the FWHM is directly related to the parameter \(c\) that determines the breadth of the curve. The specific relationship is given by the formula \(W = 2\sqrt{2\ln 2} \cdot c\), illustrating that the FWHM depends on \(c\) and provides a practical measure of spread or dispersion.

FWHM is useful across various fields, such as analyzing spectral lines in physics or measuring resolution in image processing. It gives a consistent way to compare the widths of different peaks, which is essential for interpreting data results.

• FWHM provides a standard measurement that enhances comparability between peaks.
• In Gaussian functions, FWHM helps relate the width parameter \(c\) to a physical measure.
• A smaller FWHM indicates a steeper, narrower peak, and vice versa for a wider FWHM.

Area Under the Curve

The area under the curve of a Gaussian function is a crucial concept that signifies the total accumulation of values over a range. Mathematically, this area is found by integrating the function, and for a Gaussian, it links directly to the function's key parameters.

In the case of the Gaussian function, the integral over all \(\omega\) is given by \(\int_{-\infty}^{\infty} a \cdot e^{-\frac{(\omega - b)^2}{2c^2}} \mathrm{d}\omega = a \cdot c \sqrt{2\pi}\). This expression tells us how the area depends on both the peak height \(a\) and the width \(c\).

For an experimental peak fitted to a Gaussian shape, a simplified expression \(1.064 \cdot H \cdot W \approx H \cdot W\) is often used, where \(H\) is the peak height and \(W\) the FWHM. This approximation helps in practical applications and provides a scalable measure of the area relative to basic observable quantities.

• Area under the curve indicates total intensity or accumulation of data.
• It's essential for understanding the scale or quantity represented by a Gaussian peak.
• In practical terms, approximating using \(H \cdot W\) offers a useful analytical tool.