### Full Width at Half Maximum (FWHM) & Quadratic equation

Full width at half maximum (FWHM) for a curve is used in many scientific experimental data analyses. For instance, in X–ray diffraction analysis of crystal data, it is used to find the grain size from the curve.

FWHM is the width
of the curve or simply the difference in the corresponding two ‘x’ data values
at y_{max}/2.

y_{max}
refers the highest value of ‘y’ or ‘y’ value for the maximum peak height.

__Example__

Set of ‘x’ and ‘y’ data and the corresponding curve is given below:

(View this Page for back reference.)

x y

0.0 0.000

0.1 0.146

0.2 0.284

0.3 0.414

0.4 0.536

0.5 0.650

0.6 0.756

0.7 0.854

0.8 0.944

0.9 1.026

1.0 1.100

1.1 1.166

1.2 1.224

1.3 1.274

1.4 1.316

1.5 1.350

1.6 1.376

1.7 1.394

1.8 1.404

1.9 1.406

2.0 1.400

2.1 1.386

2.2 1.364

2.3 1.334

2.4 1.296

2.5 1.250

2.6 1.196

2.7 1.134

2.8 1.064

2.9 0.986

3.0 0.900

3.1 0.806

3.2 0.704

3.3 0.594

3.4 0.476

3.5 0.350

3.6 0.216

3.7 0.074

3.8 -0.076

The given data fits in the following
equation: y = -0.4x^{2} + 1.5x.

In the above data
the maximum value of y (y_{max})is 1.406.

Hence, y_{max}/2
= 1.406/2 = 0.703.

Now, FWHM is the difference between the two ‘x’ values corresponding to this 0.703.

Let’s find out the
two ‘x’ values corresponding to this y_{max}.

Since the curve fits in the equation,

y = -0.4x^{2} + 1.5x substitute y
as 0.703.

0.703 = –0.4x^{2}
+ 1.5x

To find the value of ‘x’ for this ‘y’ value, quadratic equation can be used.

(View this Page for back reference about quadratic equation.)

Rearrange the equation:

–0.4x^{2}
+ 1.5x – 0.703 = 0

It is in quadratic equation form:

ax^{2} +
bx + c = 0

where ‘a’, ‘b’ and ‘c’ are constants.

The solution to this quadratic equation is given by the following quadratic formula:

Here b^{2}
– 4ac is called as discriminant.

If b^{2} –
4ac > 0, then the equation has two solutions.

If b^{2} –
4ac = 0, then the equation has one solution.

If b^{2} –
4ac < 0, then the equation has no real solutions.

Using quadratic equation, it is possible to find the value of ‘x’, if ‘a’, ‘b’ and ‘c’ are known.

In this, a = –0.4; b = 1.5; c = –0.703.

On substituting these values,

* *

* *

So ‘x’ has two values.

If these two ‘x’ values are substituted in the curve fitting equation,

y = -0.4x^{2} + 1.5x, then y =
0.703.

y = [-0.4 × 0.54906^{2}] + [1.5 × 0.54906] = 0.703.

y = [-0.4 × 3.20094^{2}] + [1.5 × 3.20094] = 0.703.

So, two ‘x’ values for the corresponding y_{max}/2
are: x_{2} = 3.2 and x_{1} = 0.55.

So, FWHM value for this curve is calculated
as: (x_{2} – x_{1}) = (3.2 – 0.55) = 2.65.

(Refer the curve above.)