# Calculation of silver nanoparticle size using Mie Theory

I have synthesized silver nano-particles and I wish to determine the approximate particle size using the wavelength that was most absorbed according to its absorption spectrum.

I find out that Mie theory provides a good approximation for this, (Eq 4 in this paper). However, I am clueless as to how to use the equation. In other words, how should I use the $\lambda_{max}$ obtained through the experiment in the provided equation?

• Wait isn't Equation 4 $\gamma (R) = \gamma _0 + (Av_f)/R$? Where's the $λ_{max}$? – JSCoder says Reinstate Monica Feb 23 '18 at 13:21
• Yes, it is. However, the first answer on the link suggests to use it. researchgate.net/post/… – mathnoob123 Feb 23 '18 at 13:24
• Oh, the application took pretty long to load. So I didn't see that at first! Thanks! – JSCoder says Reinstate Monica Feb 23 '18 at 13:35
• So any insights on my problem? – mathnoob123 Feb 23 '18 at 13:37
• Not yet, but I’ll work on this in math class. Middle school is so easy :) – JSCoder says Reinstate Monica Feb 23 '18 at 13:41

## Plug it in to the math equation

Ok, so we have this equation: $γ(R)=γ_0+(Av_f)/R$

First, we substitute the values into the equation that we know. $γ(R) = (5\cdot10^{12}) + (3/4\cdot1.39\cdot10^6)/R$. Simplified, this is: $γ(R) = (5\cdot10^{12}) + (1.0425 \cdot 10^6)/R$

$γ(R)$ can be measured in your experiment. Then we can solve your equation for $R...$ $\large{R = \fbox{$\frac{1.0425 \cdot 10^6}{γ(R) - 5\cdot10^{12}}$}}$

Plug in your value of $γ(R)$, and you're there!*

*In other words, how should I use the λmax obtained through the experiment in the provided equation?

The equation only mentions $\gamma (R)$, I don't know how to help you there**

**Zhe: @JavaScriptCoder For these types of questions, the first thing to ask is "do you work in a lab?" If not, why are you doing experiments outside of a lab, and if so, why is it that no one in your lab has any expertise on this?

• Here you go. I don't know what $λ_{max}$ is. – JSCoder says Reinstate Monica Feb 23 '18 at 16:46
• I know the constants (provided in the paper) and I know enough math to manipulate the equation in the provided form. This answer does not answer my question at all. – mathnoob123 Feb 23 '18 at 16:46
• Yes, I know, but I think $γ(R)$ can be measured by the scattering of the silver nanoparticles. I don't know how to find $γ(R)$ from $λ_{max}$ – JSCoder says Reinstate Monica Feb 23 '18 at 16:48