The diameter of laser synthesized colloidal nanoparticles can be derived from a volume of ablated material and absorbance at surface plasmon resonance wavelength. The origin of such nanoparticles’ diameter is studied in this work. The study is based on Beer-Lambert law where nanoparticle size-distribution and UV-Vis photoabsorbance data are used. Theoretically modeled diameter is verified by independent integral approach proving that simple model derived in our previous work is by most means usable for determination of diameter of laser synthesized nanoparticles in colloidal solution with sizes less than SPR wavelength. Here is also presented the mathematical proof which indicates that in monodisperse limit, under assumption that size-distribution of synthesized nanoparticles can be described by log-normal function, diameter obtained by simple analytical model is in fact mode diameter (most frequent diameter) of size-distribution. Furthermore, it is shown that mode diameter of size-distributions can be obtained from volume average diameter of nanoparticles over experimentally determined diameter distribution as well as from number average diameter of nanoparticles over the same diameter distribution by multiplication of those diameters with shift factors depended only on standard deviation of log-normal distribution. This is verified for size-distributions obtained independently by AFM, TEM and DLS techniques and is proposed to be universal property of log-normal distribution function.

激光合成胶体纳米粒子的直径可以从烧蚀材料的体积和表面等离子共振波长处的吸光度得出。在这项工作中研究了这种纳米粒子直径的起源。该研究基于 Beer-Lambert 定律，其中使用了纳米颗粒尺寸分布和 UV-Vis 光吸收数据。通过独立积分方法验证了理论建模的直径，证明在我们之前的工作中得出的简单模型在大多数情况下可用于确定尺寸小于 SPR 波长的胶体溶液中激光合成纳米粒子的直径。这里还提出了数学证明，表明在单分散极限下，假设合成纳米粒子的尺寸分布可以用对数正态函数来描述，通过简单分析模型得到的直径实际上是尺寸分布的众数直径（最常见的直径）。此外，它表明，尺寸分布的众数直径可以通过实验确定的直径分布上的纳米粒子的体积平均直径以及通过将那些直径与位移因子相乘而从相同直径分布上的纳米粒子的数均直径获得。关于对数正态分布的标准差。这对通过 AFM、TEM 和 DLS 技术独立获得的尺寸分布进行了验证，并被认为是对数正态分布函数的普遍性质。结果表明，尺寸分布的众数直径可以通过实验确定的直径分布上的纳米粒子的体积平均直径以及相同直径分布上的纳米粒子的数均直径通过将这些直径与仅取决于标准的位移因子相乘而获得对数正态分布的偏差。这对通过 AFM、TEM 和 DLS 技术独立获得的尺寸分布进行了验证，并被认为是对数正态分布函数的普遍性质。结果表明，尺寸分布的众数直径可以通过实验确定的直径分布上的纳米粒子的体积平均直径以及相同直径分布上的纳米粒子的数均直径通过将这些直径与仅取决于标准的位移因子相乘而获得对数正态分布的偏差。这对通过 AFM、TEM 和 DLS 技术独立获得的尺寸分布进行了验证，并被认为是对数正态分布函数的普遍性质。