*6.3. Lorentz Force in Nanoparticle Manipulation*

To discuss the mechanism of nanoparticle manipulation in a TEM, the interaction between electrons and nanoparticles on the specimen stage in the magnetic field was analysed. The TEM used in this study was 200 keV JEM-2010, which has a pole piece with a magnetic field of 10<sup>4</sup> Gauss from top to bottom, where the specimen stage is located slightly above the centre plane. In Figure 4, the electron trajectory is illustrated schematically by Horiuchi et al. [4], and the Lorentz forces F<sup>1</sup> and F<sup>2</sup> arise from the magnetic components Br and Bz, respectively, with spirally running electrons. The specimen stage is located between planes 1 and 2, and Al, W, Pt, and Cu nanoparticles experience both a tangential force to rotate and revolve and a centripetal force to migrate, bond, and embed. The momentum transfer from electrons to nanoparticles is the source of this movement.

The Lorentz force exerted on one electron, Fe, was roughly estimated by Equation (2)

$$\mathbf{F\_e = m\_e v\_e}^2 / \mathbf{r} \tag{2}$$

where m<sup>e</sup> is the mass of one static electron as 9.1094 <sup>×</sup> <sup>10</sup>−<sup>31</sup> kg, v<sup>e</sup> is the velocity of electrons, considering the relativistic effects at 200 kV as v<sup>e</sup> = v<sup>200</sup> 1.3914 = 2.900 <sup>×</sup> <sup>10</sup><sup>8</sup> m/s, and r is the distance between the nanoparticle and the irradiation centre. When assuming the experimental case of Al nanoparticles shown in Figure 5, r = 60 nm, the Lorentz force from one electron F<sup>e</sup> was 1.277 <sup>×</sup> <sup>10</sup>−<sup>6</sup> N. The total Lorentz force, F, to the Al nanoparticle of 20 nm in diameter with an irradiation time of 1200 s at 10<sup>20</sup> e/cm<sup>2</sup> s was estimated to be 9.63 <sup>×</sup> <sup>10</sup><sup>5</sup> N. Although this is the maximum value, which occurs when electrons travel from plane 1 to 2 and the driving force of nanoparticle manipulation changes the direction from tangential to centripetal, it is too high for nanoparticle movement. The author proposes the following reasons: although the electron density was measured on the fluorescent plate beneath the stage, accelerated electron velocity decreased while travelling inside a TEM, and electrons lost their kinetic energy through ionisation of the wall by their impact. The Lorentz force decreased by at least 1/100. The existence of a friction force between nanoparticles and a substrate carbon film could also be one of the causes. The effective cross-section of the nanoparticle might be considered, which decreases the impact of the electrons.

The Lorentz force for nanoparticle manipulation is also valid for W, Pt, and Cu, as shown in Figures 23, 24, and 26. Although the time to bond is different depending on the density of the weight, electron irradiation focusing to the localised region will be a candidate technology for fabricating circuits or functional dots by nanoparticle arrays.
