## 1 Introduction  
The electron beam powder bed fusion (PBF-EB) is a well-established method in additive manufacturing (AM) [1, 2, 3]. The PBF-EB process can be divided into four steps: lowering the build platform, powder application, preheating the surface with a defocused electron beam, and melting with a focused electron beam. A detailed discussion of the PBF-EB process can be found in [1, 3]. One of the great benefits of AM technology is a higher degree of design freedom compared to conventional manufacturing technologies [1, 3] Hence, the final goal of most AM processes is the production of complex parts. To manufacture a complex part, suitable process parameters have to be identified. Therefore, the scanning parameters beam power, beam velocity, and line offset are tested, mostly on cuboids with a constant scan length. Depending on the chosen parameters, a trailing or a persistent melt pool can form. The melt pool is classified as trailing if the melt pool is already solidified when the electron beam reaches the same position as the subsequent melt line. Contrarily, the melt pool is called persistent when the melt pool is still liquid. For a more detailed description, the reader is referred to [4]. Subsequently, the samples are examined for defects and swelling and a processing window is established, as illustrated for Ti-6 Al-4V by [5; 6; 7]. More detailed investigations revealed, that the choice of the scanning parameter can manipulate the microstructure within the processing window. For instance, the microstructure of Ni-based superalloys can be adjusted to equiaxed, columnar, or single-crystal by controlling the melt pool geometry and dimensions via different scanning strategies [8; 9; 10; 11]. A further example is the aluminum evaporation in titanium aluminides during the PBF-EB process, also controlled by the melt pool [12; 13]. In conclusion, the melt pool is the determining factor for part properties such as defects and microstructure. Consequently, a uniform melt pool with constant solidification conditions at every position is necessary to produce a homogeneous microstructure in a complex part. A common approach to achieving a constant melt pool is using constant thermal conditions and constant energy input for melting. Firstly, to ensure constant thermal conditions, the beam velocity \(v_{B}\) is adjusted to the scan length \(l_{shat}\) so that the mean return time \(t_{r}\) is constant (see Equation (1)). The mean return time \(t_{r}\) is defined as the average time the electron beam needs to come back to adjacent points of two subsequent scan lines. This ensures that the temperature field during melting remains on average constant, independently of the scan length for a cross-snake hatch pattern [4]. It is to note that for a constant line offset \(l_{o}\), the lateral velocity \(v_{ht}\), which is the velocity of the melt pool perpendicular to the electron beam scanning direction (see Figure 1c), stays unchanged if the return time \(t_{r}\) is also constant (Equation (2)). \[t_{r}=\frac{l_{s}}{v_{B}} \tag{1}\]  
\[v_{lat}=\frac{l_{o}}{t_{r}}=\frac{v_{B}*l_{o}}{l_{s}} \tag{2}\]  
As a measure for the energy input the line energy \(E_{L}\) and the area energy \(E_{A}\) are introduced (see Equations (3) and (4)). To account for the increasing and decreasing beam velocity \(v_{B}\) due to longer and shorter scan length \(l_{s}\), the beam current \(l_{B}\) and, respectively, the beam power \(P\) is adjusted to keep the line energy \(E_{L}\) constant. Since there is no PBF-EB system available which can adjust the acceleration voltage \(U_{B}\) during the process, this is the only way to keep the line energy \(E_{L}\) steady. Moreover, if the line offset \(l_{o}\) is fixed, the line energy \(E_{L}\) and the area energy \(E_{A}\) are proportional, and hence the area energy \(E_{A}\) is kept constant as well.