# Lattice Design of an Intermediate-Energy Electron Storage Ring Dedicated to Materials Research
## 1 Introduction  
Light sources are powerful tools for the development of human civilization, and every advance in them has greatly enhanced people's ability to understand and control the unknown world, and given a strong impetus to the development of science and technology. X-ray light sources are indispensable cutting-edge instruments for observing the internal structure of objects, and for probing and understanding the microscopic structure and dynamic processes within matter at the molecular and atomic scales. The storage ring light source is one of the world's leading synchrotron radiation facilities today and is a cost-effective, highly reliable, and stable user facility. The storage ring light source provides high flux, high average brilliance X-ray with a broad, continuously tunable photon energy spectrum from infrared to hard X-ray, accommodates dozens of beamlines and experimental stations, and allows a wide range of experiments to be performed simultaneously at each station [1]. Figure 1 shows a schematic diagram of a typical storage ring light source, which consists of a full-energy injector, a beam transport line, and a storage ring. The storage ring light source has become an indispensable experimental tool in various fields of science and technology, such as applications in frontier research in physics, chemistry, materials science, environmental science, biology, and life science. Brilliance is defined as the ratio of the photon spectral flux to the volume of the convolution between the electron beam and the photon beam in the two transverse dimensions. Brilliance is one of the most important performance parameters of a storage ring light source. The storage ring light source has gone through three generations of development and is currently under active development towards the fourth-generation light source, the diffraction-limited storage ring (DLSR), whose average brilliance is about 2-3 orders of magnitude higher than that of the third-generation storage ring light source [1; 2]. The DLSR improves the brilliance of the storage ring light source mainly by reducing the emittance of the electron beam to approach or reach the diffraction limit of X-rays. The emittance of the electron beam can be evaluated using the following simple formula:  
\[\varepsilon_{x0}=\frac{C_{\phi}\gamma^{2}}{J_{x}}F\theta^{3}, \tag{1}\]  
where \(C_{\phi}=3.83\times 10^{-13}\) m, \(\gamma\) is the Lorentz factor, \(J_{x}\) is the horizontal damping partition number, \(\theta\) is the bending angle of each bending magnet, and \(F\) is a dimensionless numerical factor related to the lattice structure of the storage ring. Equation (1) assumes that all dipoles are of the same specification, i.e., have the same length and deflection radius. Given a fixed beam energy, the most effective method of reducing the emittance is to reduce the bending angle of each bending magnet, i.e., to increase the number of bending magnets in each cell, which is the basic idea of the multi-bend achromat (MBA) structure. Currently, four DLSRs based on the MBA structure are in operation, namely MAX-IV [3; 4], Sirius [5], ESRF-EBS [6; 7], and APS-U [8]. In addition, many MBA storage rings are under commissioning, construction, or planning, including HEPS [9], PETRA-IV [10], SPring-8-II [11], Korea-4 GSR [12], Diamond-II [13], SAPS [14], SSRL-X [15], SOLEIL II [16], SLS 2.0 [17], Elettra 2.0 [18], HALF [19], ALS-U [20], etc. Except for HEPS, Korea-4 GSR, SAPS, SSRL-X, MAX IV, Sirius, and HALF, which are green-field light sources, all other light sources are upgraded from the old third-generation facilities to the DLSRs. The lattice structures of these facilities can be mainly divided into two categories: one is the hybrid MBA and the other is the higher-order achromat (HOA) MBA. The schematic diagram of the MBA lattice structures and dispersion function trends of hybrid MBA and HOA  
Figure 1: Schematic diagram of a storage ring light source.