Diffuse radio emission in galaxy clusters is known to be related to cluster mass and cluster dynamical state. We collect the observed fluxes of radio halos, relics, and mini-halos for a sample of galaxy clusters from the literature, and calculate their radio powers. We then obtain the values of cluster mass or mass proxies from previous observations, and also obtain the various dynamical parameters of these galaxy clusters from optical and X-ray data. The radio powers of relics, halos, and mini-halos are correlated with the cluster masses or mass proxies, as found by previous authors, with the correlations concerning giant radio halos being, in general, the strongest ones. We found that the inclusion of dynamical parameters as the third dimension can significantly reduce the data scatter for the scaling relations, especially for radio halos. We therefore conclude that the substructures in X-ray images of galaxy clusters and the irregular distributions of optical brightness of member galaxies can be used to quantitatively characterize the shock waves and turbulence in the intracluster medium responsible for re-accelerating particles to generate the observed diffuse radio emission. The power of radio halos and relics is correlated with cluster mass proxies and dynamical parameters in the form of a fundamental plane.

The dynamical state of galaxy clusters describes if clusters are relaxed dynamically or in a merging process of subclusters. By using archival images from the Chandra X-ray Observatory, we derive a set of parameters to describe the dynamical state for 964 galaxy clusters. Three widely used indicators for dynamical state, the concentration index c, the centroid shift ω and the power ratio P3/P0 are calculated in the circular central region with a radius of 500 kpc. We also derive two adaptive parameters, the profile parameter κ and the asymmetry factor α, in the best fitted elliptical region. The morphology index δ is then defined by combining these two adaptive parameters, which indicates the dynamical state of galaxy clusters and has good correlations to the concentration index c, the centroid shift ω, the power ratio P3/P0, and the optical relaxation factor Γ. For a large sample of clusters, the dynamical parameters are continuously distributed from the disturbed to relaxed states with a peak in the between, rather than the bimodal distribution for the two states. We find that the newly derived morphology index δ works for the similar fundamental plane between the radio power, cluster mass and the dynamical state for clusters with diffuse radio giant-halos and mini-halos. The offset between masses estimated from the Sunyaev-Zeld́ovich effect and X-ray images depends on dynamical parameters. All dynamical parameters for galaxy clusters derived from the Chandra archival images are available on http://zmtt.bao.ac.cn/galaxy_clusters/dyXimages/.