Recently, some researchers attempt to find a relationship between the evolution of rare events and temporal-spatial patterns of social media activities. Their studies verify that the relationship exists in both time and spatial domains. However, few of them can accurately deduce a time point when social media activities are highly affected by a rare event. Thus, it is difficult to characterize an accurate temporal pattern of social media during the evolution of a rare event. This work proposes an innovative method to characterize the evolution of a rare event by analyzing social media activities. We find that there is a time difference between the event and social media activities in a time domain. This is conducive to investigate the temporal pattern of social media activities. The proposed method focuses on the intensity of information volume by adopting a clustering algorithm. Our case study focuses on a hurricane named Sandy in 2012. Twitter data collected around it is used to verify the effectiveness of the method. The results not only verify that a rare event and social media activities have strong correlation, but also reveal that they have a time difference. This work provides an effective and reliable method to find a temporal pattern of social media when a rare event occurs.