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Highlights • A new finite difference scheme for space fractional equations is constructed. • The scheme is unconditionally stable for time dependent problems. • The scheme is fourth order accurate in space. • The same idea could be used to construct even higher order difference methods. Abstract A weighted and shifted difference formula is constructed based on the Lubich operators, which gives a forth-order and unconditionally stable difference scheme for the Cauchy problem of space fractional diffusion equations. The novelty of the proposed method here is that only four weighted parameters are required, compared to eight parameters used in the previous work, to achieve the fourth-order accuracy and to ensure the stability at the same time. To verify the efficiency of the proposed scheme, several numerical experiments for both one-dimensional and two-dimensional fractional diffusion problems are provided. [ABSTRACT FROM AUTHOR]