The crossover function for the specific heat of a disordered material is constructed to order E by solving the renormalisation group equations for the m n model (m + 0, n > 1).From this an effective exponent is obtained which provides a local measure of the degree of singularity of the specific heat in the critical region. It is seen that as the critical region is approached from above the critical temperature the effective exponent behaves initially as if no disorder were present. But as the critical temperature is more closely approached the effective exponent crosses over to the value predicted for the disordered material.