A set of simple and accurate formulae for the first four moments of nuclear and electronic energy losses is proposed. A new variable is introduced to include the finite maximum-impact-parameter effect in the nuclear stopping process, which is assumed to be infinite in most studies. A critical energy at which the electronic energy loss is equal to the nuclear energy loss is also defined. It determines whether the nuclear or the electronic stopping process is the dominant mechanism in terms of incident-ion energy. The critical energy increases for heavy ions implanted in heavy target materials during the first moment of energy loss. The second moment of electronic energy loss is important only for light ions implanted at high ion energies. The third and fourth moments of nuclear energy loss are much larger than those of the electronic energy loss for all ion-target combinations. Theoretical predications of the projected ranges and range stragglings for gold ions implanted in carbon films are close to the experimental data when these proposed four moments of nuclear and electronic energy losses are considered.

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Skewness of implanted ion profiles is determined by extending the Biersack model to higher-order terms. Nuclear and electronic energy losses up to the fourth moment are included in the skewness equations to yield better results for any ion-target combination. The feasibility of the proposed skewness equations is examined using profiles of various ions implanted into target materials of up to two constituents. Calculated skewness values correlate well with experimental data. Kurtosis values calculated using Biersack's fitting formula also closely correspond to experimental data.

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