Abstarct: In this research, establish the first axiomatic theory for diversification indices using six intuitive axioms, non-negativity, location invariance, scale invariance, rationality, normalization and continuity, together with risk measures. The unique class of indices satisfying these axioms, called the diversification quotients (DQ) and are defined baxsed on a parametric family of risk measures. DQ has many attractive properties, and it can address several theoretical and practical limitations of existing indices. In particular, for the popular risk measures Value-at-Risk and Expected Shortfall, DQ provides formulas. It is efficient to optimize in portfolio selection and it can properly capture tail heaviness and common shocks which are neglected by traditional diversification indices. When illustrated with financial data, DQ is intuitive to interpret, and its performance is competitive against other diversification indices. Validation of the research has been done using real data for currency and gold market and R software. The risk measures of different assets (dollar, euro, Imami coin, half coin and quarter coin) have been calculated and compared with SD, ES and VaR criteria. Also, portfolios of assets with three different approaches, including the Markowitz model, portfolio of Equally Weighted (EW) assets and Buy-and-Hold (BH) portfolio, were formed baxsed on the information of the previous 365 days, during the research form June 23, 2013, to February 20, 2024. Finally, the performance of three types of baskets and diversification indices have been calculated. Comparing performance results with portfolio diversification shows that optimal portfolio diversification is effective on portfolio performance.