We were able to enhance the purity of the intended substance.
We were able to achieve the desired size of the crystal.
この場合、次のような言い方ができますよ。
ーWe were able to enhance the purity of the intended substance.
「我々は目的の物質の純度を高めることができた」
the intended XX で「目的のXX」と言えます。
ーWe were able to achieve the desired size of the crystal.
「我々は目的のサイズの結晶を獲得することができた」
the desired XX で「目的のXX」と言えます。
ご参考まで!