  We investigate the distribution of $\alpha p$ modulo one in quadratic number
fields $\mathbb{K}$ with class number one, where $p$ is restricted to prime
elements in the ring of integers of $\mathbb{K}$. Here we improve the relevant
exponent $1/4$ obtained by the first and third named authors for imaginary
quadratic number fields~\cite{BT} and by the first and second named authors for
real quadratic number fields~\cite{BM} to $7/22$. This generalizes a result of
Harman ~\cite{HarZi} who obtained the same exponent $7/22$ for $\mathbb{Q}(i)$
by extending his method which gave this exponent for
$\mathbb{Q}$~\cite{harman1996on-the-distribu}. Our proof is based on an
extension of his sieve method to arbitrary number fields. Moreover, we need an
asymptotic evaluation of certain smooth sums over prime ideals appearing
in~\cite{BM}, for which we use analytic properties of Hecke L-functions with
Gr\"o\ss encharacters.
