Specifying priors in rstanarm for hierarchical model

Question

We are given the model $$ \begin{align*} y_{ij} & \sim \mathsf{Normal}(\alpha_j + \beta x_i, \sigma^2)\\ \alpha_j & \sim \mathsf{Normal}(\gamma_0 + \gamma_1 u_j, \tau^2) \end{align*} $$ with priors $$ \begin{align*} \beta, \gamma_0, \gamma_1 & \sim \mathsf{Normal}(0, 0.5^2)\\ \sigma, \tau & \sim \mathsf{Normal}_+(0, 0.25^2) \end{align*} $$ where we have $j$ counties and $i$ observations. I'm having trouble specifying all of my priors for rstanarm. What I have so far is

library(rstan)
library(rstanarm)
data(radon)

fit_lmer <- stan_glmer(log_radon ~ floor + (1 | county), 
                       data = radon,
                       prior_intercept = normal(0, 0.5),
                       prior_aux = normal(0, 0.25)) 

I'm just not entirely sure I properly specified everything.