WASHINGTON – For a nation of millions of broken hearts grasping for something, anything, to prevent yet another schoolhouse slaughter, the solution sounds simple:

If we could just get the mentally ill the proper care, then the nation would be spared the agony of one more Newtown.

The trouble is, that’s too simple a solution, and one that’s unlikely to work on its own, psychiatrists say.

Yes, they say, access to mental health services is not as easy as it should be. There’s a shortage of psychiatrists in Buffalo, for example, and insurance problems in many places that often mean psychiatric patients don’t get the care they need.

Yet other confounding facts prove that better mental health services alone are not the preventative medicine for mass shootings like the one that claimed the lives of 20 school children and eight adults, including the killer, in Connecticut on Friday.

For one thing, while we don’t know if the Connecticut killer, 20-year-old Adam Lanza, was under psychiatric care, we know that some previous mass murderers were – and that it did nothing to stem the bloodshed.

What’s more, psychiatrists say it’s nearly impossible to identify who among the mentally ill is truly dangerous, and that any effort to do so will only further stigmatize a category of illness where many people already shun treatment out of embarrassment and fear.

And there’s one last important fact: The vast majority of mentally ill people are not prone to violence.

Add it all up, and psychiatrists say that improving access to mental health services is likely to be only one of the measures the nation must take if it really wants to get serious about preventing bloodbaths like the one that happened at Sandy Hook Elementary School.

“My fear is always that people will write this off as just about another mentally ill person,” said Dr. Charles P. Ewing, a forensic psychologist and attorney at the University at Buffalo who specializes in the psychology of crime. “It’s about way more than mental illness.”

In fact, at this point we can’t even be sure that Lanza was officially diagnosed as mentally ill. In a story that’s been marked by misinformation from the start, all we know is that former classmates have said they believe that Lanza had Asperger syndrome, a form of autism marked by social awkwardness and odd use of language.

While we don’t know many details of Lanza’s mental state, we do know that many people inside and outside the medical profession are calling for better access to mental health services in the wake of the Connecticut massacre.

That’s especially true in Buffalo, where a shortage of psychiatrists worries many medical professionals.

Dr. Steven L. Dubovsky, professor and chairman of psychiatry at the University at Buffalo Medical School, said that the shortage stems in part from the fact that some local insurers reimburse those psychiatrists in a comparatively paltry way.

For example, some insurers reimburse psychiatrists only $5 more for a 45-minute patient visit than for a one- to 20-minute patient visit.

The result: Local psychiatric patients often don’t get the treatment they need from the right specialist – and often end up in the emergency room with problems that otherwise might have been kept under control, Dubovsky said.

“It doesn’t work well for people with serious problems,” he said.

Then again, access to the best psychiatric care is not a problem limited to Buffalo.

Dr. Claudia M. Gold, a pediatrician, writer and director of the Early Childhood Social Emotional Health program at Newton-Wellesley Hospital near Boston, said research strongly suggests that early treatment is very important for young children with behavioral problems that could mushroom into violence later.

Yet she runs into a problem when she tries to refer young patients and their parents to top mental health professionals.

“The best therapists often don’t take insurance,” she said.

Once troubled youngsters become adults, the mental health system can leave people untreated for another reason.

“One complicating factor is that in order to get treatment, people have to want to get treatment – unless they are seen as an immediate danger to themselves or others,” said Amanda Nickerson, a psychologist and associate professor at UB’s Graduate School of Education. “And even that is difficult to determine.”

What’s more, there’s no guarantee that psychiatric treatment will stop any one mad mind from lashing out. After all, James Holmes was seeing a University of Colorado psychiatrist before he shot 12 people at a showing of the Batman movie “The Dark Knight Rises” in Colorado in July.

And Seung-Hui Cho, who killed 32 people and then himself at Virginia Tech University in 2007, was under court-ordered psychiatric treatment that did nothing to prevent his rampage.

“Most of these people are getting some sort of treatment,” Dubovsky said.

Obviously, in the Colorado and Virginia Tech episodes, that treatment was by no means adequate.

If you think you should institutionalize any mental patient prone to any level of violence, the numbers are indeed daunting. Dr. Alec Buchanan, an associate professor of psychiatry at Yale University, has calculated that you would have to lock up between five and 10 troubled people to have a decent chance of preventing a single one from acting violently.

Even then, you would likely miss some dangerous individuals, noted Dr. Marvin Swartz, who served with Buchanan on an American Psychiatric Association Work Group on Violence Risk earlier this year that concluded: While psychiatrists can often identify circumstances associated with an increased likelihood of violent behavior, they cannot predict dangerousness with definitive accuracy.”

The problem, Ewing said, is that finding the violent among the mentally ill is “totally unpredictable.”

What’s more, violence is actually rare among the mentally ill. Research shows that between 92 and 96 percent of mental patients do not have violent tendencies, said Dr. Daniel Antonius, an assistant professor of psychiatry at the University at Buffalo.

“Their relative contribution to violence is so small,” Antonius said.

So if locking up the mentally ill won’t solve the problem, what will work better?

“I think we have to look at how the seriously mentally ill have access to weapons like this,” said Ewing, who is a SUNY distinguished service professor and vice dean for academic affairs at the UB Law School. “Obviously they do. And doing so, they can do great damage.”

Swartz agreed, noting that academic research shows gun control to be a better deterrent to violence than any attempt to identify dangerous individuals before they strike.

“The more efficient way to reduce the rates of violence is to make guns less available,” he said.