Enum dictionary issue: will this work?

Here's the code. The enum constants should get added to a private map
with a public, unmodifiable view as they are created. It definitely
won't work if the map put is right in the constructor. Will this sort of
thing work as written, with a static method called that initializes the
map if it's null? Or will the map just get clobbered back to null after
the enum constants are all constructed? And if not, will the
unmodifiable view be constructed correctly?

import java.util.Collections;
import java.util.HashMap;
import java.util.Map;

package thing;

public enum Thing () {

     FOO ("foo"),

     BAR ("bar");

     private String name;

     private static Map<String,Thing> thingMap;

     public static final Map<String,Thing> things =
             Collections.unmodifiableMap(thingMap);

     Thing (String name) {
         this.name = name;
         put(name, this);
     }

     private static void put (String name, Thing thing) {
         if (thingMap == null) {
             thingMap = new HashMap<String,Thing>();
         }
         thingMap.put(name, thing);
     }
}