What was found
AIRCRAFT AT CRUISE ALTITUDE, FL450, JUST WEST OF THE ROCKY MOUNTAINS WHEN WE ENCOUNTERED EITHER CLEAR-AIR TURBULENCE OR MOUNTAIN WAVE. WE OBSERVED THE AIRSPEED SLOWLY DECREASING, FOLLOWED BY THE AUTOTHROTTLES DISENGAGING. THEN WE OBSERVED THAT NR 1 ENG EXPERIENCED A ROLLBACK, FADEC HAD ENGAGED THE IGNITION ON THE NUMBER 1 ENGINE. THE PIC THEN SELECTED IGN FOR BOTH ENGINES USING THE IGNITION SELECTOR AS WE INITIATED A DESCENT AND UPON OBSERVING ITT HAD INCREASED ON NR 1 ENG TO 922C WE ELECTED TO SHUT DOWN NR 1 ENG IAW THE APPLICABLE PROCEDURES. DURING THE DESCENT THROUGH FL340, STARTED THE APU, IT OPERATED NORMALLY. DIVERTED TO KLNK, CONTINUED DESCENDING TO FL240 & COMPLETED THE ENG NR 1 RESTART PROCEDURE USING THE ABNORMAL PROCEDURES CHECKLIST. THE RESTART WAS SUCCESSFUL. FOLLOWING STARTER CUTOUT, MONITORED ENGINE NR 1 VIBRATION AND ITT, BOTH OF WHICH WERE INITIALLY WITHIN NORMAL OPERATING LIMITS. THE ENGINE NR 1 VIBRATION INDICATION INCREASED FROM APPROXIMATELY 0.12 AND CONTINUED TO RISE STEADILY TO 0.80 AT WHICH POINT, ELECTED TO SHUTDOWN & SECURE ENG NR 1 A SECOND TIME. DECLARED AN EMERGENCY & CONTINUED TO KLNK WHERE THERE WAS MAINT ON THE FIELD. ARCFT LANDED WITHOUT FURTHER INCIDENT. THERE WERE 5 SOULS ON BOARD AND APPROXIMATELY 90 MINUTES OF FUEL REMAINING AT LANDING.