HAWKER 800XP FLT CREW IS GIVEN AN ALT ALERT BY THE CTLR DURING AN APCH TO BFI.

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LEAVING 9000 FT FOR 3000 FT; I TOLD THE COPLT THAT WE NEEDED TO ADHERE TO THE 'STERILE COCKPIT' POLICY. I TOLD HIM HE NEEDS TO WORK ON THAT. I COULD SENSE THE TENSION MY REMARK MADE ON HIM. MY THOUGHTS WERE DIRECTED TOWARDS HIS FEELINGS. WE WERE ON TOP OF A VERY SCATTERED LAYER OF LOW CLOUDS. THE ARPT WAS NOT VISIBLE; BUT ALL OF THE FAMILIAR LANDMARKS WERE IN SIGHT. CONDITIONS AT THE ARPT WERE NOTHING TO WORRY ABOUT; SO COMPLACENCY SET IN ALONG WITH THE LACK OF PROPER APCH BRIEF. THE AUTOPLT WAS ON AND WE WERE TRACKING THE LOC INBOUND TO THE RWY. PRIOR TO BEING CLRED FOR THE APCH; WE WERE CLRED FROM 9000 FT TO 3000 FT. WHILE LEVEL AT 3000 FT AND A FEW MI FROM THE FINAL APCH FIX; THE GS DSNDED 1 DOT BELOW. I WAS WAITING FOR THE 'CLRED FOR THE APCH' CALL PRIOR TO GS INTERCEPT; BUT IT WAS DELAYED. THE ACFT WAS IN THE LNDG CONFIGN UP TO 25 DEG FLAPS. WHEN THE CLRNC FOR THE APCH WAS GIVEN; I HAD TO MANUALLY 'PITCH' THE AIRPLANE TO CAPTURE THE GS BELOW ME. THE AIRPLANE WAS PROBABLY DOING 500-1000 FPM TO GET DOWN TO THE GS. THE COPLT SET THE ALT ALERTER FOR THE APCH. I BELIEVE IT WAS SET TO 300 FT SINCE THE MINIMUMS WERE 250 FT. AS WE STARTED DOWN; I WAS DISTR AND LOOKED BACK TO SEE THE VERT GUIDANCE WAS MISSING! GS INFO WAS GONE! I TURNED MY ATTN TOWARDS THE COPLT'S CTL YOKE TO LOOK AT THE APCH MINIMUMS FOR THE LOC MINIMUMS. I ASKED 'WHAT ARE THE LOC ONLY MINIMUMS?' AT THIS TIME BOTH OF US WERE LOOKING DOWN; AND AWAY FROM THE FLT INSTS; AND CONCENTRATING ON WHAT THE NEW MINIMUMS WOULD BE. SINCE THERE WAS SOME CONFUSION; AND BEING FAMILIAR WITH THE APCH AND THE AREA; I FIGURED WE WOULD BE SAFE TO THE STANDARD 'NON-PRECISION' STRAIGHT IN APCH MINIMUMS OF 500-700 FT. THE COPLT HAD ONLY THE ILS RWY 13R DISPLAYED AND STORED IN THE CLIP ON THE YOKE. THE ILS RWY 13R MINIMUM SECTION SHOWED THAT WITH THE GS 'OUT;' WE WOULD HAVE TO FLY THE VOR/DME APCH. AFTER REALIZING WE HAD TO GET ANOTHER APCH PLATE OUT; AND REPROGRAM; I TRIED REPROGRAMMING THE ILS; HOPING IT WOULD WORK. DURING THIS CONFUSION; THE AIRPLANE WAS IN LNDG CONFIGN AND DSNDING TO CAPTURE THE GS; BUT THE GS WAS NOW ABOVE US! APCH CTL ASKED US AND SAID THEY SHOWED US 800-1000 FT LOW. JUST PRIOR TO THE FINAL APCH FIX; THE MINIMUM ALT WAS 2200 FT. SINCE WE WERE ABOVE THE GS AND CLRED FOR THE APCH; OUR 'ALT ALERTER' WAS SET BELOW THE 2200 FT FOR THE AREA WE WERE IN. AS SOON AS THE CTLR CALLED US; I IMMEDIATELY ADDED PWR TO STOP THE DSCNT AND START CLBING. WITHIN A FEW SECONDS; WE HAD A 'TA/RA;' 'TFC;' REDUCE VERT SPD; FOLLOWED BY 'PULL UP.!' SHORTLY AFTER FOLLOWED BY 'CLR OF TFC.' SHORTLY FOLLOWED BY SEEING THE RWY; AND BEING CLRED TO LAND. AFTER PAX DEPARTED; THE COPLT AND I SAT IN THE AIRPLANE FOR AT LEAST 30 MINS DISCUSSING WHAT HAD HAPPENED. WE COULD NOT FIGURE OUT WHY WE WOULD GET A 'TA/RA' IN AN AREA THAT WE WERE ON THE LOC (COUPLED) AND CLRED FOR THE APCH; SO CLOSE TO THE ARPT! I RECEIVED A CALL FROM THE TWR SUPVR AND TOLD HIM OF OUR CONFUSION IN THE COCKPIT. I TOLD HIM AFTER ALL THE YRS I HAVE BEEN FLYING; THAT THIS WAS THE WORST MISTAKE ANY PLT COULD MAKE. I WAS HUMBLED TO THE MAX! I KNOW NOW THAT COCKPIT 'FRICTION' BTWN PLTS CAN BE DEADLY. I KNOW NOW TO BRIEF THE APCH BETTER AND HAVE ALL APCH PLATES WHERE WE CAN CHANGE AT A MOMENTS NOTICE. I WILL NOT ALLOW AN ALT ALERTER TO BE SET TO LESS THAN THE SECTOR ALT MINIMUM UNTIL WE ARE IN ANOTHER SECTOR'S MINIMUM (REGARDLESS WHETHER WE ARE CLRED FOR THE APCH)!